Group #19151502 — subgroups

Subgroups — #19151502

Order 5696.

186 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂²×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13083553	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#15667919	C₂	1	1	yes	maximal, characteristic
5	#15667919	C₂	1	1	yes	maximal, characteristic
6	#15667919	C₂	1	1	yes	maximal, characteristic
7	#95×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₇₈	—	2	2	no	—
9	C₂×Dic₁₇₈	—	2	2	no	—
10	C₂×Dic₁₇₈	C₄	1	4	yes	complemented
11	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
12	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
15	#13083553	—	2	2	no	—
16	C₄×Dic₈₉	—	2	2	no	—
17	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
20	C₂³×C₈₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₈₉	—	2	1	no	—
22	C₂×C₄×C₈₉	—	2	1	no	—
23	C₂×C₄×C₈₉	—	2	1	no	—
24	C₂×C₄×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈₉	—	2	1	no	—
26	C₂×C₄×C₈₉	—	2	1	no	—
27	C₂×C₄×C₈₉	—	2	1	no	—
28	C₂×C₄×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₈₉	Q₈	1	2	yes	complemented
30	C₂×Dic₈₉	—	2	1	no	—
31	C₂×Dic₈₉	—	2	1	no	—
32	C₂×Dic₈₉	—	2	1	no	—
33	C₂×Dic₈₉	—	2	1	no	—
34	C₂×Dic₈₉	—	2	1	no	—
35	C₂×Dic₈₉	—	2	1	no	—
36	C₂×Dic₈₉	—	2	1	no	—
37	C₂×Dic₈₉	C₂×C₄	1	2	yes	—
38	C₂×Dic₈₉	C₂×C₄	1	2	yes	—
39	C₂×Dic₈₉	D₄	1	2	yes	—
40	C₂×Dic₈₉	D₄	1	2	yes	—
41	C₂×Dic₈₉	Q₈	1	2	yes	complemented
42	Dic₁₇₈	—	2	8	no	—
43	C₂²×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₈₉	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₈₉	#41	1	1	yes	characteristic
49	C₂²×C₈₉	#41	1	1	yes	characteristic
50	C₄×C₈₉	—	2	2	no	—
51	C₄×C₈₉	—	2	2	no	—
52	C₄×C₈₉	—	4	1	no	—
53	Dic₈₉	—	2	2	no	—
54	Dic₈₉	—	2	2	no	—
55	Dic₈₉	—	4	1	no	—
56	Dic₈₉	C₂²⋊C₄	1	4	yes	—
57	C₂×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈₉	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₈₉	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₈₉	#124	1	1	yes	characteristic
63	C₂×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₈₉	#391	1	1	yes	characteristic, complemented, Sylow
65	#391	—	89	1	no	maximal, Sylow
66	C₂²×Q₈	—	89	1	no	—
67	C₂×C₄⋊C₄	—	89	1	no	—
68	C₂×C₄²	—	89	1	no	—
69	#95	—	89	1	no	—
70	#95	—	89	1	no	—
71	#95	—	89	1	no	—
72	#95	D₈₉	1	1	yes	characteristic
73	C₂×Q₈	—	89	4	no	—
74	C₂×Q₈	—	178	2	no	—
75	C₂×Q₈	—	178	2	no	—
76	C₂²×C₄	—	89	1	no	—
77	C₂²×C₄	—	89	1	no	—
78	C₂²×C₄	—	89	1	no	—
79	C₂²×C₄	—	89	1	no	—
80	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
83	C₄⋊C₄	—	178	2	no	—
84	C₄²	—	178	2	no	—
85	C₂³	C₂²×D₈₉	1	1	yes	characteristic
86	Q₈	—	178	8	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	89	2	no	—
95	C₂×C₄	—	89	2	no	—
96	C₂×C₄	—	89	2	no	—
97	C₂×C₄	—	89	2	no	—
98	C₂×C₄	—	89	2	no	—
99	C₂×C₄	—	178	1	no	—
100	C₂×C₄	—	178	1	no	—
101	C₂×C₄	—	178	1	no	—
102	C₂×C₄	—	178	1	no	—
103	C₂×C₄	—	178	1	no	—
104	C₂×C₄	—	178	1	no	—
105	C₂×C₄	—	178	1	no	—
106	C₂×C₄	Dic₁₇₈	1	2	yes	complemented
107	C₂×C₄	C₄×D₈₉	1	2	yes	—
108	C₂²	C₂×C₄×D₈₉	1	1	yes	characteristic
109	C₂²	D₄×D₈₉	1	1	yes	characteristic
110	C₂²	D₄×D₈₉	1	1	yes	characteristic
111	C₂²	Q₈×D₈₉	1	1	yes	characteristic
112	C₂²	C₂×Dic₁₇₈	1	1	yes	characteristic
113	C₂²	#13083573	1	1	yes	characteristic
114	C₂²	#13083574	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	1	no	—
118	C₄	—	89	4	no	—
119	C₄	—	356	1	no	—
120	C₄	—	178	2	no	—
121	C₄	—	178	2	no	—
122	C₂	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
123	C₂	#15667977	1	1	yes	characteristic
124	C₂	C₄×Dic₁₇₈	1	1	yes	characteristic
125	C₂	#15667989	1	1	yes	characteristic
126	C₂	#15667990	1	1	yes	characteristic
127	C₂	#15667998	1	1	yes	characteristic
128	C₂	#15668011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₈

C₂

yes

maximal, characteristic

C₂×#13083553

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₉

C₂

yes

maximal, characteristic

#15667919

C₂

yes

maximal, characteristic

#15667919

C₂

yes

maximal, characteristic

#15667919

C₂

yes

maximal, characteristic

#95×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₄

yes

complemented

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂²

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₂³×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄×C₈₉

Q₈

yes

complemented

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×C₄

yes

—

C₂×Dic₈₉

C₂×C₄

yes

—

C₂×Dic₈₉

D₄

yes

—

C₂×Dic₈₉

D₄

yes

—

C₂×Dic₈₉

Q₈

yes

complemented

Dic₁₇₈

—

C₂²×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₈₉

C₂×Q₈

yes

characteristic

C₂²×C₈₉

C₂×Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂²⋊C₄

yes

—

C₂×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈₉

C₄×Q₈

yes

characteristic

C₂×C₈₉

C₄×Q₈

yes

characteristic

C₂×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₈₉

#124

yes

characteristic

C₂×C₈₉

C₄⋊Q₈

yes

characteristic

C₈₉

#391

yes

characteristic, complemented, Sylow

#391

—

maximal, Sylow

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₂³

C₂²×D₈₉

yes

characteristic

Q₈

—

178

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

100

C₂×C₄

—

178

—

101

C₂×C₄

—

178

—

102

C₂×C₄

—

178

—

103

C₂×C₄

—

178

—

104

C₂×C₄

—

178

—

105

C₂×C₄

—

178

—

106

C₂×C₄

Dic₁₇₈

yes

complemented

107

C₂×C₄

C₄×D₈₉

yes

—

108

C₂²

C₂×C₄×D₈₉

yes

characteristic

109

C₂²

D₄×D₈₉

yes

characteristic

110

C₂²

D₄×D₈₉

yes

characteristic

111

C₂²

Q₈×D₈₉

yes

characteristic

112

C₂²

C₂×Dic₁₇₈

yes

characteristic

113

C₂²

#13083573

yes

characteristic

114

C₂²

#13083574

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

356

—

120

C₄

—

178

—

121

C₄

—

178

—

122

C₂

C₂²⋊C₄×D₈₉

yes

characteristic

123

C₂

#15667977

yes

characteristic

124

C₂

C₄×Dic₁₇₈

yes

characteristic

125

C₂

#15667989

yes

characteristic

126

C₂

#15667990

yes

characteristic

127

C₂

#15667998

yes

characteristic

128

C₂

#15668011

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄
Fitting	#95×C₈₉	C₂
socle	C₂³×C₈₉	C₂³
hypercenter	#95	D₈₉
hypocenter	C₈₉	#391
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂²×C₄

Fitting

#95×C₈₉

C₂

socle

C₂³×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
10. C₂×Dic₁₇₈	C₄	semidirect
29. C₂×C₄×C₈₉	Q₈	semidirect
41. C₂×Dic₈₉	Q₈	semidirect
64. C₈₉	#391	semidirect
106. C₂×C₄	Dic₁₇₈	semidirect

H ≅ G/N

type

10. C₂×Dic₁₇₈

C₄

semidirect

29. C₂×C₄×C₈₉

Q₈

semidirect

41. C₂×Dic₈₉

Q₈

semidirect

64. C₈₉

#391

semidirect

106. C₂×C₄

Dic₁₇₈

semidirect

Subgroups — #19151502

Conjugacy classes of subgroups

Complements