Subgroups — #41×#6923 #16840828

Order 3904.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6923	C₂	1	2	yes	maximal, complemented
2	D₄×#6923	C₂	1	6	yes	maximal, complemented
3	#14519856	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×#6923	C₂	1	3	yes	maximal, complemented
5	#41×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6923	C₂²	1	6	yes	—
7	#41×C₆₁	C₄	1	1	yes	characteristic, complemented
8	#11758226	C₄	1	1	yes	characteristic, complemented
9	Q₈×D₆₁	C₂²	1	1	yes	characteristic
10	#11758224	C₄	1	3	yes	complemented
11	D₄×D₆₁	C₂²	1	3	yes	—
12	#11758222	C₄	1	3	yes	complemented
13	C₂×C₄×D₆₁	C₂²	1	3	yes	—
14	C₂²⋊#6923	C₂²	1	6	yes	—
15	C₄⋊#6923	C₂²	1	6	yes	—
16	C₄×#6923	C₂²	1	1	yes	characteristic
17	C₄×#6923	C₂²	1	3	yes	—
18	C₄×#6923	C₂²	1	6	yes	—
19	C₂²×D₆₁	C₂³	1	3	yes	—
20	C₂×#6923	—	2	12	no	—
21	C₂×#6923	C₂³	1	6	yes	—
22	C₂×#6923	C₂³	1	2	yes	—
23	Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
24	D₄×C₆₁	C₂×C₄	1	3	yes	complemented
25	C₂×C₄×C₆₁	C₂×C₄	1	3	yes	complemented
26	C₂²⋊D₆₁	C₂×C₄	1	6	yes	complemented
27	C₂×Dic₆₁	C₂×C₄	1	3	yes	complemented
28	D₂₄₄	C₂×C₄	1	3	yes	complemented
29	C₄×D₆₁	C₂×C₄	1	3	yes	complemented
30	C₄×D₆₁	C₂×C₄	1	3	yes	complemented
31	C₄×D₆₁	C₂³	1	1	yes	characteristic
32	C₄×D₆₁	C₂³	1	3	yes	—
33	Dic₁₂₂	C₂×C₄	1	3	yes	complemented
34	C₂²×C₆₁	C₂²×C₄	1	3	yes	—
35	C₂×D₆₁	—	2	6	no	—
36	C₂×D₆₁	C₂²×C₄	1	3	yes	—
37	C₂×D₆₁	C₂⁴	1	1	yes	characteristic
38	#6923	—	2	6	no	—
39	#6923	#41	1	4	yes	direct product factor
40	C₄×C₆₁	C₂²×C₄	1	3	yes	—
41	C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	Dic₆₁	C₂²×C₄	1	3	yes	—
43	Dic₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₆₁	—	2	3	no	—
45	C₂×C₆₁	C₂³×C₄	1	1	yes	characteristic
46	D₆₁	—	2	3	no	—
47	D₆₁	C₂×#41	1	2	yes	—
48	C₄×#41	—	61	1	no	maximal, Sylow
49	C₆₁	C₄×#41	1	1	yes	characteristic, complemented, Sylow
50	C₂×#41	—	61	1	no	—
51	C₄×Q₈	—	61	2	no	—
52	C₄×D₄	—	61	6	no	—
53	#117	—	61	3	no	—
54	C₂×C₄²	—	61	3	no	—
55	#41	—	61	1	no	—
56	#41	—	61	3	no	—
57	#41	—	61	3	no	—
58	#41	#6923	1	1	yes	characteristic, direct product factor
59	C₂×Q₈	—	61	1	no	—
60	C₂×D₄	—	61	3	no	—
61	C₂²×C₄	—	61	3	no	—
62	C₂²×C₄	—	61	6	no	—
63	C₄⋊C₄	—	61	6	no	—
64	C₂²⋊C₄	—	61	6	no	—
65	C₄²	—	61	1	no	—
66	C₄²	—	61	3	no	—
67	C₄²	—	61	6	no	—
68	C₂³	—	61	3	no	—
69	Q₈	—	61	3	no	—
70	Q₈	C₂×#6923	1	1	yes	characteristic, complemented
71	D₄	—	61	6	no	—
72	D₄	—	61	3	no	—
73	D₄	C₂×#6923	1	3	yes	complemented
74	C₂×C₄	—	61	2	no	—
75	C₂×C₄	—	61	3	no	—
76	C₂×C₄	—	61	3	no	—
77	C₂×C₄	—	61	3	no	—
78	C₂×C₄	—	61	3	no	—
79	C₂×C₄	—	61	1	no	—
80	C₂×C₄	—	61	6	no	—
81	C₂×C₄	—	122	12	no	—
82	C₂×C₄	C₂×#6923	1	3	yes	complemented
83	C₂²	—	61	1	no	—
84	C₂²	—	61	3	no	—
85	C₂²	—	122	6	no	—
86	C₂²	C₂²×#6923	1	3	yes	—
87	C₄	—	61	4	no	—
88	C₄	—	61	1	no	—
89	C₄	—	61	3	no	—
90	C₄	—	122	6	no	—
91	C₄	C₂²×#6923	1	3	yes	—
92	C₄	C₂²×#6923	1	1	yes	characteristic
93	C₂	—	2	3	no	—
94	C₂	—	61	2	no	—
95	C₂	—	122	3	no	—
96	C₂	C₂³×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#6923
commutator, socle	C₂×C₆₁	C₂³×C₄
Frattini	C₂	C₂³×#6923
Fitting	#41×C₆₁	C₄
prime radical	#41×D₆₁	C₂
cosocle	C₂×D₆₁	C₂⁴
hypercenter	#41	#6923
hypocenter	C₆₁	C₄×#41
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. Q₈×#6923	C₂	semidirect
2. D₄×#6923	C₂	semidirect
3. #14519856	C₂	semidirect
4. C₂×C₄×#6923	C₂	semidirect
7. #41×C₆₁	C₄	semidirect
8. #11758226	C₄	semidirect
10. #11758224	C₄	semidirect
12. #11758222	C₄	semidirect
23. Q₈×C₆₁	C₂×C₄	semidirect
24. D₄×C₆₁	C₂×C₄	semidirect
25. C₂×C₄×C₆₁	C₂×C₄	semidirect
26. C₂²⋊D₆₁	C₂×C₄	semidirect
27. C₂×Dic₆₁	C₂×C₄	semidirect
28. D₂₄₄	C₂×C₄	semidirect
29. C₄×D₆₁	C₂×C₄	semidirect
30. C₄×D₆₁	C₂×C₄	semidirect
33. Dic₁₂₂	C₂×C₄	semidirect
39. #6923	#41	direct (N × H)
49. C₆₁	C₄×#41	semidirect
58. #41	#6923	direct (N × H)
70. Q₈	C₂×#6923	semidirect
73. D₄	C₂×#6923	semidirect
82. C₂×C₄	C₂×#6923	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6923	2
D₄×#6923	2
#14519856	2
C₂×C₄×#6923	2
#41×D₆₁	2
C₄×#41	61