Subgroups — #28416340

Order 14912.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₆₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₃₃	C₂	1	1	yes	maximal, characteristic
3	#21418442	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₆₆	—	2	2	no	—
6	C₂×Dic₄₆₆	—	2	2	no	—
7	C₂×Dic₄₆₆	C₄	1	4	yes	complemented
8	C₂²×Dic₂₃₃	C₂²	1	2	yes	—
9	C₂²×Dic₂₃₃	C₂²	1	2	yes	—
10	C₄×Dic₂₃₃	—	2	2	no	—
11	C₂²×C₄×C₂₃₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₃₃	—	2	2	no	—
14	Dic₄₆₆	—	2	8	no	—
15	C₂×Dic₂₃₃	—	2	2	no	—
16	C₂×Dic₂₃₃	—	2	2	no	—
17	C₂×Dic₂₃₃	—	2	2	no	—
18	C₂×Dic₂₃₃	—	2	2	no	—
19	C₂×Dic₂₃₃	C₂×C₄	1	4	yes	—
20	C₂×Dic₂₃₃	Q₈	1	4	yes	complemented
21	C₂³×C₂₃₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₃₃	—	2	2	no	—
23	C₂×C₄×C₂₃₃	—	2	4	no	—
24	C₂×C₄×C₂₃₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₃₃	D₄	1	2	yes	—
26	C₂×C₄×C₂₃₃	D₄	1	2	yes	—
27	C₂²×C₂₃₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₂₃₃	C₂×Q₈	1	2	yes	—
31	C₂²×C₂₃₃	#41	1	2	yes	—
32	C₄×C₂₃₃	—	4	2	no	—
33	C₄×C₂₃₃	C₂²⋊C₄	1	4	yes	—
34	Dic₂₃₃	—	2	4	no	—
35	Dic₂₃₃	—	2	4	no	—
36	C₂×C₂₃₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₂₃₃	C₄×Q₈	1	2	yes	—
38	C₂×C₂₃₃	C₂²⋊Q₈	1	2	yes	—
39	C₂×C₂₃₃	#124	1	1	yes	characteristic
40	C₂×C₂₃₃	C₄⋊Q₈	1	1	yes	characteristic
41	C₂₃₃	#391	1	1	yes	characteristic, complemented, Sylow
42	#391	—	233	1	no	maximal, Sylow
43	C₂²×Q₈	—	233	1	no	—
44	C₂×C₄⋊C₄	D₂₃₃	1	1	yes	characteristic
45	C₂×C₄²	—	233	1	no	—
46	#95	—	233	4	no	—
47	C₂×Q₈	—	466	2	no	—
48	C₂×Q₈	—	466	2	no	—
49	C₂×Q₈	—	233	4	no	—
50	C₂²×C₄	—	233	2	no	—
51	C₂²×C₄	—	233	2	no	—
52	C₂²×C₄	C₂×D₂₃₃	1	2	yes	—
53	C₂²×C₄	C₂×D₂₃₃	1	1	yes	characteristic
54	C₄⋊C₄	—	2	2	no	—
55	C₄²	—	466	2	no	—
56	C₂³	C₂²×D₂₃₃	1	1	yes	characteristic
57	Q₈	—	466	8	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	466	2	no	—
61	C₂×C₄	—	466	2	no	—
62	C₂×C₄	—	466	2	no	—
63	C₂×C₄	—	466	2	no	—
64	C₂×C₄	—	233	4	no	—
65	C₂×C₄	—	233	4	no	—
66	C₂×C₄	C₄×D₂₃₃	1	2	yes	—
67	C₂×C₄	C₂²⋊D₂₃₃	1	2	yes	—
68	C₂×C₄	D₉₃₂	1	2	yes	—
69	C₂²	C₂×C₄×D₂₃₃	1	1	yes	characteristic
70	C₂²	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
71	C₂²	C₂×D₉₃₂	1	1	yes	characteristic
72	C₂²	Q₈×D₂₃₃	1	2	yes	—
73	C₂²	#16836593	1	2	yes	—
74	C₄	—	4	2	no	—
75	C₄	—	466	4	no	—
76	C₄	—	466	4	no	—
77	C₄	#16836571	1	4	yes	—
78	C₂	C₂×#16836571	1	1	yes	characteristic
79	C₂	#21418500	1	2	yes	—
80	C₂	#21418515	1	2	yes	—
81	C₂	#21418522	1	1	yes	characteristic
82	C₂	#21418533	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃₃
commutator	C₂²×C₂₃₃	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₂₃₃	C₂
socle, prime radical, cosocle	C₂³×C₂₃₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₂₃₃
hypocenter	C₂₃₃	#391
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂×Dic₄₆₆	C₄	semidirect
20. C₂×Dic₂₃₃	Q₈	semidirect
41. C₂₃₃	#391	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₆₆	2
C₂×C₄×Dic₂₃₃	2
#21418442	2
C₂×C₄⋊C₄×C₂₃₃	2
#391	233