Subgroups — #41×#4223 #14581402

Order 2624.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#4223	C₂	1	2	yes	maximal, complemented
2	D₄×#4223	C₂	1	6	yes	maximal, complemented
3	#13050243	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×#4223	C₂	1	3	yes	maximal, complemented
5	#41×D₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#4223	C₂²	1	6	yes	—
7	#41×C₄₁	C₄	1	1	yes	characteristic, complemented
8	#10620310	C₄	1	1	yes	characteristic, complemented
9	Q₈×D₄₁	C₂²	1	1	yes	characteristic
10	#10620308	C₄	1	3	yes	complemented
11	D₄×D₄₁	C₂²	1	3	yes	—
12	#10620306	C₄	1	3	yes	complemented
13	C₂×C₄×D₄₁	C₂²	1	3	yes	—
14	C₂²⋊#4223	C₂²	1	6	yes	—
15	C₄⋊#4223	C₂²	1	6	yes	—
16	C₄×#4223	C₂²	1	1	yes	characteristic
17	C₄×#4223	C₂²	1	3	yes	—
18	C₄×#4223	C₂²	1	6	yes	—
19	C₂²×D₄₁	C₂³	1	3	yes	—
20	C₂×#4223	—	2	12	no	—
21	C₂×#4223	C₂³	1	6	yes	—
22	C₂×#4223	C₂³	1	2	yes	—
23	Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
24	D₄×C₄₁	C₂×C₄	1	3	yes	complemented
25	C₂×C₄×C₄₁	C₂×C₄	1	3	yes	complemented
26	C₂²⋊D₄₁	C₂×C₄	1	6	yes	complemented
27	C₂×Dic₄₁	C₂×C₄	1	3	yes	complemented
28	D₁₆₄	C₂×C₄	1	3	yes	complemented
29	C₄×D₄₁	C₂×C₄	1	3	yes	complemented
30	C₄×D₄₁	C₂×C₄	1	3	yes	complemented
31	C₄×D₄₁	C₂³	1	1	yes	characteristic
32	C₄×D₄₁	C₂³	1	3	yes	—
33	Dic₈₂	C₂×C₄	1	3	yes	complemented
34	C₂²×C₄₁	C₂²×C₄	1	3	yes	—
35	C₂×D₄₁	—	2	6	no	—
36	C₂×D₄₁	C₂²×C₄	1	3	yes	—
37	C₂×D₄₁	C₂⁴	1	1	yes	characteristic
38	#4223	—	2	6	no	—
39	#4223	#41	1	4	yes	direct product factor
40	C₄×C₄₁	C₂²×C₄	1	3	yes	—
41	C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
42	Dic₄₁	C₂²×C₄	1	3	yes	—
43	Dic₄₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄₁	—	2	3	no	—
45	C₂×C₄₁	C₂³×C₄	1	1	yes	characteristic
46	D₄₁	—	2	3	no	—
47	D₄₁	C₂×#41	1	2	yes	—
48	C₄×#41	—	41	1	no	maximal, Sylow
49	C₄₁	C₄×#41	1	1	yes	characteristic, complemented, Sylow
50	C₂×#41	—	41	1	no	—
51	C₄×Q₈	—	41	2	no	—
52	C₄×D₄	—	41	6	no	—
53	#117	—	41	3	no	—
54	C₂×C₄²	—	41	3	no	—
55	#41	—	41	1	no	—
56	#41	—	41	3	no	—
57	#41	—	41	3	no	—
58	#41	#4223	1	1	yes	characteristic, direct product factor
59	C₂×Q₈	—	41	1	no	—
60	C₂×D₄	—	41	3	no	—
61	C₂²×C₄	—	41	3	no	—
62	C₂²×C₄	—	41	6	no	—
63	C₄⋊C₄	—	41	6	no	—
64	C₂²⋊C₄	—	41	6	no	—
65	C₄²	—	41	1	no	—
66	C₄²	—	41	3	no	—
67	C₄²	—	41	6	no	—
68	C₂³	—	41	3	no	—
69	Q₈	—	41	3	no	—
70	Q₈	C₂×#4223	1	1	yes	characteristic, complemented
71	D₄	—	41	6	no	—
72	D₄	—	41	3	no	—
73	D₄	C₂×#4223	1	3	yes	complemented
74	C₂×C₄	—	41	2	no	—
75	C₂×C₄	—	41	3	no	—
76	C₂×C₄	—	41	3	no	—
77	C₂×C₄	—	41	3	no	—
78	C₂×C₄	—	41	1	no	—
79	C₂×C₄	—	41	3	no	—
80	C₂×C₄	—	41	6	no	—
81	C₂×C₄	—	82	12	no	—
82	C₂×C₄	C₂×#4223	1	3	yes	complemented
83	C₂²	—	41	1	no	—
84	C₂²	—	41	3	no	—
85	C₂²	—	82	6	no	—
86	C₂²	C₂²×#4223	1	3	yes	—
87	C₄	—	41	4	no	—
88	C₄	—	41	1	no	—
89	C₄	—	41	3	no	—
90	C₄	—	82	6	no	—
91	C₄	C₂²×#4223	1	3	yes	—
92	C₄	C₂²×#4223	1	1	yes	characteristic
93	C₂	—	2	3	no	—
94	C₂	—	41	2	no	—
95	C₂	—	82	3	no	—
96	C₂	C₂³×#4223	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#4223
commutator, socle	C₂×C₄₁	C₂³×C₄
Frattini	C₂	C₂³×#4223
Fitting	#41×C₄₁	C₄
prime radical	#41×D₄₁	C₂
cosocle	C₂×D₄₁	C₂⁴
hypercenter	#41	#4223
hypocenter	C₄₁	C₄×#41
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. Q₈×#4223	C₂	semidirect
2. D₄×#4223	C₂	semidirect
3. #13050243	C₂	semidirect
4. C₂×C₄×#4223	C₂	semidirect
7. #41×C₄₁	C₄	semidirect
8. #10620310	C₄	semidirect
10. #10620308	C₄	semidirect
12. #10620306	C₄	semidirect
23. Q₈×C₄₁	C₂×C₄	semidirect
24. D₄×C₄₁	C₂×C₄	semidirect
25. C₂×C₄×C₄₁	C₂×C₄	semidirect
26. C₂²⋊D₄₁	C₂×C₄	semidirect
27. C₂×Dic₄₁	C₂×C₄	semidirect
28. D₁₆₄	C₂×C₄	semidirect
29. C₄×D₄₁	C₂×C₄	semidirect
30. C₄×D₄₁	C₂×C₄	semidirect
33. Dic₈₂	C₂×C₄	semidirect
39. #4223	#41	direct (N × H)
49. C₄₁	C₄×#41	semidirect
58. #41	#4223	direct (N × H)
70. Q₈	C₂×#4223	semidirect
73. D₄	C₂×#4223	semidirect
82. C₂×C₄	C₂×#4223	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#4223	2
D₄×#4223	2
#13050243	2
C₂×C₄×#4223	2
#41×D₄₁	2
C₄×#41	41