Subgroups — #16836459

Order 3712.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272221	C₂	1	8	yes	maximal, complemented
2	C₂×#11745214	C₂	1	6	yes	maximal, complemented
3	C₂×#11745310	C₂	1	1	yes	maximal, characteristic
4	C₂×#41×C₂₉	C₄	1	1	yes	characteristic
5	#11745310	C₄	1	4	yes	—
6	#11745310	C₂²	1	4	yes	—
7	C₂×#90777	C₄	1	3	yes	—
8	C₂²×#6597	C₂²	1	3	yes	—
9	#11745214	C₂²	1	4	yes	—
10	#11745214	C₂²	1	12	yes	complemented
11	C₂×#90770	C₂²	1	12	yes	complemented
12	#41×C₂₉	C₈	1	4	yes	—
13	#41×C₂₉	C₂×C₄	1	4	yes	—
14	C₂×Q₈×C₂₉	C₈	1	1	yes	characteristic
15	C₂×D₄×C₂₉	C₈	1	3	yes	—
16	C₂²×C₄×C₂₉	C₂×C₄	1	3	yes	—
17	#90777	C₂×C₄	1	12	yes	—
18	C₂×#6597	C₂×C₄	1	3	yes	—
19	C₂×#6597	C₂×C₄	1	6	yes	—
20	C₂×#6597	C₂³	1	1	yes	characteristic
21	C₂×#6597	C₂³	1	6	yes	—
22	#90770	C₂³	1	8	yes	—
23	C₂³×C₂₉	C₂×C₈	1	3	yes	—
24	Q₈×C₂₉	C₂×C₈	1	4	yes	—
25	D₄×C₂₉	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₂₉	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₂₉	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₉	C₂²×C₄	1	6	yes	—
30	#6597	C₂²×C₄	1	1	yes	characteristic
31	#6597	C₂²×C₄	1	6	yes	—
32	#6597	C₂⁴	1	1	yes	characteristic
33	#3407	—	29	1	no	maximal, Sylow
34	C₂²×C₂₉	—	4	3	no	—
35	C₂²×C₂₉	C₂²×C₈	1	1	yes	characteristic
36	C₂²×C₂₉	C₂²×C₈	1	6	yes	—
37	C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₂₉	C₂²×C₈	1	6	yes	—
39	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
40	C₂×#131	—	29	1	no	—
41	#504	—	29	8	no	—
42	C₂×M₃₂	—	29	6	no	—
43	C₂×C₂₉	—	2	1	no	—
44	C₂×C₂₉	—	2	6	no	—
45	C₂×C₂₉	C₂³×C₈	1	1	yes	characteristic
46	C₂×#41	#1181	1	1	yes	characteristic
47	#131	—	29	4	no	—
48	#131	—	29	4	no	—
49	C₂×M₁₆	—	29	3	no	—
50	C₂²×C₈	—	29	3	no	—
51	M₃₂	—	29	4	no	—
52	M₃₂	—	29	12	no	—
53	C₂×C₁₆	—	29	12	no	—
54	C₂₉	#3407	1	1	yes	characteristic, complemented, Sylow
55	#41	#6599	1	4	yes	—
56	#41	C₂×#1181	1	4	yes	—
57	C₂×Q₈	#6599	1	1	yes	characteristic
58	C₂×D₄	#6599	1	3	yes	—
59	C₂²×C₄	C₂×#1181	1	3	yes	—
60	M₁₆	—	29	12	no	—
61	C₂×C₈	—	29	6	no	—
62	C₂×C₈	—	29	1	no	—
63	C₂×C₈	—	29	3	no	—
64	C₂×C₈	—	29	6	no	—
65	C₁₆	—	29	8	no	—
66	C₂³	C₂×#6599	1	3	yes	—
67	Q₈	C₂×#6599	1	4	yes	—
68	D₄	C₂×#6599	1	12	yes	—
69	C₂×C₄	C₂×#6599	1	3	yes	—
70	C₂×C₄	C₂×#6599	1	6	yes	—
71	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
72	C₂×C₄	C₂²×#1181	1	6	yes	—
73	C₈	—	29	1	no	—
74	C₈	—	29	1	no	—
75	C₈	—	29	6	no	—
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×#6599	1	1	yes	characteristic
78	C₂²	C₂²×#6599	1	6	yes	—
79	C₄	C₂²×#6599	1	1	yes	characteristic
80	C₄	C₂²×#6599	1	6	yes	—
81	C₄	C₂³×#1181	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×#6599	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#1181
commutator, socle	C₂×C₂₉	C₂³×C₈
Fitting, prime radical	C₂×#41×C₂₉	C₄
cosocle	#6597	C₂⁴
hypercenter	C₂×#41	#1181
hypocenter	C₂₉	#3407
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #14272221	C₂	semidirect
2. C₂×#11745214	C₂	semidirect
10. #11745214	C₂²	semidirect
11. C₂×#90770	C₂²	semidirect
54. C₂₉	#3407	semidirect

Maximal subgroups

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

subgroup	index
#14272221	2
C₂×#11745214	2
C₂×#11745310	2
#3407	29