Subgroups — C₆₇×#22475 #30701726

Order 17152.

1168 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1985	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1870	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1869	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1818	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1729	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	#529×C₆₇	—	2	8	no	—
13	C₂×#126×C₆₇	C₄	1	2	yes	—
14	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
15	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#123×C₆₇	C₄	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#435×C₆₇	—	2	8	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
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	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#97×C₆₇	C₂²	1	2	yes	—
32	#336×C₆₇	C₂²	1	1	yes	characteristic
33	#336×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×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	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂×M₁₆×C₆₇	—	4	2	no	—
49	C₂²×C₈×C₆₇	—	2	4	no	—
50	C₂²×C₈×C₆₇	—	2	4	no	—
51	C₂²×C₈×C₆₇	—	4	2	no	—
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	#126×C₆₇	—	2	8	no	—
57	#125×C₆₇	—	2	4	no	—
58	#124×C₆₇	—	2	4	no	—
59	#123×C₆₇	—	2	8	no	—
60	C₂²⋊Q₈×C₆₇	—	2	4	no	—
61	#121×C₆₇	—	2	4	no	—
62	C₄×Q₈×C₆₇	—	2	4	no	—
63	C₄×D₄×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	2	4	no	—
65	C₄×D₄×C₆₇	D₄	1	8	yes	—
66	#117×C₆₇	—	2	4	no	—
67	#117×C₆₇	—	2	4	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₆₇	—	2	4	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	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	—	2	4	no	—
86	C₂²⋊C₈×C₆₇	—	2	4	no	—
87	C₂²⋊C₈×C₆₇	—	2	4	no	—
88	C₂²⋊C₈×C₆₇	—	2	4	no	—
89	C₂²⋊C₈×C₆₇	—	4	1	no	—
90	C₂²⋊C₈×C₆₇	—	4	1	no	—
91	#97×C₆₇	—	2	4	no	—
92	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	4	1	no	—
96	C₂×Q₈×C₆₇	—	4	1	no	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	4	1	no	—
100	C₂×D₄×C₆₇	—	4	1	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	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	2	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	4	1	no	—
115	C₂²×C₄×C₆₇	—	4	1	no	—
116	C₂²×C₄×C₆₇	—	4	1	no	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	M₁₆×C₆₇	—	4	8	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	2	2	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	4	8	no	—
141	C₂×C₈×C₆₇	#41	1	4	yes	—
142	C₄⋊C₄×C₆₇	—	2	4	no	—
143	C₄⋊C₄×C₆₇	—	2	4	no	—
144	C₄⋊C₄×C₆₇	—	2	4	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	4	no	—
147	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	4	no	—
151	C₂²⋊C₄×C₆₇	—	2	4	no	—
152	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	—	2	2	no	—
155	C₄²×C₆₇	—	2	2	no	—
156	C₄²×C₆₇	—	2	2	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	C₂×D₄	1	4	yes	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	2	1	no	—
162	C₂³×C₆₇	—	2	2	no	—
163	C₂³×C₆₇	—	2	1	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
172	C₂³×C₆₇	#131	1	2	yes	—
173	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
174	Q₈×C₆₇	—	4	4	no	—
175	D₄×C₆₇	—	4	4	no	—
176	D₄×C₆₇	—	4	8	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	2	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	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	4	4	no	—
203	C₂×C₄×C₆₇	—	4	4	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
209	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
211	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₆₇	—	2	4	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₈×C₆₇	—	4	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	2	2	no	—
220	C₂²×C₆₇	—	2	4	no	—
221	C₂²×C₆₇	—	2	2	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	4	1	no	—
224	C₂²×C₆₇	—	4	1	no	—
225	C₂²×C₆₇	—	4	1	no	—
226	C₂²×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	—	4	2	no	—
229	C₂²×C₆₇	—	4	1	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
233	C₂²×C₆₇	#520	1	1	yes	characteristic
234	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
235	C₂²×C₆₇	#568	1	1	yes	characteristic
236	C₂²×C₆₇	#568	1	1	yes	characteristic
237	C₂²×C₆₇	#568	1	1	yes	characteristic
238	C₄×C₆₇	—	2	4	no	—
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	2	2	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	—	4	1	no	—
243	C₄×C₆₇	—	4	2	no	—
244	C₄×C₆₇	—	4	1	no	—
245	C₄×C₆₇	—	4	2	no	—
246	C₄×C₆₇	#546	1	2	yes	—
247	C₄×C₆₇	#547	1	2	yes	—
248	#22475	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂×C₆₇	—	4	4	no	—
250	C₂×C₆₇	—	4	2	no	—
251	C₂×C₆₇	#2339	1	1	yes	characteristic
252	C₂×C₆₇	#2879	1	1	yes	characteristic
253	C₂×C₆₇	#2880	1	1	yes	characteristic
254	C₂×C₆₇	#2967	1	1	yes	characteristic
255	C₂×C₆₇	#2968	1	1	yes	characteristic
256	C₂×C₆₇	#2992	1	1	yes	characteristic
257	C₂×C₆₇	#2995	1	1	yes	characteristic
258	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
259	C₂×#435	C₂×C₆₇	1	2	yes	complemented
260	#1985	C₂×C₆₇	1	2	yes	complemented
261	#1870	C₂×C₆₇	1	2	yes	complemented
262	#1869	C₂×C₆₇	1	2	yes	complemented
263	#1818	C₂×C₆₇	1	2	yes	complemented
264	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#1729	C₂×C₆₇	1	2	yes	complemented
267	C₆₇	#22475	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
269	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
270	#529	—	2	8	no	—
271	C₂×#126	C₄×C₆₇	1	2	yes	—
272	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
273	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
274	C₂×#123	C₄×C₆₇	1	2	yes	—
275	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
276	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
277	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
279	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#117	C₂²×C₆₇	1	2	yes	—
281	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
282	#435	—	2	8	no	—
283	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
284	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
285	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
286	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
287	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×#97	C₂²×C₆₇	1	2	yes	—
290	#336	C₂²×C₆₇	1	2	yes	—
291	#336	C₂²×C₆₇	1	1	yes	characteristic
292	#336	C₂²×C₆₇	1	2	yes	—
293	#336	C₂²×C₆₇	1	2	yes	—
294	#336	C₂²×C₆₇	1	2	yes	—
295	#336	C₂²×C₆₇	1	2	yes	—
296	#336	C₂²×C₆₇	1	1	yes	characteristic
297	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
299	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
300	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
301	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
302	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×M₁₆	—	2	4	no	—
305	C₂×M₁₆	—	2	4	no	—
306	C₂×M₁₆	—	4	2	no	—
307	C₂²×C₈	—	2	4	no	—
308	C₂²×C₈	—	2	4	no	—
309	C₂²×C₈	—	4	2	no	—
310	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
311	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
312	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
313	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
314	#126	—	2	8	no	—
315	#125	—	2	4	no	—
316	#124	—	2	4	no	—
317	#123	—	2	8	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	#121	—	2	4	no	—
320	C₄×Q₈	—	2	4	no	—
321	C₄×D₄	—	2	4	no	—
322	C₄×D₄	—	2	4	no	—
323	C₄×D₄	D₄×C₆₇	1	8	yes	—
324	#117	—	2	4	no	—
325	#117	—	2	4	no	—
326	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
327	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
328	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
329	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
331	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
333	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
334	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
336	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
337	C₂×C₄²	—	2	4	no	—
338	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
340	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
342	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
343	C₄⋊C₈	—	2	4	no	—
344	C₂²⋊C₈	—	2	4	no	—
345	C₂²⋊C₈	—	2	4	no	—
346	C₂²⋊C₈	—	2	4	no	—
347	C₂²⋊C₈	—	4	1	no	—
348	C₂²⋊C₈	—	4	1	no	—
349	#97	—	2	4	no	—
350	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
351	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	4	1	no	—
354	C₂×Q₈	—	4	1	no	—
355	C₂×D₄	—	2	4	no	—
356	C₂×D₄	—	2	4	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	—	4	1	no	—
359	C₂×D₄	—	4	2	no	—
360	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	4	1	no	—
373	C₂²×C₄	—	4	1	no	—
374	C₂²×C₄	—	4	1	no	—
375	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
376	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
377	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
378	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
379	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
380	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
384	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
385	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
386	M₁₆	—	4	8	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	4	8	no	—
399	C₂×C₈	#41×C₆₇	1	4	yes	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
406	C₂²⋊C₄	—	2	4	no	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	2	4	no	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	2	2	no	—
416	C₄²	C₂×D₄×C₆₇	1	4	yes	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	1	no	—
425	C₂³	—	2	1	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
430	C₂³	#131×C₆₇	1	2	yes	—
431	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
432	Q₈	—	4	4	no	—
433	D₄	—	4	4	no	—
434	D₄	—	4	8	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
467	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
468	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
469	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
470	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
471	C₈	—	2	4	no	—
472	C₈	—	4	2	no	—
473	C₈	—	4	2	no	—
474	C₈	—	4	2	no	—
475	C₂²	—	2	4	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	1	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	2	no	—
489	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
490	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
491	C₂²	#520×C₆₇	1	1	yes	characteristic
492	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
493	C₂²	#568×C₆₇	1	1	yes	characteristic
494	C₂²	#568×C₆₇	1	1	yes	characteristic
495	C₂²	#568×C₆₇	1	1	yes	characteristic
496	C₄	—	2	4	no	—
497	C₄	—	2	2	no	—
498	C₄	—	2	2	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	1	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	2	no	—
504	C₄	#546×C₆₇	1	2	yes	—
505	C₄	#547×C₆₇	1	2	yes	—
506	C₂	—	4	4	no	—
507	C₂	—	4	2	no	—
508	C₂	C₆₇×#2339	1	1	yes	characteristic
509	C₂	C₆₇×#2879	1	1	yes	characteristic
510	C₂	C₆₇×#2880	1	1	yes	characteristic
511	C₂	C₆₇×#2967	1	1	yes	characteristic
512	C₂	C₆₇×#2968	1	1	yes	characteristic
513	C₂	C₆₇×#2992	1	1	yes	characteristic
514	C₂	C₆₇×#2995	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×#121×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
2. C₂×#435×C₆₇	C₂	semidirect
3. C₆₇×#1985	C₂	semidirect
4. C₆₇×#1870	C₂	semidirect
5. C₆₇×#1869	C₂	semidirect
6. C₆₇×#1818	C₂	semidirect
7. C₆₇×#1797	C₂	semidirect
8. C₆₇×#1749	C₂	semidirect
9. C₆₇×#1729	C₂	semidirect
248. #22475	C₆₇	direct (N × H)
259. C₂×#435	C₂×C₆₇	semidirect
260. #1985	C₂×C₆₇	semidirect
261. #1870	C₂×C₆₇	semidirect
262. #1869	C₂×C₆₇	semidirect
263. #1818	C₂×C₆₇	semidirect
264. #1797	C₂×C₆₇	semidirect
265. #1749	C₂×C₆₇	semidirect
266. #1729	C₂×C₆₇	semidirect
267. C₆₇	#22475	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₂×#435×C₆₇	2
C₆₇×#1985	2
C₆₇×#1870	2
C₆₇×#1869	2
C₆₇×#1818	2
C₆₇×#1797	2
C₆₇×#1749	2
C₆₇×#1729	2
#22475	67

Subgroups — C67×#22475 #30701726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22475 #30701726