Group #19152325 — subgroups

Subgroups — #19152325

Order 5696.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	1	yes	maximal, characteristic
2	#15668011	C₂	1	2	yes	maximal
3	#15668003	C₂	1	1	yes	maximal, characteristic
4	#15668002	C₂	1	2	yes	maximal
5	#15668002	C₂	1	2	yes	maximal
6	C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
8	#15667977	C₂	1	2	yes	maximal
9	#15667977	C₂	1	1	yes	maximal, characteristic
10	#15667977	C₂	1	1	yes	maximal, characteristic
11	#125×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₇₈	C₂²	1	2	yes	—
13	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₈₉	C₂²	1	2	yes	—
18	#13083553	C₂²	1	1	yes	characteristic
19	#13083553	C₂²	1	2	yes	—
20	#13083553	C₂²	1	2	yes	—
21	#13083553	C₂²	1	2	yes	—
22	#13083553	C₂²	1	1	yes	characteristic
23	#13083553	C₂²	1	2	yes	—
24	#13083553	C₂²	1	2	yes	—
25	C₄×Dic₈₉	C₂²	1	2	yes	—
26	C₄×Dic₈₉	C₂²	1	2	yes	—
27	C₄×Dic₈₉	C₂²	1	2	yes	—
28	C₄×Dic₈₉	C₂²	1	1	yes	characteristic
29	C₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
31	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂³	1	2	yes	—
38	C₂×C₄×C₈₉	C₂³	1	2	yes	—
39	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
41	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₈₉	C₂³	1	2	yes	—
43	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₈₉	C₂³	1	2	yes	—
45	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
46	Dic₁₇₈	—	2	4	no	—
47	Dic₁₇₈	—	2	2	no	—
48	Dic₁₇₈	Q₈	1	4	yes	complemented
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	C₄×C₈₉	C₂×Q₈	1	2	yes	—
55	Dic₈₉	—	2	2	no	—
56	Dic₈₉	—	2	1	no	—
57	Dic₈₉	—	2	1	no	—
58	Dic₈₉	—	2	1	no	—
59	Dic₈₉	C₂×Q₈	1	4	yes	—
60	Dic₈₉	#41	1	2	yes	—
61	C₂×C₈₉	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₈₉	C₂×#41	1	1	yes	characteristic
63	C₂×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
64	C₈₉	#557	1	1	yes	characteristic, complemented, Sylow
65	#557	—	89	1	no	maximal, Sylow
66	C₄⋊Q₈	—	89	2	no	—
67	C₄⋊Q₈	—	89	1	no	—
68	#125	—	89	2	no	—
69	#125	—	89	2	no	—
70	#125	—	89	1	no	—
71	#125	D₈₉	1	1	yes	characteristic
72	C₄×Q₈	—	89	1	no	—
73	C₄×Q₈	—	89	2	no	—
74	C₄×Q₈	—	89	1	no	—
75	C₄×Q₈	—	89	1	no	—
76	C₄×Q₈	—	89	1	no	—
77	C₂×Q₈	—	89	1	no	—
78	C₂×Q₈	—	89	2	no	—
79	C₂×Q₈	—	89	1	no	—
80	C₄⋊C₄	—	89	1	no	—
81	C₄⋊C₄	—	89	2	no	—
82	C₄⋊C₄	—	89	2	no	—
83	C₄⋊C₄	—	89	1	no	—
84	C₄⋊C₄	—	89	1	no	—
85	C₄⋊C₄	—	89	2	no	—
86	C₄⋊C₄	—	89	2	no	—
87	C₄⋊C₄	—	89	1	no	—
88	C₄⋊C₄	—	89	2	no	—
89	C₄⋊C₄	—	89	2	no	—
90	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
91	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
92	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
93	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
94	C₄²	—	89	2	no	—
95	C₄²	—	89	2	no	—
96	C₄²	—	89	1	no	—
97	C₄²	—	89	2	no	—
98	C₄²	—	89	1	no	—
99	C₄²	C₂×D₈₉	1	1	yes	characteristic
100	Q₈	—	89	4	no	—
101	Q₈	—	178	4	no	—
102	Q₈	—	178	2	no	—
103	C₂×C₄	—	89	2	no	—
104	C₂×C₄	—	89	1	no	—
105	C₂×C₄	—	89	1	no	—
106	C₂×C₄	—	89	1	no	—
107	C₂×C₄	—	89	1	no	—
108	C₂×C₄	—	89	2	no	—
109	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₈₉	1	2	yes	—
111	C₂×C₄	C₂²×D₈₉	1	2	yes	—
112	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
114	C₂²	C₂³×D₈₉	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	89	4	no	—
120	C₄	—	89	2	no	—
121	C₄	—	178	1	no	—
122	C₄	—	178	2	no	—
123	C₄	—	178	1	no	—
124	C₄	—	178	1	no	—
125	C₄	Q₈×D₈₉	1	2	yes	—
126	C₂	C₂×Q₈×D₈₉	1	1	yes	characteristic
127	C₂	#41×D₈₉	1	1	yes	characteristic
128	C₂	#15668059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668011

C₂

yes

maximal, characteristic

#15668011

C₂

yes

maximal

#15668003

C₂

yes

maximal, characteristic

#15668002

C₂

yes

maximal

#15668002

C₂

yes

maximal

C₄×Dic₁₇₈

C₂

yes

maximal, characteristic

C₄×Dic₁₇₈

C₂

yes

maximal, characteristic

#15667977

C₂

yes

maximal

#15667977

C₂

yes

maximal, characteristic

#15667977

C₂

yes

maximal, characteristic

#125×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₁₇₈

C₂²

yes

—

C₂×Dic₁₇₈

C₂²

yes

characteristic

C₂×Dic₁₇₈

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

—

#13083553

C₂²

yes

characteristic

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

characteristic

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

characteristic

C₄×Dic₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

characteristic

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

Q₈

yes

complemented

C₂²×C₈₉

C₂⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂×Q₈

yes

—

Dic₈₉

#41

yes

—

C₂×C₈₉

C₂²×Q₈

yes

characteristic

C₂×C₈₉

C₂×#41

yes

characteristic

C₂×C₈₉

2₋¹⁺⁴

yes

characteristic

C₈₉

#557

yes

characteristic, complemented, Sylow

#557

—

maximal, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

D₈₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

—

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

C₂×D₈₉

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₈₉

yes

characteristic

100

Q₈

—

101

Q₈

—

178

—

102

Q₈

—

178

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×D₈₉

yes

characteristic

110

C₂×C₄

C₂²×D₈₉

yes

—

111

C₂×C₄

C₂²×D₈₉

yes

—

112

C₂×C₄

C₂²×D₈₉

yes

characteristic

113

C₂×C₄

C₂²×D₈₉

yes

characteristic

114

C₂²

C₂³×D₈₉

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

178

—

122

C₄

—

178

—

123

C₄

—

178

—

124

C₄

—

178

—

125

C₄

Q₈×D₈₉

yes

—

126

C₂

C₂×Q₈×D₈₉

yes

characteristic

127

C₂

#41×D₈₉

yes

characteristic

128

C₂

#15668059

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#125×C₈₉	C₂
hypercenter	#125	D₈₉
hypocenter	C₈₉	#557
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

#125×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
48. Dic₁₇₈	Q₈	semidirect
64. C₈₉	#557	semidirect

H ≅ G/N

type

48. Dic₁₇₈

Q₈

semidirect

64. C₈₉

#557

semidirect

Subgroups — #19152325

Conjugacy classes of subgroups

Complements