Subgroups — #30864708

Order 17152.

670 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570732	C₂	1	1	yes	maximal, characteristic
2	#22569147	C₂	1	2	yes	maximal
3	#22561184	C₂	1	2	yes	maximal
4	#22560961	C₂	1	2	yes	maximal
5	#22560958	C₂	1	1	yes	maximal, characteristic
6	#22560955	C₂	1	1	yes	maximal, characteristic
7	#22560382	C₂	1	1	yes	maximal, characteristic
8	#22560093	C₂	1	1	yes	maximal, characteristic
9	#22560089	C₂	1	2	yes	maximal
10	#97×Dic₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1773	C₂	1	1	yes	maximal, characteristic
12	#16863949	C₂²	1	2	yes	—
13	#16863804	C₄	1	4	yes	complemented
14	#16863803	C₄	1	4	yes	complemented
15	#16863526	—	2	4	no	—
16	#16863524	—	2	4	no	—
17	C₂×#14523979	C₂²	1	1	yes	characteristic
18	C₂×#14523978	C₂²	1	2	yes	—
19	#16863434	C₂²	1	1	yes	characteristic
20	#16863432	C₂²	1	2	yes	—
21	#16863424	C₂²	1	2	yes	—
22	#16863423	C₂²	1	2	yes	—
23	#16863419	C₂²	1	2	yes	—
24	#16863351	C₂²	1	2	yes	—
25	C₂×#14523954	C₂²	1	1	yes	characteristic
26	C₂×#14523953	C₂²	1	1	yes	characteristic
27	C₂×#14523953	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
30	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
31	#16863157	C₂²	1	2	yes	—
32	#16863156	C₂²	1	2	yes	—
33	#16863155	C₂²	1	2	yes	—
34	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
35	#427×C₆₇	C₂²	1	1	yes	characteristic
36	#405×C₆₇	C₂²	1	2	yes	—
37	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
38	#339×C₆₇	C₂²	1	2	yes	—
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	#14524042	C₂×C₄	1	4	yes	—
43	#14524034	C₂×C₄	1	4	yes	—
44	#14524020	—	2	4	no	—
45	C₄×Dic₁₃₄	—	2	4	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	#14523979	—	2	2	no	—
51	#14523978	—	2	4	no	—
52	#14523977	C₂×C₄	1	2	yes	—
53	#14523976	C₂×C₄	1	4	yes	—
54	#14523975	C₂×C₄	1	2	yes	—
55	#14523974	C₂×C₄	1	4	yes	—
56	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
57	Dic₆₇⋊C₈	—	2	4	no	—
58	Dic₆₇⋊C₈	—	2	4	no	—
59	#14523954	—	2	2	no	—
60	#14523953	—	2	2	no	—
61	#14523953	C₂×C₄	1	2	yes	—
62	#14523953	C₂×C₄	1	2	yes	—
63	#14523953	D₄	1	4	yes	—
64	C₈×Dic₆₇	—	2	2	no	—
65	C₈×Dic₆₇	—	2	2	no	—
66	C₈×Dic₆₇	Q₈	1	4	yes	complemented
67	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	#117×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	#108×C₆₇	—	2	2	no	—
73	#107×C₆₇	C₂×C₄	1	2	yes	—
74	#106×C₆₇	C₂×C₄	1	2	yes	—
75	#97×C₆₇	—	2	2	no	—
76	#97×C₆₇	Q₈	1	4	yes	complemented
77	C₄×C₈×C₆₇	—	2	2	no	—
78	C₂×Dic₁₃₄	—	2	4	no	—
79	C₂²×Dic₆₇	—	2	1	no	—
80	C₂²×Dic₆₇	#41	1	2	yes	—
81	#11761612	—	2	2	no	—
82	#11761612	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	2	1	no	—
84	C₂×C₆₇⋊C₈	—	2	1	no	—
85	C₂×C₆₇⋊C₈	—	2	2	no	—
86	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
88	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₆₇⋊C₈	#41	1	2	yes	—
90	C₄⋊Dic₆₇	—	2	4	no	—
91	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
92	Dic₆₇⋊C₄	—	2	4	no	—
93	Dic₆₇⋊C₄	—	2	4	no	—
94	C₂²⋊Dic₆₇	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	2	1	no	—
97	C₄×Dic₆₇	—	4	4	no	—
98	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
99	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
100	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₆₇	—	2	2	no	—
104	M₁₆×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
110	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₆₇	#41	1	2	yes	—
113	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	2	1	no	—
117	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
118	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₆₇	—	2	4	no	—
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	2	4	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	C₄×D₄	1	4	yes	—
133	Dic₁₃₄	—	8	4	no	—
134	C₈×C₆₇	—	4	2	no	—
135	C₈×C₆₇	C₄×Q₈	1	2	yes	—
136	C₈×C₆₇	C₄×Q₈	1	2	yes	—
137	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
138	C₆₇⋊C₈	—	2	2	no	—
139	C₆₇⋊C₈	—	4	2	no	—
140	C₆₇⋊C₈	C₄×Q₈	1	4	yes	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₆₇	#533	1	1	yes	characteristic
144	C₂²×C₆₇	#533	1	1	yes	characteristic
145	C₄×C₆₇	—	4	2	no	—
146	C₄×C₆₇	—	4	4	no	—
147	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₄×C₆₇	#532	1	1	yes	characteristic
151	Dic₆₇	—	2	4	no	—
152	Dic₆₇	—	2	4	no	—
153	Dic₆₇	—	4	4	no	—
154	#21736	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₆₇	#2956	1	2	yes	—
159	#3162	—	67	1	no	—
160	#2961	—	67	2	no	—
161	#1994	—	67	2	no	—
162	#1946	—	67	2	no	—
163	#1945	—	67	1	no	—
164	#1945	—	67	1	no	—
165	#1849	—	67	1	no	—
166	#1773	—	67	1	no	—
167	#1773	D₆₇	1	1	yes	characteristic
168	#1772	—	67	2	no	—
169	C₄×#97	—	67	1	no	—
170	C₆₇	#21736	1	1	yes	characteristic, complemented, Sylow
171	#533	—	67	2	no	—
172	#501	—	67	4	no	—
173	#501	—	67	4	no	—
174	#446	—	134	4	no	—
175	#446	—	134	4	no	—
176	C₂×#108	—	67	2	no	—
177	C₂×#108	—	67	1	no	—
178	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
179	#427	—	67	1	no	—
180	#427	—	67	2	no	—
181	#427	C₂×D₆₇	1	1	yes	characteristic
182	#424	—	67	2	no	—
183	#424	—	67	2	no	—
184	#423	—	67	2	no	—
185	#405	—	67	2	no	—
186	#405	C₂×D₆₇	1	2	yes	—
187	C₂×#97	—	67	1	no	—
188	C₂×#97	—	67	1	no	—
189	C₂×#97	—	67	1	no	—
190	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
191	C₂×C₄×C₈	—	67	1	no	—
192	C₂×C₄×C₈	—	67	1	no	—
193	C₄³	—	67	1	no	—
194	#339	—	67	2	no	—
195	#339	—	67	2	no	—
196	#339	—	67	2	no	—
197	#339	C₂×D₆₇	1	2	yes	—
198	C₂×M₁₆	—	67	2	no	—
199	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
200	C₂²×C₈	—	67	1	no	—
201	C₂²×C₈	—	67	1	no	—
202	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
204	C₄⋊Q₈	—	67	4	no	—
205	#125	—	67	4	no	—
206	C₂²⋊Q₈	—	134	4	no	—
207	C₄×Q₈	—	134	4	no	—
208	#117	—	67	2	no	—
209	#117	C₂²×D₆₇	1	2	yes	—
210	C₂×C₄²	—	67	1	no	—
211	C₂×C₄²	—	67	1	no	—
212	C₂×C₄²	—	134	2	no	—
213	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
214	#108	—	2	2	no	—
215	#108	—	134	4	no	—
216	#108	—	134	2	no	—
217	#107	—	67	2	no	—
218	#107	—	67	4	no	—
219	#107	C₄×D₆₇	1	2	yes	—
220	#106	—	67	2	no	—
221	#106	—	67	4	no	—
222	#106	C₄×D₆₇	1	2	yes	—
223	C₄⋊C₈	—	134	4	no	—
224	C₄⋊C₈	—	134	4	no	—
225	C₄⋊C₈	—	134	4	no	—
226	#97	—	2	2	no	—
227	#97	—	67	2	no	—
228	#97	—	67	2	no	—
229	#97	—	67	4	no	—
230	#97	—	134	2	no	—
231	#97	—	134	2	no	—
232	#97	Dic₁₃₄	1	4	yes	complemented
233	C₄×C₈	—	2	2	no	—
234	C₄×C₈	—	67	4	no	—
235	C₄×C₈	—	134	2	no	—
236	C₄×C₈	—	134	2	no	—
237	C₂×Q₈	—	134	4	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	67	2	no	—
240	C₂²×C₄	—	134	1	no	—
241	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	134	2	no	—
245	M₁₆	—	134	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	67	2	no	—
248	C₂×C₈	—	67	2	no	—
249	C₂×C₈	—	67	2	no	—
250	C₂×C₈	—	67	2	no	—
251	C₂×C₈	—	134	2	no	—
252	C₂×C₈	—	134	1	no	—
253	C₂×C₈	—	134	1	no	—
254	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
256	C₂×C₈	D₄×D₆₇	1	2	yes	—
257	C₂×C₈	Q₈×D₆₇	1	2	yes	—
258	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
259	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
260	C₂×C₈	#11761628	1	2	yes	—
261	C₄⋊C₄	—	67	4	no	—
262	C₄⋊C₄	—	134	4	no	—
263	C₄⋊C₄	—	134	4	no	—
264	C₄⋊C₄	—	134	4	no	—
265	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	134	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	268	4	no	—
271	C₄²	—	67	2	no	—
272	C₄²	—	67	2	no	—
273	C₄²	—	67	2	no	—
274	C₄²	—	134	2	no	—
275	C₄²	—	134	1	no	—
276	C₄²	C₂×Dic₁₃₄	1	2	yes	—
277	C₂³	#41×D₆₇	1	1	yes	characteristic
278	Q₈	—	536	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	268	2	no	—
283	C₂×C₄	—	67	4	no	—
284	C₂×C₄	—	134	4	no	—
285	C₂×C₄	—	134	4	no	—
286	C₂×C₄	—	134	2	no	—
287	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
289	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
291	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	268	2	no	—
295	C₈	—	67	4	no	—
296	C₈	—	134	2	no	—
297	C₈	C₄×Dic₁₃₄	1	2	yes	—
298	C₈	C₄×Dic₁₃₄	1	2	yes	—
299	C₈	#14524042	1	4	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
302	C₂²	#16863949	1	1	yes	characteristic
303	C₂²	#16863950	1	1	yes	characteristic
304	C₄	—	4	2	no	—
305	C₄	—	4	4	no	—
306	C₄	—	268	4	no	—
307	C₄	—	134	4	no	—
308	C₄	—	134	4	no	—
309	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
310	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
311	C₄	C₂×#14524042	1	1	yes	characteristic
312	C₄	#16863943	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	C₄×#14524042	1	1	yes	characteristic
316	C₂	#22569117	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1773	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1773	D₆₇
hypocenter	C₆₇	#21736
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
13. #16863804	C₄	semidirect
14. #16863803	C₄	semidirect
66. C₈×Dic₆₇	Q₈	semidirect
76. #97×C₆₇	Q₈	semidirect
170. C₆₇	#21736	semidirect
232. #97	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570732	2
#22569147	2
#22561184	2
#22560961	2
#22560958	2
#22560955	2
#22560382	2
#22560093	2
#22560089	2
#97×Dic₆₇	2
C₆₇×#1773	2
#21736	67