Subgroups — #21447849

Order 7808.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16839944	C₂	1	3	yes	maximal
2	#16839561	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#14519825	C₂²	1	1	yes	characteristic
5	#14519808	—	2	3	no	—
6	#14519705	—	2	3	no	—
7	#14519696	C₂²	1	3	yes	—
8	C₄⋊Q₈×C₆₁	—	2	3	no	—
9	C₄⋊Q₈×C₆₁	C₂²	1	3	yes	—
10	C₄×Q₈×C₆₁	—	2	3	no	—
11	C₂×Q₈×C₆₁	—	2	1	no	—
12	C₂×Q₈×C₆₁	—	2	1	no	—
13	C₂×Q₈×C₆₁	D₄	1	3	yes	—
14	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
15	#11758226	—	4	1	no	—
16	Q₈×D₆₁	—	4	1	no	—
17	#11758222	—	2	3	no	—
18	C₄⋊C₄×C₆₁	—	2	3	no	—
19	C₄⋊C₄×C₆₁	—	2	6	no	—
20	C₄²×C₆₁	—	2	3	no	—
21	C₄²×C₆₁	D₄	1	3	yes	—
22	#11758202	—	4	3	no	—
23	#11758201	—	4	3	no	—
24	#11758194	—	2	3	no	—
25	Q₈×C₆₁	—	2	4	no	—
26	Q₈×C₆₁	—	4	1	no	—
27	Q₈×C₆₁	—	4	3	no	—
28	Q₈×C₆₁	—	8	1	no	—
29	C₂×C₄×C₆₁	—	2	3	no	—
30	C₂×C₄×C₆₁	—	2	3	no	—
31	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
32	C₂²⋊D₆₁	—	4	1	no	—
33	D₂₄₄	—	2	3	no	—
34	C₄×D₆₁	—	4	3	no	—
35	Dic₁₂₂	—	2	3	no	—
36	#92418	—	4	3	no	—
37	C₂²×C₆₁	#120	1	1	yes	characteristic
38	C₂×D₆₁	—	4	1	no	—
39	C₄×C₆₁	—	2	6	no	—
40	C₄×C₆₁	—	2	3	no	—
41	C₄×C₆₁	—	4	3	no	—
42	Dic₆₁	—	4	1	no	—
43	#2205	—	61	1	no	maximal, Sylow
44	C₂×C₆₁	—	2	1	no	—
45	C₂×C₆₁	B(4,𝔽₂)	1	1	yes	characteristic
46	D₆₁	—	8	1	no	—
47	Q₈²	D₆₁	1	1	yes	characteristic, complemented
48	#456	—	61	3	no	—
49	#356	—	61	3	no	—
50	C₆₁	#2205	1	1	yes	characteristic, complemented, Sylow
51	2₋¹⁺⁴	—	61	1	no	—
52	#137	—	122	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂×D₆₁	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	122	3	no	—
57	#101	—	61	3	no	—
58	#41	—	122	3	no	—
59	#41	—	244	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	244	1	no	—
63	C₂×Q₈	C₂²⋊D₆₁	1	3	yes	—
64	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
65	Q₁₆	—	244	3	no	—
66	SD₁₆	—	244	3	no	—
67	M₁₆	—	122	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₆₁	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	3	no	—
75	Q₈	—	8	1	no	—
76	Q₈	—	122	3	no	—
77	D₄	—	122	3	no	—
78	D₄	—	244	1	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	244	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
83	C₈	—	244	3	no	—
84	C₂²	—	244	1	no	—
85	C₂²	#14519746	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	244	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	488	1	no	—
92	C₂	#16839949	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16839949
commutator	C₂×Q₈×C₆₁	C₂³
Frattini	C₂×Q₈	C₂²×D₆₁
Fitting	Q₈²×C₆₁	C₂
socle	C₂×C₆₁	B(4,𝔽₂)
hypercenter	Q₈²	D₆₁
hypocenter	C₆₁	#2205
LCS	G ▷ C₂×Q₈×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #16839561	C₂	semidirect
3. Q₈²×C₆₁	C₂	semidirect
47. Q₈²	D₆₁	semidirect
50. C₆₁	#2205	semidirect