Finite Group Databasebeta

Subgroups — #30603756

Order 15616.

290 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 156 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	#21462176	C₂	1	2	yes	maximal
2	#21460732	C₂	1	2	yes	maximal, complemented
3	#21460731	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11758187	C₄	1	4	yes	—
6	C₂²⋊#11758187	C₄	1	4	yes	—
7	C₄×#11758187	C₂²	1	4	yes	—
8	C₂×C₄×#92418	C₄	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₈×#92418	C₄	1	1	yes	characteristic
11	C₈×#92418	C₄	1	1	yes	characteristic
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₈×#92418	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
15	C₂×#11758187	C₂×C₄	1	8	yes	—
16	C₂²×#92418	C₈	1	2	yes	—
17	C₂²×#92418	C₂×C₄	1	2	yes	—
18	C₄×#92418	C₈	1	2	yes	—
19	C₄×#92418	C₂×C₄	1	1	yes	characteristic
20	C₄×#92418	C₂×C₄	1	2	yes	—
21	C₄×#92418	C₂×C₄	1	1	yes	characteristic
22	C₄×#92418	C₂×C₄	1	1	yes	characteristic
23	C₄×#92418	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	C₈	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	C₈	1	2	yes	—
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₆₁	—	2	4	no	—
33	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
35	C₄²×C₆₁	C₄²	1	1	yes	characteristic
36	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×#92418	—	2	4	no	—
40	C₂×#92418	—	2	4	no	—
41	C₂×#92418	C₄²	1	2	yes	—
42	C₂×#92418	C₂×C₈	1	2	yes	—
43	C₂×#92418	C₂×C₈	1	2	yes	—
44	C₂×#92418	C₂²×C₄	1	2	yes	—
45	#11758187	—	2	8	no	—
46	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
53	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₆₁	—	2	2	no	—
57	C₈×C₆₁	M₃₂	1	4	yes	—
58	#92418	—	2	2	no	—
59	#92418	—	2	2	no	—
60	#92418	—	2	2	no	—
61	#92418	#131	1	4	yes	—
62	#11411	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	1	no	—
64	C₂²×C₆₁	—	2	1	no	—
65	C₂²×C₆₁	#346	1	2	yes	—
66	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	#346	1	2	yes	—
70	C₄×C₆₁	#405	1	2	yes	—
71	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
72	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
73	#2162	—	61	2	no	—
74	C₂×C₈²	—	61	1	no	—
75	#1312	—	61	2	no	—
76	#1311	—	61	2	no	—
77	C₂×C₆₁	—	2	2	no	—
78	C₂×C₆₁	#1450	1	1	yes	characteristic
79	C₂×C₆₁	C₂×#346	1	1	yes	characteristic
80	C₂×C₆₁	C₄×M₃₂	1	1	yes	characteristic
81	C₂×C₄×C₈	—	61	1	no	—
82	C₂×C₄×C₈	—	61	1	no	—
83	C₂×C₄×C₈	#6923	1	1	yes	characteristic
84	C₄⋊C₁₆	—	61	4	no	—
85	C₂²⋊C₁₆	—	61	4	no	—
86	C₄×C₁₆	—	61	4	no	—
87	C₈²	—	61	1	no	—
88	C₈²	—	61	1	no	—
89	C₈²	—	61	1	no	—
90	C₈²	—	61	1	no	—
91	C₆₁	#11411	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	61	2	no	—
93	C₂²×C₈	—	61	2	no	—
94	C₂²×C₈	#92420	1	2	yes	—
95	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
96	C₂×C₁₆	—	61	8	no	—
97	C₄×C₈	—	61	2	no	—
98	C₄×C₈	—	61	2	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	—	61	1	no	—
101	C₄×C₈	—	61	1	no	—
102	C₄×C₈	—	61	1	no	—
103	C₄×C₈	#92420	1	2	yes	—
104	C₄×C₈	C₂×#6923	1	1	yes	characteristic
105	C₄×C₈	C₂×#6923	1	1	yes	characteristic
106	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
107	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
108	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	61	2	no	—
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	—	61	2	no	—
113	C₂×C₈	—	61	2	no	—
114	C₂×C₈	—	122	4	no	—
115	C₂×C₈	—	122	4	no	—
116	C₂×C₈	C₂×#92420	1	2	yes	—
117	C₂×C₈	C₂×#92420	1	2	yes	—
118	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
119	C₄²	C₄×#6923	1	1	yes	characteristic
120	C₄²	C₂×#92420	1	1	yes	characteristic
121	C₄²	C₂²×#6923	1	1	yes	characteristic
122	C₁₆	—	122	8	no	—
123	C₂³	C₄×#92420	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×#92420	1	1	yes	characteristic
127	C₂×C₄	C₄×#92420	1	1	yes	characteristic
128	C₂×C₄	C₄×#92420	1	1	yes	characteristic
129	C₂×C₄	#14519852	1	4	yes	—
130	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
131	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
132	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	61	4	no	—
135	C₈	—	122	2	no	—
136	C₈	—	122	2	no	—
137	C₈	—	122	2	no	—
138	C₈	#14519851	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#16840651	1	2	yes	—
142	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	#16840651	1	2	yes	—
146	C₄	#16840725	1	2	yes	—
147	C₄	C₂×#14519851	1	2	yes	—
148	C₄	C₂×#14519852	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#21461022	1	1	yes	characteristic
151	C₂	C₂×#16840651	1	1	yes	characteristic
152	C₂	C₄×#14519851	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6923
commutator, socle	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₆₁	C₄
prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₄×#92418	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#11411
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #21460732	C₂	semidirect
3. #21460731	C₂	semidirect
91. C₆₁	#11411	semidirect

Maximal subgroups

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

subgroup	index
#21462176	2
#21460732	2
#21460731	2
C₂×C₈×#92418	2
#11411	61