Subgroups — C₆₇×#60206 #30739457

Order 17152.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3455	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2967	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2966	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2961	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2930	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2929	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2923	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#131×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#131×C₆₇	C₂²	1	2	yes	—
15	C₂×#131×C₆₇	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#542×C₆₇	C₄	1	4	yes	—
19	#541×C₆₇	C₄	1	2	yes	—
20	#540×C₆₇	C₄	1	2	yes	—
21	#539×C₆₇	C₄	1	4	yes	—
22	#538×C₆₇	C₄	1	4	yes	—
23	#533×C₆₇	C₂²	1	1	yes	characteristic
24	#532×C₆₇	C₂²	1	1	yes	characteristic
25	#529×C₆₇	C₂²	1	8	yes	—
26	#525×C₆₇	C₂²	1	2	yes	—
27	C₄×#41×C₆₇	C₂²	1	2	yes	—
28	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
29	#446×C₆₇	C₂²	1	4	yes	complemented
30	#446×C₆₇	C₂²	1	8	yes	complemented
31	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
32	#436×C₆₇	C₂²	1	8	yes	—
33	#436×C₆₇	C₂²	1	4	yes	complemented
34	#435×C₆₇	C₂²	1	8	yes	—
35	#435×C₆₇	C₂²	1	8	yes	—
36	#435×C₆₇	C₂²	1	8	yes	complemented
37	C₈×D₄×C₆₇	C₂²	1	8	yes	—
38	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
39	#433×C₆₇	C₂²	1	8	yes	—
40	#433×C₆₇	C₂²	1	8	yes	complemented
41	#432×C₆₇	C₂²	1	8	yes	complemented
42	#424×C₆₇	C₂²	1	2	yes	—
43	#424×C₆₇	C₂²	1	2	yes	—
44	#423×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	#408×C₆₇	C₂²	1	2	yes	—
47	#408×C₆₇	C₂²	1	2	yes	—
48	#407×C₆₇	C₂²	1	2	yes	—
49	#405×C₆₇	C₂²	1	4	yes	—
50	#405×C₆₇	C₂²	1	4	yes	—
51	#405×C₆₇	C₂²	1	2	yes	—
52	#405×C₆₇	C₂²	1	2	yes	—
53	#405×C₆₇	C₂²	1	4	yes	complemented
54	#405×C₆₇	C₂²	1	4	yes	complemented
55	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
56	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
57	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
58	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
59	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
60	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
61	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	complemented
62	C₂×#41×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	#131×C₆₇	—	2	8	no	—
65	#131×C₆₇	—	2	8	no	—
66	C₂×M₁₆×C₆₇	—	2	4	no	—
67	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
70	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₆₇	—	2	4	no	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
83	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
84	#126×C₆₇	C₂×C₄	1	8	yes	—
85	#126×C₆₇	C₂×C₄	1	8	yes	—
86	#125×C₆₇	C₂×C₄	1	8	yes	—
87	#125×C₆₇	C₂×C₄	1	2	yes	—
88	#124×C₆₇	C₂×C₄	1	8	yes	—
89	#124×C₆₇	C₂×C₄	1	2	yes	—
90	#123×C₆₇	C₂×C₄	1	4	yes	—
91	#123×C₆₇	C₂×C₄	1	8	yes	—
92	#123×C₆₇	C₂×C₄	1	8	yes	—
93	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
94	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
95	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
96	#121×C₆₇	C₂×C₄	1	4	yes	—
97	#121×C₆₇	C₂×C₄	1	8	yes	—
98	#121×C₆₇	C₂×C₄	1	4	yes	—
99	#120×C₆₇	C₂×C₄	1	4	yes	—
100	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
101	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
102	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
103	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
104	C₄×D₄×C₆₇	C₂³	1	8	yes	—
105	C₄×D₄×C₆₇	C₂³	1	4	yes	—
106	#117×C₆₇	C₂×C₄	1	2	yes	—
107	#117×C₆₇	C₂×C₄	1	2	yes	—
108	#117×C₆₇	C₂³	1	4	yes	—
109	#117×C₆₇	C₂³	1	2	yes	—
110	#117×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
116	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
117	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
118	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
119	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
120	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
121	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
122	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
123	#97×C₆₇	C₂³	1	4	yes	—
124	#97×C₆₇	C₂³	1	4	yes	—
125	#97×C₆₇	C₂³	1	4	yes	—
126	#97×C₆₇	C₂³	1	4	yes	—
127	#97×C₆₇	C₂³	1	4	yes	—
128	C₄×C₈×C₆₇	C₂³	1	2	yes	—
129	C₄×C₈×C₆₇	C₂³	1	2	yes	—
130	C₄×C₈×C₆₇	C₂³	1	4	yes	—
131	C₄×C₈×C₆₇	C₂³	1	4	yes	—
132	C₄×C₈×C₆₇	C₂³	1	2	yes	—
133	C₄×C₈×C₆₇	C₂³	1	2	yes	—
134	C₄×C₈×C₆₇	C₂³	1	4	yes	—
135	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	#41×C₆₇	—	2	4	no	—
137	#41×C₆₇	—	2	4	no	—
138	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
141	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
143	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
154	M₁₆×C₆₇	—	2	4	no	—
155	M₁₆×C₆₇	—	2	4	no	—
156	M₁₆×C₆₇	—	2	8	no	—
157	M₁₆×C₆₇	—	2	8	no	—
158	M₁₆×C₆₇	—	2	8	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	4	no	—
162	C₂×C₈×C₆₇	—	2	4	no	—
163	C₂×C₈×C₆₇	—	2	8	no	—
164	C₂×C₈×C₆₇	—	2	8	no	—
165	C₂×C₈×C₆₇	#41	1	4	yes	—
166	C₂×C₈×C₆₇	#41	1	4	yes	—
167	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
168	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
169	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
170	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
171	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
173	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
174	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
175	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
176	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
177	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
178	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
179	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
180	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
181	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
182	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
183	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
184	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
185	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
186	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
187	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
188	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
189	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
190	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
191	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
193	C₄²×C₆₇	C₂⁴	1	2	yes	—
194	C₄²×C₆₇	C₂⁴	1	4	yes	—
195	C₄²×C₆₇	C₂⁴	1	2	yes	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
199	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
200	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
201	Q₈×C₆₇	—	2	4	no	—
202	Q₈×C₆₇	—	4	4	no	—
203	D₄×C₆₇	—	2	8	no	—
204	D₄×C₆₇	—	2	4	no	—
205	D₄×C₆₇	—	4	4	no	—
206	D₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	8	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
217	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
220	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
224	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
225	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
226	C₈×C₆₇	—	2	2	no	—
227	C₈×C₆₇	—	2	2	no	—
228	C₈×C₆₇	—	2	4	no	—
229	C₈×C₆₇	—	2	4	no	—
230	C₈×C₆₇	C₂×#41	1	2	yes	—
231	C₈×C₆₇	C₂×#41	1	2	yes	—
232	C₈×C₆₇	C₂×#41	1	4	yes	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	2	2	no	—
235	C₂²×C₆₇	—	2	4	no	—
236	C₂²×C₆₇	—	2	8	no	—
237	C₂²×C₆₇	—	2	4	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	C₄×#41	1	2	yes	—
240	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	2	4	no	—
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	8	no	—
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	C₄×#41	1	2	yes	—
248	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
249	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
250	#60206	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	2	2	no	—
252	C₂×C₆₇	—	4	4	no	—
253	C₂×C₆₇	—	4	2	no	—
254	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
255	C₂×C₆₇	#3573	1	2	yes	—
256	#3455	C₂×C₆₇	1	1	yes	characteristic
257	#2967	C₂×C₆₇	1	8	yes	complemented
258	#2966	C₂×C₆₇	1	8	yes	complemented
259	#2961	C₂×C₆₇	1	1	yes	characteristic, complemented
260	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
261	#2932	C₂×C₆₇	1	1	yes	characteristic, complemented
262	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#2930	C₂×C₆₇	1	2	yes	complemented
264	#2929	C₂×C₆₇	1	2	yes	complemented
265	#2923	C₂×C₆₇	1	2	yes	complemented
266	#2921	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₄×#131	C₂×C₆₇	1	2	yes	complemented
268	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
269	C₆₇	#60206	1	1	yes	characteristic, direct product factor, Sylow
270	C₂×#131	C₂²×C₆₇	1	2	yes	—
271	C₂×#131	C₂²×C₆₇	1	2	yes	—
272	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
273	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
274	#542	C₄×C₆₇	1	4	yes	—
275	#541	C₄×C₆₇	1	2	yes	—
276	#540	C₄×C₆₇	1	2	yes	—
277	#539	C₄×C₆₇	1	4	yes	—
278	#538	C₄×C₆₇	1	4	yes	—
279	#533	C₂²×C₆₇	1	1	yes	characteristic
280	#532	C₂²×C₆₇	1	1	yes	characteristic
281	#529	C₂²×C₆₇	1	8	yes	—
282	#525	C₂²×C₆₇	1	2	yes	—
283	C₄×#41	C₂²×C₆₇	1	2	yes	—
284	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
285	#446	C₂²×C₆₇	1	4	yes	complemented
286	#446	C₂²×C₆₇	1	8	yes	complemented
287	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
288	#436	C₂²×C₆₇	1	8	yes	—
289	#436	C₂²×C₆₇	1	4	yes	complemented
290	#435	C₂²×C₆₇	1	8	yes	—
291	#435	C₂²×C₆₇	1	8	yes	—
292	#435	C₂²×C₆₇	1	8	yes	complemented
293	C₈×D₄	C₂²×C₆₇	1	8	yes	—
294	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
295	#433	C₂²×C₆₇	1	8	yes	—
296	#433	C₂²×C₆₇	1	8	yes	complemented
297	#432	C₂²×C₆₇	1	8	yes	complemented
298	#424	C₂²×C₆₇	1	2	yes	—
299	#424	C₂²×C₆₇	1	2	yes	—
300	#423	C₂²×C₆₇	1	1	yes	characteristic
301	#423	C₂²×C₆₇	1	1	yes	characteristic
302	#408	C₂²×C₆₇	1	2	yes	—
303	#408	C₂²×C₆₇	1	2	yes	—
304	#407	C₂²×C₆₇	1	2	yes	—
305	#405	C₂²×C₆₇	1	4	yes	—
306	#405	C₂²×C₆₇	1	4	yes	—
307	#405	C₂²×C₆₇	1	2	yes	—
308	#405	C₂²×C₆₇	1	2	yes	—
309	#405	C₂²×C₆₇	1	4	yes	complemented
310	#405	C₂²×C₆₇	1	4	yes	complemented
311	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
312	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
313	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	complemented
318	C₂×#41	C₂³×C₆₇	1	2	yes	—
319	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
320	#131	—	2	8	no	—
321	#131	—	2	8	no	—
322	C₂×M₁₆	—	2	4	no	—
323	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
324	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
325	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
326	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
327	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
328	C₂²×C₈	—	2	4	no	—
329	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
331	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
332	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
334	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
335	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
337	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
338	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
339	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
340	#126	C₂×C₄×C₆₇	1	8	yes	—
341	#126	C₂×C₄×C₆₇	1	8	yes	—
342	#125	C₂×C₄×C₆₇	1	8	yes	—
343	#125	C₂×C₄×C₆₇	1	2	yes	—
344	#124	C₂×C₄×C₆₇	1	8	yes	—
345	#124	C₂×C₄×C₆₇	1	2	yes	—
346	#123	C₂×C₄×C₆₇	1	4	yes	—
347	#123	C₂×C₄×C₆₇	1	8	yes	—
348	#123	C₂×C₄×C₆₇	1	8	yes	—
349	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
350	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
351	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
352	#121	C₂×C₄×C₆₇	1	4	yes	—
353	#121	C₂×C₄×C₆₇	1	8	yes	—
354	#121	C₂×C₄×C₆₇	1	4	yes	—
355	#120	C₂×C₄×C₆₇	1	4	yes	—
356	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
357	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
358	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
359	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
360	C₄×D₄	C₂³×C₆₇	1	8	yes	—
361	C₄×D₄	C₂³×C₆₇	1	4	yes	—
362	#117	C₂×C₄×C₆₇	1	2	yes	—
363	#117	C₂×C₄×C₆₇	1	2	yes	—
364	#117	C₂³×C₆₇	1	4	yes	—
365	#117	C₂³×C₆₇	1	2	yes	—
366	#117	C₂³×C₆₇	1	4	yes	—
367	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
368	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
369	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
372	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
373	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
374	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
375	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
376	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
377	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
378	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
379	#97	C₂³×C₆₇	1	4	yes	—
380	#97	C₂³×C₆₇	1	4	yes	—
381	#97	C₂³×C₆₇	1	4	yes	—
382	#97	C₂³×C₆₇	1	4	yes	—
383	#97	C₂³×C₆₇	1	4	yes	—
384	C₄×C₈	C₂³×C₆₇	1	2	yes	—
385	C₄×C₈	C₂³×C₆₇	1	2	yes	—
386	C₄×C₈	C₂³×C₆₇	1	2	yes	—
387	C₄×C₈	C₂³×C₆₇	1	2	yes	—
388	C₄×C₈	C₂³×C₆₇	1	4	yes	—
389	C₄×C₈	C₂³×C₆₇	1	4	yes	—
390	C₄×C₈	C₂³×C₆₇	1	4	yes	—
391	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
392	#41	—	2	4	no	—
393	#41	—	2	4	no	—
394	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
395	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
396	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
397	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
398	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
399	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
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	2	yes	—
406	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
410	M₁₆	—	2	4	no	—
411	M₁₆	—	2	4	no	—
412	M₁₆	—	2	8	no	—
413	M₁₆	—	2	8	no	—
414	M₁₆	—	2	8	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	4	no	—
418	C₂×C₈	—	2	4	no	—
419	C₂×C₈	—	2	8	no	—
420	C₂×C₈	—	2	8	no	—
421	C₂×C₈	#41×C₆₇	1	4	yes	—
422	C₂×C₈	#41×C₆₇	1	4	yes	—
423	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
424	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
425	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
426	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
427	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
428	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
429	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
430	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
431	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
432	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
434	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
435	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
436	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
437	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
438	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
439	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
440	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
441	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
442	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
443	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
444	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
445	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
446	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
447	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
448	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
449	C₄²	C₂⁴×C₆₇	1	2	yes	—
450	C₄²	C₂⁴×C₆₇	1	2	yes	—
451	C₄²	C₂⁴×C₆₇	1	4	yes	—
452	C₂³	—	2	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
455	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
456	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
457	Q₈	—	2	4	no	—
458	Q₈	—	4	4	no	—
459	D₄	—	2	8	no	—
460	D₄	—	2	4	no	—
461	D₄	—	4	4	no	—
462	D₄	—	4	8	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	8	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
473	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
474	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
475	C₂×C₄	C₂³×C₄×C₆₇	1	8	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	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
481	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	2	no	—
484	C₈	—	2	4	no	—
485	C₈	—	2	4	no	—
486	C₈	C₂×#41×C₆₇	1	2	yes	—
487	C₈	C₂×#41×C₆₇	1	2	yes	—
488	C₈	C₂×#41×C₆₇	1	4	yes	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	C₄×#41×C₆₇	1	2	yes	—
496	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
497	C₄	—	2	4	no	—
498	C₄	—	2	2	no	—
499	C₄	—	2	8	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	4	no	—
503	C₄	C₄×#41×C₆₇	1	2	yes	—
504	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
505	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
506	C₂	—	2	2	no	—
507	C₂	—	4	2	no	—
508	C₂	—	4	4	no	—
509	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
510	C₂	C₆₇×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	C₆₇×#3455	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2967	C₂	semidirect
3. C₆₇×#2966	C₂	semidirect
4. C₆₇×#2961	C₂	semidirect
5. C₆₇×#2960	C₂	semidirect
6. C₆₇×#2932	C₂	semidirect
7. C₆₇×#2931	C₂	semidirect
8. C₆₇×#2930	C₂	semidirect
9. C₆₇×#2929	C₂	semidirect
10. C₆₇×#2923	C₂	semidirect
11. C₆₇×#2921	C₂	semidirect
12. C₄×#131×C₆₇	C₂	semidirect
13. C₂×#405×C₆₇	C₂	semidirect
29. #446×C₆₇	C₂²	semidirect
30. #446×C₆₇	C₂²	semidirect
31. C₈×Q₈×C₆₇	C₂²	semidirect
33. #436×C₆₇	C₂²	semidirect
36. #435×C₆₇	C₂²	semidirect
38. C₈×D₄×C₆₇	C₂²	semidirect
40. #433×C₆₇	C₂²	semidirect
41. #432×C₆₇	C₂²	semidirect
53. #405×C₆₇	C₂²	semidirect
54. #405×C₆₇	C₂²	semidirect
56. C₄×M₁₆×C₆₇	C₂²	semidirect
61. C₂×C₄×C₈×C₆₇	C₂²	semidirect
250. #60206	C₆₇	direct (N × H)
257. #2967	C₂×C₆₇	semidirect
258. #2966	C₂×C₆₇	semidirect
259. #2961	C₂×C₆₇	semidirect
260. #2960	C₂×C₆₇	semidirect
261. #2932	C₂×C₆₇	semidirect
262. #2931	C₂×C₆₇	semidirect
263. #2930	C₂×C₆₇	semidirect
264. #2929	C₂×C₆₇	semidirect
265. #2923	C₂×C₆₇	semidirect
266. #2921	C₂×C₆₇	semidirect
267. C₄×#131	C₂×C₆₇	semidirect
268. C₂×#405	C₂×C₆₇	semidirect
269. C₆₇	#60206	direct (N × H)
285. #446	C₂²×C₆₇	semidirect
286. #446	C₂²×C₆₇	semidirect
287. C₈×Q₈	C₂²×C₆₇	semidirect
289. #436	C₂²×C₆₇	semidirect
292. #435	C₂²×C₆₇	semidirect
294. C₈×D₄	C₂²×C₆₇	semidirect
296. #433	C₂²×C₆₇	semidirect
297. #432	C₂²×C₆₇	semidirect
309. #405	C₂²×C₆₇	semidirect
310. #405	C₂²×C₆₇	semidirect
312. C₄×M₁₆	C₂²×C₆₇	semidirect
317. C₂×C₄×C₈	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3455	2
C₆₇×#2967	2
C₆₇×#2966	2
C₆₇×#2961	2
C₆₇×#2960	2
C₆₇×#2932	2
C₆₇×#2931	2
C₆₇×#2930	2
C₆₇×#2929	2
C₆₇×#2923	2
C₆₇×#2921	2
C₄×#131×C₆₇	2
C₂×#405×C₆₇	2
#60206	67

Subgroups — C67×#60206 #30739457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60206 #30739457