Subgroups — C₂×C₆₇×#2969 #30739452

Order 17152.

2014 conjugacy classes of subgroups.

The subgroup lattice has 522 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×#2969	C₂	1	16	yes	maximal, direct product factor
3	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#433×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#405×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
17	#529×C₆₇	C₄	1	8	yes	—
18	#529×C₆₇	C₂²	1	8	yes	—
19	C₂×#126×C₆₇	C₄	1	2	yes	—
20	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
21	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
22	C₂×#123×C₆₇	C₄	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
24	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×#117×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	#446×C₆₇	C₂²	1	8	yes	complemented
31	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
32	#436×C₆₇	C₂²	1	8	yes	complemented
33	#436×C₆₇	C₂²	1	16	yes	complemented
34	#435×C₆₇	C₂²	1	16	yes	complemented
35	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
36	#433×C₆₇	C₂²	1	16	yes	complemented
37	#431×C₆₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
42	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
43	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
45	#405×C₆₇	C₂²	1	16	yes	complemented
46	C₄×M₁₆×C₆₇	C₂²	1	8	yes	complemented
47	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#97×C₆₇	C₂²	1	2	yes	—
50	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
53	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
54	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×M₁₆×C₆₇	—	2	4	no	—
62	C₂×M₁₆×C₆₇	—	2	4	no	—
63	C₂×M₁₆×C₆₇	—	2	8	no	—
64	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
67	C₂²×C₈×C₆₇	—	2	8	no	—
68	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#126×C₆₇	C₂×C₄	1	16	yes	—
76	#125×C₆₇	C₂×C₄	1	8	yes	—
77	#124×C₆₇	C₂×C₄	1	8	yes	—
78	#123×C₆₇	C₂×C₄	1	16	yes	—
79	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
80	#121×C₆₇	C₂×C₄	1	8	yes	—
81	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
82	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
83	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
84	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
85	C₄×D₄×C₆₇	C₂³	1	4	yes	—
86	C₄×D₄×C₆₇	C₂³	1	8	yes	—
87	#117×C₆₇	C₂×C₄	1	8	yes	—
88	#117×C₆₇	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
100	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
103	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
107	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
108	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
109	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
110	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
111	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
112	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
114	#97×C₆₇	C₂³	1	4	yes	—
115	#97×C₆₇	C₂³	1	4	yes	—
116	#97×C₆₇	C₂³	1	8	yes	—
117	C₄×C₈×C₆₇	C₂³	1	4	yes	—
118	C₄×C₈×C₆₇	C₂³	1	4	yes	—
119	C₄×C₈×C₆₇	C₂³	1	8	yes	—
120	C₄×C₈×C₆₇	C₂³	1	8	yes	—
121	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Q₈×C₆₇	—	2	4	no	—
124	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	4	4	no	—
127	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	4	2	no	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
150	M₁₆×C₆₇	—	2	8	no	—
151	M₁₆×C₆₇	—	2	8	no	—
152	M₁₆×C₆₇	—	2	16	no	—
153	C₂×C₈×C₆₇	—	2	4	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	4	no	—
157	C₂×C₈×C₆₇	—	2	16	no	—
158	C₂×C₈×C₆₇	#41	1	4	yes	—
159	C₂×C₈×C₆₇	#41	1	4	yes	—
160	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
161	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
162	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
163	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
164	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
165	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
168	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
170	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
174	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
175	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
176	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
177	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
178	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
179	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
180	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
181	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
182	C₄²×C₆₇	C₂⁴	1	2	yes	—
183	C₄²×C₆₇	C₂⁴	1	4	yes	—
184	C₄²×C₆₇	C₂⁴	1	2	yes	—
185	C₂³×C₆₇	—	2	4	no	—
186	C₂³×C₆₇	—	2	2	no	—
187	C₂³×C₆₇	—	2	2	no	—
188	C₂³×C₆₇	—	2	4	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
192	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
193	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
194	Q₈×C₆₇	—	2	8	no	—
195	D₄×C₆₇	—	2	8	no	—
196	D₄×C₆₇	—	4	8	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	8	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	#131	1	4	yes	—
213	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
217	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
219	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
220	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
223	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
224	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
225	C₈×C₆₇	—	2	4	no	—
226	C₈×C₆₇	—	2	2	no	—
227	C₈×C₆₇	—	2	4	no	—
228	C₈×C₆₇	—	2	2	no	—
229	C₈×C₆₇	C₂×#41	1	4	yes	—
230	C₈×C₆₇	C₂×#41	1	4	yes	—
231	C₂²×C₆₇	—	2	4	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	2	4	no	—
235	C₂²×C₆₇	—	2	2	no	—
236	C₂²×C₆₇	—	2	4	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	4	8	no	—
239	C₂²×C₆₇	C₄×#41	1	2	yes	—
240	C₂²×C₆₇	C₂×#131	1	2	yes	—
241	C₂²×C₆₇	#568	1	2	yes	—
242	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	4	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	2	2	no	—
249	C₄×C₆₇	—	2	4	no	—
250	C₄×C₆₇	C₄×#41	1	2	yes	—
251	C₄×C₆₇	C₂×#131	1	4	yes	—
252	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
253	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
254	C₂×#2969	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₆₇	—	4	2	no	—
256	C₂×C₆₇	—	4	4	no	—
257	C₂×C₆₇	#2969	1	4	yes	direct product factor
258	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
259	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
260	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
261	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
262	#2969	C₂×C₆₇	1	16	yes	direct product factor
263	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
265	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
266	C₂×#436	C₂×C₆₇	1	2	yes	complemented
267	C₂×#435	C₂×C₆₇	1	2	yes	complemented
268	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
269	C₂×#433	C₂×C₆₇	1	2	yes	complemented
270	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₂×#405	C₂×C₆₇	1	2	yes	complemented
272	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
273	C₆₇	C₂×#2969	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
275	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
276	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
277	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
278	#529	C₄×C₆₇	1	8	yes	—
279	#529	C₂²×C₆₇	1	8	yes	—
280	C₂×#126	C₄×C₆₇	1	2	yes	—
281	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
282	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
283	C₂×#123	C₄×C₆₇	1	2	yes	—
284	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
285	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
286	C₂×C₄×Q₈	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	C₂×#117	C₂²×C₆₇	1	2	yes	—
290	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
291	#446	C₂²×C₆₇	1	8	yes	complemented
292	C₈×Q₈	C₂²×C₆₇	1	8	yes	complemented
293	#436	C₂²×C₆₇	1	8	yes	complemented
294	#436	C₂²×C₆₇	1	16	yes	complemented
295	#435	C₂²×C₆₇	1	16	yes	complemented
296	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
297	#433	C₂²×C₆₇	1	16	yes	complemented
298	#431	C₂²×C₆₇	1	8	yes	complemented
299	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
302	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
305	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
306	#405	C₂²×C₆₇	1	16	yes	complemented
307	C₄×M₁₆	C₂²×C₆₇	1	8	yes	complemented
308	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×#97	C₂²×C₆₇	1	2	yes	—
311	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
314	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
315	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
316	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
318	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
319	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
321	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
322	C₂×M₁₆	—	2	4	no	—
323	C₂×M₁₆	—	2	4	no	—
324	C₂×M₁₆	—	2	8	no	—
325	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
326	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
327	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
328	C₂²×C₈	—	2	8	no	—
329	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
330	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
331	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
334	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
336	#126	C₂×C₄×C₆₇	1	16	yes	—
337	#125	C₂×C₄×C₆₇	1	8	yes	—
338	#124	C₂×C₄×C₆₇	1	8	yes	—
339	#123	C₂×C₄×C₆₇	1	16	yes	—
340	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
341	#121	C₂×C₄×C₆₇	1	8	yes	—
342	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
343	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
344	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
345	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
346	C₄×D₄	C₂³×C₆₇	1	4	yes	—
347	C₄×D₄	C₂³×C₆₇	1	8	yes	—
348	#117	C₂×C₄×C₆₇	1	8	yes	—
349	#117	C₂³×C₆₇	1	8	yes	—
350	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
351	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
352	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
354	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
356	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
359	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
360	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
361	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
362	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
363	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
364	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
366	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
368	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
369	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
370	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
371	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
372	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
373	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
374	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
375	#97	C₂³×C₆₇	1	4	yes	—
376	#97	C₂³×C₆₇	1	4	yes	—
377	#97	C₂³×C₆₇	1	8	yes	—
378	C₄×C₈	C₂³×C₆₇	1	4	yes	—
379	C₄×C₈	C₂³×C₆₇	1	4	yes	—
380	C₄×C₈	C₂³×C₆₇	1	8	yes	—
381	C₄×C₈	C₂³×C₆₇	1	8	yes	—
382	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
383	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	4	4	no	—
388	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
389	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
398	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
399	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
401	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
402	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
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	4	yes	—
406	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
408	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
411	M₁₆	—	2	8	no	—
412	M₁₆	—	2	8	no	—
413	M₁₆	—	2	16	no	—
414	C₂×C₈	—	2	4	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	4	no	—
418	C₂×C₈	—	2	16	no	—
419	C₂×C₈	#41×C₆₇	1	4	yes	—
420	C₂×C₈	#41×C₆₇	1	4	yes	—
421	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
422	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
423	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
424	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
425	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
426	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
427	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
428	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
429	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
430	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
431	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
432	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
434	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
435	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
436	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
437	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
438	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
439	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
440	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
441	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
442	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
443	C₄²	C₂⁴×C₆₇	1	2	yes	—
444	C₄²	C₂⁴×C₆₇	1	2	yes	—
445	C₄²	C₂⁴×C₆₇	1	4	yes	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	4	no	—
452	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
453	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
454	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
455	Q₈	—	2	8	no	—
456	D₄	—	2	8	no	—
457	D₄	—	4	8	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	8	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	8	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	#131×C₆₇	1	4	yes	—
474	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
475	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
476	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
477	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
478	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
479	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
480	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
481	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
482	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
483	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
484	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
485	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
486	C₈	—	2	4	no	—
487	C₈	—	2	2	no	—
488	C₈	—	2	2	no	—
489	C₈	—	2	4	no	—
490	C₈	C₂×#41×C₆₇	1	4	yes	—
491	C₈	C₂×#41×C₆₇	1	4	yes	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	4	no	—
496	C₂²	—	2	4	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	8	no	—
500	C₂²	C₄×#41×C₆₇	1	2	yes	—
501	C₂²	C₂×#131×C₆₇	1	2	yes	—
502	C₂²	#568×C₆₇	1	2	yes	—
503	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
504	C₄	—	2	2	no	—
505	C₄	—	2	4	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	4	no	—
509	C₄	—	2	4	no	—
510	C₄	—	2	2	no	—
511	C₄	C₄×#41×C₆₇	1	2	yes	—
512	C₄	C₂×#131×C₆₇	1	4	yes	—
513	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
514	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
515	C₂	—	4	2	no	—
516	C₂	—	4	4	no	—
517	C₂	C₆₇×#2969	1	4	yes	direct product factor
518	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
519	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic
520	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×#2969	C₂	direct (N × H)
3. C₂×#446×C₆₇	C₂	semidirect
4. C₂×C₈×Q₈×C₆₇	C₂	semidirect
5. C₂×#436×C₆₇	C₂	semidirect
6. C₂×#436×C₆₇	C₂	semidirect
7. C₂×#435×C₆₇	C₂	semidirect
8. C₂×C₈×D₄×C₆₇	C₂	semidirect
9. C₂×#433×C₆₇	C₂	semidirect
10. C₂×#431×C₆₇	C₂	semidirect
11. C₂×#405×C₆₇	C₂	semidirect
12. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
30. #446×C₆₇	C₂²	semidirect
31. C₈×Q₈×C₆₇	C₂²	semidirect
32. #436×C₆₇	C₂²	semidirect
33. #436×C₆₇	C₂²	semidirect
34. #435×C₆₇	C₂²	semidirect
35. C₈×D₄×C₆₇	C₂²	semidirect
36. #433×C₆₇	C₂²	semidirect
37. #431×C₆₇	C₂²	semidirect
45. #405×C₆₇	C₂²	semidirect
46. C₄×M₁₆×C₆₇	C₂²	semidirect
254. C₂×#2969	C₆₇	direct (N × H)
257. C₂×C₆₇	#2969	direct (N × H)
262. #2969	C₂×C₆₇	direct (N × H)
263. C₂×#446	C₂×C₆₇	semidirect
264. C₂×C₈×Q₈	C₂×C₆₇	semidirect
265. C₂×#436	C₂×C₆₇	semidirect
266. C₂×#436	C₂×C₆₇	semidirect
267. C₂×#435	C₂×C₆₇	semidirect
268. C₂×C₈×D₄	C₂×C₆₇	semidirect
269. C₂×#433	C₂×C₆₇	semidirect
270. C₂×#431	C₂×C₆₇	semidirect
271. C₂×#405	C₂×C₆₇	semidirect
272. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
273. C₆₇	C₂×#2969	direct (N × H)
291. #446	C₂²×C₆₇	semidirect
292. C₈×Q₈	C₂²×C₆₇	semidirect
293. #436	C₂²×C₆₇	semidirect
294. #436	C₂²×C₆₇	semidirect
295. #435	C₂²×C₆₇	semidirect
296. C₈×D₄	C₂²×C₆₇	semidirect
297. #433	C₂²×C₆₇	semidirect
298. #431	C₂²×C₆₇	semidirect
306. #405	C₂²×C₆₇	semidirect
307. C₄×M₁₆	C₂²×C₆₇	semidirect
517. C₂	C₆₇×#2969	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×#2969	2
C₂×#446×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×#436×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#433×C₆₇	2
C₂×#431×C₆₇	2
C₂×#405×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#2969	67

Subgroups — C2×C67×#2969 #30739452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2969 #30739452