Subgroups — #17989198

Order 4672.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14556586	C₂	1	4	yes	maximal
2	#14556583	C₂	1	1	yes	maximal, characteristic, complemented
3	#14556581	C₂	1	4	yes	maximal, complemented
4	#14556567	C₂	1	2	yes	maximal
5	Q₈×Dic₇₃	C₂	1	2	yes	maximal, complemented
6	#14556548	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	C₄⋊Dic₇₃	C₂²	1	4	yes	—
11	#11924007	C₂²	1	2	yes	—
12	#11924007	C₂²	1	4	yes	—
13	#11924007	C₂²	1	4	yes	—
14	#11924005	C₂²	1	2	yes	—
15	#11924005	C₂²	1	4	yes	—
16	C₄×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₇₃	C₂²	1	4	yes	—
18	C₂×Q₈×C₇₃	C₂²	1	2	yes	—
19	C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
20	C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×D₇₃	C₂³	1	1	yes	characteristic
22	Q₈×C₇₃	—	2	4	no	—
23	C₂×C₄×C₇₃	C₂³	1	2	yes	—
24	C₂×C₄×C₇₃	C₂³	1	4	yes	—
25	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₇₃	C₂³	1	2	yes	—
27	C₂×Dic₇₃	C₂³	1	4	yes	—
28	C₂×Dic₇₃	C₂³	1	1	yes	characteristic
29	C₄×D₇₃	—	2	2	no	—
30	C₂²×C₇₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₇₃	—	2	1	no	—
32	C₂×D₇₃	—	2	2	no	—
33	C₄×C₇₃	—	2	4	no	—
34	C₄×C₇₃	—	2	2	no	—
35	C₄×C₇₃	#41	1	2	yes	—
36	Dic₇₃	—	2	1	no	—
37	Dic₇₃	—	2	2	no	—
38	Dic₇₃	—	2	4	no	—
39	C₂×C₇₃	C₂×#41	1	1	yes	characteristic
40	C₂×C₇₃	2₋¹⁺⁴	1	2	yes	—
41	D₇₃	—	4	1	no	—
42	C₇₃	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	73	1	no	maximal, Sylow
44	C₄⋊Q₈	D₇₃	1	1	yes	characteristic, complemented
45	#126	—	73	4	no	—
46	#125	—	73	4	no	—
47	#125	—	73	1	no	—
48	C₂²⋊Q₈	—	73	2	no	—
49	C₄×Q₈	—	73	2	no	—
50	#117	—	73	1	no	—
51	C₂×Q₈	C₂×D₇₃	1	2	yes	—
52	C₂²×C₄	—	73	1	no	—
53	C₄⋊C₄	—	73	2	no	—
54	C₄⋊C₄	—	73	4	no	—
55	C₄⋊C₄	—	73	4	no	—
56	C₄⋊C₄	—	73	2	no	—
57	C₄⋊C₄	—	73	4	no	—
58	C₄⋊C₄	C₂×D₇₃	1	4	yes	—
59	C₂²⋊C₄	—	73	2	no	—
60	C₂²⋊C₄	—	73	4	no	—
61	C₄²	—	73	1	no	—
62	C₄²	—	73	4	no	—
63	C₄²	C₂×D₇₃	1	1	yes	characteristic
64	C₂³	—	73	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	73	2	no	—
67	C₂×C₄	—	73	4	no	—
68	C₂×C₄	—	73	1	no	—
69	C₂×C₄	—	146	2	no	—
70	C₂×C₄	C₂²×D₇₃	1	4	yes	—
71	C₂×C₄	C₂²×D₇₃	1	2	yes	—
72	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
73	C₂²	—	146	1	no	—
74	C₂²	—	146	2	no	—
75	C₂²	C₂³×D₇₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	146	1	no	—
79	C₄	—	146	2	no	—
80	C₄	—	146	4	no	—
81	C₄	#11924027	1	2	yes	—
82	C₂	—	292	1	no	—
83	C₂	C₂×#11924027	1	1	yes	characteristic
84	C₂	#14556639	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₃
commutator, socle, cosocle	C₂²×C₇₃	C₂⁴
Fitting	C₄⋊Q₈×C₇₃	C₂
prime radical	C₂²×D₇₃	C₂³
hypercenter	C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#541
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #14556583	C₂	semidirect
3. #14556581	C₂	semidirect
5. Q₈×Dic₇₃	C₂	semidirect
7. C₄⋊Q₈×C₇₃	C₂	semidirect
42. C₇₃	#541	semidirect
44. C₄⋊Q₈	D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#14556586	2
#14556583	2
#14556581	2
#14556567	2
Q₈×Dic₇₃	2
#14556548	2
C₄⋊Q₈×C₇₃	2
#541	73