Subgroups — C₆₇×#22852 #30702103

Order 17152.

1266 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#407×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₄×#120×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1955	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1923	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1870	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1870	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1851	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1753	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	C₂×#120×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	#436×C₆₇	—	2	8	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	#407×C₆₇	—	2	8	no	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
29	#386×C₆₇	C₄	1	1	yes	characteristic
30	#386×C₆₇	C₄	1	2	yes	—
31	#385×C₆₇	C₄	1	2	yes	—
32	#385×C₆₇	C₄	1	1	yes	characteristic
33	#379×C₆₇	C₄	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#336×C₆₇	C₂²	1	2	yes	complemented
37	#336×C₆₇	C₂²	1	2	yes	complemented
38	#336×C₆₇	C₂²	1	2	yes	complemented
39	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	—	2	4	no	—
43	C₂³×C₄×C₆₇	—	2	4	no	—
44	C₂³×C₄×C₆₇	—	4	1	no	—
45	C₂³×C₄×C₆₇	—	4	1	no	—
46	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₇	—	2	4	no	—
52	C₂×M₁₆×C₆₇	—	2	4	no	—
53	C₂×M₁₆×C₆₇	—	2	4	no	—
54	C₂×M₁₆×C₆₇	—	2	4	no	—
55	C₂×M₁₆×C₆₇	—	2	4	no	—
56	C₂×M₁₆×C₆₇	—	4	2	no	—
57	C₂×M₁₆×C₆₇	—	4	2	no	—
58	C₂×M₁₆×C₆₇	—	4	1	no	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	#120×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	2	4	no	—
65	C₄×D₄×C₆₇	—	2	4	no	—
66	C₄×D₄×C₆₇	—	4	1	no	—
67	C₄×D₄×C₆₇	—	4	1	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
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	2	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₆₇	—	2	4	no	—
81	C₄⋊C₈×C₆₇	—	4	1	no	—
82	C₂²⋊C₈×C₆₇	—	2	4	no	—
83	C₂²⋊C₈×C₆₇	—	2	4	no	—
84	C₂²⋊C₈×C₆₇	—	2	4	no	—
85	C₂²⋊C₈×C₆₇	—	2	4	no	—
86	C₂²⋊C₈×C₆₇	—	4	1	no	—
87	C₂²⋊C₈×C₆₇	D₄	1	8	yes	complemented
88	#97×C₆₇	—	4	1	no	—
89	C₄×C₈×C₆₇	—	2	4	no	—
90	#95×C₆₇	C₂×C₄	1	2	yes	—
91	#95×C₆₇	C₂×C₄	1	2	yes	—
92	#95×C₆₇	C₂×C₄	1	2	yes	—
93	C₂⁴×C₆₇	—	2	4	no	—
94	C₂⁴×C₆₇	—	2	2	no	—
95	C₂⁴×C₆₇	—	4	1	no	—
96	C₂⁴×C₆₇	—	4	2	no	—
97	C₂⁴×C₆₇	—	4	1	no	—
98	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	4	no	—
104	C₂×D₄×C₆₇	—	4	1	no	—
105	C₂×D₄×C₆₇	—	4	2	no	—
106	C₂×D₄×C₆₇	—	4	1	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	4	1	no	—
123	C₂²×C₄×C₆₇	—	4	2	no	—
124	C₂²×C₄×C₆₇	—	4	4	no	—
125	C₂²×C₄×C₆₇	—	4	8	no	—
126	C₂²×C₄×C₆₇	—	4	4	no	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
135	C₂²×C₄×C₆₇	#41	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	M₁₆×C₆₇	—	4	4	no	—
138	M₁₆×C₆₇	—	4	8	no	—
139	M₁₆×C₆₇	—	4	8	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
151	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
152	C₄⋊C₄×C₆₇	—	2	4	no	—
153	C₄⋊C₄×C₆₇	—	4	1	no	—
154	C₂²⋊C₄×C₆₇	—	2	4	no	—
155	C₂²⋊C₄×C₆₇	—	2	4	no	—
156	C₂²⋊C₄×C₆₇	—	2	4	no	—
157	C₂²⋊C₄×C₆₇	—	4	1	no	—
158	C₂²⋊C₄×C₆₇	—	4	2	no	—
159	C₂²⋊C₄×C₆₇	—	4	1	no	—
160	C₂²⋊C₄×C₆₇	—	4	1	no	—
161	C₂²⋊C₄×C₆₇	M₁₆	1	4	yes	complemented
162	C₄²×C₆₇	—	2	2	no	—
163	C₄²×C₆₇	—	2	2	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	—	2	2	no	—
167	C₂³×C₆₇	—	2	2	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	2	1	no	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	—	2	1	no	—
179	C₂³×C₆₇	—	2	1	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	4	4	no	—
182	C₂³×C₆₇	—	4	8	no	—
183	C₂³×C₆₇	—	4	8	no	—
184	C₂³×C₆₇	—	4	4	no	—
185	C₂³×C₆₇	—	4	4	no	—
186	C₂³×C₆₇	C₄×D₄	1	4	yes	—
187	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
188	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
189	D₄×C₆₇	—	4	8	no	—
190	D₄×C₆₇	—	4	4	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	8	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	4	2	no	—
217	C₂×C₄×C₆₇	—	4	1	no	—
218	C₂×C₄×C₆₇	—	4	1	no	—
219	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
220	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
223	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
225	C₈×C₆₇	—	2	4	no	—
226	C₈×C₆₇	—	2	4	no	—
227	C₈×C₆₇	—	4	2	no	—
228	C₈×C₆₇	—	4	2	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	2	4	no	—
231	C₂²×C₆₇	—	2	4	no	—
232	C₂²×C₆₇	—	2	4	no	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	4	16	no	—
235	C₂²×C₆₇	—	4	1	no	—
236	C₂²×C₆₇	—	4	2	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	2	no	—
240	C₂²×C₆₇	—	4	1	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	1	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	4	1	no	—
246	C₂²×C₆₇	—	4	1	no	—
247	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
248	C₂²×C₆₇	#518	1	1	yes	characteristic
249	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
250	C₂²×C₆₇	#568	1	1	yes	characteristic
251	C₂²×C₆₇	#568	1	2	yes	—
252	C₄×C₆₇	—	2	4	no	—
253	C₄×C₆₇	—	2	4	no	—
254	C₄×C₆₇	—	4	1	no	—
255	C₄×C₆₇	—	4	1	no	—
256	C₄×C₆₇	—	4	2	no	—
257	C₄×C₆₇	—	4	1	no	—
258	C₄×C₆₇	—	4	1	no	—
259	C₄×C₆₇	D₄²	1	1	yes	characteristic
260	C₄×C₆₇	#546	1	2	yes	—
261	C₄×C₆₇	#551	1	1	yes	characteristic
262	#22852	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂×C₆₇	—	4	4	no	—
264	C₂×C₆₇	—	4	2	no	—
265	C₂×C₆₇	—	8	1	no	—
266	C₂×C₆₇	#2358	1	1	yes	characteristic
267	C₂×C₆₇	#2930	1	2	yes	—
268	C₂×C₆₇	D₄×M₁₆	1	2	yes	—
269	C₂×C₆₇	#2975	1	1	yes	characteristic
270	C₂×C₆₇	#2978	1	1	yes	characteristic
271	C₂×#436	C₂×C₆₇	1	2	yes	complemented
272	C₂×#407	C₂×C₆₇	1	2	yes	complemented
273	C₄×#120	C₂×C₆₇	1	1	yes	characteristic
274	#1955	C₂×C₆₇	1	1	yes	characteristic, complemented
275	#1923	C₂×C₆₇	1	2	yes	complemented
276	#1870	C₂×C₆₇	1	2	yes	complemented
277	#1870	C₂×C₆₇	1	1	yes	characteristic, complemented
278	#1851	C₂×C₆₇	1	2	yes	complemented
279	#1753	C₂×C₆₇	1	2	yes	complemented
280	C₆₇	#22852	1	1	yes	characteristic, direct product factor, Sylow
281	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
282	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
283	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
284	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
285	C₂×#120	C₄×C₆₇	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
287	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
288	#436	—	2	8	no	—
289	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
290	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
291	#407	—	2	8	no	—
292	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
293	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
294	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
295	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
296	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
297	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
298	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic, complemented
299	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
300	#386	C₄×C₆₇	1	1	yes	characteristic
301	#386	C₄×C₆₇	1	2	yes	—
302	#385	C₄×C₆₇	1	2	yes	—
303	#385	C₄×C₆₇	1	1	yes	characteristic
304	#379	C₄×C₆₇	1	1	yes	characteristic
305	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
306	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
307	#336	C₂²×C₆₇	1	2	yes	complemented
308	#336	C₂²×C₆₇	1	2	yes	complemented
309	#336	C₂²×C₆₇	1	2	yes	complemented
310	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
311	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
312	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₂³×C₄	—	2	4	no	—
314	C₂³×C₄	—	2	4	no	—
315	C₂³×C₄	—	4	1	no	—
316	C₂³×C₄	—	4	1	no	—
317	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
318	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
322	C₂×M₁₆	—	2	4	no	—
323	C₂×M₁₆	—	2	4	no	—
324	C₂×M₁₆	—	2	4	no	—
325	C₂×M₁₆	—	2	4	no	—
326	C₂×M₁₆	—	2	4	no	—
327	C₂×M₁₆	—	4	2	no	—
328	C₂×M₁₆	—	4	2	no	—
329	C₂×M₁₆	—	4	1	no	—
330	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
331	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
332	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
333	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
334	#120	—	2	4	no	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	4	no	—
337	C₄×D₄	—	4	1	no	—
338	C₄×D₄	—	4	1	no	—
339	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
340	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
341	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
342	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
343	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
344	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
346	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
347	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
350	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
351	C₄⋊C₈	—	2	4	no	—
352	C₄⋊C₈	—	4	1	no	—
353	C₂²⋊C₈	—	2	4	no	—
354	C₂²⋊C₈	—	2	4	no	—
355	C₂²⋊C₈	—	2	4	no	—
356	C₂²⋊C₈	—	2	4	no	—
357	C₂²⋊C₈	—	4	1	no	—
358	C₂²⋊C₈	D₄×C₆₇	1	8	yes	complemented
359	#97	—	4	1	no	—
360	C₄×C₈	—	2	4	no	—
361	#95	C₂×C₄×C₆₇	1	2	yes	—
362	#95	C₂×C₄×C₆₇	1	2	yes	—
363	#95	C₂×C₄×C₆₇	1	2	yes	—
364	C₂⁴	—	2	4	no	—
365	C₂⁴	—	2	2	no	—
366	C₂⁴	—	4	1	no	—
367	C₂⁴	—	4	2	no	—
368	C₂⁴	—	4	1	no	—
369	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	—	4	2	no	—
377	C₂×D₄	—	4	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	4	1	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	4	no	—
396	C₂²×C₄	—	4	8	no	—
397	C₂²×C₄	—	4	4	no	—
398	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
399	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
401	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
404	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
405	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
406	C₂²×C₄	#41×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
408	M₁₆	—	4	4	no	—
409	M₁₆	—	4	8	no	—
410	M₁₆	—	4	8	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
422	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	4	1	no	—
429	C₂²⋊C₄	—	4	1	no	—
430	C₂²⋊C₄	—	4	1	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₂²⋊C₄	M₁₆×C₆₇	1	4	yes	complemented
433	C₄²	—	2	2	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	1	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	1	no	—
449	C₂³	—	2	1	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	4	no	—
453	C₂³	—	4	8	no	—
454	C₂³	—	4	8	no	—
455	C₂³	—	4	4	no	—
456	C₂³	—	4	4	no	—
457	C₂³	C₄×D₄×C₆₇	1	4	yes	—
458	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
459	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
460	D₄	—	4	8	no	—
461	D₄	—	4	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	4	8	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
491	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
493	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
494	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
495	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
496	C₈	—	2	4	no	—
497	C₈	—	2	4	no	—
498	C₈	—	4	2	no	—
499	C₈	—	4	2	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	4	16	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	1	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
519	C₂²	#518×C₆₇	1	1	yes	characteristic
520	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
521	C₂²	#568×C₆₇	1	1	yes	characteristic
522	C₂²	#568×C₆₇	1	2	yes	—
523	C₄	—	2	4	no	—
524	C₄	—	2	4	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	1	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	1	no	—
530	C₄	D₄²×C₆₇	1	1	yes	characteristic
531	C₄	#546×C₆₇	1	2	yes	—
532	C₄	#551×C₆₇	1	1	yes	characteristic
533	C₂	—	4	4	no	—
534	C₂	—	4	2	no	—
535	C₂	—	8	1	no	—
536	C₂	C₆₇×#2358	1	1	yes	characteristic
537	C₂	C₆₇×#2930	1	2	yes	—
538	C₂	D₄×M₁₆×C₆₇	1	2	yes	—
539	C₂	C₆₇×#2975	1	1	yes	characteristic
540	C₂	C₆₇×#2978	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	C₂×#120×C₆₇	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×#436×C₆₇	C₂	semidirect
2. C₂×#407×C₆₇	C₂	semidirect
4. C₆₇×#1955	C₂	semidirect
5. C₆₇×#1923	C₂	semidirect
6. C₆₇×#1870	C₂	semidirect
7. C₆₇×#1870	C₂	semidirect
8. C₆₇×#1851	C₂	semidirect
9. C₆₇×#1753	C₂	semidirect
13. C₂²×M₁₆×C₆₇	C₂²	semidirect
18. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
19. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
25. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
26. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
27. C₂×#97×C₆₇	C₂²	semidirect
28. C₂×C₄×C₈×C₆₇	C₂²	semidirect
36. #336×C₆₇	C₂²	semidirect
37. #336×C₆₇	C₂²	semidirect
38. #336×C₆₇	C₂²	semidirect
87. C₂²⋊C₈×C₆₇	D₄	semidirect
161. C₂²⋊C₄×C₆₇	M₁₆	semidirect
262. #22852	C₆₇	direct (N × H)
271. C₂×#436	C₂×C₆₇	semidirect
272. C₂×#407	C₂×C₆₇	semidirect
274. #1955	C₂×C₆₇	semidirect
275. #1923	C₂×C₆₇	semidirect
276. #1870	C₂×C₆₇	semidirect
277. #1870	C₂×C₆₇	semidirect
278. #1851	C₂×C₆₇	semidirect
279. #1753	C₂×C₆₇	semidirect
280. C₆₇	#22852	direct (N × H)
284. C₂²×M₁₆	C₂²×C₆₇	semidirect
289. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
290. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
296. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
297. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
298. C₂×#97	C₂²×C₆₇	semidirect
299. C₂×C₄×C₈	C₂²×C₆₇	semidirect
307. #336	C₂²×C₆₇	semidirect
308. #336	C₂²×C₆₇	semidirect
309. #336	C₂²×C₆₇	semidirect
358. C₂²⋊C₈	D₄×C₆₇	semidirect
432. 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×#436×C₆₇	2
C₂×#407×C₆₇	2
C₄×#120×C₆₇	2
C₆₇×#1955	2
C₆₇×#1923	2
C₆₇×#1870	2
C₆₇×#1851	2
C₆₇×#1753	2
#22852	67

Subgroups — C67×#22852 #30702103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22852 #30702103