Subgroups — C₂²×#11745214 #16836453

Order 3712.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄	C₂×#1181
commutator	C₂×C₂₉	C₂³×C₈
Frattini	C₄	C₂³×#1181
Fitting	C₂³×C₄×C₂₉	C₄
socle	C₂³×C₂₉	C₂×C₈
prime radical	C₂⁴×C₂₉	C₈
cosocle	#6597	C₂⁴
hypercenter	C₂³×C₄	#1181
hypocenter	C₂₉	C₂²×M₃₂
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11745214	C₂	direct (N × H)
2. C₂²×#90770	C₂	semidirect
9. #11745214	C₂²	direct (N × H)
10. C₂×#90770	C₂²	semidirect
21. #90770	C₂³	semidirect
35. C₂²×C₂₉	M₃₂	semidirect
45. C₂×C₂₉	C₂×M₃₂	semidirect
54. C₂₉	C₂²×M₃₂	semidirect
78. C₂²	#11745214	direct (N × H)
85. C₂	C₂×#11745214	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11745214	2
C₂²×#90770	2
C₂³×#6597	2
C₂²×M₃₂	29

Subgroups — C22×#11745214 #16836453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11745214 #16836453