Finite Group Databasebeta

Subgroups — #27187265

Order 12928.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20266181	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692320	C₂	1	4	yes	maximal, complemented
3	C₄²×#87938	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265728	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108084	C₂²	1	2	yes	—
6	C₂×C₄×#87938	—	2	2	no	—
7	C₂×C₄×#87938	C₂²	1	2	yes	—
8	C₄⋊#11717581	—	2	2	no	—
9	#15692320	—	2	8	no	—
10	C₂×#13108075	C₂²	1	2	yes	—
11	#15692252	C₄	1	1	yes	characteristic, complemented
12	#15692244	C₄	1	2	yes	complemented
13	#15692230	—	2	2	no	—
14	#15692217	—	2	2	no	—
15	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic, complemented
17	C₂²×#87938	—	2	2	no	—
18	#13108084	—	2	4	no	—
19	C₂×#11717581	—	2	2	no	—
20	C₄×#87938	—	2	8	no	—
21	C₄×#87938	—	2	2	no	—
22	C₄×#87938	D₄	1	4	yes	complemented
23	#13108075	—	2	4	no	—
24	#13108075	—	2	4	no	—
25	C₂×D₄₀₄	—	2	2	no	—
26	C₂×D₄₀₄	—	2	2	no	—
27	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₀₁	—	2	2	no	—
29	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
30	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
31	#13108051	—	2	4	no	—
32	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₀₁	D₄	1	2	yes	—
34	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
35	C₄⋊C₄×C₁₀₁	—	2	2	no	—
36	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₁₀₁	—	2	2	no	—
38	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×#87938	—	2	8	no	—
40	C₂×#87938	—	2	4	no	—
41	Q₈×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
47	D₄₀₄	—	2	4	no	—
48	D₄₀₄	—	4	4	no	—
49	D₄₀₄	—	4	4	no	—
50	C₄×D₁₀₁	—	2	4	no	—
51	C₄×D₁₀₁	—	4	2	no	—
52	C₄×D₁₀₁	—	4	2	no	—
53	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
54	C₄×D₁₀₁	#41	1	2	yes	—
55	#11717581	—	4	2	no	—
56	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₁₀₁	—	2	2	no	—
58	C₂×D₁₀₁	—	4	2	no	—
59	C₂×D₁₀₁	—	4	4	no	—
60	C₂×D₁₀₁	#121	1	2	yes	—
61	#87938	—	2	8	no	—
62	C₄×C₁₀₁	—	4	2	no	—
63	C₄×C₁₀₁	#104	1	4	yes	complemented
64	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
65	Dic₁₀₁	—	2	2	no	—
66	Dic₁₀₁	#124	1	1	yes	characteristic
67	Dic₁₀₁	#127	1	1	yes	characteristic
68	C₂×C₁₀₁	#390	1	1	yes	characteristic
69	C₂×C₁₀₁	C₂×#104	1	2	yes	—
70	D₁₀₁	—	2	2	no	—
71	D₁₀₁	—	8	2	no	—
72	#1962	—	101	1	no	maximal, Sylow
73	C₁₀₁	#1962	1	1	yes	characteristic, complemented, Sylow
74	#532	—	101	1	no	—
75	#423	—	101	1	no	—
76	C₂×#104	—	101	4	no	—
77	C₄³	—	101	1	no	—
78	C₂×#41	—	101	2	no	—
79	C₂×M₁₆	—	101	2	no	—
80	C₄⋊Q₈	#87938	1	1	yes	characteristic, complemented
81	#127	—	101	1	no	—
82	#124	—	101	2	no	—
83	#121	—	202	2	no	—
84	C₄×D₄	—	202	2	no	—
85	C₂×C₄²	—	101	1	no	—
86	C₂×C₄²	—	101	2	no	—
87	C₂×C₄²	—	202	2	no	—
88	C₄⋊C₈	—	202	2	no	—
89	#104	—	202	8	no	—
90	#41	—	202	4	no	—
91	#41	—	202	4	no	—
92	C₂×Q₈	C₂×#87938	1	2	yes	—
93	C₂×D₄	—	101	2	no	—
94	C₂×D₄	—	202	2	no	—
95	C₂×D₄	—	202	2	no	—
96	C₂²×C₄	—	101	1	no	—
97	C₂²×C₄	—	101	2	no	—
98	C₂²×C₄	—	202	2	no	—
99	C₂²×C₄	—	202	2	no	—
100	M₁₆	—	202	4	no	—
101	C₂×C₈	—	202	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	202	4	no	—
104	C₄²	—	101	2	no	—
105	C₄²	—	101	1	no	—
106	C₄²	—	101	4	no	—
107	C₄²	—	202	2	no	—
108	C₄²	—	202	8	no	—
109	C₄²	C₂×#87938	1	1	yes	characteristic
110	C₂³	—	101	1	no	—
111	C₂³	—	202	2	no	—
112	Q₈	—	2	4	no	—
113	D₄	—	404	4	no	—
114	D₄	—	404	4	no	—
115	D₄	—	202	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	101	1	no	—
118	C₂×C₄	—	101	2	no	—
119	C₂×C₄	—	101	2	no	—
120	C₂×C₄	—	101	2	no	—
121	C₂×C₄	—	404	2	no	—
122	C₂×C₄	—	404	2	no	—
123	C₂×C₄	—	202	4	no	—
124	C₂×C₄	—	202	8	no	—
125	C₂×C₄	—	202	4	no	—
126	C₂×C₄	C₂²⋊#87938	1	2	yes	—
127	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
128	C₈	—	404	2	no	—
129	C₂²	—	101	2	no	—
130	C₂²	—	404	2	no	—
131	C₂²	—	404	4	no	—
132	C₂²	—	202	2	no	—
133	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
134	C₄	—	4	2	no	—
135	C₄	—	101	1	no	—
136	C₄	—	101	1	no	—
137	C₄	—	202	8	no	—
138	C₄	—	202	2	no	—
139	C₄	#15692320	1	4	yes	complemented
140	C₄	D₄×#87938	1	2	yes	—
141	C₂	—	808	2	no	—
142	C₂	—	202	2	no	—
143	C₂	#20266129	1	1	yes	characteristic
144	C₂	C₂×#15692320	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	C₄⋊Q₈×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	#20265728	C₂
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	C₄⋊Q₈	#87938
hypocenter	C₁₀₁	#1962
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20266181	C₂	semidirect
2. C₂×#15692320	C₂	semidirect
3. C₄²×#87938	C₂	semidirect
11. #15692252	C₄	semidirect
12. #15692244	C₄	semidirect
16. C₄⋊Q₈×C₁₀₁	C₄	semidirect
22. C₄×#87938	D₄	semidirect
63. C₄×C₁₀₁	#104	semidirect
73. C₁₀₁	#1962	semidirect
80. C₄⋊Q₈	#87938	semidirect
139. C₄	#15692320	semidirect

Maximal subgroups

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

subgroup	index
#20266181	2
C₂×#15692320	2
C₄²×#87938	2
#20265728	2
#1962	101