Group #16818570 — subgroups

Subgroups — #16818570

Order 3712.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#14271537	C₂	1	1	yes	maximal, characteristic
2	#14271363	C₂	1	2	yes	maximal
3	#14270912	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#6597	C₂	1	2	yes	maximal, complemented
5	Q₈²×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×C₄⋊Q₈	—	2	2	no	—
7	C₂₉×C₄⋊Q₈	—	2	2	no	—
8	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₂₉	—	2	2	no	—
10	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
11	C₂×#90784	—	2	2	no	—
12	C₂×#90784	—	2	2	no	—
13	#11745236	C₂²	1	1	yes	characteristic
14	#11745157	—	2	2	no	—
15	C₄⋊#6597	C₂²	1	2	yes	—
16	C₄×#6597	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₉	—	2	2	no	—
18	C₂×Q₈×C₂₉	D₄	1	2	yes	—
19	C₂×Q₈×C₂₉	D₄	1	2	yes	—
20	C₂×D₁₁₆	—	2	1	no	—
21	C₂×D₁₁₆	—	2	2	no	—
22	C₄⋊C₄×C₂₉	—	2	4	no	—
23	C₄⋊C₄×C₂₉	—	2	2	no	—
24	C₄⋊C₄×C₂₉	—	2	2	no	—
25	C₄⋊C₄×C₂₉	D₄	1	2	yes	—
26	C₄²×C₂₉	—	2	2	no	—
27	C₄²×C₂₉	—	2	2	no	—
28	C₄²×C₂₉	C₂³	1	1	yes	characteristic
29	#90784	—	4	4	no	—
30	#90784	—	4	4	no	—
31	C₂×#6597	—	2	1	no	—
32	C₂×#6597	—	2	2	no	—
33	C₂²×D₂₉	—	4	1	no	—
34	Q₈×C₂₉	—	2	2	no	—
35	Q₈×C₂₉	—	4	2	no	—
36	Q₈×C₂₉	—	8	2	no	—
37	Q₈×C₂₉	SD₁₆	1	4	yes	complemented
38	C₂×C₄×C₂₉	—	2	2	no	—
39	C₂×C₄×C₂₉	—	2	2	no	—
40	C₂×C₄×C₂₉	—	2	2	no	—
41	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
43	D₁₁₆	—	4	2	no	—
44	D₁₁₆	—	4	2	no	—
45	D₁₁₆	—	4	2	no	—
46	#6597	—	2	2	no	—
47	#6597	—	4	2	no	—
48	#1623	—	29	1	no	maximal, Sylow
49	C₂²×C₂₉	#120	1	1	yes	characteristic
50	C₂×D₂₉	—	8	2	no	—
51	C₂×D₂₉	—	8	1	no	—
52	C₄×C₂₉	—	2	4	no	—
53	C₄×C₂₉	—	2	1	no	—
54	C₄×C₂₉	—	4	2	no	—
55	C₄×C₂₉	—	4	2	no	—
56	C₄×C₂₉	C₂×SD₁₆	1	2	yes	—
57	C₄×C₂₉	#137	1	2	yes	—
58	Q₈²	D₂₉	1	1	yes	characteristic, complemented
59	#492	—	29	1	no	—
60	#460	—	29	2	no	—
61	#333	—	29	1	no	—
62	Q₈⋊C₈	—	29	2	no	—
63	C₂×C₂₉	#448	1	2	yes	—
64	C₂×C₂₉	#453	1	1	yes	characteristic
65	D₂₉	—	16	1	no	—
66	C₂×SD₁₆	—	58	2	no	—
67	C₂×SD₁₆	—	58	2	no	—
68	C₄⋊Q₈	—	2	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	C₂×D₂₉	1	1	yes	characteristic
71	#127	—	29	1	no	—
72	C₄×Q₈	—	2	2	no	—
73	C₄×Q₈	C₂×D₂₉	1	2	yes	—
74	C₄⋊C₈	—	29	2	no	—
75	#102	—	58	2	no	—
76	C₄×C₈	—	29	1	no	—
77	C₂₉	#1623	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₂²⋊D₂₉	1	2	yes	—
80	C₂×Q₈	C₂²⋊D₂₉	1	2	yes	—
81	C₂×D₄	—	58	2	no	—
82	C₂×D₄	—	58	1	no	—
83	SD₁₆	—	116	4	no	—
84	SD₁₆	—	116	4	no	—
85	C₂×C₈	—	58	1	no	—
86	C₂×C₈	—	58	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	C₂²⋊D₂₉	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂²×D₂₉	1	1	yes	characteristic
94	C₂³	—	116	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	#90784	1	4	yes	complemented
99	D₄	—	116	2	no	—
100	D₄	—	116	2	no	—
101	D₄	—	116	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
107	C₈	—	58	2	no	—
108	C₈	—	116	2	no	—
109	C₂²	—	232	2	no	—
110	C₂²	—	232	1	no	—
111	C₂²	#11745315	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×#90784	1	2	yes	—
117	C₄	#11745304	1	2	yes	—
118	C₂	—	464	1	no	—
119	C₂	#14271302	1	2	yes	—
120	C₂	#14271331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271537

C₂

yes

maximal, characteristic

#14271363

C₂

yes

maximal

#14270912

C₂

yes

maximal, characteristic, complemented

Q₈⋊#6597

C₂

yes

maximal, complemented

Q₈²×C₂₉

C₂

yes

maximal, characteristic, complemented

C₂₉×C₄⋊Q₈

—

C₂₉×C₄⋊Q₈

—

C₂₉×C₄⋊Q₈

C₂²

yes

characteristic

C₄×Q₈×C₂₉

—

C₄×Q₈×C₂₉

C₂²

yes

—

C₂×#90784

—

C₂×#90784

—

#11745236

C₂²

yes

characteristic

#11745157

—

C₄⋊#6597

C₂²

yes

—

C₄×#6597

C₂²

yes

characteristic

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

D₄

yes

—

C₂×Q₈×C₂₉

D₄

yes

—

C₂×D₁₁₆

—

C₂×D₁₁₆

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

D₄

yes

—

C₄²×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

C₂³

yes

—

—

—

—

C₂²×D₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

SD₁₆

yes

complemented

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

D₁₁₆

—

D₁₁₆

—

D₁₁₆

—

—

—

—

C₂²×C₂₉

yes

C₂×D₂₉

—

C₂×D₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×SD₁₆

yes

—

C₄×C₂₉

#137

yes

—

Q₈²

D₂₉

yes

characteristic, complemented

#492

—

#460

—

#333

—

Q₈⋊C₈

—

C₂×C₂₉

#448

yes

—

C₂×C₂₉

#453

yes

characteristic

D₂₉

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₉

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₉

yes

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂₉

#1623

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₉

yes

—

C₂×Q₈

C₂²⋊D₂₉

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

116

—

SD₁₆

—

116

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₉

yes

characteristic

C₂³

—

116

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#90784

yes

complemented

D₄

—

116

—

100

D₄

—

116

—

101

D₄

—

116

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

107

C₈

—

108

C₈

—

116

—

109

C₂²

—

232

—

110

C₂²

—

232

—

111

C₂²

#11745315

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×#90784

yes

—

117

C₄

#11745304

yes

—

118

C₂

—

464

—

119

C₂

#14271302

yes

—

120

C₂

#14271331

yes

characteristic

	subgroup	G/H
center	C₂²	#11745315
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	Q₈²×C₂₉	C₂
socle	C₂²×C₂₉	#120
prime radical	#11745236	C₂²
hypercenter	Q₈²	D₂₉
hypocenter	C₂₉	#1623
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#11745315

commutator, cosocle

C₄²×C₂₉

C₂³

Frattini

C₄²

C₂²×D₂₉

Fitting

Q₈²×C₂₉

C₂

C₂²×C₂₉

C₂²

Q₈²

D₂₉

hypocenter

C₂₉

#1623

LCS

G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
3. #14270912	C₂	semidirect
4. Q₈⋊#6597	C₂	semidirect
5. Q₈²×C₂₉	C₂	semidirect
37. Q₈×C₂₉	SD₁₆	semidirect
58. Q₈²	D₂₉	semidirect
77. C₂₉	#1623	semidirect
98. Q₈	#90784	semidirect

H ≅ G/N

type

3. #14270912

C₂

semidirect

4. Q₈⋊#6597

C₂

semidirect

5. Q₈²×C₂₉

C₂

semidirect

37. Q₈×C₂₉

SD₁₆

semidirect

58. Q₈²

D₂₉

semidirect

77. C₂₉

#1623

semidirect

98. Q₈

#90784

semidirect

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

subgroup	index
#14271537	2
#14271363	2
#14270912	2
Q₈⋊#6597	2
Q₈²×C₂₉	2
#1623	29

subgroup

index

Q₈²×C₂₉

Subgroups — #16818570

Conjugacy classes of subgroups

Complements

Maximal subgroups