Subgroups — C₂×C₆₇×#2968 #30739451

Order 17152.

2026 conjugacy classes of subgroups.

The subgroup lattice has 478 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	C₆₇×#2968	C₂	1	16	yes	maximal, direct product factor
3	C₂×#446×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#433×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#405×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
12	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	#529×C₆₇	C₄	1	8	yes	—
15	#529×C₆₇	C₂²	1	8	yes	—
16	C₂×#126×C₆₇	C₄	1	2	yes	—
17	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#123×C₆₇	C₄	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
21	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#446×C₆₇	C₂²	1	16	yes	complemented
28	#435×C₆₇	C₂²	1	16	yes	complemented
29	#435×C₆₇	C₂²	1	16	yes	complemented
30	C₈×D₄×C₆₇	C₂²	1	16	yes	complemented
31	#433×C₆₇	C₂²	1	16	yes	complemented
32	#431×C₆₇	C₂²	1	8	yes	complemented
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
40	#405×C₆₇	C₂²	1	16	yes	complemented
41	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#97×C₆₇	C₂²	1	2	yes	—
44	C₂×#97×C₆₇	C₂²	1	2	yes	—
45	C₂×#97×C₆₇	C₂²	1	8	yes	complemented
46	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
48	C₂²×Q₈×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	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×M₁₆×C₆₇	—	2	8	no	—
56	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
57	C₂²×C₈×C₆₇	—	2	8	no	—
58	C₂²×C₈×C₆₇	—	2	8	no	—
59	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	#126×C₆₇	C₂×C₄	1	16	yes	—
66	#125×C₆₇	C₂×C₄	1	8	yes	—
67	#124×C₆₇	C₂×C₄	1	8	yes	—
68	#123×C₆₇	C₂×C₄	1	16	yes	—
69	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
70	#121×C₆₇	C₂×C₄	1	8	yes	—
71	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
72	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
73	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
74	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
75	C₄×D₄×C₆₇	C₂³	1	8	yes	—
76	C₄×D₄×C₆₇	C₂³	1	4	yes	—
77	#117×C₆₇	C₂×C₄	1	8	yes	—
78	#117×C₆₇	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
97	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
98	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
99	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
100	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
101	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
102	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
103	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
104	#97×C₆₇	C₂³	1	4	yes	—
105	#97×C₆₇	C₂³	1	4	yes	—
106	#97×C₆₇	C₂³	1	8	yes	—
107	#97×C₆₇	C₂³	1	8	yes	—
108	C₄×C₈×C₆₇	C₂³	1	8	yes	—
109	C₄×C₈×C₆₇	C₂³	1	8	yes	—
110	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Q₈×C₆₇	—	4	2	no	—
113	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	4	2	no	—
116	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
117	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	4	2	no	—
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	2	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	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₆₇	—	2	16	no	—
139	C₂×C₈×C₆₇	—	2	4	no	—
140	C₂×C₈×C₆₇	—	2	4	no	—
141	C₂×C₈×C₆₇	—	2	4	no	—
142	C₂×C₈×C₆₇	—	2	16	no	—
143	C₂×C₈×C₆₇	—	2	16	no	—
144	C₂×C₈×C₆₇	#41	1	8	yes	—
145	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
146	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
147	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
148	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
149	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
150	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
152	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
153	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
154	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
155	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
156	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
157	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
158	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
159	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
160	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
161	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
162	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
163	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
164	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
165	C₄²×C₆₇	C₂⁴	1	2	yes	—
166	C₄²×C₆₇	C₂⁴	1	4	yes	—
167	C₄²×C₆₇	C₂⁴	1	2	yes	—
168	C₂³×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	4	no	—
171	C₂³×C₆₇	—	2	4	no	—
172	C₂³×C₆₇	—	2	4	no	—
173	C₂³×C₆₇	#131	1	4	yes	—
174	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
175	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
176	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
177	Q₈×C₆₇	—	4	4	no	—
178	D₄×C₆₇	—	2	16	no	—
179	D₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	8	no	—
191	C₂×C₄×C₆₇	—	2	4	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	4	4	no	—
195	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
196	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
198	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
199	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
200	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
201	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
202	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
203	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
205	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
206	C₈×C₆₇	—	2	4	no	—
207	C₈×C₆₇	—	2	4	no	—
208	C₈×C₆₇	—	2	4	no	—
209	C₈×C₆₇	C₂×#41	1	8	yes	—
210	C₂²×C₆₇	—	2	2	no	—
211	C₂²×C₆₇	—	2	8	no	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	4	no	—
214	C₂²×C₆₇	—	2	4	no	—
215	C₂²×C₆₇	—	2	4	no	—
216	C₂²×C₆₇	—	4	8	no	—
217	C₂²×C₆₇	C₄×#41	1	2	yes	—
218	C₂²×C₆₇	C₂×#131	1	4	yes	—
219	C₂²×C₆₇	C₂×#131	1	2	yes	—
220	C₂²×C₆₇	#568	1	2	yes	—
221	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	2	4	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	4	no	—
230	C₄×C₆₇	C₄×#41	1	2	yes	—
231	C₄×C₆₇	C₂²×#41	1	2	yes	—
232	C₂×#2968	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂×C₆₇	—	2	4	no	—
234	C₂×C₆₇	—	4	4	no	—
235	C₂×C₆₇	#2968	1	4	yes	direct product factor
236	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
237	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
238	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
239	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
240	#2968	C₂×C₆₇	1	16	yes	direct product factor
241	C₂×#446	C₂×C₆₇	1	2	yes	complemented
242	C₂×#435	C₂×C₆₇	1	2	yes	complemented
243	C₂×#435	C₂×C₆₇	1	2	yes	complemented
244	C₂×C₈×D₄	C₂×C₆₇	1	2	yes	complemented
245	C₂×#433	C₂×C₆₇	1	2	yes	complemented
246	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
247	C₂×#405	C₂×C₆₇	1	2	yes	complemented
248	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
249	C₆₇	C₂×#2968	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
251	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
252	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
253	#529	C₄×C₆₇	1	8	yes	—
254	#529	C₂²×C₆₇	1	8	yes	—
255	C₂×#126	C₄×C₆₇	1	2	yes	—
256	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
257	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
258	C₂×#123	C₄×C₆₇	1	2	yes	—
259	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
260	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
261	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
263	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#117	C₂²×C₆₇	1	2	yes	—
265	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
266	#446	C₂²×C₆₇	1	16	yes	complemented
267	#435	C₂²×C₆₇	1	16	yes	complemented
268	#435	C₂²×C₆₇	1	16	yes	complemented
269	C₈×D₄	C₂²×C₆₇	1	16	yes	complemented
270	#433	C₂²×C₆₇	1	16	yes	complemented
271	#431	C₂²×C₆₇	1	8	yes	complemented
272	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
273	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
274	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
275	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
278	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
279	#405	C₂²×C₆₇	1	16	yes	complemented
280	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
282	C₂×#97	C₂²×C₆₇	1	2	yes	—
283	C₂×#97	C₂²×C₆₇	1	2	yes	—
284	C₂×#97	C₂²×C₆₇	1	8	yes	complemented
285	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
286	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
287	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
289	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
290	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
291	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
293	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×M₁₆	—	2	8	no	—
295	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
296	C₂²×C₈	—	2	8	no	—
297	C₂²×C₈	—	2	8	no	—
298	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
299	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
300	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
301	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
302	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
303	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
304	#126	C₂×C₄×C₆₇	1	16	yes	—
305	#125	C₂×C₄×C₆₇	1	8	yes	—
306	#124	C₂×C₄×C₆₇	1	8	yes	—
307	#123	C₂×C₄×C₆₇	1	16	yes	—
308	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
309	#121	C₂×C₄×C₆₇	1	8	yes	—
310	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
311	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
312	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
313	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
314	C₄×D₄	C₂³×C₆₇	1	8	yes	—
315	C₄×D₄	C₂³×C₆₇	1	4	yes	—
316	#117	C₂×C₄×C₆₇	1	8	yes	—
317	#117	C₂³×C₆₇	1	8	yes	—
318	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
319	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
320	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
321	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
322	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
324	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
325	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
326	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
327	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
328	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
329	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
330	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
331	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
332	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
334	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
336	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
337	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
338	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
339	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
340	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
341	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
342	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
343	#97	C₂³×C₆₇	1	4	yes	—
344	#97	C₂³×C₆₇	1	4	yes	—
345	#97	C₂³×C₆₇	1	8	yes	—
346	#97	C₂³×C₆₇	1	8	yes	—
347	C₄×C₈	C₂³×C₆₇	1	8	yes	—
348	C₄×C₈	C₂³×C₆₇	1	8	yes	—
349	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
350	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
351	C₂×Q₈	—	4	2	no	—
352	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
353	C₂×D₄	—	2	8	no	—
354	C₂×D₄	—	4	2	no	—
355	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
356	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	4	2	no	—
362	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
364	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
369	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
372	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
373	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
374	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
375	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
377	M₁₆	—	2	16	no	—
378	C₂×C₈	—	2	4	no	—
379	C₂×C₈	—	2	4	no	—
380	C₂×C₈	—	2	4	no	—
381	C₂×C₈	—	2	16	no	—
382	C₂×C₈	—	2	16	no	—
383	C₂×C₈	#41×C₆₇	1	8	yes	—
384	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
385	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
386	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
387	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
388	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
389	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
390	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
391	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
392	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
393	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
394	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
395	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
396	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
397	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
398	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
399	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
400	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
401	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
402	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
403	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
404	C₄²	C₂⁴×C₆₇	1	2	yes	—
405	C₄²	C₂⁴×C₆₇	1	4	yes	—
406	C₄²	C₂⁴×C₆₇	1	2	yes	—
407	C₂³	—	2	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	4	no	—
410	C₂³	—	2	4	no	—
411	C₂³	—	2	4	no	—
412	C₂³	#131×C₆₇	1	4	yes	—
413	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
414	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
415	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
416	Q₈	—	4	4	no	—
417	D₄	—	2	16	no	—
418	D₄	—	4	4	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	8	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	8	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	4	4	no	—
434	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
435	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
436	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
437	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
438	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
439	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
440	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
441	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
442	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
444	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
445	C₈	—	2	4	no	—
446	C₈	—	2	4	no	—
447	C₈	—	2	4	no	—
448	C₈	C₂×#41×C₆₇	1	8	yes	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	8	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	2	4	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	4	8	no	—
456	C₂²	C₄×#41×C₆₇	1	2	yes	—
457	C₂²	C₂×#131×C₆₇	1	4	yes	—
458	C₂²	C₂×#131×C₆₇	1	2	yes	—
459	C₂²	#568×C₆₇	1	2	yes	—
460	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
461	C₄	—	2	2	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	4	no	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	2	4	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	4	no	—
469	C₄	C₄×#41×C₆₇	1	2	yes	—
470	C₄	C₂²×#41×C₆₇	1	2	yes	—
471	C₂	—	2	4	no	—
472	C₂	—	4	4	no	—
473	C₂	C₆₇×#2968	1	4	yes	direct product factor
474	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
475	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic
476	C₂	C₂×#568×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#121×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2968	C₂	direct (N × H)
3. C₂×#446×C₆₇	C₂	semidirect
4. C₂×#435×C₆₇	C₂	semidirect
5. C₂×#435×C₆₇	C₂	semidirect
6. C₂×C₈×D₄×C₆₇	C₂	semidirect
7. C₂×#433×C₆₇	C₂	semidirect
8. C₂×#431×C₆₇	C₂	semidirect
9. C₂×#405×C₆₇	C₂	semidirect
10. C₂²×#97×C₆₇	C₂	semidirect
27. #446×C₆₇	C₂²	semidirect
28. #435×C₆₇	C₂²	semidirect
29. #435×C₆₇	C₂²	semidirect
30. C₈×D₄×C₆₇	C₂²	semidirect
31. #433×C₆₇	C₂²	semidirect
32. #431×C₆₇	C₂²	semidirect
40. #405×C₆₇	C₂²	semidirect
45. C₂×#97×C₆₇	C₂²	semidirect
232. C₂×#2968	C₆₇	direct (N × H)
235. C₂×C₆₇	#2968	direct (N × H)
240. #2968	C₂×C₆₇	direct (N × H)
241. C₂×#446	C₂×C₆₇	semidirect
242. C₂×#435	C₂×C₆₇	semidirect
243. C₂×#435	C₂×C₆₇	semidirect
244. C₂×C₈×D₄	C₂×C₆₇	semidirect
245. C₂×#433	C₂×C₆₇	semidirect
246. C₂×#431	C₂×C₆₇	semidirect
247. C₂×#405	C₂×C₆₇	semidirect
248. C₂²×#97	C₂×C₆₇	semidirect
249. C₆₇	C₂×#2968	direct (N × H)
266. #446	C₂²×C₆₇	semidirect
267. #435	C₂²×C₆₇	semidirect
268. #435	C₂²×C₆₇	semidirect
269. C₈×D₄	C₂²×C₆₇	semidirect
270. #433	C₂²×C₆₇	semidirect
271. #431	C₂²×C₆₇	semidirect
279. #405	C₂²×C₆₇	semidirect
284. C₂×#97	C₂²×C₆₇	semidirect
473. C₂	C₆₇×#2968	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
C₆₇×#2968	2
C₂×#446×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#433×C₆₇	2
C₂×#431×C₆₇	2
C₂×#405×C₆₇	2
C₂²×#97×C₆₇	2
C₂×#2968	67

Subgroups — C2×C67×#2968 #30739451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2968 #30739451