Group #19152227 — subgroups

Subgroups — #19152227

Order 5696.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#15668011	C₂	1	2	yes	maximal, complemented
2	#15668010	C₂	1	2	yes	maximal, complemented
3	#15668002	C₂	1	2	yes	maximal, complemented
4	#15668002	C₂	1	2	yes	maximal, complemented
5	#15667991	C₂	1	1	yes	maximal, characteristic
6	#15667989	C₂	1	2	yes	maximal
7	#15667968	C₂	1	2	yes	maximal
8	C₂×#13083553	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₁₇₈	C₂²	1	2	yes	—
11	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic
12	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₈₉	C₂²	1	2	yes	—
15	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
16	#13083553	C₂²	1	2	yes	—
17	#13083553	C₂²	1	2	yes	—
18	#13083553	C₂²	1	2	yes	—
19	#13083553	C₂²	1	2	yes	—
20	#13083553	C₂²	1	2	yes	—
21	#13083553	C₂²	1	2	yes	—
22	#13083553	C₂²	1	2	yes	—
23	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
24	C₄×Dic₈₉	C₂²	1	2	yes	—
25	C₄×Dic₈₉	C₂²	1	2	yes	—
26	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
31	C₂³×C₈₉	C₂³	1	1	yes	characteristic
32	Q₈×C₈₉	—	4	1	no	—
33	C₂×C₄×C₈₉	—	4	1	no	—
34	C₂×C₄×C₈₉	C₂³	1	2	yes	—
35	C₂×C₄×C₈₉	C₂³	1	2	yes	—
36	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×Dic₈₉	—	4	1	no	—
39	C₂×Dic₈₉	Q₈	1	4	yes	complemented
40	C₂×Dic₈₉	C₂³	1	2	yes	—
41	C₂×Dic₈₉	C₂³	1	2	yes	—
42	C₂×Dic₈₉	C₂³	1	2	yes	—
43	C₂×Dic₈₉	C₂³	1	2	yes	—
44	Dic₁₇₈	—	4	1	no	—
45	Dic₁₇₈	—	4	2	no	—
46	C₂²×C₈₉	—	2	1	no	—
47	C₂²×C₈₉	—	2	1	no	—
48	C₂²×C₈₉	C₂×Q₈	1	2	yes	—
49	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₄×C₈₉	—	2	1	no	—
51	C₄×C₈₉	—	2	2	no	—
52	C₄×C₈₉	—	2	1	no	—
53	C₄×C₈₉	—	2	2	no	—
54	Dic₈₉	—	2	2	no	—
55	Dic₈₉	—	2	2	no	—
56	Dic₈₉	—	2	2	no	—
57	Dic₈₉	C₂×Q₈	1	4	yes	—
58	C₂×C₈₉	—	2	2	no	—
59	C₂×C₈₉	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
62	C₈₉	#544	1	1	yes	characteristic, complemented, Sylow
63	#544	—	89	1	no	maximal, Sylow
64	C₄⋊Q₈	—	89	2	no	—
65	C₄⋊Q₈	—	89	2	no	—
66	#125	—	89	2	no	—
67	#125	—	89	2	no	—
68	C₂²⋊Q₈	—	89	1	no	—
69	C₂²⋊Q₈	—	89	2	no	—
70	C₂²⋊Q₈	D₈₉	1	1	yes	characteristic
71	#117	—	89	2	no	—
72	C₂×C₄⋊C₄	—	89	1	no	—
73	C₂×Q₈	—	89	1	no	—
74	C₂×Q₈	—	89	2	no	—
75	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
76	C₂²×C₄	—	89	1	no	—
77	C₂²×C₄	—	89	1	no	—
78	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
79	C₄⋊C₄	—	89	2	no	—
80	C₄⋊C₄	—	89	2	no	—
81	C₄⋊C₄	—	89	2	no	—
82	C₄⋊C₄	—	89	2	no	—
83	C₄⋊C₄	—	89	2	no	—
84	C₄⋊C₄	—	89	1	no	—
85	C₄⋊C₄	—	89	2	no	—
86	C₄⋊C₄	—	89	2	no	—
87	C₄⋊C₄	—	89	2	no	—
88	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
90	C₂²⋊C₄	—	89	2	no	—
91	C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
92	C₄²	—	89	2	no	—
93	C₄²	—	89	2	no	—
94	C₂³	C₂²×D₈₉	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	356	1	no	—
97	Q₈	—	356	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	89	4	no	—
100	C₂×C₄	—	89	2	no	—
101	C₂×C₄	—	89	2	no	—
102	C₂×C₄	—	89	2	no	—
103	C₂×C₄	—	89	2	no	—
104	C₂×C₄	—	356	1	no	—
105	C₂×C₄	C₂²×D₈₉	1	2	yes	—
106	C₂×C₄	C₂²×D₈₉	1	2	yes	—
107	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₈₉	1	2	yes	—
112	C₂²	C₂³×D₈₉	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	89	4	no	—
118	C₄	—	178	2	no	—
119	C₄	—	178	2	no	—
120	C₄	—	178	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₈₉	1	1	yes	characteristic
123	C₂	#15668057	1	1	yes	characteristic
124	C₂	#15668059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668011

C₂

yes

maximal, complemented

#15668010

C₂

yes

maximal, complemented

#15668002

C₂

yes

maximal, complemented

#15668002

C₂

yes

maximal, complemented

#15667991

C₂

yes

maximal, characteristic

#15667989

C₂

yes

maximal

#15667968

C₂

yes

maximal

C₂×#13083553

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₁₇₈

C₂²

yes

—

C₂×Dic₁₇₈

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

characteristic

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

C₂²⋊Dic₈₉

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

—

C₂×Q₈×C₈₉

C₂²

yes

characteristic

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂³×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

Q₈

yes

complemented

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×Q₈

yes

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂×Q₈

yes

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×Q₈

yes

characteristic

C₂×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂×C₈₉

2₋¹⁺⁴

yes

characteristic

C₈₉

#544

yes

characteristic, complemented, Sylow

#544

—

maximal, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₈₉

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

C₂×D₈₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₈₉

yes

—

C₄²

—

C₄²

—

C₂³

C₂²×D₈₉

yes

characteristic

Q₈

—

Q₈

—

356

—

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

356

—

105

C₂×C₄

C₂²×D₈₉

yes

—

106

C₂×C₄

C₂²×D₈₉

yes

—

107

C₂×C₄

C₂²×D₈₉

yes

characteristic

108

C₂×C₄

C₂²×D₈₉

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₈₉

yes

—

112

C₂²

C₂³×D₈₉

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

178

—

119

C₄

—

178

—

120

C₄

—

178

—

121

C₂

—

122

C₂

C₂×Q₈×D₈₉

yes

characteristic

123

C₂

#15668057

yes

characteristic

124

C₂

#15668059

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	C₂²⋊Q₈×C₈₉	C₂
hypercenter	C₂²⋊Q₈	D₈₉
hypocenter	C₈₉	#544
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

C₂²⋊Q₈×C₈₉

C₂

hypercenter

C₂²⋊Q₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #15668011	C₂	semidirect
2. #15668010	C₂	semidirect
3. #15668002	C₂	semidirect
4. #15668002	C₂	semidirect
39. C₂×Dic₈₉	Q₈	semidirect
62. C₈₉	#544	semidirect

H ≅ G/N

type

1. #15668011

C₂

semidirect

2. #15668010

C₂

semidirect

3. #15668002

C₂

semidirect

4. #15668002

C₂

semidirect

39. C₂×Dic₈₉

Q₈

semidirect

62. C₈₉

#544

semidirect

Subgroups — #19152227

Conjugacy classes of subgroups

Complements