Finite Group Databasebeta

Subgroups — #30668201

Order 16768.

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	#22553713	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553687	C₂	1	1	yes	maximal, characteristic
3	#22553114	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553113	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553112	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553087	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₀₄₈	C₂²	1	1	yes	characteristic
9	C₂×#14523498	—	2	1	no	—
10	#16862242	—	2	1	no	—
11	#16862237	—	2	1	no	—
12	C₂×C₁₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16862180	C₂²	1	1	yes	characteristic
14	#16862177	—	2	1	no	—
15	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
18	#107×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×D₅₂₄	—	2	1	no	—
20	C₂×C₂²⋊D₁₃₁	—	4	1	no	—
21	C₂×C₂²⋊D₁₃₁	—	4	1	no	—
22	D₁₀₄₈	—	2	1	no	—
23	D₁₀₄₈	—	2	1	no	—
24	#14523498	—	4	2	no	—
25	C₂×C₁₃₁⋊C₈	D₄	1	1	yes	characteristic
26	C₄⋊Dic₁₃₁	—	2	1	no	—
27	C₄⋊Dic₁₃₁	—	4	1	no	—
28	Dic₁₃₁⋊C₄	—	4	1	no	—
29	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
30	C₁₃₁⋊C₁₆	—	2	1	no	—
31	C₂×D₄×C₁₃₁	—	2	1	no	—
32	D₈×C₁₃₁	—	2	1	no	—
33	D₈×C₁₃₁	—	2	1	no	—
34	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₁	—	2	1	no	—
36	C₁₆×C₁₃₁	—	2	1	no	—
37	C₂²×D₁₃₁	—	4	1	no	—
38	D₅₂₄	—	4	1	no	—
39	D₅₂₄	—	4	1	no	—
40	C₂²⋊D₁₃₁	—	8	2	no	—
41	C₂²⋊D₁₃₁	—	8	2	no	—
42	C₂×Dic₁₃₁	—	4	1	no	—
43	C₂×Dic₁₃₁	D₈	1	1	yes	characteristic
44	C₂×Dic₁₃₁	D₈	1	1	yes	characteristic
45	C₁₃₁⋊C₈	SD₁₆	1	2	yes	—
46	C₂³×C₁₃₁	—	4	1	no	—
47	D₄×C₁₃₁	—	4	1	no	—
48	D₄×C₁₃₁	—	4	1	no	—
49	C₂×C₄×C₁₃₁	—	4	1	no	—
50	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₃₁	#41	1	1	yes	characteristic
52	C₈×C₁₃₁	#41	1	1	yes	characteristic
53	C₂²×C₁₃₁	—	8	1	no	—
54	C₂²×C₁₃₁	—	8	1	no	—
55	C₂²×C₁₃₁	—	8	1	no	—
56	C₂²×C₁₃₁	C₂×D₈	1	1	yes	characteristic
57	C₂×D₁₃₁	—	8	1	no	—
58	C₂×D₁₃₁	—	8	1	no	—
59	C₂×D₁₃₁	—	8	1	no	—
60	C₄×C₁₃₁	—	8	1	no	—
61	C₄×C₁₃₁	#124	1	1	yes	characteristic
62	C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
63	Dic₁₃₁	—	2	1	no	—
64	Dic₁₃₁	—	2	1	no	—
65	Dic₁₃₁	—	8	1	no	—
66	C₂×C₁₃₁	—	16	1	no	—
67	C₂×C₁₃₁	#486	1	1	yes	characteristic
68	C₂×C₁₃₁	#509	1	1	yes	characteristic
69	C₂×C₁₃₁	#509	1	1	yes	characteristic
70	D₁₃₁	—	16	1	no	—
71	C₁₃₁	#2244	1	1	yes	characteristic, complemented, Sylow
72	#2244	—	131	1	no	maximal, Sylow
73	#499	—	131	1	no	—
74	#493	—	131	1	no	—
75	#357	—	131	1	no	—
76	#357	—	131	1	no	—
77	#357	—	131	1	no	—
78	#357	D₁₃₁	1	1	yes	characteristic, complemented
79	#346	—	131	1	no	—
80	C₂×D₈	—	262	1	no	—
81	C₂×D₈	—	131	1	no	—
82	C₂×D₈	C₂×D₁₃₁	1	1	yes	characteristic
83	#127	—	262	1	no	—
84	#125	—	262	1	no	—
85	C₂×C₁₆	—	131	1	no	—
86	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
87	#107	—	131	1	no	—
88	#107	C₂×D₁₃₁	1	1	yes	characteristic
89	#106	—	262	1	no	—
90	C₄×C₈	—	131	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	262	1	no	—
93	C₂×D₄	—	524	1	no	—
94	C₂×D₄	—	524	1	no	—
95	D₈	—	2	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	262	1	no	—
98	D₈	—	262	1	no	—
99	D₈	—	524	2	no	—
100	C₂×C₈	—	131	1	no	—
101	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	262	1	no	—
104	C₄⋊C₄	—	524	1	no	—
105	C₄⋊C₄	—	524	1	no	—
106	C₄²	—	131	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	262	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	524	1	no	—
111	D₄	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	524	1	no	—
114	D₄	—	524	1	no	—
115	D₄	—	1048	2	no	—
116	D₄	—	1048	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	524	1	no	—
119	C₂×C₄	—	131	1	no	—
120	C₂×C₄	—	131	1	no	—
121	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
122	C₈	—	131	2	no	—
123	C₈	#14523512	1	1	yes	characteristic
124	C₈	#14523513	1	1	yes	characteristic
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	1048	1	no	—
129	C₂²	—	1048	1	no	—
130	C₂²	—	1048	1	no	—
131	C₂²	D₈×D₁₃₁	1	1	yes	characteristic
132	C₄	—	262	1	no	—
133	C₄	—	262	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	1048	1	no	—
136	C₄	#16862232	1	1	yes	characteristic
137	C₄	SD₁₆×D₁₃₁	1	1	yes	characteristic
138	C₂	—	16	1	no	—
139	C₂	—	2096	1	no	—
140	C₂	#22553657	1	1	yes	characteristic
141	C₂	#22553746	1	1	yes	characteristic
142	C₂	#22553749	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₁
commutator, cosocle	C₂×C₈×C₁₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	#357×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₈
prime radical	#22553687	C₂
hypercenter	#357	D₁₃₁
hypocenter	C₁₃₁	#2244
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
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. #22553713	C₂	semidirect
3. #22553114	C₂	semidirect
4. #22553113	C₂	semidirect
5. #22553112	C₂	semidirect
6. #22553087	C₂	semidirect
7. #357×C₁₃₁	C₂	semidirect
71. C₁₃₁	#2244	semidirect
78. #357	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553713	2
#22553687	2
#22553114	2
#22553113	2
#22553112	2
#22553087	2
#357×C₁₃₁	2
#2244	131