Finite Group Databasebeta

Subgroups — #27173016

Order 12928.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20265607	C₂	1	1	yes	maximal, characteristic
2	#20265295	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265006	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264984	C₂	1	1	yes	maximal, characteristic, complemented
5	#20264983	C₂	1	1	yes	maximal, characteristic, complemented
6	#20264953	C₂	1	1	yes	maximal, characteristic, complemented
7	#443×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#15692277	C₂²	1	1	yes	characteristic
9	#15692261	—	2	1	no	—
10	#15692201	C₄	1	1	yes	characteristic, complemented
11	#15692199	C₄	1	1	yes	characteristic, complemented
12	#15692198	C₄	1	1	yes	characteristic, complemented
13	#15692196	C₄	1	1	yes	characteristic, complemented
14	C₂×#13108049	C₂²	1	1	yes	characteristic
15	C₂×#13108049	C₂²	1	1	yes	characteristic
16	#15692191	C₂²	1	1	yes	characteristic
17	#15692183	—	2	1	no	—
18	#135×C₁₀₁	C₂²	1	1	yes	characteristic
19	#131×C₁₀₁	—	2	1	no	—
20	#108×C₁₀₁	C₂²	1	1	yes	characteristic
21	#104×C₁₀₁	—	2	1	no	—
22	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
23	#13108074	—	2	1	no	—
24	Dic₄₀₄	C₂×C₄	1	1	yes	characteristic
25	#13108063	—	2	1	no	—
26	D₈₀₈	C₂×C₄	1	1	yes	characteristic
27	#13108058	—	2	1	no	—
28	#13108057	—	4	1	no	—
29	C₈×D₁₀₁	—	4	1	no	—
30	#13108049	—	2	1	no	—
31	#13108049	C₂×C₄	1	1	yes	characteristic
32	#13108049	C₂×C₄	1	1	yes	characteristic
33	#41×C₁₀₁	—	2	1	no	—
34	Q₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
35	SD₁₆×C₁₀₁	—	2	1	no	—
36	D₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
37	M₁₆×C₁₀₁	—	2	1	no	—
38	M₁₆×C₁₀₁	—	4	1	no	—
39	C₂×C₈×C₁₀₁	—	4	1	no	—
40	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
42	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
43	Q₈×C₁₀₁	—	2	1	no	—
44	D₄×C₁₀₁	—	2	1	no	—
45	D₄×C₁₀₁	—	4	1	no	—
46	C₂×C₄×C₁₀₁	—	2	1	no	—
47	C₂×C₄×C₁₀₁	—	4	1	no	—
48	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
49	C₂²⋊D₁₀₁	—	4	1	no	—
50	D₄₀₄	—	2	1	no	—
51	C₄×D₁₀₁	—	4	1	no	—
52	Dic₂₀₂	—	2	1	no	—
53	C₈×C₁₀₁	—	4	1	no	—
54	C₈×C₁₀₁	D₈	1	1	yes	characteristic
55	C₈×C₁₀₁	D₈	1	1	yes	characteristic
56	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
57	C₈×C₁₀₁	#41	1	1	yes	characteristic
58	#11717579	—	4	1	no	—
59	C₂²×C₁₀₁	—	4	1	no	—
60	C₂²×C₁₀₁	#135	1	1	yes	characteristic
61	C₂×D₁₀₁	—	4	1	no	—
62	C₄×C₁₀₁	—	4	1	no	—
63	C₄×C₁₀₁	—	4	1	no	—
64	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₁₀₁	C₂×D₈	1	1	yes	characteristic
66	Dic₁₀₁	—	4	1	no	—
67	C₂×C₁₀₁	—	2	1	no	—
68	C₂×C₁₀₁	—	8	1	no	—
69	C₂×C₁₀₁	C₄×D₈	1	1	yes	characteristic
70	D₁₀₁	—	8	1	no	—
71	#2179	—	101	1	no	maximal, Sylow
72	C₁₀₁	#2179	1	1	yes	characteristic, complemented, Sylow
73	#508	—	101	1	no	—
74	#443	—	101	1	no	—
75	#443	D₁₀₁	1	1	yes	characteristic, complemented
76	#368	—	101	1	no	—
77	#359	—	101	1	no	—
78	#359	—	101	1	no	—
79	#346	—	101	1	no	—
80	#135	—	101	1	no	—
81	#135	C₂×D₁₀₁	1	1	yes	characteristic
82	#131	—	2	1	no	—
83	#131	—	202	1	no	—
84	Q₃₂	—	101	1	no	—
85	SD₃₂	—	101	1	no	—
86	SD₃₂	—	101	1	no	—
87	D₁₆	—	101	1	no	—
88	C₂×C₁₆	—	101	1	no	—
89	C₂×C₁₆	—	101	1	no	—
90	#108	—	101	1	no	—
91	#108	C₂×D₁₀₁	1	1	yes	characteristic
92	#104	—	2	1	no	—
93	#104	—	202	1	no	—
94	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
95	#41	—	2	1	no	—
96	#41	—	202	1	no	—
97	Q₁₆	—	101	1	no	—
98	Q₁₆	C₄×D₁₀₁	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	202	1	no	—
101	D₈	—	101	1	no	—
102	D₈	C₄×D₁₀₁	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	4	1	no	—
105	M₁₆	—	404	1	no	—
106	M₁₆	—	202	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	404	1	no	—
109	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
112	C₁₆	—	101	1	no	—
113	C₁₆	—	101	1	no	—
114	C₁₆	—	202	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	202	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	404	1	no	—
120	D₄	—	202	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	404	1	no	—
124	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	404	1	no	—
127	C₈	#13108059	1	1	yes	characteristic
128	C₈	#13108059	1	1	yes	characteristic
129	C₈	C₂×C₄×D₁₀₁	1	1	yes	characteristic
130	C₈	#13108074	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	404	1	no	—
133	C₂²	#15692275	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	404	1	no	—
137	C₄	C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
138	C₄	C₂×#13108059	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	808	1	no	—
142	C₂	C₄×#13108059	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13108059
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	#443×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₄×D₈
prime radical	#20265607	C₂
cosocle	C₂×C₈×C₁₀₁	C₂³
hypercenter	#443	D₁₀₁
hypocenter	C₁₀₁	#2179
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #20265295	C₂	semidirect
3. #20265006	C₂	semidirect
4. #20264984	C₂	semidirect
5. #20264983	C₂	semidirect
6. #20264953	C₂	semidirect
7. #443×C₁₀₁	C₂	semidirect
10. #15692201	C₄	semidirect
11. #15692199	C₄	semidirect
12. #15692198	C₄	semidirect
13. #15692196	C₄	semidirect
72. C₁₀₁	#2179	semidirect
75. #443	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265607	2
#20265295	2
#20265006	2
#20264984	2
#20264983	2
#20264953	2
#443×C₁₀₁	2
#2179	101