Subgroups — #30612327

Order 15616.

782 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	#21462666	C₂	1	2	yes	maximal, complemented
2	#21461415	C₂	1	4	yes	maximal, complemented
3	D₈×#92420	C₂	1	4	yes	maximal, complemented
4	#21461366	C₂	1	1	yes	maximal, characteristic, complemented
5	#21461118	C₂	1	2	yes	maximal, complemented
6	#21461022	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
8	D₄×#92420	—	2	4	no	—
9	D₄×#92420	C₂²	1	8	yes	complemented
10	#16840778	C₂²	1	2	yes	—
11	#16840778	C₂²	1	1	yes	characteristic
12	#16840778	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊#92420	—	2	4	no	—
14	#16840625	C₂²	1	2	yes	complemented
15	#16840623	C₂²	1	2	yes	complemented
16	D₄⋊#92420	C₂²	1	8	yes	complemented
17	#16840603	C₂²	1	2	yes	complemented
18	C₈×#92420	C₂²	1	2	yes	complemented
19	C₂×D₄×Dic₆₁	C₂²	1	2	yes	—
20	D₈×Dic₆₁	C₄	1	2	yes	—
21	D₈×Dic₆₁	C₄	1	2	yes	—
22	D₈×Dic₆₁	C₂²	1	2	yes	—
23	D₈×Dic₆₁	C₂²	1	2	yes	—
24	C₂×#14519711	C₄	1	1	yes	characteristic
25	C₂×#14519698	C₄	1	2	yes	—
26	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
27	C₂²×D₈×C₆₁	C₄	1	1	yes	characteristic
28	C₂²×#92420	—	2	8	no	—
29	C₄⋊#92420	—	2	1	no	—
30	C₄⋊#92420	D₄	1	2	yes	—
31	C₄⋊#92420	C₂³	1	4	yes	—
32	C₂²⋊#92420	—	2	2	no	—
33	C₂²⋊#92420	—	2	1	no	—
34	C₂²⋊#92420	—	2	8	no	—
35	C₂²⋊#92420	D₄	1	2	yes	—
36	C₄×#92420	—	2	2	no	—
37	C₄×#92420	C₂³	1	2	yes	—
38	C₄×#92420	C₂³	1	2	yes	—
39	C₂³×Dic₆₁	—	2	2	no	—
40	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
41	D₄×Dic₆₁	—	2	2	no	—
42	D₄×Dic₆₁	—	2	2	no	—
43	D₄×Dic₆₁	C₂×C₄	1	4	yes	—
44	D₄×Dic₆₁	C₂³	1	4	yes	—
45	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
47	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
48	#14519711	C₂×C₄	1	2	yes	—
49	#14519711	C₂×C₄	1	2	yes	—
50	#14519698	C₂×C₄	1	4	yes	—
51	#14519698	C₂×C₄	1	4	yes	—
52	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
55	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
57	C₂×D₈×C₆₁	C₈	1	8	yes	complemented
58	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
61	C₂⁴×C₆₁	—	2	2	no	—
62	C₂×D₄×C₆₁	—	2	2	no	—
63	C₂×D₄×C₆₁	—	2	4	no	—
64	C₂×D₄×C₆₁	C₂×C₈	1	8	yes	complemented
65	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₆₁	—	2	2	no	—
68	C₂²×Dic₆₁	—	2	4	no	—
69	C₂²×Dic₆₁	—	2	4	no	—
70	C₂²×Dic₆₁	—	8	4	no	—
71	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₆₁	#41	1	1	yes	characteristic
73	C₂×#92420	—	2	4	no	—
74	C₂×#92420	—	2	2	no	—
75	C₂×#92420	—	8	8	no	—
76	C₂×#92420	C₂×D₄	1	4	yes	—
77	D₈×C₆₁	C₂×C₈	1	8	yes	complemented
78	D₈×C₆₁	C₂×C₈	1	8	yes	complemented
79	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
80	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊Dic₆₁	—	2	4	no	—
83	C₂²⋊Dic₆₁	—	2	4	no	—
84	C₄⋊Dic₆₁	—	2	2	no	—
85	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
86	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
88	C₄×Dic₆₁	#41	1	1	yes	characteristic
89	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
90	C₂×#92418	—	2	2	no	—
91	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
92	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₆₁	—	2	4	no	—
94	C₂³×C₆₁	—	2	8	no	—
95	C₂³×C₆₁	—	8	2	no	—
96	C₂³×C₆₁	—	8	2	no	—
97	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
98	D₄×C₆₁	—	2	4	no	—
99	D₄×C₆₁	—	2	8	no	—
100	D₄×C₆₁	C₂²×C₈	1	8	yes	—
101	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₆₁	#131	1	2	yes	—
104	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₆₁	—	2	2	no	—
106	C₂×Dic₆₁	—	2	4	no	—
107	C₂×Dic₆₁	—	4	1	no	—
108	C₂×Dic₆₁	—	4	8	no	—
109	C₂×Dic₆₁	—	8	4	no	—
110	C₂×Dic₆₁	—	8	2	no	—
111	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
112	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
113	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
115	#92420	—	2	4	no	—
116	#92420	—	4	2	no	—
117	#92420	—	4	4	no	—
118	C₈×C₆₁	C₂²×C₈	1	2	yes	—
119	C₈×C₆₁	C₂²×C₈	1	2	yes	—
120	#92418	—	2	1	no	—
121	#92418	—	2	1	no	—
122	#17854	—	61	1	no	maximal, Sylow
123	C₂²×C₆₁	—	2	1	no	—
124	C₂²×C₆₁	—	2	1	no	—
125	C₂²×C₆₁	—	2	8	no	—
126	C₂²×C₆₁	—	4	4	no	—
127	C₂²×C₆₁	—	4	4	no	—
128	C₂²×C₆₁	—	8	8	no	—
129	C₂²×C₆₁	C₈×D₄	1	2	yes	—
130	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
131	C₄×C₆₁	C₈×D₄	1	2	yes	—
132	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
134	Dic₆₁	—	2	1	no	—
135	Dic₆₁	—	2	1	no	—
136	Dic₆₁	—	2	1	no	—
137	Dic₆₁	—	4	4	no	—
138	Dic₆₁	#576	1	2	yes	—
139	#2972	—	61	2	no	—
140	C₂×C₄×D₈	—	61	1	no	—
141	#1580	—	61	4	no	—
142	C₈×D₈	—	61	4	no	—
143	#1564	—	61	1	no	—
144	#1478	—	61	2	no	—
145	#1450	—	61	1	no	—
146	C₂×C₆₁	—	2	2	no	—
147	C₂×C₆₁	—	4	8	no	—
148	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
149	C₂×C₆₁	#2948	1	1	yes	characteristic
150	C₂×C₆₁	#2957	1	1	yes	characteristic
151	C₂²×D₈	#6923	1	1	yes	characteristic
152	C₂×C₄×D₄	—	61	2	no	—
153	C₄×D₈	—	61	2	no	—
154	C₄×D₈	—	61	2	no	—
155	C₄×D₈	—	61	2	no	—
156	C₄×D₈	—	61	2	no	—
157	C₈×D₄	—	61	8	no	—
158	C₈×D₄	—	122	4	no	—
159	#431	—	61	1	no	—
160	#431	—	61	1	no	—
161	#431	—	61	2	no	—
162	C₂×#107	—	61	1	no	—
163	C₂×#102	—	61	2	no	—
164	C₂×C₂²⋊C₈	—	122	4	no	—
165	C₂×C₄×C₈	—	61	1	no	—
166	#335	—	61	2	no	—
167	#334	—	61	2	no	—
168	D₄⋊C₈	—	61	8	no	—
169	#322	—	61	2	no	—
170	C₈²	—	61	2	no	—
171	C₆₁	#17854	1	1	yes	characteristic, complemented, Sylow
172	C₂²×D₄	C₂×#6923	1	2	yes	—
173	C₂³×C₄	—	122	2	no	—
174	C₂×D₈	#92420	1	8	yes	complemented
175	C₂×D₈	C₂×#6923	1	2	yes	—
176	C₂×D₈	C₂×#6923	1	2	yes	—
177	C₂²×C₈	—	61	1	no	—
178	C₂²×C₈	—	122	8	no	—
179	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
180	C₄×D₄	—	61	4	no	—
181	C₄×D₄	—	61	4	no	—
182	C₄×D₄	—	122	2	no	—
183	C₄×D₄	—	122	2	no	—
184	C₂×C₄⋊C₄	—	61	2	no	—
185	C₂×C₂²⋊C₄	—	122	2	no	—
186	C₂×C₄²	—	61	1	no	—
187	#107	—	61	2	no	—
188	#107	—	61	2	no	—
189	C₄⋊C₈	—	61	2	no	—
190	C₄⋊C₈	—	61	4	no	—
191	C₄⋊C₈	—	122	1	no	—
192	#102	—	61	4	no	—
193	#102	—	61	4	no	—
194	C₂²⋊C₈	—	61	2	no	—
195	C₂²⋊C₈	—	122	8	no	—
196	C₂²⋊C₈	—	122	2	no	—
197	C₂²⋊C₈	—	122	1	no	—
198	C₄×C₈	—	61	2	no	—
199	C₄×C₈	—	61	1	no	—
200	C₄×C₈	—	61	1	no	—
201	C₄×C₈	—	61	1	no	—
202	C₄×C₈	—	61	1	no	—
203	C₄×C₈	—	61	2	no	—
204	C₄×C₈	—	122	2	no	—
205	C₂⁴	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂×#92420	1	8	yes	complemented
209	C₂×D₄	C₂²×#6923	1	4	yes	—
210	C₂²×C₄	—	61	1	no	—
211	C₂²×C₄	—	61	1	no	—
212	C₂²×C₄	—	122	4	no	—
213	C₂²×C₄	—	122	4	no	—
214	C₂²×C₄	—	122	2	no	—
215	C₂²×C₄	—	488	4	no	—
216	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
217	D₈	C₂×#92420	1	8	yes	complemented
218	D₈	C₂×#92420	1	8	yes	complemented
219	C₂×C₈	—	61	4	no	—
220	C₂×C₈	—	61	1	no	—
221	C₂×C₈	—	61	1	no	—
222	C₂×C₈	—	122	2	no	—
223	C₂×C₈	—	122	2	no	—
224	C₂×C₈	—	122	4	no	—
225	C₂×C₈	—	488	8	no	—
226	C₂×C₈	C₂×#92420	1	4	yes	complemented
227	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
228	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
229	C₄⋊C₄	—	61	4	no	—
230	C₄⋊C₄	—	122	2	no	—
231	C₂²⋊C₄	—	122	4	no	—
232	C₂²⋊C₄	—	122	4	no	—
233	C₄²	—	61	1	no	—
234	C₄²	—	61	1	no	—
235	C₄²	—	61	1	no	—
236	C₄²	—	61	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	D₄×#6923	1	1	yes	characteristic
242	D₄	—	2	8	no	—
243	D₄	—	2	4	no	—
244	D₄	C₂²×#92420	1	8	yes	—
245	C₂×C₄	—	61	1	no	—
246	C₂×C₄	—	61	1	no	—
247	C₂×C₄	—	61	1	no	—
248	C₂×C₄	—	61	1	no	—
249	C₂×C₄	—	122	2	no	—
250	C₂×C₄	—	122	4	no	—
251	C₂×C₄	—	488	4	no	—
252	C₂×C₄	—	488	2	no	—
253	C₂×C₄	—	244	8	no	—
254	C₂×C₄	—	244	1	no	—
255	C₂×C₄	D₄×#6923	1	1	yes	characteristic
256	C₂×C₄	C₂²×#92420	1	2	yes	—
257	C₂×C₄	#14519864	1	2	yes	—
258	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
259	C₈	—	122	4	no	—
260	C₈	—	122	1	no	—
261	C₈	—	122	1	no	—
262	C₈	—	244	4	no	—
263	C₈	—	244	2	no	—
264	C₈	C₂²×#92420	1	2	yes	—
265	C₈	C₂²×#92420	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	8	no	—
272	C₂²	D₄×#92420	1	2	yes	—
273	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
274	C₄	—	61	2	no	—
275	C₄	—	122	1	no	—
276	C₄	—	122	1	no	—
277	C₄	—	122	1	no	—
278	C₄	—	244	4	no	—
279	C₄	D₄×#92420	1	2	yes	—
280	C₄	C₂³×#92420	1	1	yes	characteristic
281	C₄	C₂×#14519864	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	8	no	—
284	C₂	C₂×D₄×#92420	1	1	yes	characteristic
285	C₂	#21462578	1	1	yes	characteristic
286	C₂	#21462595	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting, prime radical	C₂²×D₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂²×D₈	#6923
hypocenter	C₆₁	#17854
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
1. #21462666	C₂	semidirect
2. #21461415	C₂	semidirect
3. D₈×#92420	C₂	semidirect
4. #21461366	C₂	semidirect
5. #21461118	C₂	semidirect
6. #21461022	C₂	semidirect
9. D₄×#92420	C₂²	semidirect
14. #16840625	C₂²	semidirect
15. #16840623	C₂²	semidirect
16. D₄⋊#92420	C₂²	semidirect
17. #16840603	C₂²	semidirect
18. C₈×#92420	C₂²	semidirect
57. C₂×D₈×C₆₁	C₈	semidirect
64. C₂×D₄×C₆₁	C₂×C₈	semidirect
77. D₈×C₆₁	C₂×C₈	semidirect
78. D₈×C₆₁	C₂×C₈	semidirect
79. C₂×C₈×C₆₁	C₂×C₈	semidirect
171. C₆₁	#17854	semidirect
174. C₂×D₈	#92420	semidirect
208. C₂×D₄	C₂×#92420	semidirect
217. D₈	C₂×#92420	semidirect
218. D₈	C₂×#92420	semidirect
226. C₂×C₈	C₂×#92420	semidirect

Maximal subgroups

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

subgroup	index
#21462666	2
#21461415	2
D₈×#92420	2
#21461366	2
#21461118	2
#21461022	2
C₂×D₈×Dic₆₁	2
#17854	61