Finite Group Databasebeta

Subgroups — #27187539

Order 12928.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#20266223	C₂	1	1	yes	maximal, characteristic
2	#20266217	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266101	C₂	1	1	yes	maximal, characteristic, complemented
4	#20266096	C₂	1	1	yes	maximal, characteristic, complemented
5	#20266094	C₂	1	1	yes	maximal, characteristic
6	#20266067	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265619	C₂	1	1	yes	maximal, characteristic
8	#15692343	—	2	1	no	—
9	Q₈×#87938	—	2	1	no	—
10	#15692330	C₂²	1	1	yes	characteristic
11	#15692325	C₂²	1	1	yes	characteristic
12	#15692323	C₂²	1	1	yes	characteristic
13	#15692320	—	2	1	no	—
14	#15692318	—	2	1	no	—
15	#15692309	C₂²	1	1	yes	characteristic
16	#15692278	C₂²	1	1	yes	characteristic
17	Q₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
18	#15692201	C₄	1	1	yes	characteristic, complemented
19	#15692200	C₄	1	1	yes	characteristic, complemented
20	#15692198	C₄	1	1	yes	characteristic, complemented
21	#15692194	C₂²	1	1	yes	characteristic
22	Q₃₂×C₁₀₁	C₄	1	1	yes	characteristic, complemented
23	#13108084	—	2	1	no	—
24	#13108084	—	4	1	no	—
25	D₁₀₁⋊C₈	—	4	1	no	—
26	D₁₀₁⋊C₈	D₄	1	1	yes	characteristic
27	C₄⋊#87938	—	2	1	no	—
28	C₄⋊#87938	—	4	1	no	—
29	C₄×#87938	—	4	1	no	—
30	C₄×#87938	D₄	1	1	yes	characteristic
31	#13108077	—	2	1	no	—
32	#13108075	—	2	1	no	—
33	Q₈×D₁₀₁	—	2	1	no	—
34	Dic₄₀₄	C₂×C₄	1	1	yes	characteristic
35	#13108064	—	2	1	no	—
36	#13108061	—	2	1	no	—
37	D₈₀₈	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
39	#13108049	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
41	Q₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
43	C₂×#87938	—	2	1	no	—
44	C₂×#87938	—	4	1	no	—
45	Q₈×C₁₀₁	—	2	1	no	—
46	Q₈×C₁₀₁	—	2	1	no	—
47	D₄₀₄	—	2	1	no	—
48	D₄₀₄	—	4	1	no	—
49	C₄×D₁₀₁	—	4	1	no	—
50	C₄×D₁₀₁	—	4	1	no	—
51	C₄×D₁₀₁	C₂×D₄	1	1	yes	characteristic
52	Dic₂₀₂	—	2	1	no	—
53	Dic₂₀₂	—	4	1	no	—
54	#11717581	—	2	1	no	—
55	#11717581	—	4	1	no	—
56	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
57	#11717579	#41	1	1	yes	characteristic
58	C₂×D₁₀₁	—	4	1	no	—
59	C₂×D₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#87938	—	4	1	no	—
61	#87938	—	8	1	no	—
62	C₄×C₁₀₁	—	4	1	no	—
63	C₄×C₁₀₁	—	4	1	no	—
64	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
65	Dic₁₀₁	—	4	1	no	—
66	Dic₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₁₀₁	#442	1	1	yes	characteristic
68	D₁₀₁	—	2	1	no	—
69	D₁₀₁	—	8	1	no	—
70	#2180	—	101	1	no	maximal, Sylow
71	C₁₀₁	#2180	1	1	yes	characteristic, complemented, Sylow
72	#510	—	101	1	no	—
73	#444	—	101	1	no	—
74	#441	—	101	1	no	—
75	#365	—	101	1	no	—
76	#362	—	101	1	no	—
77	#360	—	101	1	no	—
78	#347	—	101	1	no	—
79	#135	—	101	1	no	—
80	C₂×Q₁₆	—	101	1	no	—
81	#131	—	202	1	no	—
82	C₄×Q₈	—	202	1	no	—
83	Q₃₂	—	101	1	no	—
84	Q₃₂	#87938	1	1	yes	characteristic, complemented
85	SD₃₂	—	101	1	no	—
86	SD₃₂	—	101	1	no	—
87	M₃₂	—	101	1	no	—
88	M₃₂	—	101	1	no	—
89	#108	—	101	1	no	—
90	#106	—	101	1	no	—
91	#104	—	202	1	no	—
92	#103	—	202	1	no	—
93	#97	—	101	1	no	—
94	#41	—	202	1	no	—
95	C₂×Q₈	—	202	1	no	—
96	Q₁₆	—	101	1	no	—
97	Q₁₆	—	202	1	no	—
98	Q₁₆	C₂×#87938	1	1	yes	characteristic
99	Q₁₆	C₂×#87938	1	1	yes	characteristic
100	SD₁₆	—	202	1	no	—
101	D₈	—	101	1	no	—
102	M₁₆	—	404	1	no	—
103	M₁₆	—	202	1	no	—
104	C₂×C₈	—	101	1	no	—
105	C₂×C₈	—	101	1	no	—
106	C₂×C₈	—	404	1	no	—
107	C₄⋊C₄	—	404	1	no	—
108	C₄⋊C₄	—	202	1	no	—
109	C₄²	—	101	1	no	—
110	C₄²	—	404	1	no	—
111	C₁₆	—	101	1	no	—
112	C₁₆	—	202	1	no	—
113	C₁₆	C₂×#87938	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	404	1	no	—
117	Q₈	—	202	1	no	—
118	D₄	—	404	1	no	—
119	D₄	—	202	1	no	—
120	C₂×C₄	—	101	1	no	—
121	C₂×C₄	—	404	1	no	—
122	C₂×C₄	—	404	1	no	—
123	C₂×C₄	—	404	1	no	—
124	C₂×C₄	—	202	1	no	—
125	C₈	—	101	1	no	—
126	C₈	—	404	1	no	—
127	C₈	—	202	1	no	—
128	C₈	C₂²×#87938	1	1	yes	characteristic
129	C₂²	—	101	1	no	—
130	C₂²	—	404	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	808	1	no	—
134	C₄	—	101	1	no	—
135	C₄	—	404	1	no	—
136	C₄	—	404	1	no	—
137	C₄	D₄×#87938	1	1	yes	characteristic
138	C₂	—	808	1	no	—
139	C₂	—	202	1	no	—
140	C₂	#20266218	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266218
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₈	C₂²×#87938
Fitting	Q₃₂×C₁₀₁	C₄
socle	C₂×C₁₀₁	#442
prime radical	#15692278	C₂²
cosocle	C₈×D₁₀₁	C₂³
hypercenter	Q₃₂	#87938
hypocenter	C₁₀₁	#2180
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20266217	C₂	semidirect
3. #20266101	C₂	semidirect
4. #20266096	C₂	semidirect
6. #20266067	C₂	semidirect
18. #15692201	C₄	semidirect
19. #15692200	C₄	semidirect
20. #15692198	C₄	semidirect
22. Q₃₂×C₁₀₁	C₄	semidirect
71. C₁₀₁	#2180	semidirect
84. Q₃₂	#87938	semidirect

Maximal subgroups

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

subgroup	index
#20266223	2
#20266217	2
#20266101	2
#20266096	2
#20266094	2
#20266067	2
#20265619	2
#2180	101