Finite Group Databasebeta

Subgroups — #29600893

Order 15616.

176 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 59 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	#21448093	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21444019	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444018	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444014	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443997	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1415	C₂	1	1	yes	maximal, characteristic
8	#16840166	—	2	1	no	—
9	D₈×Dic₆₁	—	2	1	no	—
10	C₂×#14519683	C₂²	1	1	yes	characteristic
11	#16839588	C₂²	1	1	yes	characteristic
12	#16839588	C₂²	1	1	yes	characteristic
13	#16839587	—	2	1	no	—
14	#16839563	—	2	1	no	—
15	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#366	C₂²	1	1	yes	characteristic
19	#14519776	—	4	1	no	—
20	D₄×Dic₆₁	—	4	1	no	—
21	C₂×#11758185	D₄	1	1	yes	characteristic
22	#14519711	—	2	1	no	—
23	#14519711	—	2	1	no	—
24	#14519710	—	4	1	no	—
25	#14519698	—	4	1	no	—
26	C₈×Dic₆₁	D₄	1	1	yes	characteristic
27	#14519683	—	2	1	no	—
28	C₂×D₈×C₆₁	—	2	1	no	—
29	D₁₆×C₆₁	—	2	1	no	—
30	D₁₆×C₆₁	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
32	#107×C₆₁	—	2	1	no	—
33	C₃₂×C₆₁	—	2	1	no	—
34	C₂×D₄×C₆₁	—	4	1	no	—
35	C₂²×Dic₆₁	—	8	1	no	—
36	C₂×Dic₁₂₂	—	8	1	no	—
37	D₈×C₆₁	—	2	2	no	—
38	D₈×C₆₁	—	4	1	no	—
39	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₆₁	—	4	1	no	—
41	C₂²⋊Dic₆₁	—	8	1	no	—
42	C₄⋊Dic₆₁	—	4	1	no	—
43	C₄⋊Dic₆₁	—	4	1	no	—
44	#11758196	—	8	1	no	—
45	C₄×Dic₆₁	D₈	1	1	yes	characteristic
46	C₂×#92418	D₈	1	1	yes	characteristic
47	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₆₁	#41	1	1	yes	characteristic
49	#11758185	—	2	1	no	—
50	C₂³×C₆₁	—	8	1	no	—
51	D₄×C₆₁	—	4	2	no	—
52	D₄×C₆₁	—	8	1	no	—
53	C₂×C₄×C₆₁	—	8	1	no	—
54	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₆₁	—	8	1	no	—
56	C₂×Dic₆₁	—	8	1	no	—
57	C₂×Dic₆₁	—	16	2	no	—
58	C₂×Dic₆₁	D₁₆	1	1	yes	characteristic
59	C₂×Dic₆₁	D₁₆	1	1	yes	characteristic
60	Dic₁₂₂	—	16	2	no	—
61	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₆₁	#137	1	1	yes	characteristic
63	#92418	—	2	1	no	—
64	#13703	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	—	8	2	no	—
66	C₂²×C₆₁	—	16	1	no	—
67	C₂²×C₆₁	—	16	1	no	—
68	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₆₁	—	16	1	no	—
70	C₄×C₆₁	#474	1	1	yes	characteristic
71	C₄×C₆₁	#510	1	1	yes	characteristic
72	Dic₆₁	—	2	1	no	—
73	Dic₆₁	—	16	1	no	—
74	Dic₆₁	—	16	1	no	—
75	Dic₆₁	D₃₂	1	2	yes	—
76	#2252	—	61	1	no	—
77	C₄×D₁₆	—	61	1	no	—
78	#1423	—	61	1	no	—
79	#1423	—	61	1	no	—
80	C₄⋊C₃₂	—	61	1	no	—
81	#1415	—	61	1	no	—
82	#1415	D₆₁	1	1	yes	characteristic
83	C₂×C₆₁	—	16	2	no	—
84	C₂×C₆₁	#2224	1	1	yes	characteristic
85	C₂×C₆₁	C₂×D₃₂	1	1	yes	characteristic
86	C₂×C₆₁	#2264	1	1	yes	characteristic
87	C₂×D₁₆	C₂×D₆₁	1	1	yes	characteristic
88	#500	—	122	1	no	—
89	C₄×D₈	—	122	1	no	—
90	C₂×C₃₂	—	61	1	no	—
91	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
92	#366	—	61	1	no	—
93	#366	—	61	1	no	—
94	#366	—	122	1	no	—
95	#366	C₂×D₆₁	1	1	yes	characteristic
96	#357	—	122	1	no	—
97	C₄×C₁₆	—	61	1	no	—
98	C₆₁	#13703	1	1	yes	characteristic, complemented, Sylow
99	C₂×D₈	—	2	1	no	—
100	C₄⋊Q₈	—	244	1	no	—
101	C₄×D₄	—	244	1	no	—
102	D₁₆	—	2	1	no	—
103	D₁₆	Dic₁₂₂	1	2	yes	complemented
104	C₂×C₁₆	—	61	1	no	—
105	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
106	#107	—	2	1	no	—
107	#107	—	122	1	no	—
108	#107	—	122	1	no	—
109	#107	—	244	1	no	—
110	#102	—	244	1	no	—
111	C₄×C₈	—	61	1	no	—
112	C₃₂	—	2	1	no	—
113	C₃₂	—	122	1	no	—
114	C₂×Q₈	—	488	1	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂²×C₄	—	488	1	no	—
117	D₈	—	2	2	no	—
118	D₈	—	4	1	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	488	1	no	—
123	C₄⋊C₄	—	244	1	no	—
124	C₄⋊C₄	—	244	1	no	—
125	C₂²⋊C₄	—	488	1	no	—
126	C₄²	—	61	1	no	—
127	C₁₆	—	122	1	no	—
128	C₁₆	C₂×Dic₁₂₂	1	1	yes	characteristic
129	C₁₆	#11758224	1	1	yes	characteristic
130	C₂³	—	8	1	no	—
131	Q₈	—	976	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	—	976	2	no	—
138	C₂×C₄	—	488	1	no	—
139	C₂×C₄	—	488	1	no	—
140	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
141	C₈	—	122	1	no	—
142	C₈	#14519754	1	1	yes	characteristic
143	C₈	#14519806	1	1	yes	characteristic
144	C₂²	—	8	2	no	—
145	C₂²	—	16	1	no	—
146	C₂²	—	16	1	no	—
147	C₂²	D₁₆×D₆₁	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	61	2	no	—
150	C₄	—	122	1	no	—
151	C₄	—	976	1	no	—
152	C₄	—	976	1	no	—
153	C₄	#16840040	1	1	yes	characteristic
154	C₄	#16840201	1	1	yes	characteristic
155	C₂	—	16	2	no	—
156	C₂	#21447954	1	1	yes	characteristic
157	C₂	D₃₂×D₆₁	1	1	yes	characteristic
158	C₂	#21448135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₆₁
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#1415	C₂
socle	C₂²×C₆₁	C₂×D₁₆
prime radical	C₂×D₁₆×C₆₁	C₂²
hypercenter	#1415	D₆₁
hypocenter	C₆₁	#13703
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

Complements

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

N	H ≅ G/N	type
1. #21448093	C₂	semidirect
3. #21444019	C₂	semidirect
4. #21444018	C₂	semidirect
5. #21444014	C₂	semidirect
30. D₁₆×C₆₁	Q₈	semidirect
98. C₆₁	#13703	semidirect
103. D₁₆	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21448093	2
D₁₆×Dic₆₁	2
#21444019	2
#21444018	2
#21444014	2
#21443997	2
C₆₁×#1415	2
#13703	61