Finite Group Databasebeta

Subgroups — #27170565

Order 12928.

150 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20265529	C₂	1	1	yes	maximal, characteristic
2	#20265463	C₂	1	1	yes	maximal, characteristic
3	#20265384	C₂	1	1	yes	maximal, characteristic
4	#20265301	C₂	1	1	yes	maximal, characteristic
5	#20264927	C₂	1	1	yes	maximal, characteristic
6	#20264924	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#13108064	—	2	1	no	—
9	#15692255	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₀₂	C₂²	1	1	yes	characteristic
11	#15692187	—	2	1	no	—
12	C₄⋊#11717579	C₂²	1	1	yes	characteristic
13	#15692184	—	2	1	no	—
14	#15692180	—	2	1	no	—
15	#15692179	—	2	1	no	—
16	#15692178	—	2	1	no	—
17	#15692178	—	2	1	no	—
18	C₄×#11717579	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₀₁	—	2	1	no	—
20	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₀₂	—	2	1	no	—
24	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
25	#13108064	—	4	2	no	—
26	C₂×#11717579	—	2	1	no	—
27	C₂×#11717579	D₄	1	1	yes	characteristic
28	C₂×#11717579	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₀₁	—	2	1	no	—
30	C₄⋊Dic₁₀₁	—	2	1	no	—
31	C₄⋊Dic₁₀₁	D₄	1	1	yes	characteristic
32	#13108053	—	2	1	no	—
33	C₄×Dic₁₀₁	—	2	1	no	—
34	C₂×Q₈×C₁₀₁	—	2	1	no	—
35	C₂×C₈×C₁₀₁	—	2	1	no	—
36	C₂×C₈×C₁₀₁	—	2	1	no	—
37	C₄⋊C₄×C₁₀₁	—	2	1	no	—
38	C₄⋊C₄×C₁₀₁	—	2	1	no	—
39	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
40	Q₈×C₁₀₁	—	4	1	no	—
41	Q₈×C₁₀₁	—	4	1	no	—
42	C₂×C₄×C₁₀₁	—	4	1	no	—
43	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
46	C₂×Dic₁₀₁	—	2	1	no	—
47	C₂×Dic₁₀₁	—	2	1	no	—
48	C₂×Dic₁₀₁	—	4	1	no	—
49	Dic₂₀₂	—	2	1	no	—
50	Dic₂₀₂	—	4	1	no	—
51	Dic₂₀₂	—	4	1	no	—
52	Dic₂₀₂	SD₁₆	1	2	yes	—
53	C₈×C₁₀₁	—	4	1	no	—
54	C₈×C₁₀₁	—	4	1	no	—
55	#11717579	—	2	1	no	—
56	#11717579	—	2	1	no	—
57	#11717579	—	4	1	no	—
58	C₂²×C₁₀₁	#121	1	1	yes	characteristic
59	C₄×C₁₀₁	—	2	1	no	—
60	C₄×C₁₀₁	—	8	1	no	—
61	C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₁₀₁	#135	1	1	yes	characteristic
63	C₄×C₁₀₁	#135	1	1	yes	characteristic
64	C₄×C₁₀₁	#137	1	1	yes	characteristic
65	Dic₁₀₁	—	2	2	no	—
66	Dic₁₀₁	—	4	1	no	—
67	Dic₁₀₁	—	8	1	no	—
68	C₂×C₁₀₁	#464	1	1	yes	characteristic
69	C₂×C₁₀₁	#465	1	1	yes	characteristic
70	C₂×C₁₀₁	#473	1	1	yes	characteristic
71	#1680	—	101	1	no	maximal, Sylow
72	C₁₀₁	#1680	1	1	yes	characteristic, complemented, Sylow
73	#487	—	101	1	no	—
74	#475	—	101	1	no	—
75	#462	—	101	1	no	—
76	C₈×Q₈	—	101	1	no	—
77	#334	—	101	1	no	—
78	#332	—	101	1	no	—
79	#332	D₁₀₁	1	1	yes	characteristic
80	C₂×Q₁₆	—	202	1	no	—
81	C₄⋊Q₈	—	101	1	no	—
82	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
83	C₄×Q₈	—	101	1	no	—
84	#106	—	202	1	no	—
85	C₄⋊C₈	—	101	1	no	—
86	C₄⋊C₈	—	202	1	no	—
87	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic
88	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic
89	#103	—	202	1	no	—
90	#103	—	202	1	no	—
91	#103	—	202	1	no	—
92	#103	—	202	1	no	—
93	C₄×C₈	—	101	1	no	—
94	C₄×C₈	—	202	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	101	1	no	—
97	C₂×Q₈	—	202	1	no	—
98	Q₁₆	—	404	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	101	1	no	—
102	C₂×C₈	—	101	1	no	—
103	C₂×C₈	—	202	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	101	1	no	—
107	C₄⋊C₄	—	202	1	no	—
108	C₄⋊C₄	—	202	1	no	—
109	C₄⋊C₄	—	202	1	no	—
110	C₄²	—	202	1	no	—
111	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	101	2	no	—
115	Q₈	—	404	1	no	—
116	Q₈	—	404	1	no	—
117	Q₈	—	202	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	404	1	no	—
120	C₂×C₄	—	202	1	no	—
121	C₂×C₄	—	202	1	no	—
122	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
123	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
124	C₂×C₄	#13108074	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	404	1	no	—
128	C₈	—	202	1	no	—
129	C₈	—	202	1	no	—
130	C₂²	#15692228	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	808	1	no	—
134	C₄	—	404	1	no	—
135	C₄	—	202	2	no	—
136	C₄	SD₁₆×D₁₀₁	1	1	yes	characteristic
137	C₄	#15692274	1	1	yes	characteristic
138	C₄	#15692276	1	1	yes	characteristic
139	C₄	#15692287	1	1	yes	characteristic
140	C₂	#20265402	1	1	yes	characteristic
141	C₂	#20265405	1	1	yes	characteristic
142	C₂	#20265451	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692228
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#332×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#121
hypercenter	#332	D₁₀₁
hypocenter	C₁₀₁	#1680
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

Complements

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

N	H ≅ G/N	type
72. C₁₀₁	#1680	semidirect

Maximal subgroups

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

subgroup	index
#20265529	2
#20265463	2
#20265384	2
#20265301	2
#20264927	2
#20264924	2
#332×C₁₀₁	2
#1680	101