Finite Group Databasebeta

Subgroups — #30605877

Order 15616.

258 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462198	C₂	1	2	yes	maximal, complemented
2	#21460848	C₂	1	1	yes	maximal, characteristic, complemented
3	#21460770	C₂	1	2	yes	maximal, complemented
4	C₈×#11758187	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444613	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₁₆	—	2	2	no	—
7	C₄⋊#11758187	C₂²	1	2	yes	—
8	#16840657	—	2	2	no	—
9	C₄×#11758187	C₂²	1	1	yes	characteristic
10	C₄×#11758187	C₂²	1	1	yes	characteristic
11	C₄×D₄₈₈	C₄	1	1	yes	characteristic
12	#16839827	C₂²	1	2	yes	—
13	#16839489	C₄	1	2	yes	—
14	C₈×#92418	C₂²	1	1	yes	characteristic
15	C₆₁×#335	C₄	1	1	yes	characteristic
16	D₆₁⋊C₁₆	—	4	4	no	—
17	C₂×#11758187	—	2	1	no	—
18	C₂×#11758187	—	2	2	no	—
19	C₂×#11758187	D₄	1	1	yes	characteristic
20	C₂×#11758187	D₄	1	1	yes	characteristic
21	C₂×D₄₈₈	C₈	1	1	yes	characteristic
22	C₂×C₈×D₆₁	—	2	2	no	—
23	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
24	#14519711	C₈	1	1	yes	characteristic
25	#14519699	C₈	1	2	yes	—
26	#14519689	—	2	2	no	—
27	C₄×#92418	C₂×C₄	1	1	yes	characteristic
28	C₄×#92418	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
32	C₂×C₄×D₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	#11758198	—	2	2	no	—
38	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	C₂×D₄	1	1	yes	characteristic
41	C₂×#92418	#41	1	1	yes	characteristic
42	D₄₈₈	C₁₆	1	4	yes	complemented
43	C₈×D₆₁	—	4	2	no	—
44	C₈×D₆₁	—	4	2	no	—
45	#11758187	—	2	2	no	—
46	#11758187	—	2	1	no	—
47	#11758187	—	4	2	no	—
48	#11758187	D₈	1	2	yes	complemented
49	C₂²×D₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
53	C₂×Dic₆₁	—	2	2	no	—
54	D₂₄₄	—	2	2	no	—
55	D₂₄₄	C₂×C₁₆	1	4	yes	complemented
56	C₄×D₆₁	—	4	2	no	—
57	C₄×D₆₁	—	4	2	no	—
58	C₈×C₆₁	—	2	1	no	—
59	C₈×C₆₁	—	4	2	no	—
60	C₈×C₆₁	C₂×C₁₆	1	2	yes	complemented
61	#92418	—	2	2	no	—
62	#92418	—	2	1	no	—
63	#92418	C₂×D₈	1	1	yes	characteristic
64	#92418	#135	1	1	yes	characteristic
65	C₁₆×D₈	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
67	C₂×D₆₁	—	2	4	no	—
68	C₂×D₆₁	—	4	2	no	—
69	C₂×D₆₁	—	4	2	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
72	C₄×C₆₁	#443	1	1	yes	characteristic
73	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
74	C₄×C₆₁	#504	1	1	yes	characteristic
75	Dic₆₁	—	4	2	no	—
76	D₄×C₁₆	—	61	2	no	—
77	C₈×D₈	—	61	1	no	—
78	#1370	—	61	1	no	—
79	D₄⋊C₁₆	—	61	2	no	—
80	C₈×C₁₆	—	61	1	no	—
81	C₂×C₆₁	C₈×D₈	1	1	yes	characteristic
82	C₂×C₆₁	D₄×C₁₆	1	1	yes	characteristic
83	C₂×C₆₁	#2170	1	1	yes	characteristic
84	D₆₁	—	4	4	no	—
85	C₂²×C₁₆	—	122	2	no	—
86	C₄×D₈	—	61	1	no	—
87	C₈×D₄	—	61	2	no	—
88	C₄⋊C₁₆	—	61	2	no	—
89	C₂²⋊C₁₆	—	122	2	no	—
90	C₄×C₁₆	—	61	1	no	—
91	C₄×C₁₆	—	61	1	no	—
92	#335	#6923	1	1	yes	characteristic
93	D₄⋊C₈	—	61	2	no	—
94	C₈²	—	61	1	no	—
95	C₆₁	C₁₆×D₈	1	1	yes	characteristic, complemented, Sylow
96	C₂×D₈	—	61	1	no	—
97	C₂²×C₈	—	122	2	no	—
98	C₄×D₄	—	61	2	no	—
99	C₂×C₁₆	—	61	1	no	—
100	C₂×C₁₆	—	61	1	no	—
101	C₂×C₁₆	—	122	1	no	—
102	C₂×C₁₆	—	122	2	no	—
103	C₂×C₁₆	—	244	4	no	—
104	#107	—	61	1	no	—
105	C₄⋊C₈	C₂×#6923	1	2	yes	—
106	#102	—	61	2	no	—
107	C₂²⋊C₈	—	122	2	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	1	no	—
110	C₄×C₈	C₂×#6923	1	1	yes	characteristic
111	C₂×D₄	—	61	2	no	—
112	C₂²×C₄	—	122	2	no	—
113	D₈	—	61	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	61	1	no	—
116	C₂×C₈	—	61	1	no	—
117	C₂×C₈	—	122	1	no	—
118	C₂×C₈	—	244	2	no	—
119	C₂×C₈	—	244	2	no	—
120	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
121	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
122	C₄⋊C₄	—	61	2	no	—
123	C₂²⋊C₄	—	122	2	no	—
124	C₄²	C₂²×#6923	1	1	yes	characteristic
125	C₁₆	—	61	2	no	—
126	C₁₆	—	122	1	no	—
127	C₁₆	—	122	2	no	—
128	C₁₆	—	244	2	no	—
129	C₂³	—	122	2	no	—
130	D₄	—	61	4	no	—
131	D₄	—	122	2	no	—
132	C₂×C₄	—	122	2	no	—
133	C₂×C₄	—	244	2	no	—
134	C₂×C₄	—	244	2	no	—
135	C₂×C₄	D₄×#6923	1	1	yes	characteristic
136	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
137	C₂×C₄	#14519865	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	2	no	—
140	C₈	—	61	1	no	—
141	C₈	—	61	1	no	—
142	C₈	—	122	2	no	—
143	C₈	—	122	1	no	—
144	C₈	D₆₁⋊C₁₆	1	2	yes	complemented
145	C₂²	—	122	4	no	—
146	C₂²	—	244	2	no	—
147	C₂²	—	244	2	no	—
148	C₂²	#16840790	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	244	2	no	—
151	C₄	D₈×#6923	1	1	yes	characteristic
152	C₄	#16840806	1	1	yes	characteristic
153	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
154	C₄	#16840821	1	1	yes	characteristic
155	C₂	—	244	4	no	—
156	C₂	#21461400	1	1	yes	characteristic
157	C₂	#21462198	1	1	yes	characteristic
158	C₂	#21462208	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#335	C₄
socle	C₂²×C₆₁	C₈×D₄
prime radical	C₂×D₄₈₈	C₈
cosocle	C₄×#92418	C₂³
hypercenter	#335	#6923
hypocenter	C₆₁	C₁₆×D₈
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
1. #21462198	C₂	semidirect
2. #21460848	C₂	semidirect
3. #21460770	C₂	semidirect
4. C₈×#11758187	C₂	semidirect
42. D₄₈₈	C₁₆	semidirect
48. #11758187	D₈	semidirect
55. D₂₄₄	C₂×C₁₆	semidirect
60. C₈×C₆₁	C₂×C₁₆	semidirect
95. C₆₁	C₁₆×D₈	semidirect
144. C₈	D₆₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#21462198	2
#21460848	2
#21460770	2
C₈×#11758187	2
#21444613	2
C₁₆×D₈	61