Subgroups — #44484155

Order 30592.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29505149	C₂	1	1	yes	maximal, characteristic
2	#29505120	C₂	1	4	yes	maximal, complemented
3	C₈×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₉₁₂	—	2	2	no	—
6	#21439957	C₂²	1	2	yes	—
7	#21439880	—	2	8	no	—
8	C₄×C₂₃₉⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₂₃₉	—	2	2	no	—
10	C₄⋊Q₈×C₂₃₉	C₂²	1	2	yes	—
11	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₂×D₉₅₆	—	2	4	no	—
13	C₂×D₉₅₆	—	2	2	no	—
14	D₁₉₁₂	—	4	4	no	—
15	C₂×C₂₃₉⋊C₈	D₄	1	4	yes	—
16	C₂×Q₈×C₂₃₉	—	2	2	no	—
17	Q₁₆×C₂₃₉	—	4	4	no	—
18	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
19	C₄⋊C₄×C₂₃₉	—	2	4	no	—
20	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
21	C₂²×D₂₃₉	—	4	2	no	—
22	D₉₅₆	—	4	4	no	—
23	D₉₅₆	—	4	8	no	—
24	C₂₃₉⋊C₈	—	2	4	no	—
25	Q₈×C₂₃₉	—	4	4	no	—
26	C₂×C₄×C₂₃₉	—	4	2	no	—
27	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
29	C₈×C₂₃₉	D₈	1	4	yes	—
30	C₂²×C₂₃₉	#127	1	1	yes	characteristic
31	C₂×D₂₃₉	—	8	2	no	—
32	C₂×D₂₃₉	—	8	4	no	—
33	C₄×C₂₃₉	—	8	2	no	—
34	C₄×C₂₃₉	C₂×D₈	1	2	yes	—
35	C₄×C₂₃₉	#135	1	4	yes	—
36	C₂×C₂₃₉	#493	1	1	yes	characteristic
37	C₂×C₂₃₉	#495	1	2	yes	—
38	D₂₃₉	—	16	2	no	—
39	C₂₃₉	#1709	1	1	yes	characteristic, complemented, Sylow
40	#1709	—	239	1	no	maximal, Sylow
41	#494	D₂₃₉	1	1	yes	characteristic, complemented
42	#493	—	239	1	no	—
43	#486	—	239	4	no	—
44	C₈²	—	239	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	478	2	no	—
47	C₄⋊Q₈	C₂×D₂₃₉	1	2	yes	—
48	#127	—	239	2	no	—
49	#102	—	478	8	no	—
50	C₄×C₈	—	239	2	no	—
51	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	478	4	no	—
54	C₂×D₄	—	478	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	956	4	no	—
57	C₂×C₈	—	239	4	no	—
58	C₂×C₈	C₂²⋊D₂₃₉	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
61	C₂³	—	956	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	956	4	no	—
64	D₄	—	956	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
68	C₈	—	478	4	no	—
69	C₈	#16838533	1	4	yes	—
70	C₂²	—	1912	2	no	—
71	C₂²	—	1912	4	no	—
72	C₂²	#21439956	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#16838533	1	2	yes	—
75	C₄	#21439983	1	4	yes	—
76	C₂	—	3824	2	no	—
77	C₂	#29505148	1	1	yes	characteristic
78	C₂	#29505157	1	2	yes	—

	subgroup	G/H
center	C₂²	#21439956
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	#494×C₂₃₉	C₂
socle	C₂²×C₂₃₉	#127
prime radical	#29505149	C₂
hypercenter	#494	D₂₃₉
hypocenter	C₂₃₉	#1709
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #29505120	C₂	semidirect
3. C₈×C₂₃₉⋊C₈	C₂	semidirect
4. #494×C₂₃₉	C₂	semidirect
39. C₂₃₉	#1709	semidirect
41. #494	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#29505149	2
#29505120	2
C₈×C₂₃₉⋊C₈	2
#494×C₂₃₉	2
#1709	239