Finite Group Databasebeta

Subgroups — #27170420

Order 12928.

262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 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	#20265919	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265564	C₂	1	2	yes	maximal, complemented
3	#20264917	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264916	C₂	1	2	yes	maximal, complemented
5	#496×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108062	—	2	2	no	—
7	C₂×#13108059	—	2	2	no	—
8	#15692244	—	2	1	no	—
9	#15692243	C₂²	1	2	yes	—
10	#15692241	—	2	2	no	—
11	#15692225	—	2	2	no	—
12	#15692223	—	2	1	no	—
13	#15692166	C₂²	1	2	yes	complemented
14	C₂×SD₁₆×C₁₀₁	—	2	1	no	—
15	C₂×D₈×C₁₀₁	—	2	1	no	—
16	#127×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
17	#124×C₁₀₁	C₂²	1	1	yes	characteristic
18	#97×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₁₀₁	—	2	1	no	—
20	C₂×Dic₂₀₂	D₄	1	2	yes	—
21	C₂×D₄₀₄	—	2	1	no	—
22	C₂×D₄₀₄	D₄	1	2	yes	—
23	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
24	#13108062	—	4	4	no	—
25	#13108059	—	4	4	no	—
26	C₂×#11717579	—	2	2	no	—
27	C₂²⋊Dic₁₀₁	—	2	1	no	—
28	#13108051	—	2	2	no	—
29	#13108051	—	2	2	no	—
30	#13108051	—	2	2	no	—
31	C₄×Dic₁₀₁	—	2	1	no	—
32	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
33	C₂×D₄×C₁₀₁	—	2	1	no	—
34	C₂×D₄×C₁₀₁	—	2	2	no	—
35	C₂×D₄×C₁₀₁	D₄	1	1	yes	characteristic
36	SD₁₆×C₁₀₁	—	4	2	no	—
37	D₈×C₁₀₁	—	4	2	no	—
38	C₂×C₈×C₁₀₁	—	2	1	no	—
39	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
40	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₁	—	2	1	no	—
42	C₂³×C₁₀₁	—	4	1	no	—
43	C₂²×D₁₀₁	—	2	2	no	—
44	C₂²×D₁₀₁	—	8	2	no	—
45	C₂²×D₁₀₁	—	8	1	no	—
46	Q₈×C₁₀₁	—	4	1	no	—
47	D₄×C₁₀₁	—	4	1	no	—
48	D₄×C₁₀₁	—	4	1	no	—
49	D₄×C₁₀₁	—	4	1	no	—
50	D₄×C₁₀₁	—	4	2	no	—
51	D₄×C₁₀₁	—	4	2	no	—
52	C₂×C₄×C₁₀₁	—	2	1	no	—
53	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
55	C₂²⋊D₁₀₁	—	8	2	no	—
56	C₂×Dic₁₀₁	—	2	2	no	—
57	D₄₀₄	—	4	2	no	—
58	D₄₀₄	—	8	1	no	—
59	Dic₂₀₂	—	4	2	no	—
60	C₈×C₁₀₁	—	2	2	no	—
61	#11717579	—	2	4	no	—
62	C₂²×C₁₀₁	—	4	2	no	—
63	C₂²×C₁₀₁	—	4	1	no	—
64	C₂²×C₁₀₁	—	8	2	no	—
65	C₂²×C₁₀₁	—	8	1	no	—
66	C₂²×C₁₀₁	#120	1	1	yes	characteristic
67	C₂×D₁₀₁	—	4	2	no	—
68	C₂×D₁₀₁	—	4	2	no	—
69	C₂×D₁₀₁	—	4	2	no	—
70	C₂×D₁₀₁	—	8	4	no	—
71	C₄×C₁₀₁	—	2	2	no	—
72	C₄×C₁₀₁	—	2	1	no	—
73	C₄×C₁₀₁	—	4	1	no	—
74	Dic₁₀₁	—	4	2	no	—
75	C₂×C₁₀₁	—	8	1	no	—
76	C₂×C₁₀₁	—	16	1	no	—
77	C₂×C₁₀₁	#453	1	2	yes	—
78	C₂×C₁₀₁	#453	1	1	yes	characteristic
79	D₁₀₁	—	8	2	no	—
80	#1655	—	101	1	no	maximal, Sylow
81	C₁₀₁	#1655	1	1	yes	characteristic, complemented, Sylow
82	H(3,𝔽₄)	—	101	1	no	—
83	#496	—	101	2	no	—
84	#496	D₁₀₁	1	1	yes	characteristic, complemented
85	#329	—	101	1	no	—
86	#329	—	101	2	no	—
87	C₂×SD₁₆	—	2	1	no	—
88	C₂×SD₁₆	—	202	2	no	—
89	C₂×D₈	—	2	1	no	—
90	C₂×D₈	—	202	2	no	—
91	#127	C₂×D₁₀₁	1	1	yes	characteristic, complemented
92	#124	—	101	2	no	—
93	#124	—	202	2	no	—
94	#124	—	202	1	no	—
95	#124	C₂×D₁₀₁	1	1	yes	characteristic
96	#120	—	202	2	no	—
97	#120	—	202	1	no	—
98	#97	—	101	2	no	—
99	#97	C₂×D₁₀₁	1	1	yes	characteristic, complemented
100	C₂⁴	—	202	1	no	—
101	C₂×Q₈	—	101	2	no	—
102	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	101	2	no	—
106	C₂×D₄	—	202	1	no	—
107	C₂×D₄	—	202	2	no	—
108	C₂×D₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	404	4	no	—
111	D₈	—	4	2	no	—
112	D₈	—	404	4	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	202	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	202	2	no	—
117	C₂²⋊C₄	—	202	1	no	—
118	C₂²⋊C₄	—	202	2	no	—
119	C₂²⋊C₄	—	202	2	no	—
120	C₄²	—	202	1	no	—
121	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
122	C₂³	—	2	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	808	2	no	—
125	C₂³	—	808	1	no	—
126	C₂³	—	202	2	no	—
127	Q₈	—	4	1	no	—
128	Q₈	—	404	2	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	2	no	—
134	D₄	—	808	1	no	—
135	D₄	—	808	2	no	—
136	D₄	—	404	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	202	2	no	—
139	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	202	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	808	4	no	—
148	C₂²	—	404	2	no	—
149	C₂²	—	404	2	no	—
150	C₂²	—	404	2	no	—
151	C₂²	#15692223	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	404	2	no	—
156	C₂	—	8	1	no	—
157	C₂	—	16	1	no	—
158	C₂	—	808	2	no	—
159	C₂	#20265343	1	2	yes	—
160	C₂	#20265344	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692223
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#496×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
hypercenter	#496	D₁₀₁
hypocenter	C₁₀₁	#1655
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #20265919	C₂	semidirect
2. #20265564	C₂	semidirect
3. #20264917	C₂	semidirect
4. #20264916	C₂	semidirect
5. #496×C₁₀₁	C₂	semidirect
13. #15692166	C₂²	semidirect
16. #127×C₁₀₁	C₂²	semidirect
18. #97×C₁₀₁	C₂²	semidirect
81. C₁₀₁	#1655	semidirect
84. #496	D₁₀₁	semidirect
91. #127	C₂×D₁₀₁	semidirect
99. #97	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265919	2
#20265564	2
#20264917	2
#20264916	2
#496×C₁₀₁	2
#1655	101