Subgroups — #37560150

Order 23872.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26045792	C₂	1	1	yes	maximal, characteristic
2	#26045790	C₂	1	2	yes	maximal
3	#26045784	C₂	1	2	yes	maximal
4	#26045782	C₂	1	4	yes	maximal
5	C₄×Dic₇₄₆	C₂	1	2	yes	maximal
6	C₄×Dic₇₄₆	C₂	1	1	yes	maximal, characteristic
7	#26045757	C₂	1	2	yes	maximal
8	C₄×Q₈×C₃₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₇₄₆	C₂²	1	2	yes	—
10	C₂×Dic₇₄₆	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₇₃	C₂²	1	2	yes	—
13	C₄⋊Dic₃₇₃	C₂²	1	2	yes	—
14	#20238589	C₂²	1	4	yes	—
15	#20238589	C₂²	1	2	yes	—
16	#20238589	C₂²	1	4	yes	—
17	#20238589	C₂²	1	4	yes	—
18	C₄×Dic₃₇₃	C₂²	1	4	yes	—
19	C₄×Dic₃₇₃	C₂²	1	2	yes	—
20	C₂×Q₈×C₃₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₃₇₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
23	C₄²×C₃₇₃	C₂²	1	2	yes	—
24	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic
25	Dic₇₄₆	—	2	4	no	—
26	Dic₇₄₆	Q₈	1	4	yes	complemented
27	C₂×Dic₃₇₃	C₂³	1	4	yes	—
28	C₂×Dic₃₇₃	C₂³	1	2	yes	—
29	C₂×Dic₃₇₃	C₂³	1	2	yes	—
30	Q₈×C₃₇₃	—	2	2	no	—
31	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
32	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
34	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
36	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
37	C₄×C₃₇₃	—	2	2	no	—
38	C₄×C₃₇₃	—	2	2	no	—
39	C₄×C₃₇₃	—	2	1	no	—
40	C₄×C₃₇₃	C₂×Q₈	1	2	yes	—
41	C₄×C₃₇₃	#41	1	2	yes	—
42	Dic₃₇₃	—	2	4	no	—
43	Dic₃₇₃	—	2	2	no	—
44	Dic₃₇₃	C₂×Q₈	1	4	yes	—
45	C₂×C₃₇₃	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₃₇₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₃₇₃	2₋¹⁺⁴	1	1	yes	characteristic
48	C₃₇₃	#557	1	1	yes	characteristic, complemented, Sylow
49	#557	—	373	1	no	maximal, Sylow
50	C₄⋊Q₈	—	373	2	no	—
51	C₄⋊Q₈	—	373	1	no	—
52	#125	—	373	4	no	—
53	#125	—	373	2	no	—
54	C₄×Q₈	—	373	2	no	—
55	C₄×Q₈	—	373	2	no	—
56	C₄×Q₈	—	373	1	no	—
57	C₄×Q₈	D₃₇₃	1	1	yes	characteristic
58	C₂×Q₈	—	373	1	no	—
59	C₂×Q₈	—	373	2	no	—
60	C₂×Q₈	C₂×D₃₇₃	1	1	yes	characteristic
61	C₄⋊C₄	—	373	2	no	—
62	C₄⋊C₄	—	373	4	no	—
63	C₄⋊C₄	—	373	1	no	—
64	C₄⋊C₄	—	373	4	no	—
65	C₄⋊C₄	—	373	2	no	—
66	C₄⋊C₄	—	373	4	no	—
67	C₄⋊C₄	—	373	2	no	—
68	C₄⋊C₄	C₂×D₃₇₃	1	1	yes	characteristic
69	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
70	C₄²	—	373	4	no	—
71	C₄²	—	373	2	no	—
72	C₄²	C₂×D₃₇₃	1	2	yes	—
73	C₄²	C₂×D₃₇₃	1	1	yes	characteristic
74	Q₈	—	2	2	no	—
75	Q₈	—	373	4	no	—
76	Q₈	—	746	4	no	—
77	C₂×C₄	—	373	2	no	—
78	C₂×C₄	—	373	2	no	—
79	C₂×C₄	—	373	4	no	—
80	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
81	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
83	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
85	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	1	no	—
89	C₄	—	373	4	no	—
90	C₄	—	746	4	no	—
91	C₄	—	746	2	no	—
92	C₄	Q₈×D₃₇₃	1	2	yes	—
93	C₄	#20238610	1	2	yes	—
94	C₂	C₂×Q₈×D₃₇₃	1	1	yes	characteristic
95	C₂	C₂×#20238610	1	1	yes	characteristic
96	C₂	#26045839	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, prime radical, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	C₄×Q₈×C₃₇₃	C₂
hypercenter	C₄×Q₈	D₃₇₃
hypocenter	C₃₇₃	#557
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
26. Dic₇₄₆	Q₈	semidirect
48. C₃₇₃	#557	semidirect

Maximal subgroups

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

subgroup	index
#26045792	2
#26045790	2
#26045784	2
#26045782	2
C₄×Dic₇₄₆	2
#26045757	2
C₄×Q₈×C₃₇₃	2
#557	373