Group #34000897 — subgroups

Subgroups — #34000897

Order 18688.

110 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 52 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	C₄⋊#14556641	C₂	1	1	yes	maximal, characteristic
2	#22625038	C₂	1	1	yes	maximal, characteristic
3	#22611840	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556641	—	2	1	no	—
5	C₂×#14556641	—	2	1	no	—
6	C₄⋊#11924003	—	2	1	no	—
7	C₄⋊#11924003	C₄	1	1	yes	characteristic
8	#17988363	—	2	1	no	—
9	C₄×#11924003	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
11	#14556641	—	2	2	no	—
12	#14556641	—	4	1	no	—
13	C₂×#11924003	—	2	1	no	—
14	C₂×#11924003	D₄	1	1	yes	characteristic
15	C₂×#11924003	D₄	1	1	yes	characteristic
16	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
17	C₄⋊C₈×C₇₃	—	2	1	no	—
18	C₄⋊C₈×C₇₃	C₈	1	1	yes	characteristic
19	C₄×C₈×C₇₃	—	2	1	no	—
20	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	#11924003	—	2	1	no	—
22	#11924003	—	4	1	no	—
23	#11924003	SD₁₆	1	1	yes	characteristic
24	#11924003	Q₁₆	1	1	yes	characteristic
25	C₂×Q₈×C₇₃	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₇₃	—	2	1	no	—
27	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	2	1	no	—
30	C₄⋊C₄×C₇₃	C₁₆	1	1	yes	characteristic
31	C₄²×C₇₃	—	2	1	no	—
32	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₇₃	—	4	1	no	—
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	2	1	no	—
36	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
39	C₈×C₇₃	—	2	1	no	—
40	C₈×C₇₃	—	4	1	no	—
41	C₈×C₇₃	#103	1	1	yes	characteristic
42	C₈×C₇₃	#104	1	1	yes	characteristic
43	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₄×C₇₃	—	2	1	no	—
45	C₄×C₇₃	—	2	1	no	—
46	C₄×C₇₃	—	4	1	no	—
47	C₄×C₇₃	—	4	1	no	—
48	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
49	C₄×C₇₃	#350	1	1	yes	characteristic
50	#7393	—	73	1	no	maximal, Sylow
51	C₂×C₇₃	Q₈⋊C₁₆	1	1	yes	characteristic
52	C₂×C₇₃	#1400	1	1	yes	characteristic
53	C₂×C₇₃	#1401	1	1	yes	characteristic
54	#2183	—	73	1	no	—
55	C₄⋊C₃₂	—	73	1	no	—
56	#1397	—	73	1	no	—
57	C₇₃	#7393	1	1	yes	characteristic, complemented, Sylow
58	C₈×Q₈	#64408	1	1	yes	characteristic
59	C₂×C₃₂	—	146	1	no	—
60	C₂×C₃₂	—	146	1	no	—
61	C₄⋊C₁₆	—	73	1	no	—
62	C₄⋊C₁₆	—	146	1	no	—
63	#346	—	146	1	no	—
64	C₄×C₁₆	—	73	1	no	—
65	C₄×Q₈	#10596833	1	1	yes	characteristic
66	C₂×C₁₆	—	73	1	no	—
67	C₂×C₁₆	—	73	1	no	—
68	C₂×C₁₆	—	146	1	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	#10596833	1	1	yes	characteristic
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	C₂×#64408	1	1	yes	characteristic
73	C₃₂	—	292	1	no	—
74	C₃₂	—	146	2	no	—
75	C₂×Q₈	#11924031	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
78	C₂×C₈	#11924037	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	#11924031	1	1	yes	characteristic
81	C₄²	—	2	1	no	—
82	C₄²	C₂×#10596833	1	1	yes	characteristic
83	C₁₆	—	292	1	no	—
84	C₁₆	—	73	1	no	—
85	C₁₆	—	73	1	no	—
86	C₁₆	—	146	1	no	—
87	Q₈	—	4	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
91	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
92	C₂×C₄	#14556665	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	1	no	—
95	C₈	#14556654	1	1	yes	characteristic
96	C₈	#14556656	1	1	yes	characteristic
97	C₂²	C₂²⋊#11924031	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	Q₈⋊#10596833	1	1	yes	characteristic
103	C₄	#17989485	1	1	yes	characteristic
104	C₂	Q₈⋊#11924031	1	1	yes	characteristic
105	C₂	#22625046	1	1	yes	characteristic
106	C₂	#22625048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#14556641

C₂

yes

maximal, characteristic

#22625038

C₂

yes

maximal, characteristic

#22611840

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

characteristic

#17988363

—

C₄×#11924003

C₂²

yes

characteristic

C₈×Q₈×C₇₃

C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11924003

D₄

yes

characteristic

C₄×Q₈×C₇₃

C₈

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

C₈

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

SD₁₆

yes

characteristic

#11924003

Q₁₆

yes

characteristic

C₂×Q₈×C₇₃

C₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₁₆

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

Q₈⋊C₈

yes

C₄×C₇₃

yes

—

C₂×C₇₃

Q₈⋊C₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₄⋊C₃₂

—

#1397

—

C₇₃

#7393

yes

characteristic, complemented, Sylow

C₈×Q₈

#64408

yes

characteristic

C₂×C₃₂

—

146

—

C₂×C₃₂

—

146

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

#346

—

146

—

C₄×C₁₆

—

C₄×Q₈

#10596833

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄⋊C₈

—

C₄⋊C₈

#10596833

yes

characteristic

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₃₂

—

292

—

C₃₂

—

146

—

C₂×Q₈

#11924031

yes

characteristic

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

#11924031

yes

characteristic

C₄²

—

C₄²

C₂×#10596833

yes

characteristic

C₁₆

—

292

—

C₁₆

—

C₁₆

—

C₁₆

—

146

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

Q₈⋊#10596833

yes

characteristic

103

C₄

#17989485

yes

characteristic

104

C₂

Q₈⋊#11924031

yes

characteristic

105

C₂

#22625046

yes

characteristic

106

C₂

#22625048

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#64408
Fitting	C₈×Q₈×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₂²⋊C₁₆
cosocle	C₄×#11924003	C₂²
hypercenter	C₈×Q₈	#64408
hypocenter	C₇₃	#7393
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₈

C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂×C₁₆

Frattini

C₄×C₈

C₂×#64408

Fitting

C₈×Q₈×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

C₂²⋊C₁₆

C₂²

C₈×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
57. C₇₃	#7393	semidirect

H ≅ G/N

type

57. C₇₃

#7393

semidirect

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

subgroup	index
C₄⋊#14556641	2
#22625038	2
#22611840	2
#7393	73

subgroup

index

Subgroups — #34000897

Conjugacy classes of subgroups

Complements

Maximal subgroups