Subgroups — #28416776

Order 14912.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418514	C₂	1	1	yes	maximal, characteristic
2	#21418468	C₂	1	1	yes	maximal, characteristic, complemented
3	#21418466	C₂	1	1	yes	maximal, characteristic, complemented
4	#21418458	C₂	1	1	yes	maximal, characteristic, complemented
5	#21418458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#14272338	C₂	1	1	yes	maximal, characteristic
7	#117×C₂₃₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₆₆	C₂²	1	1	yes	characteristic
9	C₂×#14272338	C₂²	1	1	yes	characteristic
10	C₂×#14272338	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
12	#16836573	—	2	1	no	—
13	C₂²⋊Dic₂₃₃	—	2	1	no	—
14	C₂²×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₂₃₃	—	2	1	no	—
18	C₄²×C₂₃₃	—	2	1	no	—
19	Dic₄₆₆	—	2	1	no	—
20	Dic₄₆₆	—	2	1	no	—
21	C₂×Dic₂₃₃	—	2	1	no	—
22	C₂×Dic₂₃₃	—	2	1	no	—
23	#14272338	—	2	1	no	—
24	#14272338	—	2	1	no	—
25	C₂³×C₂₃₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₃₃	—	2	1	no	—
27	C₂×C₄×C₂₃₃	—	2	1	no	—
28	C₂×C₄×C₂₃₃	—	2	1	no	—
29	C₂×C₄×C₂₃₃	—	2	1	no	—
30	C₂×C₄×C₂₃₃	D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
32	C₂²×C₂₃₃	—	2	1	no	—
33	C₂²×C₂₃₃	—	2	1	no	—
34	C₂²×C₂₃₃	—	2	1	no	—
35	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₃₃	—	2	1	no	—
37	C₄×C₂₃₃	—	4	1	no	—
38	C₄×C₂₃₃	—	4	1	no	—
39	C₄×C₂₃₃	#41	1	1	yes	characteristic
40	C₄×C₂₃₃	#41	1	1	yes	characteristic
41	Dic₂₃₃	—	4	1	no	—
42	Dic₂₃₃	—	4	1	no	—
43	C₂×C₂₃₃	—	4	1	no	—
44	C₂×C₂₃₃	#123	1	1	yes	characteristic
45	C₂×C₂₃₃	#135	1	1	yes	characteristic
46	C₂×C₂₃₃	#137	1	1	yes	characteristic
47	C₂₃₃	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	233	1	no	maximal, Sylow
49	C₂²⋊Q₈	—	233	1	no	—
50	#117	D₂₃₃	1	1	yes	characteristic
51	#107	—	233	1	no	—
52	#106	—	233	1	no	—
53	#103	—	233	1	no	—
54	#103	—	233	1	no	—
55	C₂²⋊C₈	—	233	1	no	—
56	C₂×Q₈	—	233	1	no	—
57	C₂²×C₄	C₂×D₂₃₃	1	1	yes	characteristic
58	C₂×C₈	—	233	1	no	—
59	C₂×C₈	—	233	1	no	—
60	C₄⋊C₄	—	466	1	no	—
61	C₄⋊C₄	—	233	1	no	—
62	C₄⋊C₄	C₂×D₂₃₃	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₂₃₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	466	1	no	—
66	C₄²	—	2	1	no	—
67	C₂³	C₂²⋊D₂₃₃	1	1	yes	characteristic
68	Q₈	—	466	1	no	—
69	Q₈	—	466	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	466	1	no	—
75	C₂×C₄	—	466	1	no	—
76	C₂×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
78	C₈	—	466	1	no	—
79	C₈	—	466	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	932	1	no	—
88	C₄	—	932	1	no	—
89	C₄	#16836594	1	1	yes	characteristic
90	C₄	#16836594	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#21418518	1	1	yes	characteristic
93	C₂	#21418555	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	#117×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×D₄
prime radical	C₂³×C₂₃₃	D₄
hypercenter	#117	D₂₃₃
hypocenter	C₂₃₃	#485
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. #21418468	C₂	semidirect
3. #21418466	C₂	semidirect
4. #21418458	C₂	semidirect
5. #21418458	C₂	semidirect
47. C₂₃₃	#485	semidirect

Maximal subgroups

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

subgroup	index
#21418514	2
#21418468	2
#21418466	2
#21418458	2
C₂²⋊#14272338	2
#117×C₂₃₃	2
#485	233