Subgroups — #30606728

Order 15616.

296 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊#364	C₂	1	1	yes	maximal, characteristic
2	#21462038	C₂	1	2	yes	maximal
3	#21461621	C₂	1	1	yes	maximal, characteristic
4	#21460868	C₂	1	2	yes	maximal
5	C₄×#14519716	C₂	1	1	yes	maximal, characteristic
6	D₆₁⋊M₃₂	C₂²	1	2	yes	—
7	D₆₁⋊#108	C₄	1	1	yes	characteristic
8	#16840772	C₄	1	1	yes	characteristic
9	#16840725	C₂²	1	2	yes	—
10	#16840685	C₄	1	2	yes	complemented
11	#16840684	C₄	1	2	yes	complemented
12	#16840653	—	2	2	no	—
13	#16840652	—	2	2	no	—
14	#16840633	C₄	1	2	yes	—
15	C₂×#14519716	C₂²	1	1	yes	characteristic
16	C₂×#14519716	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
18	#16839537	C₄	1	1	yes	characteristic, complemented
19	#16839537	C₄	1	1	yes	characteristic, complemented
20	C₄×#11758185	C₄	1	1	yes	characteristic, complemented
21	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
22	D₆₁⋊M₁₆	C₂×C₄	1	2	yes	—
23	#14519856	C₂×C₄	1	2	yes	—
24	#14519852	—	2	2	no	—
25	#14519851	C₂×C₄	1	4	yes	—
26	D₆₁⋊C₁₆	—	2	2	no	—
27	#14519848	—	2	1	no	—
28	#14519847	—	2	1	no	—
29	#14519846	—	2	1	no	—
30	#14519845	—	2	1	no	—
31	#14519831	—	2	2	no	—
32	C₈×#6923	—	2	2	no	—
33	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
35	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
36	#14519716	—	2	1	no	—
37	#14519716	—	2	1	no	—
38	#14519716	D₄	1	1	yes	characteristic
39	#14519716	D₄	1	1	yes	characteristic
40	#14519716	Q₈	1	1	yes	characteristic
41	#14519716	Q₈	1	1	yes	characteristic
42	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
43	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
44	C₄×#92418	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₆₁	—	2	1	no	—
51	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²⋊#6923	—	2	2	no	—
53	#11758217	—	2	2	no	—
54	C₄⋊#6923	—	2	2	no	—
55	C₄×#6923	—	2	2	no	—
56	#11758213	—	2	2	no	—
57	D₆₁⋊C₈	—	2	2	no	—
58	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₆₁	C₄²	1	1	yes	characteristic
61	C₄×Dic₆₁	—	2	1	no	—
62	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
63	C₂×#92418	C₄²	1	1	yes	characteristic
64	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
65	C₈×D₆₁	M₁₆	1	1	yes	characteristic
66	C₈×D₆₁	M₁₆	1	1	yes	characteristic
67	C₈×D₆₁	M₁₆	1	1	yes	characteristic
68	C₈×D₆₁	M₁₆	1	1	yes	characteristic
69	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
70	C₈×D₆₁	C₂×Q₈	1	1	yes	characteristic
71	C₈×D₆₁	#41	1	1	yes	characteristic
72	C₈×D₆₁	#41	1	1	yes	characteristic
73	#11758187	—	2	4	no	—
74	C₁₆×C₆₁	—	2	1	no	—
75	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
76	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
77	#11758185	—	2	1	no	—
78	#11758185	C₄⋊C₄	1	1	yes	characteristic
79	#11758185	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×D₆₁	#131	1	1	yes	characteristic
81	C₂×#6923	—	4	2	no	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
84	C₂×Dic₆₁	—	2	1	no	—
85	C₂×Dic₆₁	#131	1	1	yes	characteristic
86	C₄×D₆₁	—	4	2	no	—
87	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
88	C₄×D₆₁	C₄×Q₈	1	1	yes	characteristic
89	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
90	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
91	#92420	—	4	2	no	—
92	C₈×C₆₁	#97	1	1	yes	characteristic
93	C₈×C₆₁	#97	1	1	yes	characteristic
94	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₈×C₆₁	#117	1	1	yes	characteristic
96	#92418	#97	1	1	yes	characteristic
97	#92418	#97	1	1	yes	characteristic
98	#92418	C₄×D₄	1	1	yes	characteristic
99	#92418	C₄×Q₈	1	1	yes	characteristic
100	#12667	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	#405	1	1	yes	characteristic
102	C₂×D₆₁	—	4	1	no	—
103	C₂×D₆₁	#436	1	1	yes	characteristic
104	C₂×D₆₁	#446	1	1	yes	characteristic
105	#6923	—	8	2	no	—
106	C₄×C₆₁	—	2	2	no	—
107	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
109	Dic₆₁	—	4	1	no	—
110	Dic₆₁	#436	1	1	yes	characteristic
111	Dic₆₁	#446	1	1	yes	characteristic
112	C₂×#364	—	61	1	no	—
113	#2110	—	61	2	no	—
114	C₄×M₃₂	—	61	1	no	—
115	#1772	—	61	1	no	—
116	#1378	—	61	2	no	—
117	C₂×C₆₁	—	2	1	no	—
118	C₂×C₆₁	#1770	1	1	yes	characteristic
119	D₆₁	—	2	1	no	—
120	D₆₁	—	2	1	no	—
121	C₂×M₃₂	—	61	1	no	—
122	C₂×M₃₂	—	61	1	no	—
123	C₂×M₃₂	—	61	2	no	—
124	C₂×#108	—	61	1	no	—
125	#427	—	61	1	no	—
126	#405	—	61	2	no	—
127	C₂×C₄×C₈	—	61	1	no	—
128	#364	—	61	2	no	—
129	#364	—	61	2	no	—
130	#347	—	122	2	no	—
131	#347	—	122	2	no	—
132	#346	—	61	1	no	—
133	#346	—	61	1	no	—
134	C₄×C₁₆	—	61	1	no	—
135	C₄×C₁₆	#6923	1	1	yes	characteristic, complemented
136	#339	—	61	2	no	—
137	C₆₁	#12667	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	61	2	no	—
139	C₂²×C₈	—	61	1	no	—
140	C₂²×C₈	—	61	1	no	—
141	#117	—	61	2	no	—
142	C₂×C₄²	—	61	1	no	—
143	M₃₂	—	61	1	no	—
144	M₃₂	—	61	1	no	—
145	M₃₂	—	61	1	no	—
146	M₃₂	—	61	1	no	—
147	M₃₂	—	61	4	no	—
148	M₃₂	—	122	1	no	—
149	M₃₂	—	122	1	no	—
150	M₃₂	—	122	2	no	—
151	C₂×C₁₆	—	61	1	no	—
152	C₂×C₁₆	—	61	1	no	—
153	C₂×C₁₆	—	122	2	no	—
154	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
155	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
156	#108	—	122	1	no	—
157	#108	—	122	1	no	—
158	#107	—	122	1	no	—
159	#106	—	122	1	no	—
160	#97	—	122	2	no	—
161	C₄×C₈	—	61	1	no	—
162	C₄×C₈	—	61	1	no	—
163	C₄×C₈	—	61	1	no	—
164	C₄×C₈	—	122	2	no	—
165	C₄×C₈	C₂×#6923	1	1	yes	characteristic
166	C₂²×C₄	—	61	1	no	—
167	C₂²×C₄	—	122	1	no	—
168	M₁₆	—	122	2	no	—
169	M₁₆	—	122	2	no	—
170	C₂×C₈	—	61	1	no	—
171	C₂×C₈	—	61	1	no	—
172	C₂×C₈	—	61	1	no	—
173	C₂×C₈	—	61	1	no	—
174	C₂×C₈	—	61	1	no	—
175	C₂×C₈	—	61	1	no	—
176	C₂×C₈	—	61	1	no	—
177	C₂×C₈	—	61	1	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	1	no	—
180	C₂×C₈	—	122	2	no	—
181	C₂×C₈	C₄×#6923	1	1	yes	characteristic
182	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
183	C₄⋊C₄	—	122	2	no	—
184	C₂²⋊C₄	—	122	2	no	—
185	C₄²	—	61	1	no	—
186	C₄²	—	122	1	no	—
187	C₄²	—	122	2	no	—
188	C₄²	C₄×#6923	1	1	yes	characteristic
189	C₁₆	—	2	1	no	—
190	C₁₆	—	61	1	no	—
191	C₁₆	—	61	1	no	—
192	C₁₆	—	122	4	no	—
193	C₁₆	—	122	1	no	—
194	C₁₆	C₄⋊#6923	1	1	yes	characteristic
195	C₁₆	C₄⋊#6923	1	1	yes	characteristic
196	C₂³	—	61	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	61	1	no	—
199	C₂×C₄	—	61	1	no	—
200	C₂×C₄	—	61	1	no	—
201	C₂×C₄	—	61	1	no	—
202	C₂×C₄	—	61	1	no	—
203	C₂×C₄	—	122	1	no	—
204	C₂×C₄	—	244	2	no	—
205	C₂×C₄	—	244	2	no	—
206	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
207	C₈	—	61	1	no	—
208	C₈	—	61	1	no	—
209	C₈	—	61	1	no	—
210	C₈	—	61	1	no	—
211	C₈	—	244	2	no	—
212	C₈	#14519831	1	1	yes	characteristic
213	C₈	#14519831	1	1	yes	characteristic
214	C₈	C₂×C₄⋊#6923	1	1	yes	characteristic
215	C₈	#14519856	1	1	yes	characteristic
216	C₂²	—	61	1	no	—
217	C₂²	—	61	1	no	—
218	C₂²	—	244	1	no	—
219	C₂²	#16840725	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	61	1	no	—
222	C₄	—	61	1	no	—
223	C₄	—	488	2	no	—
224	C₄	—	244	1	no	—
225	C₄	C₄×C₄⋊#6923	1	1	yes	characteristic
226	C₄	C₂×#14519831	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	122	1	no	—
229	C₂	—	122	1	no	—
230	C₂	#21461613	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519856
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂×C₆₁	#1770
prime radical	C₂²×D₆₁	#131
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#12667
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16840685	C₄	semidirect
11. #16840684	C₄	semidirect
18. #16839537	C₄	semidirect
19. #16839537	C₄	semidirect
20. C₄×#11758185	C₄	semidirect
21. C₄×C₁₆×C₆₁	C₄	semidirect
135. C₄×C₁₆	#6923	semidirect
137. C₆₁	#12667	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊#364	2
#21462038	2
#21461621	2
#21460868	2
C₄×#14519716	2
#12667	61