Subgroups — #44486128

Order 30592.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29504821	C₂	1	1	yes	maximal, characteristic
2	#29504542	C₂	1	2	yes	maximal
3	#29504537	C₂	1	2	yes	maximal
4	C₂₃₉⋊#338	C₂	1	1	yes	maximal, characteristic
5	#424×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838532	C₂²	1	2	yes	—
7	C₂×#16838532	C₂²	1	2	yes	—
8	#21439896	—	2	2	no	—
9	C₂×M₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
11	#117×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₃₉	—	2	2	no	—
13	#16838532	—	2	2	no	—
14	#16838532	—	2	2	no	—
15	#16838532	—	2	2	no	—
16	#16838532	—	2	2	no	—
17	C₂×C₂₃₉⋊C₈	—	2	2	no	—
18	C₂×C₂₃₉⋊C₈	—	2	2	no	—
19	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
20	M₁₆×C₂₃₉	—	2	2	no	—
21	C₂×C₈×C₂₃₉	—	2	1	no	—
22	C₂×C₈×C₂₃₉	—	2	2	no	—
23	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₃₉	—	2	1	no	—
25	C₂²⋊C₄×C₂₃₉	Q₈	1	2	yes	—
26	C₄²×C₂₃₉	—	2	1	no	—
27	C₂₃₉⋊C₈	—	4	2	no	—
28	C₂₃₉⋊C₈	—	4	2	no	—
29	C₂³×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₂₃₉	—	2	2	no	—
31	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃₉	#41	1	2	yes	—
33	C₂×C₄×C₂₃₉	#41	1	2	yes	—
34	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
35	C₈×C₂₃₉	—	2	2	no	—
36	C₈×C₂₃₉	—	4	1	no	—
37	C₂²×C₂₃₉	—	4	1	no	—
38	C₂²×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
39	C₂²×C₂₃₉	#125	1	2	yes	—
40	C₄×C₂₃₉	—	4	2	no	—
41	C₄×C₂₃₉	#123	1	1	yes	characteristic
42	C₄×C₂₃₉	#124	1	1	yes	characteristic
43	C₄×C₂₃₉	#126	1	2	yes	—
44	C₂×C₂₃₉	—	2	1	no	—
45	C₂×C₂₃₉	—	2	2	no	—
46	C₂×C₂₃₉	#400	1	1	yes	characteristic
47	C₂₃₉	#2103	1	1	yes	characteristic, complemented, Sylow
48	#2103	—	239	1	no	maximal, Sylow
49	#430	—	239	1	no	—
50	#424	D₂₃₉	1	1	yes	characteristic
51	#344	—	239	2	no	—
52	#341	—	239	2	no	—
53	#338	—	239	1	no	—
54	C₂×M₁₆	—	239	2	no	—
55	C₂×M₁₆	—	239	2	no	—
56	C₂×M₁₆	C₂×D₂₃₉	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
58	#117	C₂×D₂₃₉	1	1	yes	characteristic
59	#108	—	478	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	M₁₆	—	478	2	no	—
64	M₁₆	—	478	2	no	—
65	M₁₆	—	478	2	no	—
66	M₁₆	—	478	2	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	478	2	no	—
70	C₂×C₈	—	478	2	no	—
71	C₂×C₈	C₂²⋊D₂₃₉	1	2	yes	—
72	C₄⋊C₄	—	2	1	no	—
73	C₂²⋊C₄	Dic₄₇₈	1	2	yes	—
74	C₄²	—	2	1	no	—
75	C₂³	C₂×Dic₄₇₈	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
78	C₂×C₄	#16838547	1	2	yes	—
79	C₂×C₄	#16838548	1	1	yes	characteristic
80	C₂×C₄	#16838549	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	4	1	no	—
83	C₈	—	956	2	no	—
84	C₈	—	956	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#21439939	1	1	yes	characteristic
87	C₂²	#21439951	1	2	yes	—
88	C₄	—	4	2	no	—
89	C₄	#21439941	1	1	yes	characteristic
90	C₄	#21439949	1	1	yes	characteristic
91	C₄	#21439955	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#29504711	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#21439949
commutator, cosocle	C₂²×C₄×C₂₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#424×C₂₃₉	C₂
socle	C₂×C₂₃₉	#400
prime radical	C₂³×C₂₃₉	C₂×Q₈
hypercenter	#424	D₂₃₉
hypocenter	C₂₃₉	#2103
LCS	G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
47. C₂₃₉	#2103	semidirect

Maximal subgroups

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

subgroup	index
#29504821	2
#29504542	2
#29504537	2
C₂₃₉⋊#338	2
#424×C₂₃₉	2
#2103	239