Subgroups — C₂×C₆₇×#2929 #30739384

Order 17152.

3046 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2929	C₂	1	16	yes	maximal, direct product factor
5	C₂×#435×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×C₈×D₄×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#131×C₆₇	—	2	8	no	—
14	C₂×#131×C₆₇	C₂²	1	8	yes	complemented
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
18	C₂³×C₈×C₆₇	—	2	8	no	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₆₇	C₂²	1	8	yes	complemented
24	C₂³×C₈×C₆₇	C₂²	1	2	yes	complemented
25	#525×C₆₇	C₄	1	8	yes	—
26	#525×C₆₇	C₂²	1	8	yes	—
27	C₂×#123×C₆₇	C₄	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
29	C₂×#121×C₆₇	C₄	1	2	yes	—
30	C₂×#120×C₆₇	C₄	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
32	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
33	#435×C₆₇	C₂²	1	32	yes	complemented
34	C₈×D₄×C₆₇	C₂²	1	32	yes	complemented
35	#424×C₆₇	C₂²	1	8	yes	complemented
36	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
37	#408×C₆₇	C₂²	1	8	yes	complemented
38	#407×C₆₇	C₂²	1	8	yes	complemented
39	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
40	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
41	#405×C₆₇	C₂²	1	8	yes	complemented
42	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
43	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
44	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×#41×C₆₇	—	2	4	no	—
46	C₂×#41×C₆₇	—	2	4	no	—
47	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×#41×C₆₇	C₂³	1	4	yes	—
49	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	—	2	4	no	—
54	C₂³×C₄×C₆₇	—	2	4	no	—
55	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	#131×C₆₇	—	2	32	no	—
63	C₂×M₁₆×C₆₇	—	2	4	no	—
64	C₂×M₁₆×C₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	—	2	8	no	—
66	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	8	yes	complemented
69	C₂²×C₈×C₆₇	—	2	4	no	—
70	C₂²×C₈×C₆₇	—	2	8	no	—
71	C₂²×C₈×C₆₇	—	2	8	no	—
72	C₂²×C₈×C₆₇	—	2	8	no	—
73	C₂²×C₈×C₆₇	—	2	32	no	—
74	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
75	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂³	1	8	yes	complemented
85	#123×C₆₇	C₂×C₄	1	16	yes	—
86	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	—
87	#121×C₆₇	C₂×C₄	1	16	yes	—
88	#120×C₆₇	C₂×C₄	1	16	yes	—
89	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	—
90	C₄×D₄×C₆₇	C₂³	1	16	yes	—
91	#117×C₆₇	C₂×C₄	1	4	yes	—
92	#117×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
98	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
99	C₄⋊C₈×C₆₇	C₂³	1	16	yes	complemented
100	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
101	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
102	#97×C₆₇	C₂³	1	8	yes	complemented
103	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
104	C₂⁴×C₆₇	—	2	4	no	—
105	C₂⁴×C₆₇	—	2	8	no	—
106	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
108	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
110	#41×C₆₇	—	2	16	no	—
111	#41×C₆₇	—	2	8	no	—
112	#41×C₆₇	—	2	8	no	—
113	C₂×Q₈×C₆₇	—	2	4	no	—
114	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
115	C₂×D₄×C₆₇	—	2	8	no	—
116	C₂×D₄×C₆₇	—	2	16	no	—
117	C₂×D₄×C₆₇	—	2	4	no	—
118	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
119	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
120	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	8	no	—
123	C₂²×C₄×C₆₇	—	2	16	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	8	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	8	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
131	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₆₇	#41	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
145	M₁₆×C₆₇	—	2	8	no	—
146	M₁₆×C₆₇	—	2	8	no	—
147	M₁₆×C₆₇	—	2	16	no	—
148	C₂×C₈×C₆₇	—	2	4	no	—
149	C₂×C₈×C₆₇	—	2	8	no	—
150	C₂×C₈×C₆₇	—	2	4	no	—
151	C₂×C₈×C₆₇	—	2	4	no	—
152	C₂×C₈×C₆₇	—	2	16	no	—
153	C₂×C₈×C₆₇	—	2	16	no	—
154	C₂×C₈×C₆₇	—	2	16	no	—
155	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
156	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
157	C₂×C₈×C₆₇	C₂×D₄	1	16	yes	complemented
158	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
159	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
160	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
161	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
162	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
163	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
164	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
165	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
166	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
167	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
168	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
169	C₄²×C₆₇	C₂⁴	1	4	yes	—
170	C₂³×C₆₇	—	2	32	no	—
171	C₂³×C₆₇	—	2	8	no	—
172	C₂³×C₆₇	—	2	4	no	—
173	C₂³×C₆₇	—	2	4	no	—
174	C₂³×C₆₇	—	2	4	no	—
175	C₂³×C₆₇	—	2	16	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	8	no	—
178	C₂³×C₆₇	C₄×D₄	1	4	yes	—
179	C₂³×C₆₇	#131	1	8	yes	complemented
180	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
181	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
182	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
183	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
184	Q₈×C₆₇	—	2	8	no	—
185	D₄×C₆₇	—	2	16	no	—
186	D₄×C₆₇	—	2	32	no	—
187	D₄×C₆₇	—	2	8	no	—
188	C₂×C₄×C₆₇	—	2	8	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	8	no	—
192	C₂×C₄×C₆₇	—	2	8	no	—
193	C₂×C₄×C₆₇	—	2	16	no	—
194	C₂×C₄×C₆₇	—	2	8	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	C₄×D₄	1	4	yes	—
200	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
201	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
202	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
203	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
204	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
206	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
208	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
209	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
210	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
211	C₈×C₆₇	—	2	4	no	—
212	C₈×C₆₇	—	2	4	no	—
213	C₈×C₆₇	—	2	4	no	—
214	C₈×C₆₇	C₂²×D₄	1	4	yes	—
215	C₈×C₆₇	C₂²×D₄	1	4	yes	—
216	C₂²×C₆₇	—	2	32	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	2	8	no	—
219	C₂²×C₆₇	—	2	16	no	—
220	C₂²×C₆₇	—	2	4	no	—
221	C₂²×C₆₇	—	2	4	no	—
222	C₂²×C₆₇	—	2	4	no	—
223	C₂²×C₆₇	—	2	8	no	—
224	C₂²×C₆₇	—	4	8	no	—
225	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
226	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
227	C₂²×C₆₇	C₂×#131	1	4	yes	—
228	C₂²×C₆₇	C₂×#131	1	8	yes	complemented
229	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
230	C₄×C₆₇	—	2	4	no	—
231	C₄×C₆₇	—	2	4	no	—
232	C₄×C₆₇	—	2	8	no	—
233	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
234	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
235	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
236	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
237	C₂×#2929	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂×C₆₇	—	2	4	no	—
239	C₂×C₆₇	—	2	8	no	—
240	C₂×C₆₇	—	4	4	no	—
241	C₂×C₆₇	#2929	1	4	yes	direct product factor
242	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
243	C₂×C₆₇	C₂²×#131	1	2	yes	—
244	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
245	C₂⁴×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₂×#525	C₂×C₆₇	1	1	yes	characteristic
247	#2929	C₂×C₆₇	1	16	yes	direct product factor
248	C₂×#435	C₂×C₆₇	1	4	yes	complemented
249	C₂×C₈×D₄	C₂×C₆₇	1	4	yes	complemented
250	C₂×#424	C₂×C₆₇	1	1	yes	characteristic, complemented
251	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₂×#407	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
254	C₆₇	C₂×#2929	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
256	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×#131	—	2	8	no	—
258	C₂×#131	C₂²×C₆₇	1	8	yes	complemented
259	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
261	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
262	C₂³×C₈	—	2	8	no	—
263	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
264	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
265	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
266	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
267	C₂³×C₈	C₂²×C₆₇	1	8	yes	complemented
268	C₂³×C₈	C₂²×C₆₇	1	2	yes	complemented
269	#525	C₄×C₆₇	1	8	yes	—
270	#525	C₂²×C₆₇	1	8	yes	—
271	C₂×#123	C₄×C₆₇	1	2	yes	—
272	C₂×C₂²⋊Q₈	C₄×C₆₇	1	2	yes	—
273	C₂×#121	C₄×C₆₇	1	2	yes	—
274	C₂×#120	C₄×C₆₇	1	2	yes	—
275	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
276	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
277	#435	C₂²×C₆₇	1	32	yes	complemented
278	C₈×D₄	C₂²×C₆₇	1	32	yes	complemented
279	#424	C₂²×C₆₇	1	8	yes	complemented
280	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
281	#408	C₂²×C₆₇	1	8	yes	complemented
282	#407	C₂²×C₆₇	1	8	yes	complemented
283	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
284	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
285	#405	C₂²×C₆₇	1	8	yes	complemented
286	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
287	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
288	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
289	C₂×#41	—	2	4	no	—
290	C₂×#41	—	2	4	no	—
291	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
292	C₂×#41	C₂³×C₆₇	1	4	yes	—
293	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
294	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
295	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
296	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂³×C₄	—	2	4	no	—
298	C₂³×C₄	—	2	4	no	—
299	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
300	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
301	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
302	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
303	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
304	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
305	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
306	#131	—	2	32	no	—
307	C₂×M₁₆	—	2	4	no	—
308	C₂×M₁₆	—	2	4	no	—
309	C₂×M₁₆	—	2	8	no	—
310	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
311	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
312	C₂×M₁₆	C₂³×C₆₇	1	8	yes	complemented
313	C₂²×C₈	—	2	4	no	—
314	C₂²×C₈	—	2	8	no	—
315	C₂²×C₈	—	2	8	no	—
316	C₂²×C₈	—	2	8	no	—
317	C₂²×C₈	—	2	32	no	—
318	C₂²×C₈	D₄×C₆₇	1	8	yes	complemented
319	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
320	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
321	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
322	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
323	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
324	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
325	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
327	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
328	C₂²×C₈	C₂³×C₆₇	1	8	yes	complemented
329	#123	C₂×C₄×C₆₇	1	16	yes	—
330	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	—
331	#121	C₂×C₄×C₆₇	1	16	yes	—
332	#120	C₂×C₄×C₆₇	1	16	yes	—
333	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	—
334	C₄×D₄	C₂³×C₆₇	1	16	yes	—
335	#117	C₂×C₄×C₆₇	1	4	yes	—
336	#117	C₂³×C₆₇	1	4	yes	—
337	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
338	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
339	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
340	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
341	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
342	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
343	C₄⋊C₈	C₂³×C₆₇	1	16	yes	complemented
344	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
345	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
346	#97	C₂³×C₆₇	1	8	yes	complemented
347	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
348	C₂⁴	—	2	4	no	—
349	C₂⁴	—	2	8	no	—
350	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
351	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
352	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
353	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
354	#41	—	2	16	no	—
355	#41	—	2	8	no	—
356	#41	—	2	8	no	—
357	C₂×Q₈	—	2	4	no	—
358	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
359	C₂×D₄	—	2	8	no	—
360	C₂×D₄	—	2	16	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
363	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
364	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
365	C₂²×C₄	—	2	4	no	—
366	C₂²×C₄	—	2	8	no	—
367	C₂²×C₄	—	2	16	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	8	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	4	no	—
372	C₂²×C₄	—	2	8	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
375	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
376	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
377	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
378	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
379	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
380	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
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	#41×C₆₇	1	4	yes	—
385	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
386	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
387	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
388	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
389	M₁₆	—	2	8	no	—
390	M₁₆	—	2	8	no	—
391	M₁₆	—	2	16	no	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	—	2	8	no	—
394	C₂×C₈	—	2	4	no	—
395	C₂×C₈	—	2	4	no	—
396	C₂×C₈	—	2	16	no	—
397	C₂×C₈	—	2	16	no	—
398	C₂×C₈	—	2	16	no	—
399	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
400	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
401	C₂×C₈	C₂×D₄×C₆₇	1	16	yes	complemented
402	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
403	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
404	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
405	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
406	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
407	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
408	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
409	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
410	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
411	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
412	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
413	C₄²	C₂⁴×C₆₇	1	4	yes	—
414	C₂³	—	2	32	no	—
415	C₂³	—	2	8	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	16	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	8	no	—
422	C₂³	C₄×D₄×C₆₇	1	4	yes	—
423	C₂³	#131×C₆₇	1	8	yes	complemented
424	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
425	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
426	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
427	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
428	Q₈	—	2	8	no	—
429	D₄	—	2	16	no	—
430	D₄	—	2	32	no	—
431	D₄	—	2	8	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	8	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	16	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
444	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
445	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
446	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
447	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
448	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
450	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
451	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
452	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
453	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
454	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
455	C₈	—	2	4	no	—
456	C₈	—	2	4	no	—
457	C₈	—	2	4	no	—
458	C₈	C₂²×D₄×C₆₇	1	4	yes	—
459	C₈	C₂²×D₄×C₆₇	1	4	yes	—
460	C₂²	—	2	32	no	—
461	C₂²	—	2	4	no	—
462	C₂²	—	2	4	no	—
463	C₂²	—	2	4	no	—
464	C₂²	—	2	8	no	—
465	C₂²	—	2	16	no	—
466	C₂²	—	2	8	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	8	no	—
469	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
470	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
471	C₂²	C₂×#131×C₆₇	1	4	yes	—
472	C₂²	C₂×#131×C₆₇	1	8	yes	complemented
473	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
474	C₄	—	2	8	no	—
475	C₄	—	2	4	no	—
476	C₄	—	2	4	no	—
477	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
478	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
479	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
480	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
481	C₂	—	2	4	no	—
482	C₂	—	2	8	no	—
483	C₂	—	4	4	no	—
484	C₂	C₆₇×#2929	1	4	yes	direct product factor
485	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
486	C₂	C₂²×#131×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₆₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#525×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#131×C₆₇	C₂	semidirect
2. C₂⁴×C₈×C₆₇	C₂	semidirect
4. C₆₇×#2929	C₂	direct (N × H)
5. C₂×#435×C₆₇	C₂	semidirect
6. C₂×C₈×D₄×C₆₇	C₂	semidirect
7. C₂×#424×C₆₇	C₂	semidirect
8. C₂×#408×C₆₇	C₂	semidirect
9. C₂×#407×C₆₇	C₂	semidirect
10. C₂×#405×C₆₇	C₂	semidirect
14. C₂×#131×C₆₇	C₂²	semidirect
17. C₂²×M₁₆×C₆₇	C₂²	semidirect
23. C₂³×C₈×C₆₇	C₂²	semidirect
24. C₂³×C₈×C₆₇	C₂²	semidirect
33. #435×C₆₇	C₂²	semidirect
34. C₈×D₄×C₆₇	C₂²	semidirect
35. #424×C₆₇	C₂²	semidirect
36. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
37. #408×C₆₇	C₂²	semidirect
38. #407×C₆₇	C₂²	semidirect
40. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
41. #405×C₆₇	C₂²	semidirect
42. C₂×#97×C₆₇	C₂²	semidirect
43. C₂×C₄×C₈×C₆₇	C₂²	semidirect
68. C₂×M₁₆×C₆₇	C₂³	semidirect
74. C₂²×C₈×C₆₇	D₄	semidirect
84. C₂²×C₈×C₆₇	C₂³	semidirect
99. C₄⋊C₈×C₆₇	C₂³	semidirect
101. C₂²⋊C₈×C₆₇	C₂³	semidirect
102. #97×C₆₇	C₂³	semidirect
103. C₄×C₈×C₆₇	C₂³	semidirect
157. C₂×C₈×C₆₇	C₂×D₄	semidirect
179. C₂³×C₆₇	#131	semidirect
228. C₂²×C₆₇	C₂×#131	semidirect
237. C₂×#2929	C₆₇	direct (N × H)
241. C₂×C₆₇	#2929	direct (N × H)
244. C₂²×#131	C₂×C₆₇	semidirect
245. C₂⁴×C₈	C₂×C₆₇	semidirect
247. #2929	C₂×C₆₇	direct (N × H)
248. C₂×#435	C₂×C₆₇	semidirect
249. C₂×C₈×D₄	C₂×C₆₇	semidirect
250. C₂×#424	C₂×C₆₇	semidirect
251. C₂×#408	C₂×C₆₇	semidirect
252. C₂×#407	C₂×C₆₇	semidirect
253. C₂×#405	C₂×C₆₇	semidirect
254. C₆₇	C₂×#2929	direct (N × H)
258. C₂×#131	C₂²×C₆₇	semidirect
261. C₂²×M₁₆	C₂²×C₆₇	semidirect
267. C₂³×C₈	C₂²×C₆₇	semidirect
268. C₂³×C₈	C₂²×C₆₇	semidirect
277. #435	C₂²×C₆₇	semidirect
278. C₈×D₄	C₂²×C₆₇	semidirect
279. #424	C₂²×C₆₇	semidirect
280. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
281. #408	C₂²×C₆₇	semidirect
282. #407	C₂²×C₆₇	semidirect
284. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
285. #405	C₂²×C₆₇	semidirect
286. C₂×#97	C₂²×C₆₇	semidirect
287. C₂×C₄×C₈	C₂²×C₆₇	semidirect
312. C₂×M₁₆	C₂³×C₆₇	semidirect
318. C₂²×C₈	D₄×C₆₇	semidirect
328. C₂²×C₈	C₂³×C₆₇	semidirect
343. C₄⋊C₈	C₂³×C₆₇	semidirect
345. C₂²⋊C₈	C₂³×C₆₇	semidirect
346. #97	C₂³×C₆₇	semidirect
347. C₄×C₈	C₂³×C₆₇	semidirect
401. C₂×C₈	C₂×D₄×C₆₇	semidirect
423. C₂³	#131×C₆₇	semidirect
472. C₂²	C₂×#131×C₆₇	semidirect
484. C₂	C₆₇×#2929	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₆₇	2
C₂⁴×C₈×C₆₇	2
C₂×#525×C₆₇	2
C₆₇×#2929	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#424×C₆₇	2
C₂×#408×C₆₇	2
C₂×#407×C₆₇	2
C₂×#405×C₆₇	2
C₂×#2929	67

Subgroups — C2×C67×#2929 #30739384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2929 #30739384