Finite Group Databasebeta

Subgroups — #30624475

Order 15616.

444 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 250 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	#21462234	C₂	1	8	yes	maximal, complemented
2	#21462184	C₂	1	4	yes	maximal
3	#21462180	C₂	1	2	yes	maximal
4	#21454754	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11758187	C₂²	1	4	yes	—
6	C₄⋊#11758187	C₂²	1	8	yes	—
7	C₂²⋊#11758187	C₂²	1	4	yes	—
8	#16840650	C₂²	1	4	yes	—
9	#16840650	C₂²	1	4	yes	—
10	C₄×#11758187	C₂²	1	4	yes	—
11	#16839893	C₄	1	4	yes	—
12	Q₈×#92418	C₂²	1	4	yes	—
13	#16839789	C₄	1	2	yes	—
14	#16839786	C₄	1	1	yes	characteristic
15	#16839718	C₂²	1	2	yes	—
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	C₆₁×#533	C₄	1	1	yes	characteristic
18	C₂×#11758187	C₂³	1	8	yes	—
19	C₂×#11758194	C₂×C₄	1	2	yes	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂³	1	1	yes	characteristic
22	C₄⋊#92418	C₂×C₄	1	4	yes	—
23	C₄⋊#92418	C₂×C₄	1	8	yes	—
24	#14519688	C₂×C₄	1	4	yes	—
25	C₄×#92418	C₂×C₄	1	2	yes	—
26	C₄×#92418	C₂³	1	2	yes	—
27	C₄×#92418	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₆₁	C₈	1	2	yes	—
29	#125×C₆₁	C₈	1	2	yes	—
30	C₂²⋊Q₈×C₆₁	C₈	1	4	yes	—
31	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	—
32	#117×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×Q₈×C₆₁	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
38	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
39	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
40	C₂²⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
41	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
44	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
45	#11758194	—	4	2	no	—
46	C₂×#92418	—	2	2	no	—
47	C₂×#92418	—	4	1	no	—
48	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
49	C₂×#92418	C₂²×C₄	1	2	yes	—
50	C₂×#92418	C₂²×C₄	1	4	yes	—
51	C₂×#92418	C₂⁴	1	1	yes	characteristic
52	#11758187	—	2	8	no	—
53	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₆₁	—	4	4	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	—	2	1	no	—
57	C₂×C₄×C₆₁	—	4	2	no	—
58	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
61	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
62	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
63	#92418	—	2	1	no	—
64	#92418	—	2	2	no	—
65	#92418	—	2	4	no	—
66	#92418	2₋¹⁺⁴	1	2	yes	—
67	#33501	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	—	2	2	no	—
69	C₂²×C₆₁	—	2	1	no	—
70	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	—	2	8	no	—
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	#519	1	1	yes	characteristic
76	C₄×C₆₁	#568	1	1	yes	characteristic
77	#2960	—	61	1	no	—
78	#2183	—	61	8	no	—
79	#2164	—	61	4	no	—
80	#2163	—	61	2	no	—
81	C₂×C₆₁	—	4	1	no	—
82	C₂×C₆₁	#2982	1	1	yes	characteristic
83	C₂×C₆₁	#3407	1	2	yes	—
84	#533	#6923	1	1	yes	characteristic
85	#446	—	61	4	no	—
86	C₈×Q₈	—	61	4	no	—
87	#424	—	61	2	no	—
88	#423	—	61	1	no	—
89	#405	—	61	2	no	—
90	C₂×C₄×C₈	—	61	1	no	—
91	C₄⋊C₁₆	—	61	4	no	—
92	C₄⋊C₁₆	—	61	8	no	—
93	C₂²⋊C₁₆	—	61	4	no	—
94	#346	—	61	4	no	—
95	#346	—	61	4	no	—
96	C₄×C₁₆	—	61	4	no	—
97	C₆₁	#33501	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	61	2	no	—
99	C₂²×C₈	—	61	1	no	—
100	C₂²×C₈	—	61	1	no	—
101	C₄⋊Q₈	#92420	1	2	yes	—
102	#125	#92420	1	2	yes	—
103	C₂²⋊Q₈	#92420	1	4	yes	—
104	C₄×Q₈	C₂×#6923	1	4	yes	—
105	#117	C₂×#6923	1	2	yes	—
106	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
107	C₂×C₁₆	—	61	8	no	—
108	C₄⋊C₈	—	61	4	no	—
109	C₄⋊C₈	—	61	8	no	—
110	#97	—	61	4	no	—
111	C₄×C₈	—	61	4	no	—
112	C₄×C₈	—	61	2	no	—
113	C₄×C₈	—	61	2	no	—
114	C₂×Q₈	C₂×#92420	1	4	yes	—
115	C₂²×C₄	C₂×#92420	1	2	yes	—
116	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
117	M₁₆	—	244	2	no	—
118	C₂×C₈	—	61	1	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	—	61	2	no	—
121	C₂×C₈	—	61	4	no	—
122	C₂×C₈	—	122	2	no	—
123	C₂×C₈	—	244	1	no	—
124	C₄⋊C₄	C₂×#92420	1	4	yes	—
125	C₄⋊C₄	C₂×#92420	1	8	yes	—
126	C₄⋊C₄	C₂×#92420	1	4	yes	—
127	C₂²⋊C₄	C₂×#92420	1	4	yes	—
128	C₄²	C₂×#92420	1	1	yes	characteristic
129	C₄²	C₂×#92420	1	1	yes	characteristic
130	C₄²	C₂²×#6923	1	2	yes	—
131	C₄²	C₂²×#6923	1	4	yes	—
132	C₁₆	—	122	8	no	—
133	C₂³	C₂²×#92420	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
139	C₂×C₄	C₂²×#92420	1	2	yes	—
140	C₂×C₄	C₂²×#92420	1	2	yes	—
141	C₂×C₄	C₂²×#92420	1	8	yes	—
142	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
143	C₈	—	61	2	no	—
144	C₈	—	122	1	no	—
145	C₈	—	122	2	no	—
146	C₈	—	122	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂³×#92420	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	8	no	—
153	C₄	—	2	1	no	—
154	C₄	#16840830	1	1	yes	characteristic
155	C₄	#16840842	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	#21462712	1	1	yes	characteristic
158	C₂	#21462786	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#533	C₄
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#533	#6923
hypocenter	C₆₁	#33501
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #21462234	C₂	semidirect
97. C₆₁	#33501	semidirect

Maximal subgroups

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

subgroup	index
#21462234	2
#21462184	2
#21462180	2
#21454754	2
#33501	61