Subgroups — C₆₇×#21107 #30700358

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1917	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1869	C₂	1	2	yes	maximal
7	C₆₇×#1850	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1745	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#432×C₆₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
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×#97×C₆₇	—	2	4	no	—
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#97×C₆₇	C₂²	1	2	yes	—
29	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	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	C₄	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₆₇	C₄	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	C₄³×C₆₇	C₂²	1	1	yes	characteristic
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	—	2	4	no	—
53	C₂²×C₈×C₆₇	—	2	4	no	—
54	C₂²×C₈×C₆₇	—	2	4	no	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₄×D₄×C₆₇	—	2	4	no	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
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	C₂³	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	—	2	4	no	—
90	C₂²⋊C₈×C₆₇	—	2	4	no	—
91	C₂²⋊C₈×C₆₇	—	2	4	no	—
92	C₂²⋊C₈×C₆₇	—	2	4	no	—
93	#97×C₆₇	—	2	4	no	—
94	#97×C₆₇	—	2	2	no	—
95	#97×C₆₇	—	2	2	no	—
96	#97×C₆₇	—	2	2	no	—
97	#97×C₆₇	D₄	1	4	yes	complemented
98	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
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	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	M₁₆	1	4	yes	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	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	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	1	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	—	4	1	no	—
125	C₂²×C₄×C₆₇	—	4	1	no	—
126	C₂²×C₄×C₆₇	—	4	1	no	—
127	C₂²×C₄×C₆₇	—	4	1	no	—
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	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	#41	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂×C₈×C₆₇	—	2	16	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	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
154	C₂×C₈×C₆₇	#41	1	4	yes	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₂²⋊C₄×C₆₇	—	2	2	no	—
161	C₂²⋊C₄×C₆₇	—	2	2	no	—
162	C₂²⋊C₄×C₆₇	—	2	2	no	—
163	C₂²⋊C₄×C₆₇	M₁₆	1	4	yes	—
164	C₄²×C₆₇	—	2	2	no	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	1	no	—
174	C₄²×C₆₇	—	2	1	no	—
175	C₄²×C₆₇	—	2	2	no	—
176	C₄²×C₆₇	—	4	1	no	—
177	C₄²×C₆₇	—	4	1	no	—
178	C₄²×C₆₇	C₂×D₄	1	2	yes	—
179	C₄²×C₆₇	#41	1	2	yes	—
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	1	no	—
184	C₂³×C₆₇	—	2	1	no	—
185	C₂³×C₆₇	—	2	1	no	—
186	C₂³×C₆₇	—	2	1	no	—
187	C₂³×C₆₇	—	2	1	no	—
188	C₂³×C₆₇	—	2	1	no	—
189	C₂³×C₆₇	—	2	1	no	—
190	C₂³×C₆₇	#117	1	2	yes	—
191	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
192	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
193	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
194	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
195	D₄×C₆₇	—	2	8	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	1	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	1	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
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	—	4	4	no	—
229	C₂×C₄×C₆₇	—	4	4	no	—
230	C₂×C₄×C₆₇	—	4	4	no	—
231	C₂×C₄×C₆₇	—	4	4	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	#117	1	2	yes	—
235	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
236	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
237	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
238	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
239	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
242	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
243	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
244	C₈×C₆₇	—	2	4	no	—
245	C₈×C₆₇	—	2	4	no	—
246	C₈×C₆₇	—	4	2	no	—
247	C₈×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	2	2	no	—
249	C₂²×C₆₇	—	2	2	no	—
250	C₂²×C₆₇	—	2	2	no	—
251	C₂²×C₆₇	—	2	2	no	—
252	C₂²×C₆₇	—	2	2	no	—
253	C₂²×C₆₇	—	2	2	no	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	4	1	no	—
258	C₂²×C₆₇	—	4	1	no	—
259	C₂²×C₆₇	—	4	1	no	—
260	C₂²×C₆₇	—	4	1	no	—
261	C₂²×C₆₇	#432	1	4	yes	complemented
262	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
263	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
265	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
266	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
267	C₂²×C₆₇	#568	1	1	yes	characteristic
268	C₂²×C₆₇	#568	1	1	yes	characteristic
269	C₄×C₆₇	—	2	2	no	—
270	C₄×C₆₇	—	2	2	no	—
271	C₄×C₆₇	—	2	2	no	—
272	C₄×C₆₇	—	2	2	no	—
273	C₄×C₆₇	—	2	2	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	4	1	no	—
276	C₄×C₆₇	—	4	1	no	—
277	C₄×C₆₇	—	4	1	no	—
278	C₄×C₆₇	—	4	1	no	—
279	C₄×C₆₇	—	4	1	no	—
280	C₄×C₆₇	—	4	1	no	—
281	C₄×C₆₇	#525	1	2	yes	—
282	C₄×C₆₇	#529	1	2	yes	—
283	#21107	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
284	C₂×C₆₇	—	2	4	no	—
285	C₂×C₆₇	—	4	2	no	—
286	C₂×C₆₇	#2290	1	1	yes	characteristic
287	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
288	C₂×C₆₇	#2923	1	1	yes	characteristic
289	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
290	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
291	C₂×C₆₇	#2970	1	1	yes	characteristic
292	C₂×C₆₇	#2970	1	1	yes	characteristic
293	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
294	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
295	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
296	#1924	C₂×C₆₇	1	2	yes	complemented
297	#1917	C₂×C₆₇	1	2	yes	complemented
298	#1869	C₂×C₆₇	1	2	yes	—
299	#1850	C₂×C₆₇	1	2	yes	complemented
300	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
301	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1745	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₆₇	#21107	1	1	yes	characteristic, direct product factor, Sylow
305	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
306	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
309	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
310	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
311	#432	—	2	4	no	—
312	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
313	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
314	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
315	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
316	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
317	C₂×#97	—	2	4	no	—
318	C₂×#97	C₂²×C₆₇	1	2	yes	—
319	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
320	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
323	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
324	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
325	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
326	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
327	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
328	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
329	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
330	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
331	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
332	C₄³	C₂²×C₆₇	1	1	yes	characteristic
333	#336	C₂²×C₆₇	1	2	yes	—
334	#336	C₂²×C₆₇	1	2	yes	—
335	#336	C₂²×C₆₇	1	2	yes	—
336	#336	C₂²×C₆₇	1	2	yes	—
337	#336	C₂²×C₆₇	1	2	yes	—
338	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
339	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
340	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
341	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
342	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
345	C₂²×C₈	—	2	4	no	—
346	C₂²×C₈	—	2	4	no	—
347	C₂²×C₈	—	2	4	no	—
348	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
349	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
350	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
351	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	—	2	4	no	—
354	C₄×D₄	—	2	4	no	—
355	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
360	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
363	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
364	C₂×C₄²	—	2	2	no	—
365	C₂×C₄²	—	2	2	no	—
366	C₂×C₄²	—	2	2	no	—
367	C₂×C₄²	—	2	2	no	—
368	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
370	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
381	C₄⋊C₈	—	2	4	no	—
382	C₄⋊C₈	—	2	4	no	—
383	C₂²⋊C₈	—	2	4	no	—
384	C₂²⋊C₈	—	2	4	no	—
385	C₂²⋊C₈	—	2	4	no	—
386	#97	—	2	4	no	—
387	#97	—	2	2	no	—
388	#97	—	2	2	no	—
389	#97	—	2	2	no	—
390	#97	D₄×C₆₇	1	4	yes	complemented
391	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
392	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
393	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
394	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
395	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
396	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
397	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
398	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	M₁₆×C₆₇	1	4	yes	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	4	1	no	—
418	C₂²×C₄	—	4	1	no	—
419	C₂²×C₄	—	4	1	no	—
420	C₂²×C₄	—	4	1	no	—
421	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
422	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
423	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
426	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
428	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	#41×C₆₇	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
435	C₂×C₈	—	2	16	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
447	C₂×C₈	#41×C₆₇	1	4	yes	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₂²⋊C₄	—	2	2	no	—
452	C₂²⋊C₄	—	2	2	no	—
453	C₂²⋊C₄	—	2	2	no	—
454	C₂²⋊C₄	—	2	2	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	M₁₆×C₆₇	1	4	yes	—
457	C₄²	—	2	2	no	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	2	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	2	no	—
463	C₄²	—	2	1	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	2	1	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	2	no	—
469	C₄²	—	4	1	no	—
470	C₄²	—	4	1	no	—
471	C₄²	C₂×D₄×C₆₇	1	2	yes	—
472	C₄²	#41×C₆₇	1	2	yes	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	1	no	—
479	C₂³	—	2	1	no	—
480	C₂³	—	2	1	no	—
481	C₂³	—	2	1	no	—
482	C₂³	—	2	1	no	—
483	C₂³	#117×C₆₇	1	2	yes	—
484	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
485	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
486	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
487	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
488	D₄	—	2	8	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	#117×C₆₇	1	2	yes	—
528	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
529	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
530	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
531	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
532	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
534	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
535	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
536	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
537	C₈	—	2	4	no	—
538	C₈	—	2	4	no	—
539	C₈	—	4	2	no	—
540	C₈	—	4	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	#432×C₆₇	1	4	yes	complemented
555	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
556	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
557	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
558	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
559	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
560	C₂²	#568×C₆₇	1	1	yes	characteristic
561	C₂²	#568×C₆₇	1	1	yes	characteristic
562	C₄	—	2	2	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	4	1	no	—
574	C₄	#525×C₆₇	1	2	yes	—
575	C₄	#529×C₆₇	1	2	yes	—
576	C₂	—	2	4	no	—
577	C₂	—	4	2	no	—
578	C₂	C₆₇×#2290	1	1	yes	characteristic
579	C₂	C₂×#432×C₆₇	1	1	yes	characteristic
580	C₂	C₆₇×#2923	1	1	yes	characteristic
581	C₂	D₄×M₁₆×C₆₇	1	1	yes	characteristic
582	C₂	D₄×M₁₆×C₆₇	1	1	yes	characteristic
583	C₂	C₆₇×#2970	1	1	yes	characteristic
584	C₂	C₆₇×#2970	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	C₂²×D₄×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×#432×C₆₇	C₂	semidirect
2. C₂²×#97×C₆₇	C₂	semidirect
4. C₆₇×#1924	C₂	semidirect
5. C₆₇×#1917	C₂	semidirect
7. C₆₇×#1850	C₂	semidirect
8. C₆₇×#1767	C₂	semidirect
9. C₆₇×#1749	C₂	semidirect
10. C₆₇×#1749	C₂	semidirect
11. C₆₇×#1745	C₂	semidirect
97. #97×C₆₇	D₄	semidirect
261. C₂²×C₆₇	#432	semidirect
283. #21107	C₆₇	direct (N × H)
293. C₂×#432	C₂×C₆₇	semidirect
294. C₂²×#97	C₂×C₆₇	semidirect
296. #1924	C₂×C₆₇	semidirect
297. #1917	C₂×C₆₇	semidirect
299. #1850	C₂×C₆₇	semidirect
300. #1767	C₂×C₆₇	semidirect
301. #1749	C₂×C₆₇	semidirect
302. #1749	C₂×C₆₇	semidirect
303. #1745	C₂×C₆₇	semidirect
304. C₆₇	#21107	direct (N × H)
390. #97	D₄×C₆₇	semidirect
554. C₂²	#432×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₆₇	2
C₂²×#97×C₆₇	2
C₄²×D₄×C₆₇	2
C₆₇×#1924	2
C₆₇×#1917	2
C₆₇×#1869	2
C₆₇×#1850	2
C₆₇×#1767	2
C₆₇×#1749	2
C₆₇×#1745	2
#21107	67

Subgroups — C67×#21107 #30700358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21107 #30700358