Finite Group Databasebeta

Subgroups — #29600932

Order 15616.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#21448100	C₂	1	1	yes	maximal, characteristic, complemented
2	#21447700	C₂	1	1	yes	maximal, characteristic
3	#21444025	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444024	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444016	C₂	1	1	yes	maximal, characteristic, complemented
6	#21444006	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1418	C₂	1	1	yes	maximal, characteristic
8	#16840166	—	2	1	no	—
9	#16839879	—	2	1	no	—
10	#16839596	C₂²	1	1	yes	characteristic
11	#16839592	C₂²	1	1	yes	characteristic
12	#16839590	C₂²	1	1	yes	characteristic
13	#16839589	—	2	1	no	—
14	#16839569	—	2	1	no	—
15	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#508	C₂²	1	1	yes	characteristic
17	C₆₁×M₆₄	C₂²	1	1	yes	characteristic
18	C₆₁×#367	C₂²	1	1	yes	characteristic
19	#14519784	—	4	1	no	—
20	#14519776	—	4	1	no	—
21	C₂×#11758185	D₄	1	1	yes	characteristic
22	#14519713	—	2	1	no	—
23	#14519711	—	2	1	no	—
24	#14519710	—	4	1	no	—
25	#14519704	—	4	1	no	—
26	C₈×Dic₆₁	D₄	1	1	yes	characteristic
27	#14519683	—	2	1	no	—
28	#135×C₆₁	—	2	1	no	—
29	Q₃₂×C₆₁	Q₈	1	1	yes	characteristic, complemented
30	SD₃₂×C₆₁	—	2	1	no	—
31	D₁₆×C₆₁	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
33	#107×C₆₁	—	2	1	no	—
34	C₃₂×C₆₁	—	2	1	no	—
35	#41×C₆₁	—	4	1	no	—
36	C₂×Dic₁₂₂	—	8	1	no	—
37	Q₁₆×C₆₁	—	2	1	no	—
38	SD₁₆×C₆₁	—	4	1	no	—
39	D₈×C₆₁	—	2	1	no	—
40	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₆₁	—	4	1	no	—
42	C₄⋊Dic₆₁	—	4	1	no	—
43	#11758196	—	8	1	no	—
44	C₄×Dic₆₁	D₈	1	1	yes	characteristic
45	#11758194	—	4	1	no	—
46	#11758194	—	8	1	no	—
47	C₂×#92418	—	8	1	no	—
48	C₂×#92418	D₈	1	1	yes	characteristic
49	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₁₆×C₆₁	#41	1	1	yes	characteristic
51	#11758185	—	2	1	no	—
52	Q₈×C₆₁	—	4	1	no	—
53	D₄×C₆₁	—	4	1	no	—
54	D₄×C₆₁	—	8	1	no	—
55	C₂×C₄×C₆₁	—	8	1	no	—
56	C₂×C₄×C₆₁	—	8	1	no	—
57	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₆₁	—	2	1	no	—
59	C₂×Dic₆₁	—	8	1	no	—
60	Dic₁₂₂	—	16	2	no	—
61	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₆₁	#137	1	1	yes	characteristic
63	#92418	—	8	1	no	—
64	#92418	SD₃₂	1	1	yes	characteristic
65	#92418	SD₃₂	1	1	yes	characteristic
66	#13710	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	8	1	no	—
68	C₂²×C₆₁	#509	1	1	yes	characteristic
69	C₄×C₆₁	—	8	1	no	—
70	C₄×C₆₁	—	16	1	no	—
71	C₄×C₆₁	#474	1	1	yes	characteristic
72	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
73	Dic₆₁	—	2	2	no	—
74	Dic₆₁	—	16	1	no	—
75	#2254	—	61	1	no	—
76	#2178	—	61	1	no	—
77	#1426	—	61	1	no	—
78	#1426	—	61	1	no	—
79	#1422	—	61	1	no	—
80	#1418	—	61	1	no	—
81	#1418	D₆₁	1	1	yes	characteristic
82	C₂×C₆₁	—	2	1	no	—
83	C₂×C₆₁	—	16	1	no	—
84	C₂×C₆₁	#2226	1	1	yes	characteristic
85	#508	C₂×D₆₁	1	1	yes	characteristic
86	#500	—	122	1	no	—
87	#443	—	122	1	no	—
88	M₆₄	—	61	1	no	—
89	M₆₄	C₂×D₆₁	1	1	yes	characteristic
90	#368	—	61	1	no	—
91	#367	—	61	1	no	—
92	#367	—	122	1	no	—
93	#367	C₂×D₆₁	1	1	yes	characteristic
94	#359	—	122	1	no	—
95	C₄×C₁₆	—	61	1	no	—
96	C₆₁	#13710	1	1	yes	characteristic, complemented, Sylow
97	#135	—	2	1	no	—
98	#131	—	244	1	no	—
99	C₄⋊Q₈	—	244	1	no	—
100	Q₃₂	Dic₁₂₂	1	1	yes	characteristic, complemented
101	SD₃₂	—	2	1	no	—
102	D₁₆	Dic₁₂₂	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	61	1	no	—
104	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
105	#108	—	122	1	no	—
106	#107	—	2	1	no	—
107	#107	—	122	1	no	—
108	#107	—	244	1	no	—
109	#104	—	244	1	no	—
110	C₄×C₈	—	61	1	no	—
111	C₃₂	—	2	1	no	—
112	C₃₂	—	122	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	488	1	no	—
115	Q₁₆	—	2	1	no	—
116	SD₁₆	—	4	1	no	—
117	D₈	—	2	1	no	—
118	M₁₆	—	488	1	no	—
119	M₁₆	—	244	1	no	—
120	C₂×C₈	—	61	1	no	—
121	C₂×C₈	—	488	1	no	—
122	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	488	1	no	—
125	C₄⋊C₄	—	244	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	Q₈	—	4	1	no	—
131	Q₈	—	976	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	122	1	no	—
137	C₂×C₄	—	488	1	no	—
138	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
139	C₈	—	61	1	no	—
140	C₈	—	61	1	no	—
141	C₈	—	488	1	no	—
142	C₈	#14519754	1	1	yes	characteristic
143	C₈	#14519806	1	1	yes	characteristic
144	C₂²	—	8	1	no	—
145	C₂²	#16840196	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	16	1	no	—
148	C₄	—	122	2	no	—
149	C₄	—	976	1	no	—
150	C₄	#16840040	1	1	yes	characteristic
151	C₄	SD₃₂×D₆₁	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	16	1	no	—
154	C₂	#21447964	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21447964
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#1418	C₂
socle	C₂×C₆₁	#2226
prime radical	C₆₁×#508	C₂²
hypercenter	#1418	D₆₁
hypocenter	C₆₁	#13710
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
1. #21448100	C₂	semidirect
3. #21444025	C₂	semidirect
4. #21444024	C₂	semidirect
5. #21444016	C₂	semidirect
29. Q₃₂×C₆₁	Q₈	semidirect
31. D₁₆×C₆₁	Q₈	semidirect
96. C₆₁	#13710	semidirect
100. Q₃₂	Dic₁₂₂	semidirect
102. D₁₆	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21448100	2
#21447700	2
#21444025	2
#21444024	2
#21444016	2
#21444006	2
C₆₁×#1418	2
#13710	61