Subgroups — C₆₇×#22035 #30701286

Order 17152.

1008 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×#435×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2388	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1955	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1870	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1870	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1851	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1780	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	#435×C₆₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
21	#408×C₆₇	—	2	4	no	—
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
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	2	yes	—
30	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#97×C₆₇	C₂²	1	2	yes	—
32	#395×C₆₇	C₄	1	1	yes	characteristic
33	#394×C₆₇	C₄	1	2	yes	—
34	#394×C₆₇	C₄	1	1	yes	characteristic
35	#393×C₆₇	C₄	1	1	yes	characteristic
36	#392×C₆₇	C₄	1	1	yes	characteristic
37	#392×C₆₇	C₄	1	2	yes	—
38	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	—	2	2	no	—
45	C₂×#41×C₆₇	—	4	2	no	—
46	C₂×#41×C₆₇	—	4	2	no	—
47	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	—	4	2	no	—
58	C₂×M₁₆×C₆₇	—	4	2	no	—
59	C₂²×C₈×C₆₇	—	4	2	no	—
60	C₂²×C₈×C₆₇	D₄	1	4	yes	complemented
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₆₇	—	2	2	no	—
68	C₄×D₄×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	—	4	2	no	—
70	C₄×D₄×C₆₇	—	4	2	no	—
71	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊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	2	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
82	C₄⋊C₈×C₆₇	—	2	2	no	—
83	C₄⋊C₈×C₆₇	—	4	2	no	—
84	C₂²⋊C₈×C₆₇	—	2	2	no	—
85	C₂²⋊C₈×C₆₇	—	2	2	no	—
86	C₂²⋊C₈×C₆₇	—	2	4	no	—
87	C₂²⋊C₈×C₆₇	—	2	4	no	—
88	C₂²⋊C₈×C₆₇	—	2	2	no	—
89	C₂²⋊C₈×C₆₇	—	4	2	no	—
90	C₂²⋊C₈×C₆₇	—	4	2	no	—
91	C₂²⋊C₈×C₆₇	—	4	2	no	—
92	#97×C₆₇	—	2	2	no	—
93	#97×C₆₇	—	4	2	no	—
94	#95×C₆₇	C₂×C₄	1	2	yes	—
95	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
96	#95×C₆₇	C₂×C₄	1	2	yes	—
97	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
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	#41×C₆₇	—	4	4	no	—
102	#41×C₆₇	—	4	4	no	—
103	#41×C₆₇	—	4	8	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	—	4	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	4	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	4	2	no	—
110	C₂×D₄×C₆₇	—	4	1	no	—
111	C₂×D₄×C₆₇	—	4	2	no	—
112	C₂×D₄×C₆₇	—	4	1	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	4	2	no	—
115	C₂×D₄×C₆₇	—	4	2	no	—
116	C₂×D₄×C₆₇	—	4	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₇	#41	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
139	M₁₆×C₆₇	—	4	4	no	—
140	M₁₆×C₆₇	—	4	4	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	1	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	1	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	—	4	4	no	—
154	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	4	2	no	—
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₂²⋊C₄×C₆₇	—	4	2	no	—
161	C₂²⋊C₄×C₆₇	—	4	2	no	—
162	C₂²⋊C₄×C₆₇	—	4	2	no	—
163	C₂²⋊C₄×C₆₇	—	4	2	no	—
164	C₂²⋊C₄×C₆₇	—	4	2	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	—	2	1	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	—	2	1	no	—
173	C₂³×C₆₇	—	2	1	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	C₄×D₄	1	2	yes	—
186	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
187	Q₈×C₆₇	—	4	4	no	—
188	D₄×C₆₇	—	4	4	no	—
189	D₄×C₆₇	—	4	4	no	—
190	D₄×C₆₇	—	4	8	no	—
191	D₄×C₆₇	—	4	8	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	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	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	2	2	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	—	2	2	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	2	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	2	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
223	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
224	C₈×C₆₇	—	2	2	no	—
225	C₈×C₆₇	—	2	2	no	—
226	C₈×C₆₇	—	4	1	no	—
227	C₈×C₆₇	—	4	1	no	—
228	C₈×C₆₇	—	4	2	no	—
229	C₈×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	4	2	no	—
234	C₂²×C₆₇	—	4	2	no	—
235	C₂²×C₆₇	—	4	2	no	—
236	C₂²×C₆₇	—	4	1	no	—
237	C₂²×C₆₇	—	4	1	no	—
238	C₂²×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	1	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	4	2	no	—
246	C₂²×C₆₇	—	4	2	no	—
247	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
248	C₂²×C₆₇	#518	1	2	yes	—
249	C₂²×C₆₇	#568	1	1	yes	characteristic
250	C₂²×C₆₇	#568	1	1	yes	characteristic
251	C₂²×C₆₇	#568	1	2	yes	—
252	C₄×C₆₇	—	2	2	no	—
253	C₄×C₆₇	—	4	1	no	—
254	C₄×C₆₇	—	4	2	no	—
255	C₄×C₆₇	—	4	1	no	—
256	C₄×C₆₇	—	4	1	no	—
257	C₄×C₆₇	—	4	2	no	—
258	C₄×C₆₇	—	4	2	no	—
259	C₄×C₆₇	#534	1	1	yes	characteristic
260	C₄×C₆₇	#535	1	1	yes	characteristic
261	C₄×C₆₇	#538	1	2	yes	—
262	#22035	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂×C₆₇	—	4	2	no	—
264	C₂×C₆₇	—	8	2	no	—
265	C₂×C₆₇	—	8	1	no	—
266	C₂×C₆₇	#2321	1	1	yes	characteristic
267	C₂×C₆₇	#2930	1	1	yes	characteristic
268	C₂×C₆₇	#2932	1	1	yes	characteristic
269	C₂×C₆₇	#2978	1	2	yes	—
270	C₂×C₆₇	#2980	1	2	yes	—
271	C₂×#435	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
273	#2388	C₂×C₆₇	1	1	yes	characteristic
274	#1955	C₂×C₆₇	1	2	yes	complemented
275	#1924	C₂×C₆₇	1	2	yes	complemented
276	#1924	C₂×C₆₇	1	2	yes	complemented
277	#1870	C₂×C₆₇	1	1	yes	characteristic, complemented
278	#1870	C₂×C₆₇	1	2	yes	complemented
279	#1851	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#1780	C₂×C₆₇	1	1	yes	characteristic, complemented
281	#1752	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₆₇	#22035	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
284	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
285	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
286	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
289	#435	—	2	4	no	—
290	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
292	#408	—	2	4	no	—
293	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
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	1	yes	characteristic
297	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
299	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
300	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
301	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×#97	C₂²×C₆₇	1	2	yes	—
303	#395	C₄×C₆₇	1	1	yes	characteristic
304	#394	C₄×C₆₇	1	2	yes	—
305	#394	C₄×C₆₇	1	1	yes	characteristic
306	#393	C₄×C₆₇	1	1	yes	characteristic
307	#392	C₄×C₆₇	1	1	yes	characteristic
308	#392	C₄×C₆₇	1	2	yes	—
309	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
310	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
311	#336	C₂²×C₆₇	1	2	yes	—
312	#336	C₂²×C₆₇	1	2	yes	—
313	#336	C₂²×C₆₇	1	2	yes	—
314	C₂×#41	—	2	2	no	—
315	C₂×#41	—	2	2	no	—
316	C₂×#41	—	4	2	no	—
317	C₂×#41	—	4	2	no	—
318	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
319	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
320	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
321	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
322	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
325	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
326	C₂×M₁₆	—	2	2	no	—
327	C₂×M₁₆	—	2	2	no	—
328	C₂×M₁₆	—	4	2	no	—
329	C₂×M₁₆	—	4	2	no	—
330	C₂²×C₈	—	4	2	no	—
331	C₂²×C₈	D₄×C₆₇	1	4	yes	complemented
332	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
333	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
335	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	4	2	no	—
341	C₄×D₄	—	4	2	no	—
342	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄⋊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	2	yes	—
348	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
349	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
350	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
351	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
352	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	—	4	2	no	—
355	C₂²⋊C₈	—	2	2	no	—
356	C₂²⋊C₈	—	2	2	no	—
357	C₂²⋊C₈	—	2	4	no	—
358	C₂²⋊C₈	—	2	4	no	—
359	C₂²⋊C₈	—	2	2	no	—
360	C₂²⋊C₈	—	4	2	no	—
361	C₂²⋊C₈	—	4	2	no	—
362	C₂²⋊C₈	—	4	2	no	—
363	#97	—	2	2	no	—
364	#97	—	4	2	no	—
365	#95	C₂×C₄×C₆₇	1	2	yes	—
366	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
367	#95	C₂×C₄×C₆₇	1	2	yes	—
368	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
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	#41	—	4	4	no	—
373	#41	—	4	4	no	—
374	#41	—	4	8	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	4	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	1	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	4	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
401	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
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	C₂²×C₄×C₆₇	1	1	yes	characteristic
406	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
408	C₂²×C₄	#41×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
410	M₁₆	—	4	4	no	—
411	M₁₆	—	4	4	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	1	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	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	4	4	no	—
425	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
426	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	4	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₂²⋊C₄	—	4	2	no	—
433	C₂²⋊C₄	—	4	2	no	—
434	C₂²⋊C₄	—	4	2	no	—
435	C₂²⋊C₄	—	4	2	no	—
436	C₄²	—	2	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	1	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	1	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	C₄×D₄×C₆₇	1	2	yes	—
457	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
458	Q₈	—	4	4	no	—
459	D₄	—	4	4	no	—
460	D₄	—	4	8	no	—
461	D₄	—	4	4	no	—
462	D₄	—	4	8	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	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	2	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	—	2	2	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
493	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
494	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
495	C₈	—	2	2	no	—
496	C₈	—	2	2	no	—
497	C₈	—	4	1	no	—
498	C₈	—	4	2	no	—
499	C₈	—	4	1	no	—
500	C₈	—	4	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	1	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
519	C₂²	#518×C₆₇	1	2	yes	—
520	C₂²	#568×C₆₇	1	1	yes	characteristic
521	C₂²	#568×C₆₇	1	1	yes	characteristic
522	C₂²	#568×C₆₇	1	2	yes	—
523	C₄	—	2	2	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	1	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	#534×C₆₇	1	1	yes	characteristic
531	C₄	#535×C₆₇	1	1	yes	characteristic
532	C₄	#538×C₆₇	1	2	yes	—
533	C₂	—	4	2	no	—
534	C₂	—	8	2	no	—
535	C₂	—	8	1	no	—
536	C₂	C₆₇×#2321	1	1	yes	characteristic
537	C₂	C₆₇×#2930	1	1	yes	characteristic
538	C₂	C₆₇×#2932	1	1	yes	characteristic
539	C₂	C₆₇×#2978	1	2	yes	—
540	C₂	C₆₇×#2980	1	2	yes	—

	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	C₆₇×#2388	C₂
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×#435×C₆₇	C₂	semidirect
2. C₂×#408×C₆₇	C₂	semidirect
4. C₆₇×#1955	C₂	semidirect
5. C₆₇×#1924	C₂	semidirect
6. C₆₇×#1924	C₂	semidirect
7. C₆₇×#1870	C₂	semidirect
8. C₆₇×#1870	C₂	semidirect
9. C₆₇×#1851	C₂	semidirect
10. C₆₇×#1780	C₂	semidirect
11. C₆₇×#1752	C₂	semidirect
60. C₂²×C₈×C₆₇	D₄	semidirect
262. #22035	C₆₇	direct (N × H)
271. C₂×#435	C₂×C₆₇	semidirect
272. C₂×#408	C₂×C₆₇	semidirect
274. #1955	C₂×C₆₇	semidirect
275. #1924	C₂×C₆₇	semidirect
276. #1924	C₂×C₆₇	semidirect
277. #1870	C₂×C₆₇	semidirect
278. #1870	C₂×C₆₇	semidirect
279. #1851	C₂×C₆₇	semidirect
280. #1780	C₂×C₆₇	semidirect
281. #1752	C₂×C₆₇	semidirect
282. C₆₇	#22035	direct (N × H)
331. C₂²×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#435×C₆₇	2
C₂×#408×C₆₇	2
C₆₇×#2388	2
C₆₇×#1955	2
C₆₇×#1924	2
C₆₇×#1870	2
C₆₇×#1851	2
C₆₇×#1780	2
C₆₇×#1752	2
#22035	67

Subgroups — C67×#22035 #30701286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22035 #30701286