Subgroups — #28417315

Order 14912.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16836603	C₂	1	2	yes	maximal
2	C₂×#16836573	C₂	1	1	yes	maximal, characteristic
3	#16836603	—	2	2	no	—
4	#16836603	C₄	1	4	yes	complemented
5	C₂×#14272345	—	2	2	no	—
6	C₂²×Dic₂₃₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₃₃	C₂²	1	1	yes	characteristic
8	#16836573	—	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	—	4	1	no	—
13	C₂×Dic₂₃₃	C₂×C₄	1	2	yes	—
14	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
15	C₂×Dic₂₃₃	D₄	1	2	yes	—
16	C₂×Dic₂₃₃	Q₈	1	1	yes	characteristic
17	C₂³×C₂₃₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃₃	—	2	1	no	—
19	C₂×C₄×C₂₃₃	—	4	1	no	—
20	C₂²×C₂₃₃	—	2	1	no	—
21	C₂²×C₂₃₃	—	2	1	no	—
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	C₄×C₂₃₃	—	4	1	no	—
26	Dic₂₃₃	—	4	1	no	—
27	Dic₂₃₃	C₂²⋊C₄	1	2	yes	—
28	Dic₂₃₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₂₃₃	—	2	1	no	—
30	C₂×C₂₃₃	—	2	1	no	—
31	C₂×C₂₃₃	#95	1	1	yes	characteristic
32	C₂×C₂₃₃	#100	1	1	yes	characteristic
33	C₂×C₂₃₃	#101	1	1	yes	characteristic
34	C₂₃₃	#343	1	1	yes	characteristic, complemented, Sylow
35	#343	—	233	1	no	maximal, Sylow
36	C₂×M₁₆	—	233	2	no	—
37	C₂×C₄⋊C₄	—	233	1	no	—
38	C₂²×C₄	—	233	1	no	—
39	C₂²×C₄	—	233	1	no	—
40	C₂²×C₄	#11745400	1	1	yes	characteristic
41	M₁₆	—	466	2	no	—
42	M₁₆	—	233	4	no	—
43	C₂×C₈	—	466	2	no	—
44	C₄⋊C₄	—	466	2	no	—
45	C₂³	C₂×#11745400	1	1	yes	characteristic
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	932	1	no	—
49	C₂×C₄	—	466	1	no	—
50	C₂×C₄	—	233	2	no	—
51	C₂×C₄	—	233	2	no	—
52	C₂×C₄	—	233	1	no	—
53	C₂×C₄	—	233	1	no	—
54	C₈	—	466	4	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	C₄×#11745400	1	1	yes	characteristic
58	C₂²	C₂²⋊#11745400	1	1	yes	characteristic
59	C₂²	C₄⋊#11745400	1	1	yes	characteristic
60	C₄	—	4	1	no	—
61	C₄	—	932	1	no	—
62	C₄	—	233	2	no	—
63	C₄	—	233	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	2	1	no	—
66	C₂	(C₂×#11745400)⋊C₄	1	1	yes	characteristic
67	C₂	#21418595	1	1	yes	characteristic
68	C₂	#21418596	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11745400
commutator	C₂²×C₂₃₃	C₄²
Frattini	C₂³	C₂×#11745400
Fitting	C₂²×C₄×C₂₃₃	C₄
socle	C₂²×C₂₃₃	C₂²⋊C₄
prime radical	C₂³×C₂₃₃	C₂×C₄
cosocle	C₂²×Dic₂₃₃	C₂²
hypercenter	C₂²×C₄	#11745400
hypocenter	C₂₃₃	#343
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #16836603	C₄	semidirect
34. C₂₃₃	#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16836603	2
C₂×#16836573	2
#343	233