Group #41012122 — subgroups

Subgroups — #41012122

Order 27008.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×D₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#27213202	C₂	1	2	yes	maximal
3	#27212895	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212894	C₂	1	2	yes	maximal, complemented
5	#333×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₂₁₁	C₄	1	2	yes	—
7	#20269977	C₂²	1	1	yes	characteristic
8	#20269976	—	2	2	no	—
9	#20269975	C₄	1	2	yes	—
10	C₄⋊C₄×D₂₁₁	—	2	2	no	—
11	C₄²×D₂₁₁	C₂²	1	1	yes	characteristic
12	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	2	yes	—
13	(C₂×D₂₁₁)⋊C₈	—	2	2	no	—
14	#20269887	—	2	2	no	—
15	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₁₁	C₂²	1	2	yes	—
17	C₂×Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₄₂₂	C₂×C₄	1	2	yes	—
19	Q₈×D₂₁₁	—	4	4	no	—
20	C₂×C₄×D₂₁₁	—	2	2	no	—
21	C₂×C₄×D₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₂₁₁	D₄	1	2	yes	—
23	C₂×C₂₁₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₂₁₁	—	2	2	no	—
25	Dic₂₁₁⋊C₄	—	2	4	no	—
26	C₄×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₂₁₁	D₄	1	2	yes	—
28	C₂×Q₈×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₁₁	—	2	2	no	—
30	C₄⋊C₄×C₂₁₁	—	2	2	no	—
31	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
32	C₂²×D₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₄₂₂	—	4	4	no	—
34	Dic₄₂₂	—	4	2	no	—
35	C₂×Dic₂₁₁	—	2	2	no	—
36	C₂×Dic₂₁₁	C₂²⋊C₄	1	2	yes	—
37	C₂×Dic₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₂₁₁	—	2	2	no	—
39	C₄×D₂₁₁	—	2	2	no	—
40	C₄×D₂₁₁	—	2	2	no	—
41	C₄×D₂₁₁	—	4	4	no	—
42	C₂₁₁⋊C₈	—	4	2	no	—
43	Q₈×C₂₁₁	—	4	2	no	—
44	C₂×C₄×C₂₁₁	—	2	2	no	—
45	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
47	C₈×C₂₁₁	—	4	2	no	—
48	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₂₁₁	—	2	2	no	—
50	C₂×D₂₁₁	#100	1	2	yes	—
51	C₄×C₂₁₁	—	2	1	no	—
52	C₄×C₂₁₁	—	4	2	no	—
53	C₄×C₂₁₁	#137	1	2	yes	—
54	C₄×C₂₁₁	#137	1	2	yes	—
55	Dic₂₁₁	—	2	1	no	—
56	Dic₂₁₁	—	2	2	no	—
57	Dic₂₁₁	—	4	2	no	—
58	C₂×C₂₁₁	C₂×#100	1	1	yes	characteristic
59	C₂×C₂₁₁	#419	1	2	yes	—
60	D₂₁₁	—	2	2	no	—
61	C₂₁₁	#1558	1	1	yes	characteristic, complemented, Sylow
62	#1558	—	211	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	211	1	no	—
64	#432	—	211	2	no	—
65	#333	—	211	1	no	—
66	#333	—	211	2	no	—
67	#333	D₂₁₁	1	1	yes	characteristic, complemented
68	C₂²×Q₈	—	211	2	no	—
69	C₄⋊Q₈	—	422	2	no	—
70	C₄⋊Q₈	—	211	2	no	—
71	C₄⋊Q₈	—	211	1	no	—
72	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	422	2	no	—
74	C₂×C₄²	—	211	1	no	—
75	C₄⋊C₈	—	211	2	no	—
76	C₄⋊C₈	C₂×D₂₁₁	1	2	yes	—
77	C₂²⋊C₈	—	422	2	no	—
78	#97	—	422	2	no	—
79	C₂×Q₈	—	844	4	no	—
80	C₂×Q₈	—	422	2	no	—
81	C₂×Q₈	—	211	2	no	—
82	C₂×Q₈	C₄×D₂₁₁	1	2	yes	—
83	C₂²×C₄	—	422	2	no	—
84	C₂²×C₄	—	211	1	no	—
85	C₂²×C₄	—	211	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	422	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	422	4	no	—
90	C₄⋊C₄	—	422	2	no	—
91	C₄²	—	211	1	no	—
92	C₄²	—	211	2	no	—
93	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
94	C₂³	—	211	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	844	2	no	—
97	Q₈	—	844	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	844	4	no	—
100	C₂×C₄	—	422	2	no	—
101	C₂×C₄	—	422	2	no	—
102	C₂×C₄	—	422	2	no	—
103	C₂×C₄	—	422	2	no	—
104	C₂×C₄	—	211	2	no	—
105	C₂×C₄	—	211	1	no	—
106	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	844	2	no	—
110	C₂²	—	422	2	no	—
111	C₂²	—	211	2	no	—
112	C₂²	C₂²⋊C₄×D₂₁₁	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	844	2	no	—
116	C₄	—	422	1	no	—
117	C₄	—	422	2	no	—
118	C₄	#20270006	1	2	yes	—
119	C₄	#20270009	1	2	yes	—
120	C₂	—	422	2	no	—
121	C₂	#100×D₂₁₁	1	1	yes	characteristic
122	C₂	#27213153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₂₁₁

C₂

yes

maximal, characteristic

#27213202

C₂

yes

maximal

#27212895

C₂

yes

maximal, characteristic, complemented

#27212894

C₂

yes

maximal, complemented

#333×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₂₁₁

C₄

yes

—

#20269977

C₂²

yes

characteristic

#20269976

—

#20269975

C₄

yes

—

C₄⋊C₄×D₂₁₁

—

C₄²×D₂₁₁

C₂²

yes

characteristic

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

—

(C₂×D₂₁₁)⋊C₈

—

#20269887

—

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

C₂²

yes

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

Q₈×D₂₁₁

—

C₂×C₄×D₂₁₁

—

C₂×C₄×D₂₁₁

C₂×C₄

yes

characteristic

C₂×C₄×D₂₁₁

D₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

Dic₂₁₁⋊C₄

—

C₄×Dic₂₁₁

C₂×C₄

yes

characteristic

C₄×Dic₂₁₁

D₄

yes

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂²×D₂₁₁

C₂²⋊C₄

yes

characteristic

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₂²⋊C₄

yes

—

C₂×Dic₂₁₁

C₂²⋊C₄

yes

characteristic

C₄×D₂₁₁

—

C₄×D₂₁₁

—

C₄×D₂₁₁

—

C₄×D₂₁₁

—

C₂₁₁⋊C₈

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₁₁

—

C₂×D₂₁₁

#100

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#137

yes

—

C₄×C₂₁₁

#137

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

C₂×#100

yes

characteristic

C₂×C₂₁₁

#419

yes

—

D₂₁₁

—

C₂₁₁

#1558

yes

characteristic, complemented, Sylow

#1558

—

211

maximal, Sylow

C₂×C₄⋊Q₈

—

211

—

#432

—

211

—

#333

—

211

—

#333

—

211

—

#333

D₂₁₁

yes

characteristic, complemented

C₂²×Q₈

—

211

—

C₄⋊Q₈

—

422

—

C₄⋊Q₈

—

211

—

C₄⋊Q₈

—

211

—

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

422

—

C₂×C₄²

—

211

—

C₄⋊C₈

—

211

—

C₄⋊C₈

C₂×D₂₁₁

yes

—

C₂²⋊C₈

—

422

—

#97

—

422

—

C₂×Q₈

—

844

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂×Q₈

C₄×D₂₁₁

yes

—

C₂²×C₄

—

422

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂×C₈

—

C₂×C₈

—

422

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄²

—

211

—

C₄²

—

211

—

C₄²

C₂²×D₂₁₁

yes

characteristic

C₂³

—

211

—

Q₈

—

Q₈

—

844

—

Q₈

—

844

—

C₂×C₄

—

C₂×C₄

—

844

—

100

C₂×C₄

—

422

—

101

C₂×C₄

—

422

—

102

C₂×C₄

—

422

—

103

C₂×C₄

—

422

—

104

C₂×C₄

—

211

—

105

C₂×C₄

—

211

—

106

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

107

C₂×C₄

D₄×D₂₁₁

yes

—

108

C₈

—

109

C₈

—

844

—

110

C₂²

—

422

—

111

C₂²

—

211

—

112

C₂²

C₂²⋊C₄×D₂₁₁

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

844

—

116

C₄

—

422

—

117

C₄

—

422

—

118

C₄

#20270006

yes

—

119

C₄

#20270009

yes

—

120

C₂

—

422

—

121

C₂

#100×D₂₁₁

yes

characteristic

122

C₂

#27213153

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#333×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₁₁	C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	#333	D₂₁₁
hypocenter	C₂₁₁	#1558
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#333×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₁₁

C₂²⋊C₄

cosocle

C₄²×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #27212895	C₂	semidirect
4. #27212894	C₂	semidirect
5. #333×C₂₁₁	C₂	semidirect
61. C₂₁₁	#1558	semidirect
67. #333	D₂₁₁	semidirect

H ≅ G/N

type

3. #27212895

C₂

semidirect

4. #27212894

C₂

semidirect

5. #333×C₂₁₁

C₂

semidirect

61. C₂₁₁

#1558

semidirect

67. #333

D₂₁₁

semidirect

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

subgroup	index
C₄⋊Q₈×D₂₁₁	2
#27213202	2
#27212895	2
#27212894	2
#333×C₂₁₁	2
#1558	211

subgroup

index

C₄⋊Q₈×D₂₁₁

#333×C₂₁₁

211

Subgroups — #41012122

Conjugacy classes of subgroups

Complements

Maximal subgroups