Subgroups — #34013316

Order 18688.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924031	C₂	1	6	yes	maximal
2	#22609618	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924031	C₂²	1	4	yes	—
4	C₄⋊#11924031	—	2	12	no	—
5	C₂×#14556502	C₄	1	3	yes	—
6	C₂×C₄×#10596831	C₂²	1	3	yes	—
7	#376×C₇₃	C₄	1	1	yes	characteristic
8	C₂×#11924031	—	2	12	no	—
9	C₂²×#10596831	C₂×C₄	1	3	yes	—
10	C₂²×#10596831	C₂³	1	1	yes	characteristic
11	#14556502	—	2	6	no	—
12	C₄×#10596831	D₄	1	6	yes	—
13	C₄×#10596831	Q₈	1	6	yes	—
14	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
15	#95×C₇₃	C₈	1	4	yes	—
16	#11924031	—	4	4	no	—
17	C₂×#10596831	—	2	6	no	—
18	C₂×#10596831	C₄⋊C₄	1	6	yes	—
19	C₂×#10596831	C₂×D₄	1	3	yes	—
20	C₂×#10596831	C₂×Q₈	1	3	yes	—
21	C₂²×C₄×C₇₃	C₂×C₈	1	6	yes	—
22	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
23	C₄²×C₇₃	C₄⋊C₄	1	6	yes	—
24	C₄²×C₇₃	M₁₆	1	6	yes	—
25	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₇₃	—	2	12	no	—
27	C₂×C₄×C₇₃	C₄⋊C₈	1	12	yes	—
28	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
30	#10596831	—	2	6	no	—
31	#10596831	#127	1	1	yes	characteristic
32	#10596831	C₄⋊Q₈	1	3	yes	—
33	C₂²×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
34	C₂²×C₇₃	#504	1	4	yes	—
35	C₄×C₇₃	—	2	12	no	—
36	C₄×C₇₃	#384	1	1	yes	characteristic
37	C₄×C₇₃	#423	1	3	yes	—
38	#12899	—	73	1	no	maximal, Sylow
39	C₂×C₇₃	#1842	1	1	yes	characteristic
40	C₂×C₇₃	#2151	1	6	yes	—
41	C₂×C₄⋊C₁₆	—	73	6	no	—
42	#1726	—	73	1	no	—
43	C₇₃	#12899	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₁₆	—	73	4	no	—
45	C₂×#97	—	73	3	no	—
46	C₂×C₄×C₈	—	73	3	no	—
47	#376	#64408	1	1	yes	characteristic
48	C₄⋊C₁₆	—	146	12	no	—
49	C₂²×C₈	—	73	1	no	—
50	C₂²×C₈	—	73	3	no	—
51	C₂×C₄²	C₂×#64408	1	3	yes	—
52	C₂×C₁₆	—	146	12	no	—
53	#97	—	146	6	no	—
54	C₄×C₈	—	73	6	no	—
55	C₄×C₈	—	73	6	no	—
56	#95	#10596833	1	4	yes	—
57	C₂²×C₄	C₂×#10596833	1	6	yes	—
58	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
59	C₂×C₈	—	73	3	no	—
60	C₂×C₈	—	73	3	no	—
61	C₂×C₈	—	73	6	no	—
62	C₂×C₈	—	146	6	no	—
63	C₄²	C₄⋊#64408	1	6	yes	—
64	C₄²	#11924038	1	6	yes	—
65	C₁₆	—	292	4	no	—
66	C₂³	C₂²×#10596833	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#10596833	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
70	C₂×C₄	C₂×#11924038	1	3	yes	—
71	C₈	—	73	1	no	—
72	C₈	—	73	3	no	—
73	C₈	—	146	6	no	—
74	C₂²	C₂×C₄⋊#10596833	1	3	yes	—
75	C₂²	#17989645	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#17989533	1	1	yes	characteristic
78	C₄	#17989591	1	3	yes	—
79	C₂	#22625863	1	1	yes	characteristic
80	C₂	#22626266	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator, socle, prime radical	C₂³×C₇₃	C₂²×C₈
Fitting	#376×C₇₃	C₄
cosocle	C₂²×#10596831	C₂³
hypercenter	#376	#64408
hypocenter	C₇₃	#12899
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
43. C₇₃	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924031	2
#22609618	2
#12899	73