Subgroups — #30624124

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840772	C₂	1	2	yes	maximal
2	#21462171	C₂	1	8	yes	maximal, complemented
3	#21462170	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519856	—	2	2	no	—
6	#16840772	—	2	2	no	—
7	#16840772	—	2	2	no	—
8	#16840772	C₂²	1	4	yes	—
9	#16840772	C₂²	1	4	yes	—
10	C₂×#14519846	C₂²	1	4	yes	—
11	C₂×#14519845	—	2	2	no	—
12	#16840689	C₂²	1	8	yes	complemented
13	#16840688	C₂²	1	8	yes	complemented
14	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
15	C₂×#14519717	C₄	1	1	yes	characteristic, complemented
16	C₂×#14519716	C₄	1	2	yes	complemented
17	M₃₂×D₆₁	C₄	1	4	yes	complemented
18	M₃₂×D₆₁	C₂²	1	4	yes	—
19	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
20	C₂×M₃₂×C₆₁	C₄	1	1	yes	characteristic, complemented
21	#14519856	—	2	4	no	—
22	#14519856	—	2	4	no	—
23	#14519856	—	4	2	no	—
24	#14519856	—	4	2	no	—
25	C₂×C₄⋊#6923	—	2	4	no	—
26	C₂×C₂²⋊#6923	—	4	2	no	—
27	C₂×C₄×#6923	—	4	2	no	—
28	#14519846	—	2	4	no	—
29	#14519846	C₂³	1	8	yes	—
30	#14519845	—	2	4	no	—
31	#14519845	—	2	4	no	—
32	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
33	C₂²×#92418	Q₈	1	1	yes	characteristic
34	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₆₁	D₄	1	2	yes	—
37	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
38	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₆₁	Q₈	1	2	yes	—
40	C₂×C₈×D₆₁	C₂³	1	2	yes	—
41	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
42	#14519717	C₂×C₄	1	4	yes	complemented
43	#14519716	C₂×C₄	1	8	yes	complemented
44	C₂×#11758185	C₂×C₄	1	2	yes	complemented
45	C₁₆×D₆₁	C₂×C₄	1	4	yes	complemented
46	C₁₆×D₆₁	C₂³	1	4	yes	—
47	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₆₁	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
50	C₂³×D₆₁	D₈	1	1	yes	characteristic
51	C₂²×#6923	—	4	4	no	—
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	Q₁₆	1	1	yes	characteristic
54	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×D₆₁	D₈	1	2	yes	—
57	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
58	C₂×C₄×D₆₁	Q₁₆	1	1	yes	characteristic
59	C₂×C₄×D₆₁	Q₁₆	1	2	yes	—
60	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊#6923	—	4	4	no	—
63	C₂²⋊#6923	—	4	4	no	—
64	C₄⋊#6923	—	2	8	no	—
65	C₄⋊#6923	—	4	4	no	—
66	C₄×#6923	—	4	4	no	—
67	C₄×#6923	—	4	4	no	—
68	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
72	C₂×#92418	C₄⋊C₄	1	2	yes	—
73	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#92418	C₂×Q₈	1	2	yes	—
75	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
76	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
77	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
78	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
79	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₈×D₆₁	C₂²×C₄	1	2	yes	—
81	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
82	C₈×D₆₁	C₂×D₄	1	2	yes	—
83	C₈×D₆₁	C₂×Q₈	1	2	yes	—
84	C₈×D₆₁	C₂×Q₈	1	1	yes	characteristic
85	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
86	C₁₆×C₆₁	C₂²×C₄	1	4	yes	—
87	#11758185	C₂²×C₄	1	4	yes	—
88	C₂³×C₆₁	#107	1	1	yes	characteristic
89	C₂²×D₆₁	—	4	2	no	—
90	C₂²×D₆₁	#107	1	2	yes	—
91	C₂²×D₆₁	#107	1	1	yes	characteristic
92	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
93	C₂²×D₆₁	C₂×D₈	1	2	yes	—
94	C₂×#6923	—	4	8	no	—
95	C₂×#6923	—	8	8	no	—
96	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#107	1	2	yes	—
98	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₆₁	#107	1	2	yes	—
101	C₂×Dic₆₁	#107	1	1	yes	characteristic
102	C₂×Dic₆₁	C₂×Q₁₆	1	2	yes	—
103	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
104	C₄×D₆₁	#107	1	2	yes	—
105	C₄×D₆₁	#107	1	2	yes	—
106	C₄×D₆₁	#107	1	2	yes	—
107	C₄×D₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₄×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
110	C₄×D₆₁	C₂×D₈	1	2	yes	—
111	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
112	C₄×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
113	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
116	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
117	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
118	#92418	C₂×C₄⋊C₄	1	2	yes	—
119	#92418	C₂²×Q₈	1	1	yes	characteristic
120	#33409	—	61	1	no	maximal, Sylow
121	C₂²×C₆₁	—	4	1	no	—
122	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
123	C₂²×C₆₁	C₂×#107	1	2	yes	—
124	C₂×D₆₁	—	2	4	no	—
125	C₂×D₆₁	—	2	2	no	—
126	C₂×D₆₁	—	4	1	no	—
127	C₂×D₆₁	—	4	2	no	—
128	C₂×D₆₁	C₂×#107	1	1	yes	characteristic
129	C₂×D₆₁	C₂×#107	1	2	yes	—
130	C₂×D₆₁	C₂²×D₈	1	1	yes	characteristic
131	#6923	—	8	8	no	—
132	C₄×C₆₁	C₂×#107	1	2	yes	—
133	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
134	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	Dic₆₁	C₂×#107	1	1	yes	characteristic
136	Dic₆₁	C₂×#107	1	2	yes	—
137	Dic₆₁	C₂²×Q₁₆	1	1	yes	characteristic
138	C₂²×M₃₂	—	61	1	no	—
139	C₂×#427	—	61	2	no	—
140	#2159	—	61	8	no	—
141	#2158	—	61	4	no	—
142	C₂×C₆₁	—	2	2	no	—
143	C₂×C₆₁	—	2	1	no	—
144	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
145	D₆₁	—	2	2	no	—
146	D₆₁	—	2	1	no	—
147	D₆₁	#3414	1	2	yes	—
148	C₂³×C₈	—	61	1	no	—
149	C₂×#117	—	122	2	no	—
150	C₂×M₃₂	—	61	1	no	—
151	C₂×M₃₂	—	61	4	no	—
152	C₂×M₃₂	—	61	4	no	—
153	C₂×M₃₂	—	61	2	no	—
154	C₂×M₃₂	#6923	1	1	yes	characteristic, complemented
155	C₂²×C₁₆	—	61	2	no	—
156	#427	—	61	4	no	—
157	#427	—	61	4	no	—
158	#427	—	122	2	no	—
159	#427	—	122	2	no	—
160	C₂×#107	—	61	4	no	—
161	C₂×#106	—	122	2	no	—
162	#367	—	61	8	no	—
163	#366	—	61	8	no	—
164	C₆₁	#33409	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	61	1	no	—
166	C₂²×C₈	—	61	1	no	—
167	C₂²×C₈	—	61	2	no	—
168	C₂²×C₈	—	61	1	no	—
169	C₂²×C₈	—	61	2	no	—
170	C₂²×C₈	—	61	1	no	—
171	C₂²×C₈	—	61	2	no	—
172	C₂²×C₈	—	61	1	no	—
173	C₂²×C₈	—	61	2	no	—
174	C₂²×C₈	—	61	1	no	—
175	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
176	#117	—	122	4	no	—
177	#117	—	122	4	no	—
178	#117	—	244	2	no	—
179	#117	—	244	2	no	—
180	C₂×C₄⋊C₄	—	122	4	no	—
181	C₂×C₂²⋊C₄	—	244	2	no	—
182	C₂×C₄²	—	244	2	no	—
183	M₃₂	—	61	4	no	—
184	M₃₂	—	61	8	no	—
185	M₃₂	C₂×#6923	1	4	yes	complemented
186	C₂×C₁₆	—	61	2	no	—
187	C₂×C₁₆	—	61	4	no	—
188	C₂×C₁₆	—	61	4	no	—
189	C₂×C₁₆	C₂×#6923	1	2	yes	complemented
190	#107	—	61	8	no	—
191	#107	—	122	4	no	—
192	#106	—	122	4	no	—
193	#106	—	122	4	no	—
194	C₂⁴	—	61	1	no	—
195	C₂²×C₄	—	61	1	no	—
196	C₂²×C₄	—	61	2	no	—
197	C₂²×C₄	—	61	1	no	—
198	C₂²×C₄	—	61	1	no	—
199	C₂²×C₄	—	61	2	no	—
200	C₂²×C₄	—	61	2	no	—
201	C₂²×C₄	—	61	1	no	—
202	C₂²×C₄	—	61	2	no	—
203	C₂²×C₄	—	61	1	no	—
204	C₂²×C₄	—	244	4	no	—
205	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
206	C₂×C₈	—	61	2	no	—
207	C₂×C₈	—	61	1	no	—
208	C₂×C₈	—	61	2	no	—
209	C₂×C₈	—	61	1	no	—
210	C₂×C₈	—	61	1	no	—
211	C₂×C₈	—	61	2	no	—
212	C₂×C₈	—	61	2	no	—
213	C₂×C₈	—	61	2	no	—
214	C₂×C₈	—	61	2	no	—
215	C₂×C₈	—	61	2	no	—
216	C₂×C₈	—	61	1	no	—
217	C₂×C₈	—	61	1	no	—
218	C₂×C₈	—	61	1	no	—
219	C₂×C₈	—	61	2	no	—
220	C₂×C₈	C₄⋊#6923	1	2	yes	—
221	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
222	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
223	C₂×C₈	C₂²×#6923	1	2	yes	—
224	C₄⋊C₄	—	122	8	no	—
225	C₄⋊C₄	—	244	4	no	—
226	C₂²⋊C₄	—	244	4	no	—
227	C₂²⋊C₄	—	244	4	no	—
228	C₄²	—	244	4	no	—
229	C₄²	—	244	4	no	—
230	C₁₆	—	61	4	no	—
231	C₁₆	C₂²×#6923	1	4	yes	—
232	C₂³	—	61	1	no	—
233	C₂³	—	61	2	no	—
234	C₂³	—	61	2	no	—
235	C₂³	—	61	1	no	—
236	C₂³	—	244	2	no	—
237	C₂³	#14519846	1	1	yes	characteristic
238	C₂×C₄	—	61	2	no	—
239	C₂×C₄	—	61	1	no	—
240	C₂×C₄	—	61	2	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	61	2	no	—
243	C₂×C₄	—	61	2	no	—
244	C₂×C₄	—	61	2	no	—
245	C₂×C₄	—	61	2	no	—
246	C₂×C₄	—	61	2	no	—
247	C₂×C₄	—	61	1	no	—
248	C₂×C₄	—	61	1	no	—
249	C₂×C₄	—	61	1	no	—
250	C₂×C₄	—	61	2	no	—
251	C₂×C₄	—	61	1	no	—
252	C₂×C₄	—	488	8	no	—
253	C₂×C₄	—	244	8	no	—
254	C₂×C₄	#14519846	1	1	yes	characteristic
255	C₂×C₄	#14519846	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
258	C₈	—	61	1	no	—
259	C₈	—	61	2	no	—
260	C₈	—	61	1	no	—
261	C₈	C₂×C₄⋊#6923	1	1	yes	characteristic
262	C₈	C₂×C₄⋊#6923	1	2	yes	—
263	C₈	C₂³×#6923	1	1	yes	characteristic
264	C₂²	—	4	1	no	—
265	C₂²	—	61	1	no	—
266	C₂²	—	61	1	no	—
267	C₂²	—	61	2	no	—
268	C₂²	—	122	4	no	—
269	C₂²	—	122	2	no	—
270	C₂²	—	244	2	no	—
271	C₂²	—	244	1	no	—
272	C₂²	C₂×#14519846	1	1	yes	characteristic
273	C₂²	C₂×#14519846	1	2	yes	—
274	C₄	—	61	1	no	—
275	C₄	—	61	1	no	—
276	C₄	—	61	2	no	—
277	C₄	—	488	8	no	—
278	C₄	C₂×#14519846	1	2	yes	—
279	C₄	C₂×#14519846	1	1	yes	characteristic
280	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	61	2	no	—
284	C₂	—	122	2	no	—
285	C₂	—	122	1	no	—
286	C₂	C₂²×#14519846	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14519846
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₂×M₃₂×C₆₁	C₄
socle	C₂×C₆₁	C₂²×#107
prime radical	C₂³×D₆₁	D₈
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂×M₃₂	#6923
hypocenter	C₆₁	#33409
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #21462171	C₂	semidirect
3. #21462170	C₂	semidirect
12. #16840689	C₂²	semidirect
13. #16840688	C₂²	semidirect
15. C₂×#14519717	C₄	semidirect
16. C₂×#14519716	C₄	semidirect
17. M₃₂×D₆₁	C₄	semidirect
20. C₂×M₃₂×C₆₁	C₄	semidirect
42. #14519717	C₂×C₄	semidirect
43. #14519716	C₂×C₄	semidirect
44. C₂×#11758185	C₂×C₄	semidirect
45. C₁₆×D₆₁	C₂×C₄	semidirect
48. M₃₂×C₆₁	C₂×C₄	semidirect
49. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
154. C₂×M₃₂	#6923	semidirect
164. C₆₁	#33409	semidirect
185. M₃₂	C₂×#6923	semidirect
189. C₂×C₁₆	C₂×#6923	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840772	2
#21462171	2
#21462170	2
C₂×M₃₂×D₆₁	2
#33409	61