Finite Group Databasebeta

Subgroups — #44486397

Order 30592.

162 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#21439903	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#29504854	C₂	1	1	yes	maximal, characteristic
4	#29504599	C₂	1	1	yes	maximal, characteristic, complemented
5	#29504574	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₉₁₂	C₂²	1	1	yes	characteristic
9	Q₈×Dic₂₃₉	—	2	1	no	—
10	#21439922	—	2	1	no	—
11	#21439903	C₄	1	4	yes	complemented
12	C₂×C₂₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#21439894	C₂²	1	1	yes	characteristic
14	#21439884	—	2	1	no	—
15	#21439880	—	2	1	no	—
16	C₈×Dic₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
19	#107×C₂₃₉	C₂²	1	1	yes	characteristic
20	C₂×D₉₅₆	—	2	1	no	—
21	C₂×C₄×D₂₃₉	—	4	1	no	—
22	D₁₉₁₂	—	2	1	no	—
23	D₁₉₁₂	C₂×C₄	1	2	yes	—
24	C₂×C₂₃₉⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₂₃₉	—	2	1	no	—
26	C₄⋊Dic₂₃₉	—	4	1	no	—
27	(C₂×D₂₃₉)⋊C₄	—	4	1	no	—
28	C₄×Dic₂₃₉	—	4	1	no	—
29	C₄×Dic₂₃₉	D₄	1	1	yes	characteristic
30	C₂₃₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×Q₈×C₂₃₉	—	2	1	no	—
32	Q₁₆×C₂₃₉	—	2	1	no	—
33	Q₁₆×C₂₃₉	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₃₉	—	2	1	no	—
36	C₁₆×C₂₃₉	—	2	1	no	—
37	C₂²×D₂₃₉	—	4	1	no	—
38	D₉₅₆	—	2	2	no	—
39	D₉₅₆	—	4	1	no	—
40	C₂×Dic₂₃₉	—	4	1	no	—
41	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
42	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
43	C₄×D₂₃₉	—	8	2	no	—
44	C₂₃₉⋊C₈	—	2	1	no	—
45	Q₈×C₂₃₉	—	2	2	no	—
46	Q₈×C₂₃₉	—	4	1	no	—
47	C₂×C₄×C₂₃₉	—	4	1	no	—
48	C₂×C₄×C₂₃₉	—	4	1	no	—
49	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
50	C₈×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₂₃₉	#41	1	1	yes	characteristic
52	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
53	C₂×D₂₃₉	—	4	2	no	—
54	C₂×D₂₃₉	—	8	1	no	—
55	C₂×D₂₃₉	—	8	1	no	—
56	C₄×C₂₃₉	—	4	2	no	—
57	C₄×C₂₃₉	—	8	1	no	—
58	C₄×C₂₃₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₂₃₉	#135	1	1	yes	characteristic
60	Dic₂₃₉	—	2	1	no	—
61	Dic₂₃₉	—	8	1	no	—
62	Dic₂₃₉	SD₃₂	1	2	yes	complemented
63	C₂×C₂₃₉	C₄×D₈	1	1	yes	characteristic
64	C₂×C₂₃₉	C₂×SD₃₂	1	1	yes	characteristic
65	C₂×C₂₃₉	#508	1	1	yes	characteristic
66	D₂₃₉	—	8	2	no	—
67	C₂₃₉	C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₄×SD₃₂	—	239	1	no	maximal, Sylow
69	C₂×SD₃₂	—	239	1	no	—
70	C₄×Q₁₆	—	239	1	no	—
71	C₄×D₈	—	239	1	no	—
72	#367	—	239	1	no	—
73	#358	D₂₃₉	1	1	yes	characteristic, complemented
74	#357	—	239	1	no	—
75	C₄×C₁₆	—	239	1	no	—
76	C₂×Q₁₆	C₂×D₂₃₉	1	1	yes	characteristic
77	C₂×D₈	—	239	1	no	—
78	C₄×Q₈	—	478	1	no	—
79	C₄×D₄	—	478	1	no	—
80	SD₃₂	—	239	4	no	—
81	C₂×C₁₆	—	239	1	no	—
82	C₂×C₁₆	C₂×D₂₃₉	1	1	yes	characteristic
83	#107	—	239	1	no	—
84	#107	C₂×D₂₃₉	1	1	yes	characteristic
85	#103	—	478	1	no	—
86	#102	—	478	1	no	—
87	C₄×C₈	—	239	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	478	1	no	—
90	C₂²×C₄	—	956	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₄×D₂₃₉	1	2	yes	complemented
93	D₈	—	478	1	no	—
94	D₈	—	239	2	no	—
95	C₂×C₈	—	239	1	no	—
96	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	956	1	no	—
99	C₄⋊C₄	—	478	1	no	—
100	C₂²⋊C₄	—	956	1	no	—
101	C₄²	—	956	1	no	—
102	C₄²	—	239	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	239	2	no	—
105	C₂³	—	956	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	956	1	no	—
109	D₄	—	478	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	1912	2	no	—
113	C₂×C₄	—	956	1	no	—
114	C₂×C₄	—	239	1	no	—
115	C₂×C₄	—	239	1	no	—
116	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
117	C₈	—	478	1	no	—
118	C₈	C₂×C₄×D₂₃₉	1	1	yes	characteristic
119	C₈	#16838547	1	1	yes	characteristic
120	C₂²	—	1912	1	no	—
121	C₂²	—	1912	1	no	—
122	C₂²	—	956	2	no	—
123	C₂²	D₈×D₂₃₉	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	1	no	—
126	C₄	—	1912	1	no	—
127	C₄	—	478	1	no	—
128	C₄	—	239	2	no	—
129	C₄	(C₂²⋊D₂₃₉)⋊C₄	1	1	yes	characteristic
130	C₄	#21439981	1	1	yes	characteristic
131	C₂	—	1912	2	no	—
132	C₂	#29504852	1	1	yes	characteristic
133	C₂	SD₃₂×D₂₃₉	1	1	yes	characteristic
134	C₂	#29505205	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₃₉
commutator	C₈×C₂₃₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	#358×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₈
prime radical	C₂×D₁₉₁₂	C₂²
cosocle	C₂×C₈×C₂₃₉	C₂³
hypercenter	#358	D₂₃₉
hypocenter	C₂₃₉	C₄×SD₃₂
LCS	G ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₂₃₉	C₂	semidirect
4. #29504599	C₂	semidirect
6. C₁₆×Dic₂₃₉	C₂	semidirect
7. #358×C₂₃₉	C₂	semidirect
11. #21439903	C₄	semidirect
30. C₂₃₉⋊C₁₆	C₂×C₄	semidirect
33. Q₁₆×C₂₃₉	C₂×C₄	semidirect
62. Dic₂₃₉	SD₃₂	semidirect
67. C₂₃₉	C₄×SD₃₂	semidirect
73. #358	D₂₃₉	semidirect
92. Q₁₆	C₄×D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#21439903	2
Q₁₆×Dic₂₃₉	2
#29504854	2
#29504599	2
#29504574	2
C₁₆×Dic₂₃₉	2
#358×C₂₃₉	2
C₄×SD₃₂	239