Subgroups — #30624310

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462595	C₂	1	2	yes	maximal, complemented
2	#21462223	C₂	1	4	yes	maximal, complemented
3	#21462218	C₂	1	4	yes	maximal, complemented
4	D₆₁⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
5	#21462134	C₂	1	2	yes	maximal, complemented
6	#21462027	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16840191	C₂	1	1	yes	maximal, characteristic
8	#16840840	—	2	2	no	—
9	D₆₁⋊M₃₂	C₂²	1	1	yes	characteristic
10	#16840809	—	2	4	no	—
11	#16840804	C₂²	1	8	yes	complemented
12	D₆₁⋊#108	C₂²	1	2	yes	—
13	#16840760	—	2	2	no	—
14	#16840725	C₂²	1	1	yes	characteristic
15	#16840691	C₂²	1	2	yes	complemented
16	#16840690	C₂²	1	2	yes	complemented
17	#16840676	C₂²	1	8	yes	complemented
18	#16840649	C₂²	1	2	yes	complemented
19	C₁₆×#6923	C₂²	1	2	yes	complemented
20	C₂×#14519796	C₂²	1	2	yes	—
21	#16840191	C₄	1	2	yes	complemented
22	#16840191	C₄	1	2	yes	complemented
23	#16840191	C₂²	1	2	yes	—
24	#16840191	C₂²	1	2	yes	—
25	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic, complemented
26	C₂×#14519721	C₄	1	2	yes	complemented
27	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
28	C₂×D₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
29	#14519864	—	2	8	no	—
30	#14519864	—	4	4	no	—
31	D₆₁⋊M₁₆	—	2	2	no	—
32	D₆₁⋊M₁₆	D₄	1	1	yes	characteristic
33	C₂×#11758217	—	4	4	no	—
34	#14519856	D₄	1	1	yes	characteristic
35	#14519852	—	2	1	no	—
36	#14519851	—	2	1	no	—
37	D₆₁⋊C₁₆	C₂³	1	2	yes	—
38	#14519848	C₂³	1	4	yes	—
39	#14519847	—	2	2	no	—
40	#14519841	—	2	8	no	—
41	#14519831	D₄	1	2	yes	—
42	C₈×#6923	C₂³	1	2	yes	—
43	C₂×#11758224	—	2	2	no	—
44	#14519796	—	2	2	no	—
45	#14519796	—	2	2	no	—
46	#14519796	C₂×C₄	1	4	yes	—
47	#14519796	C₂³	1	4	yes	—
48	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
49	C₂×#11758200	—	2	2	no	—
50	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
51	Dic₄₈₈	C₂×C₄	1	2	yes	—
52	Dic₄₈₈	C₂×C₄	1	2	yes	—
53	#14519721	C₂×C₄	1	4	yes	—
54	#14519721	C₂×C₄	1	4	yes	—
55	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
56	C₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic
57	C₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic
58	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
59	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
60	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
61	D₁₆×C₆₁	C₂×C₄	1	2	yes	—
62	D₁₆×C₆₁	C₂×C₄	1	2	yes	—
63	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄×C₆₁	—	2	2	no	—
65	C₂×C₂²⋊D₆₁	—	4	2	no	—
66	C₂²×Dic₆₁	—	4	2	no	—
67	#11758224	—	2	4	no	—
68	#11758224	—	2	4	no	—
69	#11758224	—	4	2	no	—
70	#11758224	—	4	2	no	—
71	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×Dic₁₂₂	—	2	2	no	—
73	C₂²⋊#6923	—	2	1	no	—
74	#11758217	—	4	2	no	—
75	#11758217	—	4	8	no	—
76	#11758217	D₈	1	2	yes	—
77	C₂×#92420	—	4	8	no	—
78	C₄⋊#6923	—	2	1	no	—
79	C₄×#6923	C₂×D₄	1	2	yes	—
80	#11758213	—	2	4	no	—
81	#11758213	D₈	1	2	yes	—
82	D₆₁⋊C₈	—	4	2	no	—
83	D₆₁⋊C₈	C₂×D₄	1	2	yes	—
84	D₈×C₆₁	—	2	2	no	—
85	D₈×C₆₁	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	#11758200	—	2	4	no	—
88	#11758200	—	2	4	no	—
89	C₂×#92418	#41	1	1	yes	characteristic
90	Dic₂₄₄	—	2	2	no	—
91	Dic₂₄₄	C₂²×C₄	1	4	yes	—
92	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
94	C₈×D₆₁	#41	1	1	yes	characteristic
95	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
96	#11758187	—	2	2	no	—
97	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	#11758185	C₂²×C₄	1	1	yes	characteristic
100	#11758185	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₆₁	—	4	2	no	—
102	C₂²×D₆₁	#135	1	1	yes	characteristic
103	C₂×#6923	—	2	2	no	—
104	D₄×C₆₁	—	2	4	no	—
105	D₄×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂²⋊D₆₁	—	4	4	no	—
108	C₂²⋊D₆₁	—	4	4	no	—
109	C₂×Dic₆₁	—	4	2	no	—
110	C₂×Dic₆₁	—	4	4	no	—
111	C₂×Dic₆₁	—	4	4	no	—
112	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
113	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
114	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
115	C₄×D₆₁	#135	1	1	yes	characteristic
116	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
117	Dic₁₂₂	—	2	4	no	—
118	Dic₁₂₂	—	4	2	no	—
119	#92420	—	4	4	no	—
120	#92420	C₂×D₈	1	4	yes	—
121	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
123	#92418	C₄×D₄	1	1	yes	characteristic
124	#92418	C₂×#41	1	1	yes	characteristic
125	#33465	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	4	4	no	—
127	C₂²×C₆₁	—	16	2	no	—
128	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
129	C₂×D₆₁	—	4	1	no	—
130	C₂×D₆₁	C₄×D₈	1	1	yes	characteristic
131	C₂×D₆₁	C₂×#135	1	1	yes	characteristic
132	#6923	—	4	2	no	—
133	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
134	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
135	Dic₆₁	—	4	4	no	—
136	Dic₆₁	C₄×D₈	1	1	yes	characteristic
137	Dic₆₁	C₂²×D₈	1	1	yes	characteristic
138	C₂×#508	—	61	1	no	—
139	#2957	—	61	2	no	—
140	#2179	—	61	4	no	—
141	#2178	—	61	4	no	—
142	C₂×#368	—	61	1	no	—
143	#2143	—	61	2	no	—
144	#2108	—	61	1	no	—
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	8	4	no	—
147	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
148	D₆₁	—	2	1	no	—
149	D₆₁	—	2	1	no	—
150	C₂×#135	—	61	2	no	—
151	#568	—	122	2	no	—
152	#508	—	61	2	no	—
153	#508	—	61	2	no	—
154	#508	—	61	2	no	—
155	#508	—	61	2	no	—
156	C₂×Q₃₂	—	61	1	no	—
157	C₂×SD₃₂	—	61	2	no	—
158	C₂×D₁₆	#6923	1	1	yes	characteristic, complemented
159	C₂×M₃₂	—	61	1	no	—
160	C₂²×C₁₆	—	61	1	no	—
161	#444	—	122	4	no	—
162	#443	—	61	8	no	—
163	C₂×#108	—	61	2	no	—
164	#421	—	122	2	no	—
165	#405	—	61	1	no	—
166	#368	—	61	2	no	—
167	#368	—	61	2	no	—
168	#359	—	61	8	no	—
169	#346	—	61	2	no	—
170	C₄×C₁₆	—	61	2	no	—
171	C₆₁	#33465	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	122	2	no	—
173	#135	—	61	4	no	—
174	#135	—	61	4	no	—
175	#135	—	122	2	no	—
176	#135	—	122	2	no	—
177	C₂×Q₁₆	—	61	2	no	—
178	C₂×SD₁₆	—	122	2	no	—
179	C₂×D₈	C₂×#6923	1	2	yes	—
180	#131	—	122	8	no	—
181	#131	—	244	4	no	—
182	C₂×M₁₆	—	61	1	no	—
183	C₂×M₁₆	—	122	2	no	—
184	C₂×M₁₆	—	244	4	no	—
185	C₂²×C₈	—	61	1	no	—
186	#117	—	61	1	no	—
187	Q₃₂	—	61	2	no	—
188	Q₃₂	—	61	2	no	—
189	SD₃₂	—	61	4	no	—
190	SD₃₂	—	61	4	no	—
191	D₁₆	C₂×#6923	1	2	yes	—
192	D₁₆	C₂×#6923	1	2	yes	—
193	M₃₂	—	122	1	no	—
194	M₃₂	—	122	1	no	—
195	C₂×C₁₆	—	61	1	no	—
196	C₂×C₁₆	—	61	1	no	—
197	C₂×C₁₆	—	61	1	no	—
198	C₂×C₁₆	—	61	1	no	—
199	C₂×C₁₆	—	61	1	no	—
200	C₂×C₁₆	—	61	2	no	—
201	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
202	#108	—	61	4	no	—
203	#108	—	122	2	no	—
204	#104	—	122	8	no	—
205	#97	—	61	2	no	—
206	C₄×C₈	—	61	2	no	—
207	#41	—	122	4	no	—
208	#41	—	122	4	no	—
209	#41	—	244	2	no	—
210	#41	—	244	2	no	—
211	C₂×Q₈	—	122	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	244	2	no	—
214	C₂²×C₄	—	61	1	no	—
215	C₂²×C₄	—	244	2	no	—
216	Q₁₆	—	61	4	no	—
217	Q₁₆	—	122	2	no	—
218	SD₁₆	—	122	4	no	—
219	SD₁₆	—	122	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×#6923	1	4	yes	—
222	M₁₆	—	61	2	no	—
223	M₁₆	—	61	2	no	—
224	M₁₆	—	122	4	no	—
225	M₁₆	—	244	8	no	—
226	M₁₆	—	244	2	no	—
227	C₂×C₈	—	61	2	no	—
228	C₂×C₈	—	61	1	no	—
229	C₂×C₈	—	61	1	no	—
230	C₂×C₈	—	61	1	no	—
231	C₂×C₈	—	61	1	no	—
232	C₂×C₈	—	61	1	no	—
233	C₂×C₈	—	244	8	no	—
234	C₂×C₈	—	244	2	no	—
235	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
236	C₄⋊C₄	—	122	1	no	—
237	C₂²⋊C₄	—	122	1	no	—
238	C₄²	—	61	2	no	—
239	C₁₆	—	61	1	no	—
240	C₁₆	—	61	1	no	—
241	C₁₆	—	122	2	no	—
242	C₁₆	C₂²×#6923	1	1	yes	characteristic
243	C₁₆	C₂²×#6923	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	—	61	1	no	—
246	Q₈	—	122	4	no	—
247	Q₈	—	244	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	244	4	no	—
251	D₄	—	244	4	no	—
252	C₂×C₄	—	61	1	no	—
253	C₂×C₄	—	61	1	no	—
254	C₂×C₄	—	61	1	no	—
255	C₂×C₄	—	61	1	no	—
256	C₂×C₄	—	61	1	no	—
257	C₂×C₄	—	122	2	no	—
258	C₂×C₄	—	244	4	no	—
259	C₂×C₄	—	244	4	no	—
260	C₂×C₄	—	244	2	no	—
261	C₂×C₄	D₄×#6923	1	1	yes	characteristic
262	C₈	—	61	4	no	—
263	C₈	—	61	1	no	—
264	C₈	—	61	1	no	—
265	C₈	—	244	4	no	—
266	C₈	D₄×#6923	1	1	yes	characteristic
267	C₈	C₂³×#6923	1	1	yes	characteristic
268	C₂²	—	4	4	no	—
269	C₂²	—	16	2	no	—
270	C₂²	—	61	1	no	—
271	C₂²	—	61	1	no	—
272	C₂²	—	244	1	no	—
273	C₂²	D₈×#6923	1	1	yes	characteristic
274	C₄	—	61	1	no	—
275	C₄	—	61	1	no	—
276	C₄	—	244	2	no	—
277	C₄	—	244	4	no	—
278	C₄	D₈×#6923	1	1	yes	characteristic
279	C₄	C₂×D₄×#6923	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	8	4	no	—
282	C₂	—	122	1	no	—
283	C₂	—	122	1	no	—
284	C₂	C₂×D₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#6923
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₂×D₁₆×C₆₁	C₄
socle	C₂×C₆₁	C₂×C₄×D₈
prime radical	C₂×#16840191	C₂
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂×D₁₆	#6923
hypocenter	C₆₁	#33465
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #21462595	C₂	semidirect
2. #21462223	C₂	semidirect
3. #21462218	C₂	semidirect
4. D₆₁⋊#368	C₂	semidirect
5. #21462134	C₂	semidirect
6. #21462027	C₂	semidirect
11. #16840804	C₂²	semidirect
15. #16840691	C₂²	semidirect
16. #16840690	C₂²	semidirect
17. #16840676	C₂²	semidirect
18. #16840649	C₂²	semidirect
19. C₁₆×#6923	C₂²	semidirect
21. #16840191	C₄	semidirect
22. #16840191	C₄	semidirect
25. C₂×Dic₄₈₈	C₄	semidirect
26. C₂×#14519721	C₄	semidirect
28. C₂×D₁₆×C₆₁	C₄	semidirect
158. C₂×D₁₆	#6923	semidirect
171. C₆₁	#33465	semidirect

Maximal subgroups

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

subgroup	index
#21462595	2
#21462223	2
#21462218	2
D₆₁⋊#368	2
#21462134	2
#21462027	2
C₂×#16840191	2
#33465	61