Subgroups — C₆₇×#11010 #30690261

Order 17152.

1430 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#103×C₆₇	C₂	1	4	yes	maximal
3	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
6	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₆₇	—	2	8	no	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₄	1	4	yes	complemented
10	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#103×C₆₇	—	2	8	no	—
14	C₂×#103×C₆₇	—	2	8	no	—
15	C₂×#103×C₆₇	—	2	8	no	—
16	C₂×#95×C₆₇	—	2	2	no	—
17	#336×C₆₇	—	2	2	no	—
18	C₂²×Q₈×C₆₇	—	2	2	no	—
19	C₂²×Q₈×C₆₇	—	2	4	no	—
20	C₂²×Q₈×C₆₇	—	2	4	no	—
21	C₂²×Q₈×C₆₇	—	2	4	no	—
22	C₂²×Q₈×C₆₇	—	2	4	no	—
23	C₂²×Q₈×C₆₇	—	2	4	no	—
24	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
25	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
26	C₂³×C₄×C₆₇	—	2	2	no	—
27	C₂³×C₄×C₆₇	—	2	2	no	—
28	C₂³×C₄×C₆₇	D₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	4	no	—
32	C₂²×C₈×C₆₇	—	2	4	no	—
33	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	D₄	1	4	yes	—
35	C₄⋊Q₈×C₆₇	—	2	16	no	—
36	C₄⋊Q₈×C₆₇	—	2	16	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
43	C₂×C₄²×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	D₄	1	2	yes	—
46	#103×C₆₇	—	2	32	no	—
47	#95×C₆₇	—	4	4	no	—
48	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Q₈×C₆₇	—	2	4	no	—
50	C₂×Q₈×C₆₇	—	2	8	no	—
51	C₂×Q₈×C₆₇	—	2	8	no	—
52	C₂×Q₈×C₆₇	—	2	8	no	—
53	C₂×Q₈×C₆₇	—	2	8	no	—
54	C₂×Q₈×C₆₇	—	2	8	no	—
55	C₂×Q₈×C₆₇	—	2	8	no	—
56	C₂×Q₈×C₆₇	—	2	8	no	—
57	C₂×Q₈×C₆₇	—	4	4	no	—
58	C₂×Q₈×C₆₇	—	4	4	no	—
59	C₂×Q₈×C₆₇	—	4	4	no	—
60	C₂×Q₈×C₆₇	—	4	4	no	—
61	C₂×Q₈×C₆₇	—	4	4	no	—
62	C₂×Q₈×C₆₇	—	4	2	no	—
63	C₂×Q₈×C₆₇	—	4	2	no	—
64	C₂×Q₈×C₆₇	C₂²⋊C₄	1	8	yes	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	8	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	4	4	no	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	—	4	4	no	—
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
78	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
79	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	C₂×C₈×C₆₇	—	2	8	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	4	4	no	—
91	C₂×C₈×C₆₇	—	4	2	no	—
92	C₄⋊C₄×C₆₇	—	2	32	no	—
93	C₄⋊C₄×C₆₇	—	2	8	no	—
94	C₄⋊C₄×C₆₇	—	4	4	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₆₇	#120	1	2	yes	—
99	C₂³×C₆₇	#121	1	2	yes	—
100	C₂³×C₆₇	#124	1	1	yes	characteristic
101	C₂³×C₆₇	#127	1	1	yes	characteristic
102	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
103	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
104	C₂³×C₆₇	#137	1	4	yes	—
105	Q₈×C₆₇	—	2	16	no	—
106	Q₈×C₆₇	—	4	8	no	—
107	Q₈×C₆₇	—	4	8	no	—
108	Q₈×C₆₇	—	4	8	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	8	no	—
112	C₂×C₄×C₆₇	—	2	8	no	—
113	C₂×C₄×C₆₇	—	2	8	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	—	4	8	no	—
117	C₂×C₄×C₆₇	—	4	8	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	8	no	—
120	C₂×C₄×C₆₇	—	8	4	no	—
121	C₂×C₄×C₆₇	—	8	2	no	—
122	C₂×C₄×C₆₇	—	8	2	no	—
123	C₂×C₄×C₆₇	#103	1	8	yes	complemented
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₆₇	#120	1	4	yes	—
128	C₂×C₄×C₆₇	#120	1	2	yes	—
129	C₂×C₄×C₆₇	#121	1	4	yes	—
130	C₂×C₄×C₆₇	#121	1	4	yes	—
131	C₂×C₄×C₆₇	#121	1	2	yes	—
132	C₂×C₄×C₆₇	#123	1	4	yes	—
133	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
135	C₈×C₆₇	—	4	4	no	—
136	C₂²×C₆₇	#379	1	1	yes	characteristic
137	C₂²×C₆₇	#390	1	1	yes	characteristic
138	C₂²×C₆₇	#390	1	1	yes	characteristic
139	C₂²×C₆₇	#392	1	2	yes	—
140	C₂²×C₆₇	#394	1	2	yes	—
141	C₂²×C₆₇	C₂×#103	1	2	yes	—
142	C₂²×C₆₇	#419	1	2	yes	—
143	C₂²×C₆₇	#448	1	4	yes	—
144	C₂²×C₆₇	#451	1	4	yes	—
145	C₂²×C₆₇	#461	1	4	yes	—
146	C₂²×C₆₇	#462	1	4	yes	—
147	C₂²×C₆₇	#487	1	2	yes	—
148	C₂²×C₆₇	#491	1	2	yes	—
149	C₂²×C₆₇	#492	1	1	yes	characteristic
150	C₂²×C₆₇	#494	1	1	yes	characteristic
151	C₂²×C₆₇	#497	1	2	yes	—
152	C₄×C₆₇	—	2	8	no	—
153	C₄×C₆₇	—	4	8	no	—
154	C₄×C₆₇	—	8	4	no	—
155	C₄×C₆₇	#386	1	4	yes	—
156	C₄×C₆₇	#392	1	2	yes	—
157	C₄×C₆₇	#394	1	2	yes	—
158	#11010	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₆₇	#1438	1	1	yes	characteristic
160	C₂×C₆₇	#1787	1	2	yes	—
161	C₂×C₆₇	#1956	1	1	yes	characteristic
162	C₂×C₆₇	#1960	1	1	yes	characteristic
163	C₂×C₆₇	#1981	1	2	yes	—
164	C₂×C₆₇	#2013	1	2	yes	—
165	C₂×C₆₇	#2016	1	2	yes	—
166	C₂×C₆₇	#2043	1	4	yes	—
167	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
168	C₂²×#103	C₂×C₆₇	1	4	yes	—
169	C₂×#336	C₂×C₆₇	1	1	yes	characteristic
170	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
171	C₆₇	#11010	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
173	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
174	C₂×C₄⋊Q₈	—	2	8	no	—
175	C₂×C₄⋊Q₈	—	2	2	no	—
176	C₂×C₄⋊Q₈	C₄×C₆₇	1	4	yes	complemented
177	C₂²×C₄⋊C₄	—	2	2	no	—
178	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
179	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
180	C₂×#103	—	2	8	no	—
181	C₂×#103	—	2	8	no	—
182	C₂×#103	—	2	8	no	—
183	C₂×#95	—	2	2	no	—
184	#336	—	2	2	no	—
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×Q₈	—	2	4	no	—
187	C₂²×Q₈	—	2	4	no	—
188	C₂²×Q₈	—	2	4	no	—
189	C₂²×Q₈	—	2	4	no	—
190	C₂²×Q₈	—	2	4	no	—
191	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
192	C₂²×Q₈	D₄×C₆₇	1	8	yes	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	D₄×C₆₇	1	2	yes	—
196	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	D₄×C₆₇	1	4	yes	—
202	C₄⋊Q₈	—	2	16	no	—
203	C₄⋊Q₈	—	2	16	no	—
204	C₂×C₄⋊C₄	—	2	16	no	—
205	C₂×C₄⋊C₄	—	2	8	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
212	C₂×C₄²	D₄×C₆₇	1	2	yes	—
213	#103	—	2	32	no	—
214	#95	—	4	4	no	—
215	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	8	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	2	8	no	—
221	C₂×Q₈	—	2	8	no	—
222	C₂×Q₈	—	2	8	no	—
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	C₂²⋊C₄×C₆₇	1	8	yes	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
245	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
246	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
247	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
249	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	#41×C₆₇	1	2	yes	—
253	C₂²×C₄	#41×C₆₇	1	2	yes	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	4	no	—
258	C₂×C₈	—	4	2	no	—
259	C₄⋊C₄	—	2	32	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	4	no	—
264	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
265	C₂³	#120×C₆₇	1	2	yes	—
266	C₂³	#121×C₆₇	1	2	yes	—
267	C₂³	#124×C₆₇	1	1	yes	characteristic
268	C₂³	#127×C₆₇	1	1	yes	characteristic
269	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
270	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
271	C₂³	#137×C₆₇	1	4	yes	—
272	Q₈	—	2	16	no	—
273	Q₈	—	4	8	no	—
274	Q₈	—	4	8	no	—
275	Q₈	—	4	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	#103×C₆₇	1	8	yes	complemented
291	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
292	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
293	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
294	C₂×C₄	#120×C₆₇	1	4	yes	—
295	C₂×C₄	#120×C₆₇	1	2	yes	—
296	C₂×C₄	#121×C₆₇	1	4	yes	—
297	C₂×C₄	#121×C₆₇	1	4	yes	—
298	C₂×C₄	#121×C₆₇	1	2	yes	—
299	C₂×C₄	#123×C₆₇	1	4	yes	—
300	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
301	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
302	C₈	—	4	4	no	—
303	C₂²	#379×C₆₇	1	1	yes	characteristic
304	C₂²	#390×C₆₇	1	1	yes	characteristic
305	C₂²	#390×C₆₇	1	1	yes	characteristic
306	C₂²	#392×C₆₇	1	2	yes	—
307	C₂²	#394×C₆₇	1	2	yes	—
308	C₂²	C₂×#103×C₆₇	1	2	yes	—
309	C₂²	#419×C₆₇	1	2	yes	—
310	C₂²	#448×C₆₇	1	4	yes	—
311	C₂²	#451×C₆₇	1	4	yes	—
312	C₂²	#461×C₆₇	1	4	yes	—
313	C₂²	#462×C₆₇	1	4	yes	—
314	C₂²	#487×C₆₇	1	2	yes	—
315	C₂²	#491×C₆₇	1	2	yes	—
316	C₂²	#492×C₆₇	1	1	yes	characteristic
317	C₂²	#494×C₆₇	1	1	yes	characteristic
318	C₂²	#497×C₆₇	1	2	yes	—
319	C₄	—	2	8	no	—
320	C₄	—	4	8	no	—
321	C₄	—	8	4	no	—
322	C₄	#386×C₆₇	1	4	yes	—
323	C₄	#392×C₆₇	1	2	yes	—
324	C₄	#394×C₆₇	1	2	yes	—
325	C₂	C₆₇×#1438	1	1	yes	characteristic
326	C₂	C₆₇×#1787	1	2	yes	—
327	C₂	C₆₇×#1956	1	1	yes	characteristic
328	C₂	C₆₇×#1960	1	1	yes	characteristic
329	C₂	C₆₇×#1981	1	2	yes	—
330	C₂	C₆₇×#2013	1	2	yes	—
331	C₂	C₆₇×#2016	1	2	yes	—
332	C₂	C₆₇×#2043	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Q₈×C₆₇	C₄	semidirect
123. C₂×C₄×C₆₇	#103	semidirect
158. #11010	C₆₇	direct (N × H)
171. C₆₇	#11010	direct (N × H)
176. C₂×C₄⋊Q₈	C₄×C₆₇	semidirect
290. C₂×C₄	#103×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂²×#103×C₆₇	2
C₂×#336×C₆₇	2
C₆₇×C₄⋊#95	2
#11010	67

Subgroups — C67×#11010 #30690261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11010 #30690261