Finite Group Databasebeta

Subgroups — #44486788

Order 30592.

152 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 47 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	#29505173	C₂	1	1	yes	maximal, characteristic, complemented
2	#29505147	C₂	1	1	yes	maximal, characteristic
3	#29504574	C₂	1	1	yes	maximal, characteristic, complemented
4	#29504573	C₂	1	1	yes	maximal, characteristic, complemented
5	#29504572	C₂	1	1	yes	maximal, characteristic, complemented
6	#29504547	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₉₁₂	C₂²	1	1	yes	characteristic
9	C₂×#16838533	—	2	1	no	—
10	#21439956	—	2	1	no	—
11	#21439951	—	2	1	no	—
12	C₂×C₂₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#21439894	C₂²	1	1	yes	characteristic
14	#21439891	—	2	1	no	—
15	C₈×Dic₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
18	#107×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₂×D₉₅₆	—	2	1	no	—
20	C₂×C₂²⋊D₂₃₉	—	4	1	no	—
21	C₂×C₂²⋊D₂₃₉	—	4	1	no	—
22	D₁₉₁₂	—	2	1	no	—
23	D₁₉₁₂	—	2	1	no	—
24	#16838533	—	4	2	no	—
25	C₂×C₂₃₉⋊C₈	D₄	1	1	yes	characteristic
26	C₄⋊Dic₂₃₉	—	2	1	no	—
27	C₄⋊Dic₂₃₉	—	4	1	no	—
28	Dic₂₃₉⋊C₄	—	4	1	no	—
29	C₄×Dic₂₃₉	D₄	1	1	yes	characteristic
30	C₂₃₉⋊C₁₆	—	2	1	no	—
31	C₂×D₄×C₂₃₉	—	2	1	no	—
32	D₈×C₂₃₉	—	2	1	no	—
33	D₈×C₂₃₉	—	2	1	no	—
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	—	4	1	no	—
39	D₉₅₆	—	4	1	no	—
40	C₂²⋊D₂₃₉	—	8	2	no	—
41	C₂²⋊D₂₃₉	—	8	2	no	—
42	C₂×Dic₂₃₉	—	4	1	no	—
43	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
44	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
45	C₂₃₉⋊C₈	SD₁₆	1	2	yes	—
46	C₂³×C₂₃₉	—	4	1	no	—
47	D₄×C₂₃₉	—	4	1	no	—
48	D₄×C₂₃₉	—	4	1	no	—
49	C₂×C₄×C₂₃₉	—	4	1	no	—
50	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
51	C₈×C₂₃₉	#41	1	1	yes	characteristic
52	C₈×C₂₃₉	#41	1	1	yes	characteristic
53	C₂²×C₂₃₉	—	8	1	no	—
54	C₂²×C₂₃₉	—	8	1	no	—
55	C₂²×C₂₃₉	—	8	1	no	—
56	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
57	C₂×D₂₃₉	—	8	1	no	—
58	C₂×D₂₃₉	—	8	1	no	—
59	C₂×D₂₃₉	—	8	1	no	—
60	C₄×C₂₃₉	—	8	1	no	—
61	C₄×C₂₃₉	#124	1	1	yes	characteristic
62	C₄×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
63	Dic₂₃₉	—	2	1	no	—
64	Dic₂₃₉	—	2	1	no	—
65	Dic₂₃₉	—	8	1	no	—
66	C₂×C₂₃₉	—	16	1	no	—
67	C₂×C₂₃₉	#486	1	1	yes	characteristic
68	C₂×C₂₃₉	#509	1	1	yes	characteristic
69	C₂×C₂₃₉	#509	1	1	yes	characteristic
70	D₂₃₉	—	16	1	no	—
71	C₂₃₉	#2244	1	1	yes	characteristic, complemented, Sylow
72	#2244	—	239	1	no	maximal, Sylow
73	#499	—	239	1	no	—
74	#493	—	239	1	no	—
75	#357	—	239	1	no	—
76	#357	—	239	1	no	—
77	#357	—	239	1	no	—
78	#357	D₂₃₉	1	1	yes	characteristic, complemented
79	#346	—	239	1	no	—
80	C₂×D₈	—	478	1	no	—
81	C₂×D₈	—	239	1	no	—
82	C₂×D₈	C₂×D₂₃₉	1	1	yes	characteristic
83	#127	—	478	1	no	—
84	#125	—	478	1	no	—
85	C₂×C₁₆	—	239	1	no	—
86	C₂×C₁₆	C₂×D₂₃₉	1	1	yes	characteristic
87	#107	—	239	1	no	—
88	#107	C₂×D₂₃₉	1	1	yes	characteristic
89	#106	—	478	1	no	—
90	C₄×C₈	—	239	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	956	1	no	—
93	C₂×D₄	—	956	1	no	—
94	C₂×D₄	—	478	1	no	—
95	D₈	—	2	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	956	2	no	—
98	D₈	—	478	1	no	—
99	D₈	—	478	1	no	—
100	C₂×C₈	—	239	1	no	—
101	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	956	1	no	—
104	C₄⋊C₄	—	956	1	no	—
105	C₄⋊C₄	—	478	1	no	—
106	C₄²	—	239	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	478	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	956	1	no	—
111	D₄	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	1912	2	no	—
114	D₄	—	1912	2	no	—
115	D₄	—	956	1	no	—
116	D₄	—	956	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	956	1	no	—
119	C₂×C₄	—	239	1	no	—
120	C₂×C₄	—	239	1	no	—
121	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
122	C₈	—	239	2	no	—
123	C₈	#16838547	1	1	yes	characteristic
124	C₈	#16838548	1	1	yes	characteristic
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	1912	1	no	—
129	C₂²	—	1912	1	no	—
130	C₂²	—	1912	1	no	—
131	C₂²	D₈×D₂₃₉	1	1	yes	characteristic
132	C₄	—	8	1	no	—
133	C₄	—	1912	1	no	—
134	C₄	—	478	1	no	—
135	C₄	—	478	1	no	—
136	C₄	#21439946	1	1	yes	characteristic
137	C₄	SD₁₆×D₂₃₉	1	1	yes	characteristic
138	C₂	—	16	1	no	—
139	C₂	—	3824	1	no	—
140	C₂	#29505117	1	1	yes	characteristic
141	C₂	#29505206	1	1	yes	characteristic
142	C₂	#29505209	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₃₉
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	#357×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₈
prime radical	#29505147	C₂
hypercenter	#357	D₂₃₉
hypocenter	C₂₃₉	#2244
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
1. #29505173	C₂	semidirect
3. #29504574	C₂	semidirect
4. #29504573	C₂	semidirect
5. #29504572	C₂	semidirect
6. #29504547	C₂	semidirect
7. #357×C₂₃₉	C₂	semidirect
71. C₂₃₉	#2244	semidirect
78. #357	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#29505173	2
#29505147	2
#29504574	2
#29504573	2
#29504572	2
#29504547	2
#357×C₂₃₉	2
#2244	239