Subgroups — C₆₇×#22553 #30701804

Order 17152.

1212 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1923	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1851	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1817	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1779	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1753	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#431×C₆₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
21	#407×C₆₇	—	2	4	no	—
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
31	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
40	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
41	#336×C₆₇	C₂²	1	2	yes	complemented
42	#336×C₆₇	C₂²	1	2	yes	complemented
43	#336×C₆₇	C₂²	1	2	yes	complemented
44	#336×C₆₇	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	2	no	—
47	C₂³×C₄×C₆₇	—	2	4	no	—
48	C₂³×C₄×C₆₇	—	2	2	no	—
49	C₂³×C₄×C₆₇	—	4	2	no	—
50	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₆₇	—	2	4	no	—
61	C₂×M₁₆×C₆₇	—	2	4	no	—
62	C₂×M₁₆×C₆₇	—	4	2	no	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	—	2	2	no	—
78	C₂×C₄²×C₆₇	—	2	2	no	—
79	C₂×C₄²×C₆₇	—	4	2	no	—
80	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	—	2	4	no	—
89	C₄⋊C₈×C₆₇	—	4	2	no	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₂²⋊C₈×C₆₇	—	2	4	no	—
93	C₂²⋊C₈×C₆₇	—	2	2	no	—
94	C₂²⋊C₈×C₆₇	—	2	4	no	—
95	C₂²⋊C₈×C₆₇	—	4	2	no	—
96	C₂²⋊C₈×C₆₇	D₄	1	4	yes	—
97	#97×C₆₇	—	4	2	no	—
98	C₄×C₈×C₆₇	—	2	4	no	—
99	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
100	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
102	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
103	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
104	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
105	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
106	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₂⁴×C₆₇	—	2	2	no	—
108	C₂⁴×C₆₇	—	2	2	no	—
109	C₂⁴×C₆₇	—	2	2	no	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	2	no	—
112	C₂⁴×C₆₇	—	4	1	no	—
113	C₂⁴×C₆₇	—	4	1	no	—
114	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	4	1	no	—
140	C₂²×C₄×C₆₇	—	4	1	no	—
141	C₂²×C₄×C₆₇	—	4	1	no	—
142	C₂²×C₄×C₆₇	—	4	1	no	—
143	C₂²×C₄×C₆₇	—	4	1	no	—
144	C₂²×C₄×C₆₇	—	4	1	no	—
145	C₂²×C₄×C₆₇	—	4	1	no	—
146	C₂²×C₄×C₆₇	—	4	8	no	—
147	C₂²×C₄×C₆₇	—	4	8	no	—
148	C₂²×C₄×C₆₇	M₁₆	1	4	yes	complemented
149	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₆₇	#41	1	4	yes	—
162	C₂²×C₄×C₆₇	#41	1	2	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
164	M₁₆×C₆₇	—	4	8	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	—	2	2	no	—
168	C₂×C₈×C₆₇	—	2	2	no	—
169	C₂×C₈×C₆₇	—	2	2	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	2	no	—
176	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
177	C₂²⋊C₄×C₆₇	—	2	2	no	—
178	C₂²⋊C₄×C₆₇	—	2	2	no	—
179	C₂²⋊C₄×C₆₇	—	2	2	no	—
180	C₂²⋊C₄×C₆₇	—	2	2	no	—
181	C₂²⋊C₄×C₆₇	—	2	2	no	—
182	C₂²⋊C₄×C₆₇	—	2	2	no	—
183	C₂²⋊C₄×C₆₇	—	2	2	no	—
184	C₂²⋊C₄×C₆₇	M₁₆	1	4	yes	complemented
185	C₄²×C₆₇	—	2	2	no	—
186	C₄²×C₆₇	—	2	2	no	—
187	C₄²×C₆₇	—	2	1	no	—
188	C₄²×C₆₇	—	2	1	no	—
189	C₄²×C₆₇	—	2	1	no	—
190	C₄²×C₆₇	—	2	1	no	—
191	C₄²×C₆₇	—	2	2	no	—
192	C₄²×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	4	1	no	—
194	C₄²×C₆₇	—	4	1	no	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	1	no	—
197	C₂³×C₆₇	—	2	1	no	—
198	C₂³×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	—	2	8	no	—
200	C₂³×C₆₇	—	2	1	no	—
201	C₂³×C₆₇	—	2	1	no	—
202	C₂³×C₆₇	—	2	1	no	—
203	C₂³×C₆₇	—	2	1	no	—
204	C₂³×C₆₇	—	2	1	no	—
205	C₂³×C₆₇	—	2	1	no	—
206	C₂³×C₆₇	—	2	1	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	—	2	1	no	—
210	C₂³×C₆₇	—	2	1	no	—
211	C₂³×C₆₇	—	2	1	no	—
212	C₂³×C₆₇	—	4	4	no	—
213	C₂³×C₆₇	—	4	4	no	—
214	C₂³×C₆₇	—	4	4	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	4	no	—
217	C₂³×C₆₇	—	4	4	no	—
218	C₂³×C₆₇	#117	1	2	yes	—
219	C₂³×C₆₇	C₄×D₄	1	2	yes	—
220	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
221	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
222	C₂³×C₆₇	#131	1	2	yes	—
223	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	1	no	—
248	C₂×C₄×C₆₇	—	2	1	no	—
249	C₂×C₄×C₆₇	—	2	1	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	1	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	1	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	1	no	—
256	C₂×C₄×C₆₇	—	4	4	no	—
257	C₂×C₄×C₆₇	—	4	4	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	4	no	—
260	C₂×C₄×C₆₇	—	4	4	no	—
261	C₂×C₄×C₆₇	—	4	4	no	—
262	C₂×C₄×C₆₇	—	4	2	no	—
263	C₂×C₄×C₆₇	—	4	2	no	—
264	C₂×C₄×C₆₇	#117	1	2	yes	—
265	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
267	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
268	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
269	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
270	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
272	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
274	C₈×C₆₇	—	2	4	no	—
275	C₈×C₆₇	—	4	2	no	—
276	C₈×C₆₇	—	4	2	no	—
277	C₈×C₆₇	—	4	2	no	—
278	C₂²×C₆₇	—	2	2	no	—
279	C₂²×C₆₇	—	2	2	no	—
280	C₂²×C₆₇	—	2	2	no	—
281	C₂²×C₆₇	—	2	2	no	—
282	C₂²×C₆₇	—	2	2	no	—
283	C₂²×C₆₇	—	2	2	no	—
284	C₂²×C₆₇	—	2	2	no	—
285	C₂²×C₆₇	—	2	2	no	—
286	C₂²×C₆₇	—	2	2	no	—
287	C₂²×C₆₇	—	2	2	no	—
288	C₂²×C₆₇	—	2	2	no	—
289	C₂²×C₆₇	—	2	2	no	—
290	C₂²×C₆₇	—	4	16	no	—
291	C₂²×C₆₇	—	4	1	no	—
292	C₂²×C₆₇	—	4	1	no	—
293	C₂²×C₆₇	—	4	1	no	—
294	C₂²×C₆₇	—	4	1	no	—
295	C₂²×C₆₇	—	4	1	no	—
296	C₂²×C₆₇	—	4	1	no	—
297	C₂²×C₆₇	—	4	1	no	—
298	C₂²×C₆₇	—	4	1	no	—
299	C₂²×C₆₇	#435	1	4	yes	—
300	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
301	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
302	C₂²×C₆₇	#518	1	1	yes	characteristic
303	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
304	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
305	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
306	C₂²×C₆₇	#568	1	1	yes	characteristic
307	C₄×C₆₇	—	2	2	no	—
308	C₄×C₆₇	—	2	2	no	—
309	C₄×C₆₇	—	2	2	no	—
310	C₄×C₆₇	—	2	2	no	—
311	C₄×C₆₇	—	2	2	no	—
312	C₄×C₆₇	—	2	2	no	—
313	C₄×C₆₇	—	4	1	no	—
314	C₄×C₆₇	—	4	1	no	—
315	C₄×C₆₇	—	4	1	no	—
316	C₄×C₆₇	—	4	1	no	—
317	C₄×C₆₇	—	4	1	no	—
318	C₄×C₆₇	#546	1	2	yes	—
319	C₄×C₆₇	#551	1	2	yes	—
320	#22553	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
321	C₂×C₆₇	—	2	4	no	—
322	C₂×C₆₇	—	4	2	no	—
323	C₂×C₆₇	—	4	2	no	—
324	C₂×C₆₇	#2342	1	1	yes	characteristic
325	C₂×C₆₇	#2920	1	1	yes	characteristic
326	C₂×C₆₇	#2924	1	1	yes	characteristic
327	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
328	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
329	C₂×C₆₇	#2974	1	1	yes	characteristic
330	C₂×C₆₇	#2975	1	1	yes	characteristic
331	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
332	C₂×#407	C₂×C₆₇	1	1	yes	characteristic
333	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
334	#1924	C₂×C₆₇	1	2	yes	complemented
335	#1923	C₂×C₆₇	1	2	yes	complemented
336	#1851	C₂×C₆₇	1	2	yes	complemented
337	#1817	C₂×C₆₇	1	1	yes	characteristic, complemented
338	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
339	#1779	C₂×C₆₇	1	1	yes	characteristic
340	#1753	C₂×C₆₇	1	2	yes	complemented
341	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
342	C₆₇	#22553	1	1	yes	characteristic, direct product factor, Sylow
343	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
344	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
345	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
348	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
349	#431	—	2	4	no	—
350	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
351	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
352	#407	—	2	4	no	—
353	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
354	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
355	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
356	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
357	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
358	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
359	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
361	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
362	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
363	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
364	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
365	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
366	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
367	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
368	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
369	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
370	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
371	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
372	#336	C₂²×C₆₇	1	2	yes	complemented
373	#336	C₂²×C₆₇	1	2	yes	complemented
374	#336	C₂²×C₆₇	1	2	yes	complemented
375	#336	C₂²×C₆₇	1	2	yes	complemented
376	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
377	C₂³×C₄	—	2	2	no	—
378	C₂³×C₄	—	2	4	no	—
379	C₂³×C₄	—	2	2	no	—
380	C₂³×C₄	—	4	2	no	—
381	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
382	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
383	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
384	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
385	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
386	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
387	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
391	C₂×M₁₆	—	2	4	no	—
392	C₂×M₁₆	—	2	4	no	—
393	C₂×M₁₆	—	4	2	no	—
394	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
395	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
396	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
397	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
398	C₂×C₂²⋊C₄	—	2	4	no	—
399	C₂×C₂²⋊C₄	—	2	4	no	—
400	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
401	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
406	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
407	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄²	—	2	2	no	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	—	4	2	no	—
411	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
414	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
419	C₄⋊C₈	—	2	4	no	—
420	C₄⋊C₈	—	4	2	no	—
421	C₂²⋊C₈	—	2	2	no	—
422	C₂²⋊C₈	—	2	2	no	—
423	C₂²⋊C₈	—	2	4	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	—	2	4	no	—
426	C₂²⋊C₈	—	4	2	no	—
427	C₂²⋊C₈	D₄×C₆₇	1	4	yes	—
428	#97	—	4	2	no	—
429	C₄×C₈	—	2	4	no	—
430	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
431	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
432	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
433	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
434	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
435	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
436	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
437	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
438	C₂⁴	—	2	2	no	—
439	C₂⁴	—	2	2	no	—
440	C₂⁴	—	2	2	no	—
441	C₂⁴	—	2	2	no	—
442	C₂⁴	—	2	2	no	—
443	C₂⁴	—	4	1	no	—
444	C₂⁴	—	4	1	no	—
445	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
446	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
447	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	4	1	no	—
471	C₂²×C₄	—	4	1	no	—
472	C₂²×C₄	—	4	1	no	—
473	C₂²×C₄	—	4	1	no	—
474	C₂²×C₄	—	4	1	no	—
475	C₂²×C₄	—	4	1	no	—
476	C₂²×C₄	—	4	1	no	—
477	C₂²×C₄	—	4	8	no	—
478	C₂²×C₄	—	4	8	no	—
479	C₂²×C₄	M₁₆×C₆₇	1	4	yes	complemented
480	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
482	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
484	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
488	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
489	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
492	C₂²×C₄	#41×C₆₇	1	4	yes	—
493	C₂²×C₄	#41×C₆₇	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
495	M₁₆	—	4	8	no	—
496	C₂×C₈	—	2	2	no	—
497	C₂×C₈	—	2	2	no	—
498	C₂×C₈	—	2	2	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	2	2	no	—
505	C₂×C₈	—	2	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	M₁₆×C₆₇	1	4	yes	complemented
516	C₄²	—	2	2	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	1	no	—
519	C₄²	—	2	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	4	1	no	—
525	C₄²	—	4	1	no	—
526	C₂³	—	2	1	no	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	1	no	—
530	C₂³	—	2	8	no	—
531	C₂³	—	2	1	no	—
532	C₂³	—	2	1	no	—
533	C₂³	—	2	1	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	2	1	no	—
536	C₂³	—	2	1	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	1	no	—
540	C₂³	—	2	1	no	—
541	C₂³	—	2	1	no	—
542	C₂³	—	2	1	no	—
543	C₂³	—	4	4	no	—
544	C₂³	—	4	4	no	—
545	C₂³	—	4	4	no	—
546	C₂³	—	4	4	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	4	4	no	—
549	C₂³	#117×C₆₇	1	2	yes	—
550	C₂³	C₄×D₄×C₆₇	1	2	yes	—
551	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
552	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
553	C₂³	#131×C₆₇	1	2	yes	—
554	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	8	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	4	no	—
592	C₂×C₄	—	4	4	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	#117×C₆₇	1	2	yes	—
596	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
597	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
598	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
599	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
600	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
601	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
602	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
603	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
604	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
605	C₈	—	2	4	no	—
606	C₈	—	4	2	no	—
607	C₈	—	4	2	no	—
608	C₈	—	4	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	2	no	—
614	C₂²	—	2	2	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	2	2	no	—
621	C₂²	—	4	16	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	4	1	no	—
625	C₂²	—	4	1	no	—
626	C₂²	—	4	1	no	—
627	C₂²	—	4	1	no	—
628	C₂²	—	4	1	no	—
629	C₂²	—	4	1	no	—
630	C₂²	#435×C₆₇	1	4	yes	—
631	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
632	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
633	C₂²	#518×C₆₇	1	1	yes	characteristic
634	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
635	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
636	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
637	C₂²	#568×C₆₇	1	1	yes	characteristic
638	C₄	—	2	2	no	—
639	C₄	—	2	2	no	—
640	C₄	—	2	2	no	—
641	C₄	—	2	2	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	4	1	no	—
645	C₄	—	4	1	no	—
646	C₄	—	4	1	no	—
647	C₄	—	4	1	no	—
648	C₄	—	4	1	no	—
649	C₄	#546×C₆₇	1	2	yes	—
650	C₄	#551×C₆₇	1	2	yes	—
651	C₂	—	2	4	no	—
652	C₂	—	4	2	no	—
653	C₂	—	4	2	no	—
654	C₂	C₆₇×#2342	1	1	yes	characteristic
655	C₂	C₆₇×#2920	1	1	yes	characteristic
656	C₂	C₆₇×#2924	1	1	yes	characteristic
657	C₂	C₂×#435×C₆₇	1	1	yes	characteristic
658	C₂	D₄×M₁₆×C₆₇	1	1	yes	characteristic
659	C₂	C₆₇×#2974	1	1	yes	characteristic
660	C₂	C₆₇×#2975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#431×C₆₇	C₂	semidirect
4. C₆₇×#1924	C₂	semidirect
5. C₆₇×#1923	C₂	semidirect
6. C₆₇×#1851	C₂	semidirect
7. C₆₇×#1817	C₂	semidirect
8. C₆₇×#1797	C₂	semidirect
10. C₆₇×#1753	C₂	semidirect
11. C₆₇×#1749	C₂	semidirect
19. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
20. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
29. C₂×#97×C₆₇	C₂²	semidirect
30. C₂×C₄×C₈×C₆₇	C₂²	semidirect
41. #336×C₆₇	C₂²	semidirect
42. #336×C₆₇	C₂²	semidirect
43. #336×C₆₇	C₂²	semidirect
44. #336×C₆₇	C₂²	semidirect
148. C₂²×C₄×C₆₇	M₁₆	semidirect
184. C₂²⋊C₄×C₆₇	M₁₆	semidirect
320. #22553	C₆₇	direct (N × H)
331. C₂×#431	C₂×C₆₇	semidirect
334. #1924	C₂×C₆₇	semidirect
335. #1923	C₂×C₆₇	semidirect
336. #1851	C₂×C₆₇	semidirect
337. #1817	C₂×C₆₇	semidirect
338. #1797	C₂×C₆₇	semidirect
340. #1753	C₂×C₆₇	semidirect
341. #1749	C₂×C₆₇	semidirect
342. C₆₇	#22553	direct (N × H)
350. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
351. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
360. C₂×#97	C₂²×C₆₇	semidirect
361. C₂×C₄×C₈	C₂²×C₆₇	semidirect
372. #336	C₂²×C₆₇	semidirect
373. #336	C₂²×C₆₇	semidirect
374. #336	C₂²×C₆₇	semidirect
375. #336	C₂²×C₆₇	semidirect
479. C₂²×C₄	M₁₆×C₆₇	semidirect
515. C₂²⋊C₄	M₁₆×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#431×C₆₇	2
C₂×#407×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1924	2
C₆₇×#1923	2
C₆₇×#1851	2
C₆₇×#1817	2
C₆₇×#1797	2
C₆₇×#1779	2
C₆₇×#1753	2
C₆₇×#1749	2
#22553	67

Subgroups — C67×#22553 #30701804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22553 #30701804