Finite Group Databasebeta

Subgroups — #30605887

Order 15616.

258 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 76 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	#21461400	C₂	1	2	yes	maximal, complemented
2	#21460850	C₂	1	1	yes	maximal, characteristic, complemented
3	#21460774	C₂	1	2	yes	maximal, complemented
4	C₁₆×#92420	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447665	C₂	1	1	yes	maximal, characteristic
6	#16840790	—	2	2	no	—
7	#16840648	C₂²	1	1	yes	characteristic
8	#16840625	C₂²	1	2	yes	—
9	#16840610	—	2	2	no	—
10	C₈×#92420	C₂²	1	1	yes	characteristic
11	C₂×D₉₇₆	C₄	1	1	yes	characteristic
12	#16839844	C₂²	1	2	yes	—
13	#16839562	C₄	1	2	yes	—
14	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#366	C₄	1	1	yes	characteristic
16	C₂×D₆₁⋊C₈	—	4	2	no	—
17	C₂×#11758187	—	2	1	no	—
18	C₄⋊#92420	—	2	2	no	—
19	#14519834	—	4	2	no	—
20	C₄×#92420	D₄	1	1	yes	characteristic
21	C₄×#92420	D₄	1	1	yes	characteristic
22	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
23	#14519739	—	2	2	no	—
24	D₉₇₆	C₈	1	4	yes	complemented
25	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
26	#14519697	—	2	2	no	—
27	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
29	#107×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×D₂₄₄	—	2	2	no	—
31	C₂×C₄×D₆₁	—	4	2	no	—
32	C₂×#92420	—	2	1	no	—
33	C₂×#92420	—	4	2	no	—
34	C₂×#92420	D₈	1	1	yes	characteristic
35	C₂×#92420	D₈	1	1	yes	characteristic
36	D₆₁⋊C₈	—	8	4	no	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	—	2	2	no	—
39	#11758198	—	4	2	no	—
40	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×#92418	#41	1	1	yes	characteristic
42	D₄₈₈	—	2	2	no	—
43	D₄₈₈	C₂×C₈	1	4	yes	complemented
44	#11758187	—	2	2	no	—
45	C₁₆×C₆₁	C₂×C₈	1	2	yes	complemented
46	#11758185	—	2	1	no	—
47	C₂²×D₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	—	4	2	no	—
49	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
50	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₆₁	#135	1	1	yes	characteristic
52	D₂₄₄	—	2	4	no	—
53	D₂₄₄	—	4	2	no	—
54	C₄×D₆₁	—	8	2	no	—
55	C₄×D₆₁	—	8	2	no	—
56	#92420	—	2	2	no	—
57	#92420	—	2	1	no	—
58	#92420	—	8	2	no	—
59	#92420	D₁₆	1	2	yes	complemented
60	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₈×C₆₁	#131	1	1	yes	characteristic
62	#92418	—	2	1	no	—
63	C₈×D₁₆	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
65	C₂×D₆₁	—	4	4	no	—
66	C₂×D₆₁	—	8	2	no	—
67	C₂×D₆₁	—	8	2	no	—
68	C₄×C₆₁	—	8	2	no	—
69	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
70	C₄×C₆₁	#443	1	1	yes	characteristic
71	Dic₆₁	—	2	1	no	—
72	Dic₆₁	C₂×D₁₆	1	1	yes	characteristic
73	Dic₆₁	#508	1	1	yes	characteristic
74	C₄×D₁₆	—	61	1	no	—
75	C₈×D₈	—	61	2	no	—
76	#1371	—	61	1	no	—
77	#1331	—	61	2	no	—
78	C₈×C₁₆	—	61	1	no	—
79	C₂×C₆₁	C₈×D₈	1	1	yes	characteristic
80	C₂×C₆₁	C₄×D₁₆	1	1	yes	characteristic
81	C₂×C₆₁	#2178	1	1	yes	characteristic
82	D₆₁	—	8	4	no	—
83	C₂×D₁₆	—	61	1	no	—
84	C₄×D₈	—	61	2	no	—
85	C₈×D₄	—	122	2	no	—
86	#366	#6923	1	1	yes	characteristic
87	#357	—	61	2	no	—
88	C₄×C₁₆	—	61	1	no	—
89	C₄×C₁₆	—	61	1	no	—
90	#335	—	61	2	no	—
91	D₄⋊C₈	—	122	2	no	—
92	C₈²	—	61	1	no	—
93	C₆₁	C₈×D₁₆	1	1	yes	characteristic, complemented, Sylow
94	C₂×D₈	—	61	2	no	—
95	C₂²×C₈	—	244	2	no	—
96	C₄×D₄	—	122	2	no	—
97	D₁₆	—	61	4	no	—
98	C₂×C₁₆	—	61	1	no	—
99	C₂×C₁₆	—	122	1	no	—
100	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
101	#107	C₂×#6923	1	2	yes	—
102	C₄⋊C₈	—	122	2	no	—
103	#102	—	122	2	no	—
104	C₂²⋊C₈	—	244	2	no	—
105	C₄×C₈	—	61	1	no	—
106	C₄×C₈	—	61	1	no	—
107	C₄×C₈	—	61	1	no	—
108	C₂×D₄	—	122	2	no	—
109	C₂²×C₄	—	244	2	no	—
110	D₈	—	61	4	no	—
111	D₈	—	122	2	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	61	1	no	—
114	C₂×C₈	—	61	1	no	—
115	C₂×C₈	—	122	1	no	—
116	C₂×C₈	—	488	4	no	—
117	C₂×C₈	—	244	2	no	—
118	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	244	2	no	—
121	C₄²	—	61	1	no	—
122	C₁₆	—	122	2	no	—
123	C₁₆	—	122	1	no	—
124	C₁₆	D₆₁⋊C₈	1	2	yes	complemented
125	C₂³	—	244	2	no	—
126	D₄	—	122	4	no	—
127	D₄	—	244	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	61	1	no	—
130	C₂×C₄	—	61	1	no	—
131	C₂×C₄	—	488	2	no	—
132	C₂×C₄	—	488	2	no	—
133	C₂×C₄	D₄×#6923	1	1	yes	characteristic
134	C₈	—	61	2	no	—
135	C₈	—	122	1	no	—
136	C₈	—	122	2	no	—
137	C₈	—	122	1	no	—
138	C₈	—	488	2	no	—
139	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
140	C₈	#14519865	1	1	yes	characteristic
141	C₂²	—	488	2	no	—
142	C₂²	—	488	2	no	—
143	C₂²	—	244	4	no	—
144	C₂²	D₈×#6923	1	1	yes	characteristic
145	C₄	—	8	2	no	—
146	C₄	—	61	1	no	—
147	C₄	—	61	1	no	—
148	C₄	—	122	1	no	—
149	C₄	#16840790	1	1	yes	characteristic
150	C₄	#16840806	1	1	yes	characteristic
151	C₂	—	488	4	no	—
152	C₂	#21461400	1	1	yes	characteristic
153	C₂	D₁₆×#6923	1	1	yes	characteristic
154	C₂	#21462220	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#6923
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#366	C₄
socle	C₂²×C₆₁	C₄×D₈
prime radical	C₂×D₉₇₆	C₄
cosocle	C₈×Dic₆₁	C₂³
hypercenter	#366	#6923
hypocenter	C₆₁	C₈×D₁₆
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
1. #21461400	C₂	semidirect
2. #21460850	C₂	semidirect
3. #21460774	C₂	semidirect
4. C₁₆×#92420	C₂	semidirect
24. D₉₇₆	C₈	semidirect
43. D₄₈₈	C₂×C₈	semidirect
45. C₁₆×C₆₁	C₂×C₈	semidirect
59. #92420	D₁₆	semidirect
93. C₆₁	C₈×D₁₆	semidirect
124. C₁₆	D₆₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21461400	2
#21460850	2
#21460774	2
C₁₆×#92420	2
#21447665	2
C₈×D₁₆	61