Subgroups — #42121861

Order 27712.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₄₃₃	C₂	1	1	yes	maximal, characteristic
2	#28357149	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#16812358	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₄₃₃	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₄₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₃₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₄₃₃	C₄	1	1	yes	characteristic
8	C₂×#16812358	C₂²	1	2	yes	—
9	C₄×Dic₄₃₃	C₄	1	1	yes	characteristic
10	C₄×Dic₄₃₃	C₄	1	1	yes	characteristic
11	C₄×Dic₄₃₃	C₂²	1	1	yes	characteristic
12	C₄×Dic₄₃₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₃₃	C₂²	1	2	yes	—
15	C₂×Dic₄₃₃	—	2	2	no	—
16	C₂×Dic₄₃₃	C₈	1	4	yes	complemented
17	C₂×Dic₄₃₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₄₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₄₃₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₄₃₃	C₂×C₄	1	1	yes	characteristic
21	#16812358	—	2	2	no	—
22	C₂³×C₄₃₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₄₃₃	—	2	2	no	—
24	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
26	C₈×C₄₃₃	—	2	2	no	—
27	C₂²×C₄₃₃	—	2	1	no	—
28	C₂²×C₄₃₃	—	2	1	no	—
29	C₂²×C₄₃₃	C₂×C₈	1	2	yes	—
30	C₂²×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₄₃₃	—	2	1	no	—
32	C₄×C₄₃₃	#41	1	2	yes	—
33	Dic₄₃₃	—	2	1	no	—
34	Dic₄₃₃	C₂×C₈	1	2	yes	—
35	Dic₄₃₃	C₂×C₈	1	2	yes	—
36	Dic₄₃₃	M₁₆	1	2	yes	—
37	C₂×C₄₃₃	—	2	2	no	—
38	C₂×C₄₃₃	#117	1	1	yes	characteristic
39	C₂×C₄₃₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₄₃₃	C₂×M₁₆	1	1	yes	characteristic
41	C₄₃₃	#431	1	1	yes	characteristic, complemented, Sylow
42	#431	—	433	1	no	maximal, Sylow
43	C₂×C₄²	—	433	1	no	—
44	C₄⋊C₈	—	433	2	no	—
45	C₂²⋊C₈	—	433	1	no	—
46	C₂²⋊C₈	D₄₃₃	1	1	yes	characteristic
47	C₄×C₈	—	433	2	no	—
48	C₂²×C₄	—	433	1	no	—
49	C₂²×C₄	—	433	1	no	—
50	C₂²×C₄	C₂×D₄₃₃	1	1	yes	characteristic
51	C₂×C₈	—	433	2	no	—
52	C₂×C₈	C₂×D₄₃₃	1	2	yes	—
53	C₄²	—	433	1	no	—
54	C₄²	—	433	1	no	—
55	C₄²	—	433	1	no	—
56	C₄²	—	433	1	no	—
57	C₂³	C₄×D₄₃₃	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	866	2	no	—
60	C₂×C₄	—	433	1	no	—
61	C₂×C₄	—	433	4	no	—
62	C₂×C₄	—	433	1	no	—
63	C₂×C₄	—	433	1	no	—
64	C₂×C₄	—	433	1	no	—
65	C₂×C₄	C₄×D₄₃₃	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	866	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	C₈×D₄₃₃	1	2	yes	—
72	C₂²	C₂×C₄×D₄₃₃	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	866	1	no	—
75	C₄	—	433	2	no	—
76	C₄	—	433	2	no	—
77	C₄	—	433	2	no	—
78	C₄	#20294096	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	#28357176	1	1	yes	characteristic
81	C₂	C₂×C₈×D₄₃₃	1	1	yes	characteristic
82	C₂	M₁₆×D₄₃₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₃₃
commutator	C₂×C₄₃₃	C₂²×C₈
Fitting	C₂²⋊C₈×C₄₃₃	C₂
socle	C₂²×C₄₃₃	C₂²×C₄
prime radical	C₂³×C₄₃₃	C₂×C₄
cosocle	C₂×C₄×C₄₃₃	C₂³
hypercenter	C₂²⋊C₈	D₄₃₃
hypocenter	C₄₃₃	#431
LCS	G ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #28357149	C₂	semidirect
4. C₈×Dic₄₃₃	C₂	semidirect
16. C₂×Dic₄₃₃	C₈	semidirect
41. C₄₃₃	#431	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₄₃₃	2
#28357149	2
C₂²⋊#16812358	2
C₈×Dic₄₃₃	2
C₂²⋊C₈×C₄₃₃	2
#431	433