Finite Group Databasebeta

Subgroups — #27187342

Order 12928.

184 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₂×#15692330	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₀₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20266066	C₂	1	1	yes	maximal, characteristic, complemented
4	#20266065	C₂	1	1	yes	maximal, characteristic, complemented
5	#20266063	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13108077	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265134	C₂	1	1	yes	maximal, characteristic
8	#15692330	C₄	1	4	yes	complemented
9	D₁₀₁⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#13108077	C₂²	1	1	yes	characteristic
11	C₂×#13108077	C₂²	1	1	yes	characteristic
12	C₂×#13108077	C₂²	1	1	yes	characteristic
13	C₂×#13108077	C₂²	1	1	yes	characteristic
14	C₂×#13108057	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
16	#15692212	C₄	1	1	yes	characteristic
17	#15692165	C₄	1	1	yes	characteristic
18	C₄×#11717579	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₀₁	C₂²	1	1	yes	characteristic
20	#97×C₁₀₁	C₄	1	1	yes	characteristic
21	#13108077	C₂×C₄	1	2	yes	—
22	#13108077	C₂×C₄	1	2	yes	—
23	#13108077	C₂×C₄	1	2	yes	—
24	#13108077	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
26	#13108057	C₈	1	4	yes	—
27	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
28	C₂×#11717579	C₂³	1	1	yes	characteristic
29	C₈×D₁₀₁	C₂×C₄	1	2	yes	—
30	C₈×D₁₀₁	C₂×C₄	1	2	yes	—
31	#13108053	C₈	1	2	yes	—
32	#13108051	C₈	1	2	yes	—
33	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₁₀₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₁	C₂×C₈	1	2	yes	—
41	C₂×Dic₁₀₁	C₂×C₈	1	1	yes	characteristic
42	C₄×D₁₀₁	C₄²	1	2	yes	—
43	C₄×D₁₀₁	C₂×C₈	1	2	yes	—
44	C₈×C₁₀₁	C₄²	1	2	yes	—
45	C₈×C₁₀₁	C₂×C₈	1	2	yes	—
46	#11717579	C₂×C₈	1	2	yes	—
47	#11717579	C₂²×C₄	1	1	yes	characteristic
48	#11717579	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×D₁₀₁	—	2	2	no	—
51	C₂×D₁₀₁	C₄×C₈	1	2	yes	—
52	C₄×C₁₀₁	—	2	2	no	—
53	C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
55	Dic₁₀₁	—	2	2	no	—
56	Dic₁₀₁	C₄×C₈	1	2	yes	—
57	C₂×C₁₀₁	C₂×C₄×C₈	1	1	yes	characteristic
58	C₂×C₁₀₁	#504	1	2	yes	—
59	D₁₀₁	—	2	2	no	—
60	#2108	—	101	1	no	maximal, Sylow
61	C₁₀₁	#2108	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₃₂	—	101	1	no	—
63	C₂²×C₁₆	—	101	1	no	—
64	#405	—	101	1	no	—
65	#346	—	101	1	no	—
66	#346	—	101	1	no	—
67	C₄×C₁₆	—	101	1	no	—
68	C₄×C₁₆	—	101	1	no	—
69	C₂×M₁₆	—	101	1	no	—
70	C₂²×C₈	—	101	1	no	—
71	#117	—	101	1	no	—
72	M₃₂	—	101	4	no	—
73	C₂×C₁₆	—	101	1	no	—
74	C₂×C₁₆	—	101	1	no	—
75	C₂×C₁₆	—	101	1	no	—
76	C₂×C₁₆	—	101	1	no	—
77	C₂×C₁₆	—	101	4	no	—
78	#97	—	101	1	no	—
79	#97	#87938	1	1	yes	characteristic
80	C₄×C₈	—	101	1	no	—
81	C₄×C₈	—	101	1	no	—
82	C₂²×C₄	—	101	1	no	—
83	M₁₆	—	101	4	no	—
84	C₂×C₈	—	101	1	no	—
85	C₂×C₈	—	101	1	no	—
86	C₂×C₈	—	101	2	no	—
87	C₂×C₈	—	101	2	no	—
88	C₂×C₈	C₂×#87938	1	1	yes	characteristic
89	C₂×C₈	C₂×#87938	1	1	yes	characteristic
90	C₄⋊C₄	—	101	2	no	—
91	C₂²⋊C₄	—	101	2	no	—
92	C₄²	—	101	1	no	—
93	C₄²	C₂×#87938	1	1	yes	characteristic
94	C₁₆	—	101	2	no	—
95	C₁₆	—	101	2	no	—
96	C₁₆	—	101	2	no	—
97	C₁₆	—	101	2	no	—
98	C₂³	—	101	1	no	—
99	C₂×C₄	—	101	1	no	—
100	C₂×C₄	—	101	2	no	—
101	C₂×C₄	—	101	2	no	—
102	C₂×C₄	—	101	2	no	—
103	C₂×C₄	D₁₀₁⋊C₈	1	2	yes	—
104	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
105	C₈	—	101	1	no	—
106	C₈	—	101	1	no	—
107	C₈	—	101	2	no	—
108	C₈	C₄×#87938	1	2	yes	—
109	C₈	D₁₀₁⋊C₈	1	2	yes	—
110	C₂²	—	101	2	no	—
111	C₂²	—	202	2	no	—
112	C₂²	C₂×D₁₀₁⋊C₈	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	101	2	no	—
115	C₄	—	202	2	no	—
116	C₄	C₂×C₄×#87938	1	1	yes	characteristic
117	C₄	C₂×D₁₀₁⋊C₈	1	1	yes	characteristic
118	C₂	—	202	2	no	—
119	C₂	C₄×D₁₀₁⋊C₈	1	1	yes	characteristic
120	C₂	#20266236	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87938
commutator	C₂×C₁₀₁	C₂×C₄×C₈
Fitting	#97×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂²×C₈
prime radical	C₂²×D₁₀₁	C₂×C₈
cosocle	C₂×#11717579	C₂³
hypercenter	#97	#87938
hypocenter	C₁₀₁	#2108
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. #20266066	C₂	semidirect
4. #20266065	C₂	semidirect
5. #20266063	C₂	semidirect
6. C₄×#13108077	C₂	semidirect
8. #15692330	C₄	semidirect
9. D₁₀₁⋊C₁₆	C₄	semidirect
61. C₁₀₁	#2108	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692330	2
C₂×D₁₀₁⋊C₁₆	2
#20266066	2
#20266065	2
#20266063	2
C₄×#13108077	2
#20265134	2
#2108	101