Subgroups — #29596448

Order 15616.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839590	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839589	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#21445977	C₂	1	2	yes	maximal
5	C₂×C₆₁×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂×#14519711	C₂²	1	2	yes	—
8	C₂×#14519710	—	2	2	no	—
9	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839631	—	2	2	no	—
11	#16839590	—	2	2	no	—
12	#16839589	—	2	4	no	—
13	C₁₆×Dic₆₁	C₄	1	4	yes	complemented
14	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#367	—	2	2	no	—
17	C₂²×#92418	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×#11758196	—	4	4	no	—
20	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
21	C₂×#11758185	C₂×C₄	1	2	yes	—
22	C₂×#11758185	D₄	1	2	yes	—
23	C₂×#11758185	Q₈	1	2	yes	—
24	#14519711	—	2	2	no	—
25	#14519711	—	2	2	no	—
26	#14519710	—	4	4	no	—
27	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
28	C₈×Dic₆₁	Q₈	1	2	yes	—
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
33	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
34	#107×C₆₁	—	2	2	no	—
35	#107×C₆₁	—	2	2	no	—
36	C₂²×C₄×C₆₁	—	4	2	no	—
37	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₆₁	—	4	2	no	—
39	C₂²×Dic₆₁	D₈	1	2	yes	—
40	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₄⋊C₄×C₆₁	—	4	2	no	—
47	C₄⋊C₄×C₆₁	—	4	2	no	—
48	C₄⋊Dic₆₁	—	4	2	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	#11758196	—	8	8	no	—
51	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₆₁	Q₁₆	1	2	yes	—
53	C₂×#92418	C₄⋊C₄	1	2	yes	—
54	C₂×#92418	D₈	1	2	yes	—
55	C₂×#92418	Q₁₆	1	2	yes	—
56	C₁₆×C₆₁	C₄⋊C₄	1	4	yes	complemented
57	#11758185	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	—	8	2	no	—
60	C₂×C₄×C₆₁	—	8	4	no	—
61	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₆₁	—	8	2	no	—
68	C₂×Dic₆₁	—	8	4	no	—
69	C₂×Dic₆₁	#107	1	4	yes	—
70	C₂×Dic₆₁	SD₃₂	1	8	yes	—
71	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
74	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
75	#92418	#107	1	4	yes	—
76	#12906	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
78	C₂²×C₆₁	#493	1	1	yes	characteristic
79	C₂²×C₆₁	#500	1	1	yes	characteristic
80	C₂²×C₆₁	C₂×SD₃₂	1	4	yes	—
81	C₄×C₆₁	—	16	2	no	—
82	C₄×C₆₁	#384	1	1	yes	characteristic
83	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
84	C₄×C₆₁	#494	1	1	yes	characteristic
85	C₄×C₆₁	#500	1	1	yes	characteristic
86	Dic₆₁	—	16	2	no	—
87	Dic₆₁	#367	1	8	yes	complemented
88	C₂×#367	—	61	1	no	—
89	C₂×#367	—	61	2	no	—
90	C₂×#367	D₆₁	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	61	1	no	—
92	#1845	—	61	2	no	—
93	C₂×C₆₁	#1845	1	1	yes	characteristic
94	C₂×C₆₁	C₂×#367	1	2	yes	—
95	C₂×C₆₁	#2248	1	2	yes	—
96	C₂×C₆₁	#2254	1	2	yes	—
97	C₂²×C₁₆	—	61	1	no	—
98	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
99	C₂×#107	—	61	2	no	—
100	C₂×#107	—	122	2	no	—
101	C₂×#107	C₂×D₆₁	1	2	yes	—
102	C₂×C₄×C₈	—	61	1	no	—
103	#384	—	122	2	no	—
104	#367	—	2	2	no	—
105	#367	—	122	4	no	—
106	#367	—	122	2	no	—
107	C₄×C₁₆	—	61	4	no	—
108	C₆₁	#12906	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	61	1	no	—
110	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	122	2	no	—
113	C₂×C₄⋊C₄	—	244	4	no	—
114	C₂×C₄²	—	61	1	no	—
115	C₂×C₁₆	—	61	2	no	—
116	C₂×C₁₆	—	61	2	no	—
117	C₂×C₁₆	—	61	2	no	—
118	C₂×C₁₆	Dic₁₂₂	1	2	yes	—
119	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
121	#107	—	2	2	no	—
122	#107	—	2	2	no	—
123	#107	—	122	2	no	—
124	#107	—	122	2	no	—
125	#107	—	244	4	no	—
126	C₄×C₈	—	61	2	no	—
127	C₄×C₈	—	61	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	61	2	no	—
130	C₂²×C₄	—	244	2	no	—
131	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
132	C₂×C₈	—	61	2	no	—
133	C₂×C₈	—	61	2	no	—
134	C₂×C₈	—	61	2	no	—
135	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
137	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	488	8	no	—
144	C₄⋊C₄	—	244	2	no	—
145	C₄⋊C₄	—	244	2	no	—
146	C₄²	—	61	2	no	—
147	C₄²	—	61	2	no	—
148	C₁₆	—	61	4	no	—
149	C₁₆	#11758196	1	4	yes	complemented
150	C₂³	D₈×D₆₁	1	1	yes	characteristic
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	61	4	no	—
154	C₂×C₄	—	61	8	no	—
155	C₂×C₄	—	488	2	no	—
156	C₂×C₄	—	488	4	no	—
157	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
158	C₂×C₄	#14519773	1	1	yes	characteristic
159	C₂×C₄	#14519776	1	1	yes	characteristic
160	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
161	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
162	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
163	C₈	—	61	4	no	—
164	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
165	C₈	C₂×#11758196	1	1	yes	characteristic
166	C₈	#14519775	1	1	yes	characteristic
167	C₈	#14519776	1	1	yes	characteristic
168	C₂²	#107×D₆₁	1	1	yes	characteristic
169	C₂²	#16840137	1	1	yes	characteristic
170	C₂²	#16840166	1	1	yes	characteristic
171	C₂²	SD₃₂×D₆₁	1	4	yes	—
172	C₄	—	16	2	no	—
173	C₄	—	61	8	no	—
174	C₄	—	976	2	no	—
175	C₄	#16839631	1	1	yes	characteristic
176	C₄	#107×D₆₁	1	1	yes	characteristic
177	C₄	#16840140	1	1	yes	characteristic
178	C₄	#16840166	1	1	yes	characteristic
179	C₂	#21445977	1	1	yes	characteristic
180	C₂	#367×D₆₁	1	2	yes	—
181	C₂	#21448073	1	2	yes	—
182	C₂	#21448100	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#367	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#367	D₆₁
hypocenter	C₆₁	#12906
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₆₁	C₄	semidirect
56. C₁₆×C₆₁	C₄⋊C₄	semidirect
57. #11758185	C₄⋊C₄	semidirect
87. Dic₆₁	#367	semidirect
108. C₆₁	#12906	semidirect
149. C₁₆	#11758196	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839590	2
C₂×#16839589	2
C₂×C₁₆×Dic₆₁	2
#21445977	2
C₂×C₆₁×#367	2
#12906	61