Group #19152173 — subgroups

Subgroups — #19152173

Order 5696.

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

normal subgroups only

This group has 270 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	C₂×#13083575	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#15668010	C₂	1	1	yes	maximal, characteristic, complemented
4	#15668001	C₂	1	1	yes	maximal, characteristic, complemented
5	#15668000	C₂	1	1	yes	maximal, characteristic, complemented
6	#15667997	C₂	1	4	yes	maximal, complemented
7	#15667990	C₂	1	4	yes	maximal, complemented
8	#15667969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#13083575	—	2	4	no	—
11	C₂×Dic₁₇₈	C₂²	1	2	yes	—
12	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	Q₈×D₈₉	—	2	4	no	—
14	C₂×D₃₅₆	C₂²	1	2	yes	—
15	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₈₉	C₂²	1	2	yes	—
17	C₂×C₄×D₈₉	C₂²	1	2	yes	—
18	C₂×C₄×D₈₉	C₂²	1	1	yes	characteristic
19	#13083553	C₂²	1	2	yes	—
20	#13083553	C₂²	1	4	yes	—
21	#13083551	C₂²	1	2	yes	—
22	#13083551	C₂²	1	4	yes	—
23	#13083551	C₂²	1	4	yes	—
24	C₄×Dic₈₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
28	C₄²×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×D₈₉	C₂³	1	2	yes	—
30	C₂²×D₈₉	C₂³	1	1	yes	characteristic
31	Q₈×C₈₉	—	2	2	no	—
32	Q₈×C₈₉	—	2	2	no	—
33	C₂×C₄×C₈₉	C₂³	1	2	yes	—
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×Dic₈₉	C₂³	1	2	yes	—
38	C₂×Dic₈₉	C₂³	1	2	yes	—
39	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
40	D₃₅₆	—	2	4	no	—
41	D₃₅₆	—	2	2	no	—
42	C₄×D₈₉	—	2	4	no	—
43	C₄×D₈₉	—	2	4	no	—
44	C₄×D₈₉	D₄	1	4	yes	—
45	Dic₁₇₈	—	2	2	no	—
46	Dic₁₇₈	—	2	4	no	—
47	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂×D₈₉	—	2	2	no	—
49	C₂×D₈₉	—	2	2	no	—
50	C₂×D₈₉	—	2	4	no	—
51	C₂×D₈₉	C₂×D₄	1	2	yes	—
52	C₄×C₈₉	—	2	2	no	—
53	C₄×C₈₉	—	2	2	no	—
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	C₂×D₄	1	2	yes	—
56	Dic₈₉	—	2	2	no	—
57	Dic₈₉	—	2	2	no	—
58	Dic₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₈₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₈₉	2₋¹⁺⁴	1	2	yes	—
61	D₈₉	—	2	2	no	—
62	D₈₉	—	4	2	no	—
63	C₈₉	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	89	1	no	maximal, Sylow
65	C₂×#41	—	89	1	no	—
66	C₂²×Q₈	—	89	1	no	—
67	C₄⋊Q₈	—	89	1	no	—
68	C₄⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
69	#124	—	89	1	no	—
70	#124	—	89	1	no	—
71	#123	—	89	4	no	—
72	C₂²⋊Q₈	—	89	4	no	—
73	#117	—	89	1	no	—
74	#41	—	178	4	no	—
75	C₂×Q₈	—	89	2	no	—
76	C₂×Q₈	—	89	1	no	—
77	C₂×Q₈	—	178	4	no	—
78	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
80	C₂×D₄	—	89	2	no	—
81	C₂×D₄	—	89	1	no	—
82	C₂²×C₄	—	89	1	no	—
83	C₂²×C₄	—	89	2	no	—
84	C₂²×C₄	—	89	2	no	—
85	C₄⋊C₄	—	89	2	no	—
86	C₄⋊C₄	—	89	4	no	—
87	C₄⋊C₄	C₂×D₈₉	1	4	yes	—
88	C₂²⋊C₄	—	89	2	no	—
89	C₂²⋊C₄	—	89	4	no	—
90	C₂²⋊C₄	—	89	4	no	—
91	C₄²	—	89	1	no	—
92	C₄²	C₂×D₈₉	1	1	yes	characteristic
93	C₂³	—	89	1	no	—
94	C₂³	—	89	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	178	4	no	—
98	Q₈	—	178	2	no	—
99	D₄	—	178	4	no	—
100	D₄	—	178	2	no	—
101	C₂×C₄	—	89	1	no	—
102	C₂×C₄	—	89	4	no	—
103	C₂×C₄	—	89	2	no	—
104	C₂×C₄	—	89	2	no	—
105	C₂×C₄	—	178	4	no	—
106	C₂×C₄	—	178	4	no	—
107	C₂×C₄	C₂²×D₈₉	1	2	yes	—
108	C₂×C₄	C₂²×D₈₉	1	2	yes	—
109	C₂×C₄	C₂²×D₈₉	1	2	yes	—
110	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
111	C₂²	—	89	2	no	—
112	C₂²	—	178	2	no	—
113	C₂²	—	178	4	no	—
114	C₂²	—	178	2	no	—
115	C₂²	C₂³×D₈₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	89	2	no	—
120	C₄	—	178	2	no	—
121	C₄	—	178	2	no	—
122	C₄	D₄×D₈₉	1	2	yes	—
123	C₂	—	356	2	no	—
124	C₂	—	178	2	no	—
125	C₂	C₂×D₄×D₈₉	1	1	yes	characteristic
126	C₂	#15668060	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13083575

C₂

yes

maximal, characteristic

C₂×Q₈×D₈₉

C₂

yes

maximal, characteristic, complemented

#15668010

C₂

yes

maximal, characteristic, complemented

#15668001

C₂

yes

maximal, characteristic, complemented

#15668000

C₂

yes

maximal, characteristic, complemented

#15667997

C₂

yes

maximal, complemented

#15667990

C₂

yes

maximal, complemented

#15667969

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

#13083575

—

C₂×Dic₁₇₈

C₂²

yes

—

C₂×Dic₁₇₈

C₂²

yes

characteristic

Q₈×D₈₉

—

C₂×D₃₅₆

C₂²

yes

—

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂²

yes

—

C₂×C₄×D₈₉

C₂²

yes

—

C₂×C₄×D₈₉

C₂²

yes

characteristic

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄²×C₈₉

C₂²

yes

characteristic

C₂²×D₈₉

C₂³

yes

—

C₂²×D₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

characteristic

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×D₄

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₄

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂×D₄

yes

—

C₂×C₈₉

C₂²×D₄

yes

characteristic

C₂×C₈₉

2₋¹⁺⁴

yes

—

D₈₉

—

D₈₉

—

C₈₉

#536

yes

characteristic, complemented, Sylow

#536

—

maximal, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₈₉

yes

characteristic, complemented

#124

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

178

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₈₉

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

178

—

Q₈

—

178

—

D₄

—

178

—

100

D₄

—

178

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

C₂²×D₈₉

yes

—

108

C₂×C₄

C₂²×D₈₉

yes

—

109

C₂×C₄

C₂²×D₈₉

yes

—

110

C₂×C₄

C₂²×D₈₉

yes

characteristic

111

C₂²

—

112

C₂²

—

178

—

113

C₂²

—

178

—

114

C₂²

—

178

—

115

C₂²

C₂³×D₈₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

178

—

121

C₄

—

178

—

122

C₄

D₄×D₈₉

yes

—

123

C₂

—

356

—

124

C₂

—

178

—

125

C₂

C₂×D₄×D₈₉

yes

characteristic

126

C₂

#15668060

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	C₄⋊Q₈×C₈₉	C₂
hypercenter	C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#536
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

C₄⋊Q₈×C₈₉

C₂

hypercenter

C₄⋊Q₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₈₉	C₂	semidirect
3. #15668010	C₂	semidirect
4. #15668001	C₂	semidirect
5. #15668000	C₂	semidirect
6. #15667997	C₂	semidirect
7. #15667990	C₂	semidirect
8. #15667969	C₂	semidirect
9. C₄⋊Q₈×C₈₉	C₂	semidirect
63. C₈₉	#536	semidirect
68. C₄⋊Q₈	D₈₉	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₈₉

C₂

semidirect

3. #15668010

C₂

semidirect

4. #15668001

C₂

semidirect

5. #15668000

C₂

semidirect

6. #15667997

C₂

semidirect

7. #15667990

C₂

semidirect

8. #15667969

C₂

semidirect

9. C₄⋊Q₈×C₈₉

C₂

semidirect

63. C₈₉

#536

semidirect

68. C₄⋊Q₈

D₈₉

semidirect

Subgroups — #19152173

Conjugacy classes of subgroups

Complements