Subgroups — C₂×C₆₇×#2952 #30739417

Order 17152.

2806 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2952	C₂	1	16	yes	maximal, direct product factor
4	C₂×#442×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#441×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×SD₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#417×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	2	yes	—
11	C₂×#135×C₆₇	C₄	1	16	yes	complemented
12	C₂×#135×C₆₇	C₂²	1	8	yes	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#520×C₆₇	—	2	8	no	—
19	#520×C₆₇	C₂²	1	16	yes	complemented
20	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₆₇	—	2	4	no	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
23	C₂×#117×C₆₇	—	2	2	no	—
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	#442×C₆₇	C₂²	1	16	yes	complemented
28	#441×C₆₇	C₂²	1	16	yes	complemented
29	C₄×SD₁₆×C₆₇	C₂²	1	32	yes	complemented
30	C₂×#106×C₆₇	C₂²	1	2	yes	—
31	C₂×#106×C₆₇	C₂²	1	2	yes	—
32	C₂×#106×C₆₇	C₂²	1	8	yes	complemented
33	#417×C₆₇	C₂²	1	16	yes	complemented
34	C₂×#103×C₆₇	C₂²	1	4	yes	—
35	C₂×#102×C₆₇	C₂²	1	4	yes	—
36	#405×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#97×C₆₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
39	C₂×#41×C₆₇	—	2	4	no	—
40	C₂×#41×C₆₇	—	2	8	no	—
41	C₂×#41×C₆₇	C₂×C₄	1	16	yes	complemented
42	C₂×#41×C₆₇	C₂³	1	8	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	—	2	2	no	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	—	2	4	no	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	#135×C₆₇	C₂×C₄	1	64	yes	complemented
51	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
53	C₂×SD₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
54	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
55	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
56	C₂×D₈×C₆₇	C₂³	1	4	yes	—
57	C₂×M₁₆×C₆₇	—	2	4	no	—
58	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
64	C₄×Q₈×C₆₇	—	2	8	no	—
65	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
66	C₄×D₄×C₆₇	—	2	32	no	—
67	C₄×D₄×C₆₇	—	2	8	no	—
68	C₄×D₄×C₆₇	C₂³	1	16	yes	—
69	#117×C₆₇	—	2	16	no	—
70	#117×C₆₇	D₄	1	4	yes	—
71	#117×C₆₇	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
81	C₂×C₄²×C₆₇	—	2	4	no	—
82	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
83	#106×C₆₇	C₂³	1	8	yes	—
84	#106×C₆₇	C₂³	1	8	yes	—
85	#103×C₆₇	C₂³	1	16	yes	—
86	#102×C₆₇	C₂³	1	16	yes	—
87	#97×C₆₇	C₂³	1	8	yes	—
88	C₄×C₈×C₆₇	C₂³	1	8	yes	—
89	C₂⁴×C₆₇	—	2	2	no	—
90	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
91	#41×C₆₇	—	2	32	no	—
92	#41×C₆₇	—	2	16	no	—
93	#41×C₆₇	C₂²×C₄	1	32	yes	complemented
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
97	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
98	C₂×D₄×C₆₇	—	2	16	no	—
99	C₂×D₄×C₆₇	—	2	8	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
103	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	16	no	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	16	no	—
110	C₂²×C₄×C₆₇	—	8	8	no	—
111	C₂²×C₄×C₆₇	—	8	4	no	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	#41	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
119	Q₁₆×C₆₇	C₂²×C₄	1	16	yes	complemented
120	SD₁₆×C₆₇	C₂²×C₄	1	32	yes	complemented
121	D₈×C₆₇	C₂²×C₄	1	16	yes	complemented
122	M₁₆×C₆₇	—	2	8	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	complemented
127	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
128	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
129	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	16	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
134	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
135	C₂²⋊C₄×C₆₇	—	2	16	no	—
136	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
137	C₄²×C₆₇	—	2	16	no	—
138	C₄²×C₆₇	C₂×D₄	1	4	yes	—
139	C₄²×C₆₇	C₂⁴	1	4	yes	—
140	C₂³×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	2	8	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	4	4	no	—
144	C₂³×C₆₇	—	4	4	no	—
145	C₂³×C₆₇	C₄×D₄	1	4	yes	—
146	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
147	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₆₇	—	2	8	no	—
149	Q₈×C₆₇	—	2	4	no	—
150	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
151	D₄×C₆₇	—	2	32	no	—
152	D₄×C₆₇	—	2	8	no	—
153	D₄×C₆₇	—	2	4	no	—
154	D₄×C₆₇	C₂³×C₄	1	8	yes	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	32	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	4	8	no	—
161	C₂×C₄×C₆₇	—	8	16	no	—
162	C₂×C₄×C₆₇	—	8	8	no	—
163	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
164	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
165	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
166	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
167	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
171	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
172	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
173	C₈×C₆₇	—	2	4	no	—
174	C₈×C₆₇	C₂³×C₄	1	4	yes	—
175	C₈×C₆₇	C₂³×C₄	1	4	yes	—
176	C₂²×C₆₇	—	2	8	no	—
177	C₂²×C₆₇	—	2	4	no	—
178	C₂²×C₆₇	—	2	8	no	—
179	C₂²×C₆₇	—	4	16	no	—
180	C₂²×C₆₇	—	4	4	no	—
181	C₂²×C₆₇	—	4	4	no	—
182	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
183	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
184	C₂²×C₆₇	#577	1	4	yes	—
185	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
186	C₄×C₆₇	—	2	8	no	—
187	C₄×C₆₇	—	2	8	no	—
188	C₄×C₆₇	—	4	8	no	—
189	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
190	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
191	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
192	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
193	C₂×#2952	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂×C₆₇	—	2	4	no	—
195	C₂×C₆₇	—	4	8	no	—
196	C₂×C₆₇	#2952	1	4	yes	direct product factor
197	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#577	1	2	yes	—
199	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
200	C₂×#520	C₂×C₆₇	1	2	yes	complemented
201	#2952	C₂×C₆₇	1	16	yes	direct product factor
202	C₂×#442	C₂×C₆₇	1	2	yes	complemented
203	C₂×#441	C₂×C₆₇	1	2	yes	complemented
204	C₂×C₄×SD₁₆	C₂×C₆₇	1	4	yes	complemented
205	C₂²×#106	C₂×C₆₇	1	1	yes	characteristic, complemented
206	C₂×#417	C₂×C₆₇	1	2	yes	complemented
207	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
208	C₆₇	C₂×#2952	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×#41	C₂²×C₆₇	1	2	yes	—
210	C₂×#135	C₄×C₆₇	1	16	yes	complemented
211	C₂×#135	C₂²×C₆₇	1	8	yes	—
212	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
213	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
214	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
215	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
216	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
217	#520	—	2	8	no	—
218	#520	C₂²×C₆₇	1	16	yes	complemented
219	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
220	C₂×C₄×D₄	—	2	4	no	—
221	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
222	C₂×#117	—	2	2	no	—
223	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
224	C₂²×C₄⋊C₄	—	2	2	no	—
225	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
226	#442	C₂²×C₆₇	1	16	yes	complemented
227	#441	C₂²×C₆₇	1	16	yes	complemented
228	C₄×SD₁₆	C₂²×C₆₇	1	32	yes	complemented
229	C₂×#106	C₂²×C₆₇	1	2	yes	—
230	C₂×#106	C₂²×C₆₇	1	2	yes	—
231	C₂×#106	C₂²×C₆₇	1	8	yes	complemented
232	#417	C₂²×C₆₇	1	16	yes	complemented
233	C₂×#103	C₂²×C₆₇	1	4	yes	—
234	C₂×#102	C₂²×C₆₇	1	4	yes	—
235	#405	C₂²×C₆₇	1	8	yes	complemented
236	C₂×#97	C₂²×C₆₇	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
238	C₂×#41	—	2	4	no	—
239	C₂×#41	—	2	8	no	—
240	C₂×#41	C₂×C₄×C₆₇	1	16	yes	complemented
241	C₂×#41	C₂³×C₆₇	1	8	yes	—
242	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
243	C₂²×D₄	—	2	2	no	—
244	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
245	C₂³×C₄	—	2	2	no	—
246	C₂³×C₄	—	2	2	no	—
247	C₂³×C₄	—	2	4	no	—
248	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
249	#135	C₂×C₄×C₆₇	1	64	yes	complemented
250	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
251	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
252	C₂×SD₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
253	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
254	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
255	C₂×D₈	C₂³×C₆₇	1	4	yes	—
256	C₂×M₁₆	—	2	4	no	—
257	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
258	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
259	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
260	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
263	C₄×Q₈	—	2	8	no	—
264	C₄×Q₈	C₂³×C₆₇	1	16	yes	—
265	C₄×D₄	—	2	32	no	—
266	C₄×D₄	—	2	8	no	—
267	C₄×D₄	C₂³×C₆₇	1	16	yes	—
268	#117	—	2	16	no	—
269	#117	D₄×C₆₇	1	4	yes	—
270	#117	C₂³×C₆₇	1	4	yes	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	4	no	—
273	C₂×C₄⋊C₄	—	2	4	no	—
274	C₂×C₄⋊C₄	—	2	16	no	—
275	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
277	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
278	C₂×C₂²⋊C₄	—	2	4	no	—
279	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
280	C₂×C₄²	—	2	4	no	—
281	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
282	#106	C₂³×C₆₇	1	8	yes	—
283	#106	C₂³×C₆₇	1	8	yes	—
284	#103	C₂³×C₆₇	1	16	yes	—
285	#102	C₂³×C₆₇	1	16	yes	—
286	#97	C₂³×C₆₇	1	8	yes	—
287	C₄×C₈	C₂³×C₆₇	1	8	yes	—
288	C₂⁴	—	2	2	no	—
289	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
290	#41	—	2	32	no	—
291	#41	—	2	16	no	—
292	#41	C₂²×C₄×C₆₇	1	32	yes	complemented
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	4	no	—
295	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
296	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
297	C₂×D₄	—	2	16	no	—
298	C₂×D₄	—	2	8	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	4	no	—
301	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
302	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	16	no	—
305	C₂²×C₄	—	2	8	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	—	2	16	no	—
309	C₂²×C₄	—	8	8	no	—
310	C₂²×C₄	—	8	4	no	—
311	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
313	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
315	C₂²×C₄	#41×C₆₇	1	4	yes	—
316	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
318	Q₁₆	C₂²×C₄×C₆₇	1	16	yes	complemented
319	SD₁₆	C₂²×C₄×C₆₇	1	32	yes	complemented
320	D₈	C₂²×C₄×C₆₇	1	16	yes	complemented
321	M₁₆	—	2	8	no	—
322	C₂×C₈	—	2	4	no	—
323	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
324	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
325	C₂×C₈	C₂²×C₄×C₆₇	1	16	yes	complemented
326	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
327	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
328	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	16	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
333	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
334	C₂²⋊C₄	—	2	16	no	—
335	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
336	C₄²	—	2	16	no	—
337	C₄²	C₂×D₄×C₆₇	1	4	yes	—
338	C₄²	C₂⁴×C₆₇	1	4	yes	—
339	C₂³	—	2	4	no	—
340	C₂³	—	2	8	no	—
341	C₂³	—	2	4	no	—
342	C₂³	—	4	4	no	—
343	C₂³	—	4	4	no	—
344	C₂³	C₄×D₄×C₆₇	1	4	yes	—
345	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
346	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
347	Q₈	—	2	8	no	—
348	Q₈	—	2	4	no	—
349	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
350	D₄	—	2	32	no	—
351	D₄	—	2	8	no	—
352	D₄	—	2	4	no	—
353	D₄	C₂³×C₄×C₆₇	1	8	yes	—
354	C₂×C₄	—	2	8	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	32	no	—
358	C₂×C₄	—	2	8	no	—
359	C₂×C₄	—	4	8	no	—
360	C₂×C₄	—	8	16	no	—
361	C₂×C₄	—	8	8	no	—
362	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
363	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
364	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
365	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
366	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
367	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
369	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
370	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
371	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
372	C₈	—	2	4	no	—
373	C₈	C₂³×C₄×C₆₇	1	4	yes	—
374	C₈	C₂³×C₄×C₆₇	1	4	yes	—
375	C₂²	—	2	4	no	—
376	C₂²	—	2	8	no	—
377	C₂²	—	2	8	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	4	16	no	—
381	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
382	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
383	C₂²	#577×C₆₇	1	4	yes	—
384	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
385	C₄	—	2	8	no	—
386	C₄	—	2	8	no	—
387	C₄	—	4	8	no	—
388	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
389	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
390	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
391	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
392	C₂	—	2	4	no	—
393	C₂	—	4	8	no	—
394	C₂	C₆₇×#2952	1	4	yes	direct product factor
395	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
396	C₂	C₂×#577×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#520×C₆₇	C₂	semidirect
3. C₆₇×#2952	C₂	direct (N × H)
4. C₂×#442×C₆₇	C₂	semidirect
5. C₂×#441×C₆₇	C₂	semidirect
6. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
7. C₂²×#106×C₆₇	C₂	semidirect
8. C₂×#417×C₆₇	C₂	semidirect
9. C₂×#405×C₆₇	C₂	semidirect
11. C₂×#135×C₆₇	C₄	semidirect
19. #520×C₆₇	C₂²	semidirect
27. #442×C₆₇	C₂²	semidirect
28. #441×C₆₇	C₂²	semidirect
29. C₄×SD₁₆×C₆₇	C₂²	semidirect
32. C₂×#106×C₆₇	C₂²	semidirect
33. #417×C₆₇	C₂²	semidirect
36. #405×C₆₇	C₂²	semidirect
41. C₂×#41×C₆₇	C₂×C₄	semidirect
50. #135×C₆₇	C₂×C₄	semidirect
51. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
53. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
55. C₂×D₈×C₆₇	C₂×C₄	semidirect
59. C₂²×C₈×C₆₇	C₂×C₄	semidirect
93. #41×C₆₇	C₂²×C₄	semidirect
119. Q₁₆×C₆₇	C₂²×C₄	semidirect
120. SD₁₆×C₆₇	C₂²×C₄	semidirect
121. D₈×C₆₇	C₂²×C₄	semidirect
126. C₂×C₈×C₆₇	C₂²×C₄	semidirect
193. C₂×#2952	C₆₇	direct (N × H)
196. C₂×C₆₇	#2952	direct (N × H)
200. C₂×#520	C₂×C₆₇	semidirect
201. #2952	C₂×C₆₇	direct (N × H)
202. C₂×#442	C₂×C₆₇	semidirect
203. C₂×#441	C₂×C₆₇	semidirect
204. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
205. C₂²×#106	C₂×C₆₇	semidirect
206. C₂×#417	C₂×C₆₇	semidirect
207. C₂×#405	C₂×C₆₇	semidirect
208. C₆₇	C₂×#2952	direct (N × H)
210. C₂×#135	C₄×C₆₇	semidirect
218. #520	C₂²×C₆₇	semidirect
226. #442	C₂²×C₆₇	semidirect
227. #441	C₂²×C₆₇	semidirect
228. C₄×SD₁₆	C₂²×C₆₇	semidirect
231. C₂×#106	C₂²×C₆₇	semidirect
232. #417	C₂²×C₆₇	semidirect
235. #405	C₂²×C₆₇	semidirect
240. C₂×#41	C₂×C₄×C₆₇	semidirect
249. #135	C₂×C₄×C₆₇	semidirect
250. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
252. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
254. C₂×D₈	C₂×C₄×C₆₇	semidirect
258. C₂²×C₈	C₂×C₄×C₆₇	semidirect
292. #41	C₂²×C₄×C₆₇	semidirect
318. Q₁₆	C₂²×C₄×C₆₇	semidirect
319. SD₁₆	C₂²×C₄×C₆₇	semidirect
320. D₈	C₂²×C₄×C₆₇	semidirect
325. C₂×C₈	C₂²×C₄×C₆₇	semidirect
394. C₂	C₆₇×#2952	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×#135×C₆₇	2
C₂×#520×C₆₇	2
C₆₇×#2952	2
C₂×#442×C₆₇	2
C₂×#441×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂²×#106×C₆₇	2
C₂×#417×C₆₇	2
C₂×#405×C₆₇	2
C₂×#2952	67

Subgroups — C2×C67×#2952 #30739417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2952 #30739417