Group #34191346 — subgroups

Subgroups — #34191346

Order 20224.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23764223	C₂	1	3	yes	maximal
2	#23761281	C₂	1	3	yes	maximal, complemented
3	C₇₉×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14579158	C₂²	1	1	yes	characteristic
5	C₂×#14579081	—	2	6	no	—
6	C₂×#14579072	C₂²	1	3	yes	—
7	#398×C₇₉	—	2	6	no	—
8	#382×C₇₉	C₂²	1	3	yes	—
9	C₂²×D₃₁₆	—	2	3	no	—
10	C₂²×C₇₉⋊C₈	—	2	3	no	—
11	#14579158	—	2	3	no	—
12	#14579158	Q₈	1	2	yes	—
13	#14579081	—	4	12	no	—
14	#14579072	—	2	6	no	—
15	C₂×C₄⋊C₄×C₇₉	—	2	6	no	—
16	C₂×C₄⋊C₄×C₇₉	D₄	1	6	yes	—
17	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
18	#95×C₇₉	—	2	2	no	—
19	#95×C₇₉	—	2	6	no	—
20	C₂³×D₇₉	—	4	1	no	—
21	C₂×D₃₁₆	—	2	6	no	—
22	C₂×D₃₁₆	—	4	6	no	—
23	C₂×D₃₁₆	—	4	6	no	—
24	C₂×D₃₁₆	—	4	3	no	—
25	C₂×C₇₉⋊C₈	—	2	6	no	—
26	C₂×C₇₉⋊C₈	—	4	3	no	—
27	C₂×C₇₉⋊C₈	—	4	3	no	—
28	C₂²×C₄×C₇₉	—	2	6	no	—
29	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
30	C₄⋊C₄×C₇₉	—	4	6	no	—
31	C₄⋊C₄×C₇₉	—	8	6	no	—
32	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₇₉	#41	1	3	yes	—
34	C₂³×C₇₉	#120	1	1	yes	characteristic
35	C₂²×D₇₉	—	4	2	no	—
36	C₂²×D₇₉	—	8	3	no	—
37	C₂²×D₇₉	—	8	3	no	—
38	C₂×C₄×C₇₉	—	2	3	no	—
39	C₂×C₄×C₇₉	—	2	3	no	—
40	C₂×C₄×C₇₉	—	4	6	no	—
41	C₂×C₄×C₇₉	—	4	6	no	—
42	C₂×C₄×C₇₉	—	4	6	no	—
43	C₂×C₄×C₇₉	C₂²⋊Q₈	1	3	yes	—
44	C₂×C₄×C₇₉	#124	1	3	yes	—
45	D₃₁₆	—	4	12	no	—
46	D₃₁₆	—	8	6	no	—
47	C₇₉⋊C₈	—	4	6	no	—
48	C₂²×C₇₉	#393	1	1	yes	characteristic
49	C₂²×C₇₉	#453	1	3	yes	—
50	C₂²×C₇₉	#454	1	3	yes	—
51	C₂×D₇₉	—	8	6	no	—
52	C₂×D₇₉	—	16	3	no	—
53	C₂×D₇₉	—	16	1	no	—
54	C₄×C₇₉	—	2	3	no	—
55	C₄×C₇₉	—	2	3	no	—
56	C₄×C₇₉	—	8	6	no	—
57	#10360	—	79	1	no	maximal, Sylow
58	C₂×C₇₉	#1656	1	3	yes	—
59	C₂×C₇₉	#1658	1	1	yes	characteristic
60	C₂×C₇₉	#2027	1	3	yes	—
61	D₇₉	—	16	2	no	—
62	#2845	D₇₉	1	1	yes	characteristic, complemented
63	#1961	—	79	3	no	—
64	#1295	—	79	3	no	—
65	C₇₉	#10360	1	1	yes	characteristic, complemented, Sylow
66	C₂×#127	—	79	1	no	—
67	C₂×#102	—	158	6	no	—
68	C₂×#97	—	79	3	no	—
69	#398	—	2	6	no	—
70	#382	C₂×D₇₉	1	3	yes	—
71	C₂²×D₄	—	158	3	no	—
72	C₂²×C₈	—	158	3	no	—
73	#127	—	79	2	no	—
74	#127	—	158	3	no	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	6	yes	—
77	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
78	#102	—	316	12	no	—
79	#97	—	158	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	316	1	no	—
83	C₂×D₄	—	316	6	no	—
84	C₂×D₄	—	316	6	no	—
85	C₂×D₄	—	316	3	no	—
86	C₂×D₄	—	158	6	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
89	C₂×C₈	—	316	3	no	—
90	C₂×C₈	—	316	3	no	—
91	C₂×C₈	—	158	6	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₇₉	1	1	yes	characteristic
95	C₄²	#11929625	1	3	yes	—
96	C₂³	—	316	2	no	—
97	C₂³	—	632	3	no	—
98	C₂³	—	632	3	no	—
99	C₂³	#14579130	1	1	yes	characteristic
100	D₄	—	316	12	no	—
101	D₄	—	632	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#14579137	1	3	yes	—
108	C₂×C₄	#14579150	1	3	yes	—
109	C₈	—	316	6	no	—
110	C₂²	—	632	6	no	—
111	C₂²	—	1264	3	no	—
112	C₂²	—	1264	1	no	—
113	C₂²	#19098474	1	1	yes	characteristic
114	C₂²	#19098731	1	3	yes	—
115	C₂²	#19098736	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	1264	2	no	—
120	C₂	#23762693	1	3	yes	—
121	C₂	#23762702	1	1	yes	characteristic
122	C₂	#23764553	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764223

C₂

yes

maximal

#23761281

C₂

yes

maximal, complemented

C₇₉×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#14579158

C₂²

yes

characteristic

C₂×#14579081

—

C₂×#14579072

C₂²

yes

—

#398×C₇₉

—

#382×C₇₉

C₂²

yes

—

C₂²×D₃₁₆

—

C₂²×C₇₉⋊C₈

—

#14579158

—

#14579158

Q₈

yes

—

#14579081

—

#14579072

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#95×C₇₉

—

#95×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#120

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#124

yes

—

D₃₁₆

—

D₃₁₆

—

C₇₉⋊C₈

—

C₂²×C₇₉

#393

yes

characteristic

C₂²×C₇₉

#453

yes

—

C₂²×C₇₉

#454

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

#10360

—

maximal, Sylow

C₂×C₇₉

#1656

yes

—

C₂×C₇₉

#1658

yes

characteristic

C₂×C₇₉

#2027

yes

—

D₇₉

—

#2845

D₇₉

yes

characteristic, complemented

#1961

—

#1295

—

C₇₉

#10360

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

158

—

C₂×#97

—

#398

—

#382

C₂×D₇₉

yes

—

C₂²×D₄

—

158

—

C₂²×C₈

—

158

—

#127

—

#127

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#102

—

316

—

#97

—

158

—

#95

—

#95

—

C₂⁴

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

C₂×C₈

—

316

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₇₉

yes

characteristic

C₄²

#11929625

yes

—

C₂³

—

316

—

C₂³

—

632

—

C₂³

—

632

—

C₂³

#14579130

yes

characteristic

100

D₄

—

316

—

101

D₄

—

632

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#14579137

yes

—

108

C₂×C₄

#14579150

yes

—

109

C₈

—

316

—

110

C₂²

—

632

—

111

C₂²

—

1264

—

112

C₂²

—

1264

—

113

C₂²

#19098474

yes

characteristic

114

C₂²

#19098731

yes

—

115

C₂²

#19098736

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

1264

—

120

C₂

#23762693

yes

—

121

C₂

#23762702

yes

characteristic

122

C₂

#23764553

yes

—

	subgroup	G/H
center	C₂³	#14579130
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2845	C₂
socle	C₂³×C₇₉	#120
prime radical	C₂×#14579158	C₂²
hypercenter	#2845	D₇₉
hypocenter	C₇₉	#10360
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14579130

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2845

C₂

C₂³×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #23761281	C₂	semidirect
3. C₇₉×#2845	C₂	semidirect
62. #2845	D₇₉	semidirect
65. C₇₉	#10360	semidirect

H ≅ G/N

type

2. #23761281

C₂

semidirect

3. C₇₉×#2845

C₂

semidirect

62. #2845

D₇₉

semidirect

65. C₇₉

#10360

semidirect

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

subgroup	index
#23764223	2
#23761281	2
C₇₉×#2845	2
#10360	79

subgroup

index

#23764223

#23761281

C₇₉×#2845

#10360

Subgroups — #34191346

Conjugacy classes of subgroups

Complements

Maximal subgroups