Finite Group Databasebeta

Subgroups — #27187532

Order 12928.

146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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	#20266221	C₂	1	1	yes	maximal, characteristic, complemented
2	#20266219	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266106	C₂	1	1	yes	maximal, characteristic, complemented
4	#20266092	C₂	1	1	yes	maximal, characteristic, complemented
5	#20266091	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×#87938	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265608	C₂	1	1	yes	maximal, characteristic
8	#15692343	—	2	1	no	—
9	#15692342	—	2	1	no	—
10	D₁₀₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#15692326	C₂²	1	1	yes	characteristic
12	#15692326	C₂²	1	1	yes	characteristic
13	#15692320	—	2	1	no	—
14	#15692319	—	2	1	no	—
15	C₈×#87938	C₂²	1	1	yes	characteristic
16	#15692278	C₂²	1	1	yes	characteristic
17	#15692274	C₂²	1	1	yes	characteristic
18	#15692201	C₄	1	1	yes	characteristic, complemented
19	#15692200	C₄	1	1	yes	characteristic, complemented
20	#15692196	C₄	1	1	yes	characteristic, complemented
21	C₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₀₁	C₄	1	1	yes	characteristic, complemented
23	#13108084	—	2	1	no	—
24	#13108084	—	2	1	no	—
25	#13108084	—	4	1	no	—
26	#13108083	—	4	1	no	—
27	D₁₀₁⋊C₈	—	4	1	no	—
28	D₁₀₁⋊C₈	D₄	1	1	yes	characteristic
29	C₂×#11717581	—	4	1	no	—
30	C₄×#87938	D₄	1	1	yes	characteristic
31	#13108077	—	2	1	no	—
32	#13108075	—	2	1	no	—
33	#13108073	—	2	1	no	—
34	Dic₄₀₄	C₂×C₄	1	1	yes	characteristic
35	#13108062	—	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	D₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
43	C₂×#87938	—	2	1	no	—
44	Q₈×C₁₀₁	—	2	1	no	—
45	D₄×C₁₀₁	—	2	1	no	—
46	C₂²⋊D₁₀₁	—	4	1	no	—
47	C₂×Dic₁₀₁	—	4	1	no	—
48	D₄₀₄	—	2	1	no	—
49	D₄₀₄	—	4	1	no	—
50	C₄×D₁₀₁	—	4	1	no	—
51	C₄×D₁₀₁	C₂×D₄	1	1	yes	characteristic
52	Dic₂₀₂	—	2	1	no	—
53	#11717581	—	4	1	no	—
54	#11717581	—	4	1	no	—
55	#11717581	D₈	1	1	yes	characteristic
56	#11717581	D₈	1	1	yes	characteristic
57	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
58	#11717579	#41	1	1	yes	characteristic
59	C₂²×C₁₀₁	—	4	1	no	—
60	C₂×D₁₀₁	—	4	1	no	—
61	C₂×D₁₀₁	#135	1	1	yes	characteristic
62	#87938	—	2	2	no	—
63	C₄×C₁₀₁	—	4	1	no	—
64	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
65	Dic₁₀₁	—	4	1	no	—
66	Dic₁₀₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₁₀₁	—	8	1	no	—
68	C₂×C₁₀₁	C₄×D₈	1	1	yes	characteristic
69	D₁₀₁	—	2	1	no	—
70	D₁₀₁	—	8	1	no	—
71	#2178	—	101	1	no	maximal, Sylow
72	C₁₀₁	#2178	1	1	yes	characteristic, complemented, Sylow
73	#508	—	101	1	no	—
74	#443	—	101	1	no	—
75	#443	—	101	1	no	—
76	#368	—	101	1	no	—
77	#359	—	101	1	no	—
78	#359	—	101	1	no	—
79	C₄×C₁₆	—	101	1	no	—
80	#135	—	101	1	no	—
81	#135	—	101	1	no	—
82	#131	—	202	1	no	—
83	#131	—	202	1	no	—
84	Q₃₂	—	101	1	no	—
85	SD₃₂	—	101	1	no	—
86	SD₃₂	#87938	1	1	yes	characteristic, complemented
87	D₁₆	—	101	1	no	—
88	C₂×C₁₆	—	101	1	no	—
89	C₂×C₁₆	—	101	1	no	—
90	#108	—	101	1	no	—
91	#108	—	101	1	no	—
92	#104	—	202	1	no	—
93	#104	—	202	1	no	—
94	C₄×C₈	—	101	1	no	—
95	#41	—	202	1	no	—
96	#41	—	202	1	no	—
97	Q₁₆	—	101	1	no	—
98	Q₁₆	C₂×#87938	1	1	yes	characteristic
99	SD₁₆	—	202	1	no	—
100	SD₁₆	—	202	1	no	—
101	D₈	—	101	1	no	—
102	D₈	C₂×#87938	1	1	yes	characteristic
103	M₁₆	—	404	1	no	—
104	M₁₆	—	404	1	no	—
105	M₁₆	—	202	1	no	—
106	M₁₆	—	202	1	no	—
107	C₂×C₈	—	101	1	no	—
108	C₂×C₈	—	101	1	no	—
109	C₂×C₈	—	404	1	no	—
110	C₂×C₈	—	404	1	no	—
111	C₄²	—	101	1	no	—
112	C₁₆	—	101	1	no	—
113	C₁₆	—	202	1	no	—
114	C₁₆	C₂×#87938	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	202	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	404	1	no	—
119	D₄	—	404	1	no	—
120	D₄	—	202	1	no	—
121	C₂×C₄	—	101	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	—	101	1	no	—
127	C₈	—	101	1	no	—
128	C₈	—	404	1	no	—
129	C₈	—	404	1	no	—
130	C₈	C₂²×#87938	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	101	1	no	—
133	C₂²	—	404	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	101	1	no	—
136	C₄	—	404	1	no	—
137	C₄	—	202	2	no	—
138	C₄	D₄×#87938	1	1	yes	characteristic
139	C₂	—	8	1	no	—
140	C₂	—	808	1	no	—
141	C₂	—	202	1	no	—
142	C₂	D₈×#87938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#87938
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₈	C₂²×#87938
Fitting	SD₃₂×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₄×D₈
prime radical	#20265608	C₂
cosocle	C₈×D₁₀₁	C₂³
hypercenter	SD₃₂	#87938
hypocenter	C₁₀₁	#2178
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20266221	C₂	semidirect
2. #20266219	C₂	semidirect
3. #20266106	C₂	semidirect
4. #20266092	C₂	semidirect
5. #20266091	C₂	semidirect
6. C₁₆×#87938	C₂	semidirect
18. #15692201	C₄	semidirect
19. #15692200	C₄	semidirect
20. #15692196	C₄	semidirect
22. SD₃₂×C₁₀₁	C₄	semidirect
72. C₁₀₁	#2178	semidirect
86. SD₃₂	#87938	semidirect

Maximal subgroups

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

subgroup	index
#20266221	2
#20266219	2
#20266106	2
#20266092	2
#20266091	2
C₁₆×#87938	2
#20265608	2
#2178	101