Finite Group Databasebeta

Subgroups — #27170715

Order 12928.

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

normal subgroups only

This group has 140 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	#20265528	C₂	1	1	yes	maximal, characteristic
2	#20265473	C₂	1	1	yes	maximal, characteristic
3	#20265463	C₂	1	1	yes	maximal, characteristic
4	#20265307	C₂	1	1	yes	maximal, characteristic
5	#20264925	C₂	1	1	yes	maximal, characteristic
6	#20264922	C₂	1	1	yes	maximal, characteristic
7	#334×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	#15692255	C₂²	1	1	yes	characteristic
9	#15692255	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₀₂	C₂²	1	1	yes	characteristic
11	#15692189	—	2	1	no	—
12	#15692187	—	2	1	no	—
13	C₄⋊#11717579	C₂²	1	1	yes	characteristic
14	#15692184	—	2	1	no	—
15	#15692180	—	2	1	no	—
16	#15692180	—	2	1	no	—
17	#15692180	—	2	1	no	—
18	#15692180	—	2	1	no	—
19	#15692165	—	2	1	no	—
20	C₄⋊C₈×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	—	2	1	no	—
25	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
26	C₂×#11717579	—	2	1	no	—
27	C₄⋊Dic₁₀₁	—	2	1	no	—
28	C₄⋊Dic₁₀₁	—	2	1	no	—
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×C₈×C₁₀₁	—	2	1	no	—
35	C₂×C₈×C₁₀₁	—	2	1	no	—
36	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
38	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
42	C₂×Dic₁₀₁	—	2	1	no	—
43	C₂×Dic₁₀₁	—	2	1	no	—
44	C₂×Dic₁₀₁	—	4	1	no	—
45	C₂×Dic₁₀₁	—	4	1	no	—
46	Dic₂₀₂	—	2	1	no	—
47	Dic₂₀₂	—	2	1	no	—
48	Dic₂₀₂	—	4	1	no	—
49	Dic₂₀₂	—	4	1	no	—
50	Dic₂₀₂	—	8	1	no	—
51	C₈×C₁₀₁	—	2	1	no	—
52	C₈×C₁₀₁	—	2	1	no	—
53	C₈×C₁₀₁	—	4	1	no	—
54	C₈×C₁₀₁	—	4	1	no	—
55	#11717579	—	4	1	no	—
56	C₂²×C₁₀₁	#121	1	1	yes	characteristic
57	C₄×C₁₀₁	—	2	1	no	—
58	C₄×C₁₀₁	#135	1	1	yes	characteristic
59	C₄×C₁₀₁	#137	1	1	yes	characteristic
60	C₄×C₁₀₁	#137	1	1	yes	characteristic
61	C₄×C₁₀₁	#137	1	1	yes	characteristic
62	Dic₁₀₁	—	4	1	no	—
63	Dic₁₀₁	—	4	1	no	—
64	Dic₁₀₁	—	8	1	no	—
65	Dic₁₀₁	—	8	1	no	—
66	C₂×C₁₀₁	#464	1	1	yes	characteristic
67	C₂×C₁₀₁	#470	1	1	yes	characteristic
68	C₂×C₁₀₁	#471	1	1	yes	characteristic
69	#1704	—	101	1	no	maximal, Sylow
70	C₁₀₁	#1704	1	1	yes	characteristic, complemented, Sylow
71	#487	—	101	1	no	—
72	#477	—	101	1	no	—
73	#475	—	101	1	no	—
74	#446	—	101	1	no	—
75	#334	D₁₀₁	1	1	yes	characteristic
76	#333	—	101	1	no	—
77	#332	—	101	1	no	—
78	C₄⋊Q₈	—	101	1	no	—
79	C₄⋊Q₈	—	101	1	no	—
80	C₄×Q₈	—	101	1	no	—
81	#107	—	202	1	no	—
82	#106	—	202	1	no	—
83	C₄⋊C₈	—	101	1	no	—
84	C₄⋊C₈	—	202	1	no	—
85	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic
87	#103	—	202	1	no	—
88	#103	—	202	1	no	—
89	#103	—	202	1	no	—
90	#103	—	202	1	no	—
91	#97	—	202	1	no	—
92	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
93	C₂×Q₈	—	101	1	no	—
94	C₂×Q₈	—	202	1	no	—
95	C₂×Q₈	—	202	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	202	1	no	—
99	C₂×C₈	D₄₀₄	1	1	yes	characteristic
100	C₂×C₈	D₄₀₄	1	1	yes	characteristic
101	C₄⋊C₄	—	101	1	no	—
102	C₄⋊C₄	—	202	1	no	—
103	C₄⋊C₄	—	202	1	no	—
104	C₄⋊C₄	—	202	1	no	—
105	C₄⋊C₄	—	202	1	no	—
106	C₄⋊C₄	—	202	1	no	—
107	C₄²	—	202	1	no	—
108	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
109	Q₈	—	808	1	no	—
110	Q₈	—	404	1	no	—
111	Q₈	—	404	1	no	—
112	Q₈	—	202	1	no	—
113	Q₈	—	202	1	no	—
114	C₂×C₄	—	404	1	no	—
115	C₂×C₄	—	404	1	no	—
116	C₂×C₄	—	202	1	no	—
117	C₂×C₄	—	202	1	no	—
118	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
120	C₂×C₄	#13108075	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	404	1	no	—
126	C₂²	#15692230	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	808	1	no	—
129	C₄	—	808	1	no	—
130	C₄	—	404	1	no	—
131	C₄	—	404	1	no	—
132	C₄	#15692277	1	1	yes	characteristic
133	C₄	#15692284	1	1	yes	characteristic
134	C₄	#15692284	1	1	yes	characteristic
135	C₄	#15692288	1	1	yes	characteristic
136	C₂	#20265400	1	1	yes	characteristic
137	C₂	#20265436	1	1	yes	characteristic
138	C₂	#20265441	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692230
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#334×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#121
hypercenter	#334	D₁₀₁
hypocenter	C₁₀₁	#1704
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
70. C₁₀₁	#1704	semidirect

Maximal subgroups

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

subgroup	index
#20265528	2
#20265473	2
#20265463	2
#20265307	2
#20264925	2
#20264922	2
#334×C₁₀₁	2
#1704	101