Subgroups — #35233154

Order 21184.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₃₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₃₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₃₁	—	2	1	no	—
5	C₂×C₃₃₁⋊C₈	—	2	1	no	—
6	C₂×C₃₃₁⋊C₈	—	2	1	no	—
7	C₂×C₃₃₁⋊C₈	C₄	1	2	yes	complemented
8	C₄⋊Dic₃₃₁	—	2	1	no	—
9	C₄⋊Dic₃₃₁	C₄	1	2	yes	complemented
10	C₂²×C₄×C₃₃₁	—	2	1	no	—
11	C₂²×C₄×C₃₃₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₃₁	—	2	1	no	—
13	C₄⋊C₄×C₃₃₁	C₄	1	2	yes	complemented
14	C₂×Dic₃₃₁	—	2	2	no	—
15	C₂×Dic₃₃₁	—	4	1	no	—
16	C₂×Dic₃₃₁	—	4	1	no	—
17	C₃₃₁⋊C₈	—	2	2	no	—
18	C₂³×C₃₃₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₃₁	—	2	2	no	—
20	C₂×C₄×C₃₃₁	—	4	1	no	—
21	C₂×C₄×C₃₃₁	—	4	1	no	—
22	C₂×C₄×C₃₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₃₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₃₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₃₁	Q₈	1	1	yes	characteristic
28	C₂²×C₃₃₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₃₃₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₃₃₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₃₃₁	D₈	1	1	yes	characteristic
32	C₂²×C₃₃₁	SD₁₆	1	1	yes	characteristic
33	C₂²×C₃₃₁	SD₁₆	1	1	yes	characteristic
34	C₂²×C₃₃₁	Q₁₆	1	1	yes	characteristic
35	C₄×C₃₃₁	—	4	2	no	—
36	C₄×C₃₃₁	C₄²	1	1	yes	characteristic
37	C₄×C₃₃₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₃₃₁	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₃₃₁	C₄⋊C₄	1	1	yes	characteristic
40	Dic₃₃₁	—	4	2	no	—
41	C₂×C₃₃₁	#95	1	1	yes	characteristic
42	C₂×C₃₃₁	#102	1	1	yes	characteristic
43	C₂×C₃₃₁	#102	1	1	yes	characteristic
44	C₂×C₃₃₁	#103	1	1	yes	characteristic
45	C₂×C₃₃₁	#103	1	1	yes	characteristic
46	C₂×C₃₃₁	#106	1	1	yes	characteristic
47	C₂×C₃₃₁	#107	1	1	yes	characteristic
48	C₃₃₁	#340	1	1	yes	characteristic, complemented, Sylow
49	#340	—	331	1	no	maximal, Sylow
50	C₂²×C₈	—	331	1	no	—
51	C₂×C₄⋊C₄	—	331	1	no	—
52	C₂×C₄⋊C₄	D₃₃₁	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	662	1	no	—
55	C₂²×C₄	C₂×D₃₃₁	1	1	yes	characteristic
56	C₂×C₈	—	331	2	no	—
57	C₂×C₈	—	662	1	no	—
58	C₂×C₈	—	662	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	331	2	no	—
61	C₄⋊C₄	—	662	1	no	—
62	C₄⋊C₄	Dic₃₃₁	1	2	yes	complemented
63	C₂³	C₂²⋊D₃₃₁	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	1324	1	no	—
68	C₂×C₄	—	1324	1	no	—
69	C₂×C₄	—	662	2	no	—
70	C₂×C₄	C₄×D₃₃₁	1	1	yes	characteristic
71	C₂×C₄	C₄×D₃₃₁	1	1	yes	characteristic
72	C₂×C₄	C₂×Dic₃₃₁	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₃₃₁	1	1	yes	characteristic
74	C₂×C₄	D₁₃₂₄	1	1	yes	characteristic
75	C₂×C₄	Dic₆₆₂	1	1	yes	characteristic
76	C₈	—	662	2	no	—
77	C₂²	(C₂×D₃₃₁)⋊C₄	1	1	yes	characteristic
78	C₂²	C₂²⋊Dic₃₃₁	1	1	yes	characteristic
79	C₂²	Dic₃₃₁⋊C₄	1	1	yes	characteristic
80	C₂²	#19127021	1	1	yes	characteristic
81	C₂²	#19127023	1	1	yes	characteristic
82	C₂²	#19127024	1	1	yes	characteristic
83	C₂²	#19127026	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	1324	2	no	—
86	C₄	C₄×Dic₃₃₁	1	1	yes	characteristic
87	C₄	(C₂×D₃₃₁)⋊C₄	1	1	yes	characteristic
88	C₄	Dic₃₃₁⋊C₄	1	1	yes	characteristic
89	C₄	C₄⋊Dic₃₃₁	1	1	yes	characteristic
90	C₂	C₃₃₁⋊#95	1	1	yes	characteristic
91	C₂	#24902528	1	1	yes	characteristic
92	C₂	#24902529	1	1	yes	characteristic
93	C₂	#24902531	1	1	yes	characteristic
94	C₂	#24902532	1	1	yes	characteristic
95	C₂	#24902539	1	1	yes	characteristic
96	C₂	#24902541	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₃₃₁
commutator	C₄×C₃₃₁	C₄²
Frattini	C₂²×C₄	C₂×D₃₃₁
Fitting	C₂×C₄⋊C₄×C₃₃₁	C₂
socle, prime radical	C₂³×C₃₃₁	D₄
cosocle	C₂²×C₄×C₃₃₁	C₂²
hypercenter	C₂×C₄⋊C₄	D₃₃₁
hypocenter	C₃₃₁	#340
LCS	G ▷ C₄×C₃₃₁ ▷ C₂×C₃₃₁ ▷ C₃₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. C₂×C₃₃₁⋊C₈	C₄	semidirect
9. C₄⋊Dic₃₃₁	C₄	semidirect
13. C₄⋊C₄×C₃₃₁	C₄	semidirect
48. C₃₃₁	#340	semidirect
62. C₄⋊C₄	Dic₃₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₃₁⋊C₈	2
C₂×C₄⋊Dic₃₃₁	2
C₂×C₄⋊C₄×C₃₃₁	2
#340	331