Subgroups — #30864704

Order 17152.

670 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570731	C₂	1	1	yes	maximal, characteristic
2	#22569149	C₂	1	1	yes	maximal, characteristic
3	#22561187	C₂	1	1	yes	maximal, characteristic
4	#22561185	C₂	1	1	yes	maximal, characteristic
5	#22560959	C₂	1	2	yes	maximal
6	#22560954	C₂	1	2	yes	maximal
7	#22560383	C₂	1	1	yes	maximal, characteristic
8	#22560094	C₂	1	2	yes	maximal
9	#22560089	C₂	1	2	yes	maximal
10	#97×Dic₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2961	C₂	1	1	yes	maximal, characteristic
12	#16863953	C₂²	1	1	yes	characteristic
13	#16863806	C₄	1	4	yes	complemented
14	#16863802	C₄	1	4	yes	complemented
15	#16863524	—	2	4	no	—
16	C₂×#14523978	C₂²	1	2	yes	—
17	C₂×#14523978	C₂²	1	2	yes	—
18	#16863432	C₂²	1	2	yes	—
19	#16863432	C₂²	1	2	yes	—
20	#16863422	C₂²	1	2	yes	—
21	#16863420	C₂²	1	1	yes	characteristic
22	#16863351	C₂²	1	2	yes	—
23	C₂×#14523954	C₂²	1	1	yes	characteristic
24	C₂×#14523953	C₂²	1	1	yes	characteristic
25	C₂×#14523953	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
28	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
29	#16863157	C₂²	1	2	yes	—
30	#16863157	C₂²	1	2	yes	—
31	#16863155	C₂²	1	2	yes	—
32	#16863155	C₂²	1	2	yes	—
33	#533×C₆₇	C₂²	1	1	yes	characteristic
34	#446×C₆₇	—	2	4	no	—
35	#424×C₆₇	C₂²	1	2	yes	—
36	#423×C₆₇	C₂²	1	1	yes	characteristic
37	#405×C₆₇	C₂²	1	2	yes	—
38	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761612	C₂³	1	2	yes	—
40	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
42	#14524043	C₂×C₄	1	2	yes	—
43	#14524035	C₂×C₄	1	2	yes	—
44	#14524022	—	2	2	no	—
45	C₄×Dic₁₃₄	—	2	2	no	—
46	#14524002	C₂³	1	2	yes	—
47	C₂×C₄×Dic₆₇	—	2	2	no	—
48	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
50	#14523978	—	2	4	no	—
51	#14523978	—	2	4	no	—
52	#14523976	C₂×C₄	1	4	yes	—
53	#14523976	C₂×C₄	1	4	yes	—
54	#14523974	C₂×C₄	1	4	yes	—
55	#14523974	C₂×C₄	1	4	yes	—
56	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
57	Dic₆₇⋊C₈	—	2	4	no	—
58	#14523954	C₂×C₄	1	2	yes	—
59	#14523954	C₂×C₄	1	2	yes	—
60	#14523953	—	2	2	no	—
61	#14523953	—	2	2	no	—
62	#14523953	Q₈	1	4	yes	complemented
63	C₈×Dic₆₇	—	2	2	no	—
64	C₈×Dic₆₇	—	2	2	no	—
65	C₈×Dic₆₇	Q₈	1	4	yes	complemented
66	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
70	#125×C₆₇	C₂×C₄	1	2	yes	—
71	C₂²⋊Q₈×C₆₇	—	2	2	no	—
72	C₄×Q₈×C₆₇	—	2	2	no	—
73	#117×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	4	no	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	#97×C₆₇	—	2	2	no	—
78	#97×C₆₇	D₄	1	4	yes	—
79	C₄×C₈×C₆₇	—	2	2	no	—
80	C₂×Dic₁₃₄	—	2	2	no	—
81	C₂²×Dic₆₇	—	2	1	no	—
82	C₂²×Dic₆₇	—	2	1	no	—
83	#11761612	—	2	2	no	—
84	#11761612	—	2	2	no	—
85	C₂×C₆₇⋊C₈	—	2	2	no	—
86	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂×C₆₇⋊C₈	#41	1	2	yes	—
91	C₂×C₆₇⋊C₈	#41	1	2	yes	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
94	Dic₆₇⋊C₄	—	2	4	no	—
95	C₂²⋊Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	2	2	no	—
97	C₄×Dic₆₇	—	2	1	no	—
98	C₄×Dic₆₇	—	2	1	no	—
99	C₄×Dic₆₇	—	4	4	no	—
100	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
101	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	#41	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₆₇	—	2	2	no	—
106	M₁₆×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
117	C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
123	Q₈×C₆₇	—	8	2	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₆₇	—	2	4	no	—
135	C₂×Dic₆₇	—	2	2	no	—
136	C₂×Dic₆₇	—	2	4	no	—
137	C₂×Dic₆₇	—	2	2	no	—
138	C₂×Dic₆₇	—	4	2	no	—
139	Dic₁₃₄	—	8	2	no	—
140	C₈×C₆₇	—	2	2	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
143	C₆₇⋊C₈	—	4	2	no	—
144	C₆₇⋊C₈	C₄×Q₈	1	4	yes	—
145	C₆₇⋊C₈	C₄×Q₈	1	4	yes	—
146	C₂²×C₆₇	—	2	2	no	—
147	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₆₇	#533	1	1	yes	characteristic
149	C₂²×C₆₇	#533	1	1	yes	characteristic
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	4	4	no	—
153	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₄×C₆₇	#532	1	1	yes	characteristic
157	Dic₆₇	—	2	4	no	—
158	Dic₆₇	—	4	2	no	—
159	Dic₆₇	—	4	4	no	—
160	#21736	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	1	no	—
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
164	C₂×C₆₇	#2956	1	2	yes	—
165	#3162	—	67	1	no	—
166	#2961	—	67	1	no	—
167	#2961	D₆₇	1	1	yes	characteristic
168	#1994	—	67	1	no	—
169	#1994	—	67	1	no	—
170	#1946	—	67	2	no	—
171	#1945	—	67	2	no	—
172	#1849	—	67	1	no	—
173	#1773	—	67	2	no	—
174	#1772	—	67	2	no	—
175	C₄×#97	—	67	1	no	—
176	C₆₇	#21736	1	1	yes	characteristic, complemented, Sylow
177	#533	—	67	1	no	—
178	#533	C₂×D₆₇	1	1	yes	characteristic
179	#501	—	67	4	no	—
180	#501	—	67	4	no	—
181	#446	—	2	4	no	—
182	#446	—	134	4	no	—
183	C₂×#108	—	67	2	no	—
184	C₂×#108	—	67	2	no	—
185	#427	—	67	2	no	—
186	#427	—	67	2	no	—
187	#424	—	67	2	no	—
188	#424	C₂×D₆₇	1	2	yes	—
189	#423	—	67	1	no	—
190	#423	C₂×D₆₇	1	1	yes	characteristic
191	#405	—	67	2	no	—
192	#405	C₂×D₆₇	1	2	yes	—
193	C₂×#97	—	67	1	no	—
194	C₂×#97	—	67	1	no	—
195	C₂×#97	—	67	1	no	—
196	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
197	C₂×C₄×C₈	—	67	1	no	—
198	C₂×C₄×C₈	—	67	1	no	—
199	C₄³	—	67	1	no	—
200	#339	—	67	2	no	—
201	#339	—	67	2	no	—
202	#339	—	67	2	no	—
203	#339	—	67	2	no	—
204	C₂×M₁₆	—	67	2	no	—
205	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
206	C₂²×C₈	—	67	1	no	—
207	C₂²×C₈	—	67	1	no	—
208	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₄⋊Q₈	—	67	2	no	—
211	C₄⋊Q₈	C₄×D₆₇	1	2	yes	—
212	#125	—	67	2	no	—
213	#125	C₄×D₆₇	1	2	yes	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	134	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	134	2	no	—
218	#117	—	67	2	no	—
219	#117	C₂²×D₆₇	1	2	yes	—
220	C₂×C₄²	—	67	1	no	—
221	C₂×C₄²	—	67	1	no	—
222	C₂×C₄²	—	134	2	no	—
223	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
224	#108	—	134	4	no	—
225	#108	—	134	4	no	—
226	#107	—	67	4	no	—
227	#107	—	67	4	no	—
228	#106	—	67	4	no	—
229	#106	—	67	4	no	—
230	C₄⋊C₈	—	2	4	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	134	4	no	—
233	C₄⋊C₈	—	134	2	no	—
234	#97	—	2	2	no	—
235	#97	—	67	4	no	—
236	#97	—	67	2	no	—
237	#97	—	67	2	no	—
238	#97	—	134	2	no	—
239	#97	—	134	2	no	—
240	#97	C₂²⋊D₆₇	1	4	yes	—
241	C₄×C₈	—	2	2	no	—
242	C₄×C₈	—	67	4	no	—
243	C₄×C₈	—	134	2	no	—
244	C₄×C₈	—	134	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	134	2	no	—
247	C₂²×C₄	—	134	1	no	—
248	C₂²×C₄	—	134	1	no	—
249	C₂²×C₄	#11761628	1	2	yes	—
250	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	134	2	no	—
254	M₁₆	—	134	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	67	2	no	—
259	C₂×C₈	—	67	2	no	—
260	C₂×C₈	—	67	2	no	—
261	C₂×C₈	—	67	2	no	—
262	C₂×C₈	—	67	2	no	—
263	C₂×C₈	—	67	2	no	—
264	C₂×C₈	—	134	2	no	—
265	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
266	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
267	C₂×C₈	Q₈×D₆₇	1	2	yes	—
268	C₂×C₈	Q₈×D₆₇	1	2	yes	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	67	4	no	—
272	C₄⋊C₄	—	134	4	no	—
273	C₄⋊C₄	—	134	2	no	—
274	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	134	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	268	4	no	—
279	C₄²	—	67	2	no	—
280	C₄²	—	67	2	no	—
281	C₄²	—	134	2	no	—
282	C₄²	—	134	1	no	—
283	C₄²	—	134	1	no	—
284	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
287	C₂³	C₂×#11761628	1	1	yes	characteristic
288	Q₈	—	8	2	no	—
289	Q₈	—	536	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	268	2	no	—
293	C₂×C₄	—	134	4	no	—
294	C₂×C₄	—	134	2	no	—
295	C₂×C₄	—	134	4	no	—
296	C₂×C₄	—	134	2	no	—
297	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
298	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
303	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
304	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	268	2	no	—
308	C₈	—	67	4	no	—
309	C₈	—	67	4	no	—
310	C₈	#14524041	1	4	yes	—
311	C₂²	—	2	2	no	—
312	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
313	C₂²	#16863950	1	1	yes	characteristic
314	C₂²	#16863950	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	268	2	no	—
319	C₄	—	268	4	no	—
320	C₄	—	134	4	no	—
321	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
322	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
323	C₄	C₂×#14524041	1	1	yes	characteristic
324	C₄	#16863942	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	C₄×#14524041	1	1	yes	characteristic
328	C₂	#22569116	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2961	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2961	D₆₇
hypocenter	C₆₇	#21736
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #2961

Complements

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

N	H ≅ G/N	type
13. #16863806	C₄	semidirect
14. #16863802	C₄	semidirect
62. #14523953	Q₈	semidirect
65. C₈×Dic₆₇	Q₈	semidirect
176. C₆₇	#21736	semidirect

Maximal subgroups

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

subgroup	index
#22570731	2
#22569149	2
#22561187	2
#22561185	2
#22560959	2
#22560954	2
#22560383	2
#22560094	2
#22560089	2
#97×Dic₆₇	2
C₆₇×#2961	2
#21736	67