Group #34215503 — subgroups

Subgroups — #34215503

Order 20224.

1118 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 411 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	#23761662	C₂	1	8	yes	maximal
2	#23761630	C₂	1	6	yes	maximal
3	C₂×#376×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
5	C₂×C₇₉⋊#95	C₂²	1	24	yes	—
6	C₂²×C₄²×C₇₉	C₂²	1	3	yes	—
7	#376×C₇₉	C₄	1	8	yes	complemented
8	C₂×#95×C₇₉	C₂²	1	4	yes	—
9	C₂³×Dic₇₉	C₂³	1	8	yes	—
10	C₂×C₄⋊Dic₇₉	—	2	24	no	—
11	C₇₉⋊#95	—	4	24	no	—
12	C₂³×C₄×C₇₉	C₂³	1	6	yes	—
13	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₇₉	—	2	12	no	—
15	C₂×C₄²×C₇₉	C₂×C₄	1	12	yes	—
16	#95×C₇₉	C₂×C₄	1	16	yes	—
17	C₂²×Dic₇₉	—	2	24	no	—
18	C₂²×Dic₇₉	—	2	24	no	—
19	C₂²×Dic₇₉	—	2	8	no	—
20	C₄⋊Dic₇₉	—	4	16	no	—
21	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
22	C₂²×C₄×C₇₉	—	2	6	no	—
23	C₂²×C₄×C₇₉	—	2	6	no	—
24	C₂²×C₄×C₇₉	—	2	12	no	—
25	C₂²×C₄×C₇₉	—	2	12	no	—
26	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
27	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
28	C₂²×C₄×C₇₉	#41	1	12	yes	—
29	C₄²×C₇₉	—	2	24	no	—
30	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
31	C₂³×C₇₉	C₂×#41	1	6	yes	—
32	C₂³×C₇₉	H(5,𝔽₂)	1	4	yes	—
33	C₂³×C₇₉	2₋¹⁺⁴	1	4	yes	—
34	C₂×C₄×C₇₉	—	2	12	no	—
35	C₂×C₄×C₇₉	—	2	24	no	—
36	C₂×C₄×C₇₉	—	4	6	no	—
37	C₂×C₄×C₇₉	—	4	6	no	—
38	C₂×C₄×C₇₉	—	4	12	no	—
39	C₂×C₄×C₇₉	#117	1	12	yes	—
40	C₂×C₄×C₇₉	#126	1	16	yes	—
41	C₂×Dic₇₉	—	4	8	no	—
42	C₂×Dic₇₉	—	4	24	no	—
43	C₂×Dic₇₉	—	4	24	no	—
44	C₂²×C₇₉	C₂×#117	1	3	yes	—
45	C₂²×C₇₉	#520	1	4	yes	—
46	C₂²×C₇₉	C₂×#126	1	4	yes	—
47	C₂²×C₇₉	#540	1	6	yes	—
48	C₂²×C₇₉	#541	1	6	yes	—
49	C₂²×C₇₉	#542	1	12	yes	—
50	C₄×C₇₉	—	4	12	no	—
51	C₄×C₇₉	#383	1	8	yes	—
52	Dic₇₉	—	8	8	no	—
53	#14350	—	79	1	no	maximal, Sylow
54	C₂×C₇₉	C₂×#383	1	1	yes	characteristic
55	C₂×C₇₉	#2314	1	6	yes	—
56	C₂×C₇₉	#2644	1	4	yes	—
57	C₂×C₇₉	#2645	1	4	yes	—
58	C₂×#376	D₇₉	1	1	yes	characteristic
59	#1443	—	79	8	no	—
60	C₄⋊#95	—	79	6	no	—
61	C₇₉	#14350	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₄⋊C₄	—	79	4	no	—
63	C₂²×C₄²	C₂×D₇₉	1	3	yes	—
64	#376	Dic₇₉	1	8	yes	complemented
65	C₂×#95	—	79	24	no	—
66	C₂×#95	C₂×D₇₉	1	4	yes	—
67	C₂³×C₄	—	79	8	no	—
68	C₂³×C₄	C₂²×D₇₉	1	6	yes	—
69	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	158	24	no	—
71	C₂×C₄²	—	2	12	no	—
72	C₂×C₄²	C₂×Dic₇₉	1	12	yes	—
73	#95	—	316	24	no	—
74	#95	C₂×Dic₇₉	1	16	yes	—
75	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
76	C₂²×C₄	—	2	6	no	—
77	C₂²×C₄	—	2	6	no	—
78	C₂²×C₄	—	2	12	no	—
79	C₂²×C₄	—	2	12	no	—
80	C₂²×C₄	—	158	24	no	—
81	C₂²×C₄	—	158	24	no	—
82	C₂²×C₄	—	158	8	no	—
83	C₂²×C₄	C₂²×Dic₇₉	1	12	yes	—
84	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
85	C₂²×C₄	#11929624	1	12	yes	—
86	C₄⋊C₄	—	316	16	no	—
87	C₄²	—	2	24	no	—
88	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
89	C₂³	C₂×#11929624	1	6	yes	—
90	C₂³	#14579207	1	4	yes	—
91	C₂³	#14579208	1	4	yes	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	24	no	—
94	C₂×C₄	—	4	6	no	—
95	C₂×C₄	—	4	6	no	—
96	C₂×C₄	—	4	12	no	—
97	C₂×C₄	—	316	24	no	—
98	C₂×C₄	—	316	24	no	—
99	C₂×C₄	—	316	8	no	—
100	C₂×C₄	#14579120	1	12	yes	—
101	C₂×C₄	#14579155	1	16	yes	—
102	C₂²	C₂×#14579120	1	3	yes	—
103	C₂²	#19099046	1	4	yes	—
104	C₂²	C₂×#14579155	1	4	yes	—
105	C₂²	#19099168	1	6	yes	—
106	C₂²	#19099175	1	6	yes	—
107	C₂²	#19099184	1	12	yes	—
108	C₄	—	4	12	no	—
109	C₄	—	632	8	no	—
110	C₄	#19098431	1	8	yes	—
111	C₂	C₂×#19098431	1	1	yes	characteristic
112	C₂	#23766009	1	6	yes	—
113	C₂	#23769360	1	4	yes	—
114	C₂	#23769365	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23761662

C₂

yes

maximal

#23761630

C₂

yes

maximal

C₂×#376×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₇₉⋊#95

C₂²

yes

—

C₂²×C₄²×C₇₉

C₂²

yes

—

#376×C₇₉

C₄

yes

complemented

C₂×#95×C₇₉

C₂²

yes

—

C₂³×Dic₇₉

C₂³

yes

—

C₂×C₄⋊Dic₇₉

—

C₇₉⋊#95

—

C₂³×C₄×C₇₉

C₂³

yes

—

C₂³×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

#95×C₇₉

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₂⁴×C₇₉

C₂⁴

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

#41

yes

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂³×C₄

yes

characteristic

C₂³×C₇₉

C₂×#41

yes

—

C₂³×C₇₉

H(5,𝔽₂)

yes

—

C₂³×C₇₉

2₋¹⁺⁴

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#126

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂²×C₇₉

C₂×#117

yes

—

C₂²×C₇₉

#520

yes

—

C₂²×C₇₉

C₂×#126

yes

—

C₂²×C₇₉

#540

yes

—

C₂²×C₇₉

#541

yes

—

C₂²×C₇₉

#542

yes

—

C₄×C₇₉

—

C₄×C₇₉

#383

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2644

yes

—

C₂×C₇₉

#2645

yes

—

C₂×#376

D₇₉

yes

characteristic

#1443

—

C₄⋊#95

—

C₇₉

#14350

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₇₉

yes

—

#376

Dic₇₉

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₇₉

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₇₉

yes

—

C₂³×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇₉

yes

—

#95

—

316

—

#95

C₂×Dic₇₉

yes

—

C₂⁴

C₂³×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²×Dic₇₉

yes

—

C₂²×C₄

C₂²×Dic₇₉

yes

—

C₂²×C₄

#11929624

yes

—

C₄⋊C₄

—

316

—

C₄²

—

C₂³

C₂³×Dic₇₉

yes

characteristic

C₂³

C₂×#11929624

yes

—

C₂³

#14579207

yes

—

C₂³

#14579208

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

316

—

C₂×C₄

—

316

—

C₂×C₄

—

316

—

100

C₂×C₄

#14579120

yes

—

101

C₂×C₄

#14579155

yes

—

102

C₂²

C₂×#14579120

yes

—

103

C₂²

#19099046

yes

—

104

C₂²

C₂×#14579155

yes

—

105

C₂²

#19099168

yes

—

106

C₂²

#19099175

yes

—

107

C₂²

#19099184

yes

—

108

C₄

—

109

C₄

—

632

—

110

C₄

#19098431

yes

—

111

C₂

C₂×#19098431

yes

characteristic

112

C₂

#23766009

yes

—

113

C₂

#23769360

yes

—

114

C₂

#23769365

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₉
commutator	C₂³×C₇₉	C₂³×C₄
Fitting	C₂×#376×C₇₉	C₂
socle, prime radical, cosocle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂×#376	D₇₉
hypocenter	C₇₉	#14350
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₇₉

commutator

C₂³×C₇₉

C₂³×C₄

Fitting

C₂×#376×C₇₉

C₂

socle, prime radical, cosocle

C₂⁴×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
7. #376×C₇₉	C₄	semidirect
61. C₇₉	#14350	semidirect
64. #376	Dic₇₉	semidirect

H ≅ G/N

type

7. #376×C₇₉

C₄

semidirect

61. C₇₉

#14350

semidirect

64. #376

Dic₇₉

semidirect

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

subgroup	index
#23761662	2
#23761630	2
C₂×#376×C₇₉	2
#14350	79

subgroup

index

#23761662

#23761630

C₂×#376×C₇₉

#14350

Subgroups — #34215503

Conjugacy classes of subgroups

Complements

Maximal subgroups