Subgroups — #17996626

Order 4736.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14557715	C₂	1	3	yes	maximal
2	#14557332	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#11924584	C₂²	1	1	yes	characteristic
5	#11924567	—	2	3	no	—
6	#11924464	—	2	3	no	—
7	#11924455	C₂²	1	3	yes	—
8	C₄⋊Q₈×C₃₇	—	2	3	no	—
9	C₄⋊Q₈×C₃₇	C₂²	1	3	yes	—
10	C₄×Q₈×C₃₇	—	2	3	no	—
11	C₂×Q₈×C₃₇	—	2	1	no	—
12	C₂×Q₈×C₃₇	—	2	1	no	—
13	C₂×Q₈×C₃₇	D₄	1	3	yes	—
14	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
15	#10596966	—	4	1	no	—
16	Q₈×D₃₇	—	4	1	no	—
17	#10596962	—	2	3	no	—
18	C₄⋊C₄×C₃₇	—	2	3	no	—
19	C₄⋊C₄×C₃₇	—	2	6	no	—
20	C₄²×C₃₇	—	2	3	no	—
21	C₄²×C₃₇	D₄	1	3	yes	—
22	#10596942	—	4	3	no	—
23	#10596941	—	4	3	no	—
24	#10596934	—	2	3	no	—
25	Q₈×C₃₇	—	2	4	no	—
26	Q₈×C₃₇	—	4	1	no	—
27	Q₈×C₃₇	—	4	3	no	—
28	Q₈×C₃₇	—	8	1	no	—
29	C₂×C₄×C₃₇	—	2	3	no	—
30	C₂×C₄×C₃₇	—	2	3	no	—
31	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
32	C₂²⋊D₃₇	—	4	1	no	—
33	D₁₄₈	—	2	3	no	—
34	C₄×D₃₇	—	4	3	no	—
35	Dic₇₄	—	2	3	no	—
36	#64436	—	4	3	no	—
37	C₂²×C₃₇	#120	1	1	yes	characteristic
38	C₂×D₃₇	—	4	1	no	—
39	C₄×C₃₇	—	2	6	no	—
40	C₄×C₃₇	—	2	3	no	—
41	C₄×C₃₇	—	4	3	no	—
42	Dic₃₇	—	4	1	no	—
43	#2205	—	37	1	no	maximal, Sylow
44	C₂×C₃₇	—	2	1	no	—
45	C₂×C₃₇	B(4,𝔽₂)	1	1	yes	characteristic
46	D₃₇	—	8	1	no	—
47	Q₈²	D₃₇	1	1	yes	characteristic, complemented
48	#456	—	37	3	no	—
49	#356	—	37	3	no	—
50	C₃₇	#2205	1	1	yes	characteristic, complemented, Sylow
51	2₋¹⁺⁴	—	37	1	no	—
52	#137	—	74	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂×D₃₇	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	74	3	no	—
57	#101	—	37	3	no	—
58	#41	—	74	3	no	—
59	#41	—	148	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	148	1	no	—
63	C₂×Q₈	C₂²⋊D₃₇	1	3	yes	—
64	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
65	Q₁₆	—	148	3	no	—
66	SD₁₆	—	148	3	no	—
67	M₁₆	—	74	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₃₇	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	—	74	3	no	—
77	D₄	—	74	3	no	—
78	D₄	—	148	1	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	148	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
83	C₈	—	148	3	no	—
84	C₂²	—	148	1	no	—
85	C₂²	#11924505	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	148	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	296	1	no	—
92	C₂	#14557720	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14557720
commutator, cosocle	C₂×Q₈×C₃₇	C₂³
Frattini	C₂×Q₈	C₂²×D₃₇
Fitting	Q₈²×C₃₇	C₂
socle	C₂×C₃₇	B(4,𝔽₂)
prime radical	#11924584	C₂²
hypercenter	Q₈²	D₃₇
hypocenter	C₃₇	#2205
LCS	G ▷ C₂×Q₈×C₃₇ ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #14557332	C₂	semidirect
3. Q₈²×C₃₇	C₂	semidirect
47. Q₈²	D₃₇	semidirect
50. C₃₇	#2205	semidirect

Maximal subgroups

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

subgroup	index
#14557715	2
#14557332	2
Q₈²×C₃₇	2
#2205	37