Subgroups — #28416573

Order 14912.

158 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, complemented
2	C₂×#16836582	C₂	1	2	yes	maximal, complemented
3	#21418458	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#14272338	C₂	1	1	yes	maximal, characteristic, complemented
5	#121×C₂₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₆₆	—	2	4	no	—
7	C₂×Dic₄₆₆	—	2	1	no	—
8	C₂×Dic₄₆₆	C₂²	1	2	yes	—
9	C₂²×Dic₂₃₃	—	2	1	no	—
10	#16836582	—	2	4	no	—
11	C₂×#14272338	C₂²	1	2	yes	—
12	C₂×D₄×C₂₃₃	—	2	1	no	—
13	C₂×D₄×C₂₃₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₂₃₃	—	2	1	no	—
17	Dic₄₆₆	—	2	4	no	—
18	Dic₄₆₆	—	2	2	no	—
19	Dic₄₆₆	D₄	1	4	yes	complemented
20	C₂×Dic₂₃₃	—	2	4	no	—
21	C₂×Dic₂₃₃	—	2	2	no	—
22	#14272338	—	2	2	no	—
23	C₂³×C₂₃₃	—	2	1	no	—
24	C₂³×C₂₃₃	D₄	1	1	yes	characteristic
25	D₄×C₂₃₃	—	2	2	no	—
26	D₄×C₂₃₃	—	4	2	no	—
27	C₂×C₄×C₂₃₃	—	2	2	no	—
28	C₂×C₄×C₂₃₃	—	2	1	no	—
29	C₂×C₄×C₂₃₃	D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
31	C₂²×C₂₃₃	—	2	1	no	—
32	C₂²×C₂₃₃	—	2	1	no	—
33	C₂²×C₂₃₃	—	4	1	no	—
34	C₂²×C₂₃₃	—	4	1	no	—
35	C₂²×C₂₃₃	—	4	1	no	—
36	C₂²×C₂₃₃	SD₁₆	1	2	yes	complemented
37	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₂₃₃	—	2	1	no	—
39	C₄×C₂₃₃	—	4	1	no	—
40	C₄×C₂₃₃	C₂×D₄	1	2	yes	—
41	Dic₂₃₃	—	2	4	no	—
42	C₂×C₂₃₃	—	2	2	no	—
43	C₂×C₂₃₃	—	8	1	no	—
44	C₂×C₂₃₃	#120	1	1	yes	characteristic
45	C₂×C₂₃₃	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₂₃₃	#137	1	1	yes	characteristic
47	C₂₃₃	#448	1	1	yes	characteristic, complemented, Sylow
48	#448	—	233	1	no	maximal, Sylow
49	C₂²×Q₈	—	233	1	no	—
50	C₂×SD₁₆	—	233	2	no	—
51	#121	D₂₃₃	1	1	yes	characteristic
52	#103	—	233	2	no	—
53	C₂²⋊C₈	—	233	1	no	—
54	C₂×Q₈	—	466	4	no	—
55	C₂×Q₈	—	466	1	no	—
56	C₂×Q₈	—	233	2	no	—
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	C₂×D₂₃₃	1	1	yes	characteristic
59	C₂²×C₄	—	466	1	no	—
60	C₂²×C₄	C₂×D₂₃₃	1	1	yes	characteristic
61	SD₁₆	—	466	4	no	—
62	C₂×C₈	—	233	2	no	—
63	C₄⋊C₄	C₂×D₂₃₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂³	—	2	1	no	—
66	C₂³	C₂²⋊D₂₃₃	1	1	yes	characteristic
67	Q₈	—	466	4	no	—
68	Q₈	—	466	2	no	—
69	Q₈	—	233	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	4	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	466	4	no	—
75	C₂×C₄	—	466	2	no	—
76	C₂×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
78	C₈	—	466	2	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	#16836582	1	2	yes	complemented
85	C₂²	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	466	4	no	—
89	C₄	D₄×D₂₃₃	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	—	8	1	no	—
92	C₂	#21418503	1	1	yes	characteristic
93	C₂	C₂×#16836582	1	1	yes	characteristic
94	C₂	#21418565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃₃
commutator, cosocle	C₂×C₄×C₂₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₃
Fitting, prime radical	#121×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×D₄
hypercenter	#121	D₂₃₃
hypocenter	C₂₃₃	#448
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₆₆	C₂	semidirect
2. C₂×#16836582	C₂	semidirect
3. #21418458	C₂	semidirect
4. C₂²⋊#14272338	C₂	semidirect
19. Dic₄₆₆	D₄	semidirect
36. C₂²×C₂₃₃	SD₁₆	semidirect
47. C₂₃₃	#448	semidirect
84. C₂²	#16836582	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×#16836582	2
#21418458	2
C₂²⋊#14272338	2
#121×C₂₃₃	2
#448	233