Subgroups — #28417301

Order 14912.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14272345	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₃₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14272345	—	2	2	no	—
4	C₂×#14272345	—	2	2	no	—
5	C₂×#14272345	C₄	1	4	yes	complemented
6	C₂²×Dic₂₃₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₃₃	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₃₃	—	2	2	no	—
9	C₂²×C₄×C₂₃₃	C₄	1	1	yes	characteristic
10	#14272345	—	2	4	no	—
11	C₂×Dic₂₃₃	—	2	1	no	—
12	C₂×Dic₂₃₃	—	2	1	no	—
13	C₂×Dic₂₃₃	C₈	1	2	yes	complemented
14	C₂×Dic₂₃₃	C₂×C₄	1	2	yes	—
15	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
16	C₂×Dic₂₃₃	D₄	1	2	yes	—
17	C₂×Dic₂₃₃	Q₈	1	1	yes	characteristic
18	C₂³×C₂₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃₃	—	2	1	no	—
20	C₂×C₄×C₂₃₃	—	2	1	no	—
21	C₂×C₄×C₂₃₃	C₈	1	2	yes	complemented
22	C₂²×C₂₃₃	C₄²	1	1	yes	characteristic
23	C₂²×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₂₃₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₂₃₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₂₃₃	M₁₆	1	1	yes	characteristic
28	C₂²×C₂₃₃	M₁₆	1	1	yes	characteristic
29	C₄×C₂₃₃	—	2	2	no	—
30	Dic₂₃₃	—	2	2	no	—
31	Dic₂₃₃	C₂²⋊C₄	1	2	yes	—
32	Dic₂₃₃	C₄⋊C₄	1	2	yes	—
33	C₂×C₂₃₃	#95	1	1	yes	characteristic
34	C₂×C₂₃₃	C₄×C₈	1	1	yes	characteristic
35	C₂×C₂₃₃	#97	1	1	yes	characteristic
36	C₂×C₂₃₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₂₃₃	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₂₃₃	C₄⋊C₈	1	1	yes	characteristic
39	C₂×C₂₃₃	C₄⋊C₈	1	1	yes	characteristic
40	C₂₃₃	#336	1	1	yes	characteristic, complemented, Sylow
41	#336	—	233	1	no	maximal, Sylow
42	C₂²×C₈	—	233	2	no	—
43	C₂×C₄²	—	233	1	no	—
44	C₂²×C₄	—	233	1	no	—
45	C₂²×C₄	—	233	1	no	—
46	C₂²×C₄	#11745400	1	1	yes	characteristic
47	C₂×C₈	—	466	2	no	—
48	C₂×C₈	—	466	2	no	—
49	C₂×C₈	—	233	4	no	—
50	C₄²	—	466	2	no	—
51	C₂³	C₂×#11745400	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	466	1	no	—
55	C₂×C₄	—	466	1	no	—
56	C₂×C₄	—	233	2	no	—
57	C₂×C₄	—	233	2	no	—
58	C₂×C₄	—	233	2	no	—
59	C₂×C₄	—	233	1	no	—
60	C₂×C₄	—	233	1	no	—
61	C₂×C₄	#14272345	1	2	yes	complemented
62	C₈	—	466	4	no	—
63	C₂²	C₄×#11745400	1	1	yes	characteristic
64	C₂²	C₂²⋊#11745400	1	1	yes	characteristic
65	C₂²	C₄⋊#11745400	1	1	yes	characteristic
66	C₂²	C₂×#14272345	1	1	yes	characteristic
67	C₂²	#16836602	1	1	yes	characteristic
68	C₂²	#16836603	1	1	yes	characteristic
69	C₂²	#16836604	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	466	2	no	—
72	C₄	—	233	2	no	—
73	C₄	—	233	2	no	—
74	C₂	(C₂×#11745400)⋊C₄	1	1	yes	characteristic
75	C₂	C₄×#14272345	1	1	yes	characteristic
76	C₂	#21418590	1	1	yes	characteristic
77	C₂	C₂²⋊#14272345	1	1	yes	characteristic
78	C₂	#21418592	1	1	yes	characteristic
79	C₂	C₄⋊#14272345	1	1	yes	characteristic
80	C₂	#21418602	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×#11745400
commutator	C₂×C₂₃₃	C₄×C₈
Fitting	C₂²×C₄×C₂₃₃	C₄
socle, prime radical	C₂³×C₂₃₃	C₂×C₄
cosocle	C₂²×Dic₂₃₃	C₂²
hypercenter	C₂²×C₄	#11745400
hypocenter	C₂₃₃	#336
LCS	G ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
5. C₂×#14272345	C₄	semidirect
13. C₂×Dic₂₃₃	C₈	semidirect
21. C₂×C₄×C₂₃₃	C₈	semidirect
40. C₂₃₃	#336	semidirect
61. C₂×C₄	#14272345	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14272345	2
C₂×C₄×Dic₂₃₃	2
#336	233