Subgroups — #44484688

Order 30592.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29504796	C₂	1	1	yes	maximal, characteristic
2	C₂×#21439871	C₂	1	1	yes	maximal, characteristic
3	#29504536	C₂	1	4	yes	maximal
4	#423×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#16838532	C₂²	1	2	yes	—
6	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₂₃₉⋊C₈)	—	2	2	no	—
8	#21439871	C₄	1	4	yes	—
9	C₂×M₁₆×C₂₃₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₃₉	—	2	2	no	—
12	#16838532	—	2	4	no	—
13	C₂×C₂₃₉⋊C₈	—	2	2	no	—
14	C₂×C₂₃₉⋊C₈	—	2	2	no	—
15	C₂×C₂₃₉⋊C₈	—	2	2	no	—
16	C₂×C₂₃₉⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₃₉	Q₈	1	2	yes	—
18	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
19	M₁₆×C₂₃₉	—	2	4	no	—
20	C₂×C₈×C₂₃₉	—	2	2	no	—
21	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
22	C₄²×C₂₃₉	D₄	1	2	yes	—
23	C₂₃₉⋊C₈	—	2	4	no	—
24	C₂₃₉⋊C₈	—	4	2	no	—
25	C₂³×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₂₃₉	—	4	2	no	—
27	C₂×C₄×C₂₃₉	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₃₉	#41	1	2	yes	—
31	C₂×C₄×C₂₃₉	#41	1	2	yes	—
32	C₈×C₂₃₉	—	4	2	no	—
33	C₂²×C₂₃₉	—	2	2	no	—
34	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₂₃₉	#125	1	2	yes	—
36	C₄×C₂₃₉	—	2	4	no	—
37	C₄×C₂₃₉	#117	1	2	yes	—
38	C₄×C₂₃₉	#124	1	1	yes	characteristic
39	C₄×C₂₃₉	#124	1	1	yes	characteristic
40	C₂×C₂₃₉	—	2	2	no	—
41	C₂×C₂₃₉	#382	1	1	yes	characteristic
42	C₂×C₂₃₉	#430	1	2	yes	—
43	C₂₃₉	#1838	1	1	yes	characteristic, complemented, Sylow
44	#1838	—	239	1	no	maximal, Sylow
45	#423	—	239	1	no	—
46	#423	D₂₃₉	1	1	yes	characteristic
47	C₂×#97	—	239	1	no	—
48	#341	—	239	4	no	—
49	C₂×M₁₆	—	239	2	no	—
50	C₂×M₁₆	C₂×D₂₃₉	1	2	yes	—
51	C₂²×C₈	—	239	2	no	—
52	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	478	2	no	—
55	#97	—	239	4	no	—
56	C₂²×C₄	Dic₄₇₈	1	2	yes	—
57	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
58	M₁₆	—	2	4	no	—
59	M₁₆	—	478	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	478	2	no	—
62	C₂×C₈	—	478	2	no	—
63	C₂×C₈	—	478	2	no	—
64	C₂×C₈	—	239	4	no	—
65	C₄²	C₄×D₂₃₉	1	2	yes	—
66	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
67	C₂³	C₂×Dic₄₇₈	1	1	yes	characteristic
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	Dic₂₃₉⋊C₄	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
71	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
72	C₂×C₄	#16838547	1	2	yes	—
73	C₂×C₄	#16838549	1	2	yes	—
74	C₈	—	4	2	no	—
75	C₈	—	956	2	no	—
76	C₈	—	478	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
79	C₂²	#21439951	1	2	yes	—
80	C₄	—	2	4	no	—
81	C₄	#21439918	1	2	yes	—
82	C₄	#21439947	1	1	yes	characteristic
83	C₄	#21439949	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#29504637	1	1	yes	characteristic
86	C₂	#29504820	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃₉
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#423×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃₉	C₂×Q₈
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	#423	D₂₃₉
hypocenter	C₂₃₉	#1838
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
43. C₂₃₉	#1838	semidirect

Maximal subgroups

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

subgroup	index
#29504796	2
C₂×#21439871	2
#29504536	2
#423×C₂₃₉	2
#1838	239