Finite Group Databasebeta

Subgroups — #30679109

Order 16768.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#22554129	C₂	1	1	yes	maximal, characteristic, complemented
2	#22554117	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553755	C₂	1	3	yes	maximal, complemented
4	#22553750	C₂	1	3	yes	maximal, complemented
5	#22553744	C₂	1	3	yes	maximal, complemented
6	C₂×#16862191	C₂	1	3	yes	maximal, complemented
7	#504×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#16862293	—	2	1	no	—
9	#16862292	—	2	1	no	—
10	#16862279	—	2	3	no	—
11	#16862274	—	2	3	no	—
12	#16862268	C₂²	1	3	yes	—
13	#16862268	C₂²	1	3	yes	complemented
14	C₂×Dic₅₂₄	C₂²	1	3	yes	complemented
15	C₂×D₁₀₄₈	C₂²	1	3	yes	—
16	Dic₁₀₄₈	C₂²	1	3	yes	complemented
17	#16862191	C₂²	1	3	yes	complemented
18	#16862191	C₂²	1	6	yes	complemented
19	#16862191	C₂²	1	1	yes	characteristic, complemented
20	D₂₀₉₆	C₂²	1	3	yes	complemented
21	#131×C₁₃₁	C₂²	1	1	yes	characteristic
22	M₃₂×C₁₃₁	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₁₃₁	C₂²	1	3	yes	complemented
24	#14523514	—	4	1	no	—
25	#14523513	—	2	3	no	—
26	#14523513	—	2	3	no	—
27	#14523512	—	4	3	no	—
28	C₂×Dic₂₆₂	—	2	3	no	—
29	Q₈×D₁₃₁	—	4	1	no	—
30	C₂×D₅₂₄	—	2	3	no	—
31	D₄×D₁₃₁	—	4	3	no	—
32	Dic₅₂₄	—	2	3	no	—
33	Dic₅₂₄	C₂³	1	3	yes	—
34	Dic₅₂₄	C₂³	1	1	yes	characteristic
35	#14523502	—	2	3	no	—
36	#14523502	—	2	3	no	—
37	D₁₀₄₈	—	2	3	no	—
38	D₁₀₄₈	C₂³	1	3	yes	—
39	D₁₀₄₈	C₂³	1	1	yes	characteristic
40	#41×C₁₃₁	D₄	1	1	yes	characteristic
41	M₁₆×C₁₃₁	D₄	1	3	yes	—
42	C₂×C₈×C₁₃₁	C₂³	1	3	yes	—
43	C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₁₆×C₁₃₁	C₂³	1	3	yes	—
45	C₂²×D₁₃₁	—	4	3	no	—
46	Dic₂₆₂	—	2	1	no	—
47	Dic₂₆₂	—	2	3	no	—
48	Dic₂₆₂	—	4	3	no	—
49	D₅₂₄	—	2	3	no	—
50	D₅₂₄	—	2	1	no	—
51	D₅₂₄	—	4	3	no	—
52	C₂²⋊D₁₃₁	—	4	3	no	—
53	C₂²⋊D₁₃₁	—	4	3	no	—
54	C₂×Dic₁₃₁	—	4	3	no	—
55	C₄×D₁₃₁	—	4	3	no	—
56	C₄×D₁₃₁	—	4	3	no	—
57	Q₈×C₁₃₁	D₈	1	1	yes	characteristic
58	D₄×C₁₃₁	D₈	1	3	yes	—
59	C₂×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
60	C₈×C₁₃₁	C₂×D₄	1	3	yes	—
61	C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₃₁	C₂×D₈	1	3	yes	—
63	C₂×D₁₃₁	—	4	1	no	—
64	C₂×D₁₃₁	—	4	3	no	—
65	C₂×D₁₃₁	—	16	3	no	—
66	C₄×C₁₃₁	C₂×D₈	1	3	yes	—
67	C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₃₁	—	4	1	no	—
69	Dic₁₃₁	—	4	3	no	—
70	C₂×C₁₃₁	—	2	3	no	—
71	C₂×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
72	D₁₃₁	—	8	1	no	—
73	D₁₃₁	—	8	3	no	—
74	C₁₃₁	#3416	1	1	yes	characteristic, complemented, Sylow
75	#3416	—	131	1	no	maximal, Sylow
76	#578	—	131	1	no	—
77	#576	—	131	1	no	—
78	#510	—	131	3	no	—
79	#509	—	131	3	no	—
80	#508	—	131	3	no	—
81	C₂×SD₃₂	—	131	3	no	—
82	#504	D₁₃₁	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	262	1	no	—
84	H(5,𝔽₂)	—	262	1	no	—
85	#137	—	262	3	no	—
86	AGL(1,ℤ/8)	—	262	3	no	—
87	#135	—	131	3	no	—
88	#135	—	131	3	no	—
89	C₂×Q₁₆	—	131	3	no	—
90	C₂×D₈	—	131	3	no	—
91	#131	C₂×D₁₃₁	1	1	yes	characteristic
92	Q₃₂	—	131	3	no	—
93	SD₃₂	—	131	3	no	—
94	SD₃₂	—	131	6	no	—
95	SD₃₂	—	131	1	no	—
96	D₁₆	—	131	3	no	—
97	M₃₂	C₂×D₁₃₁	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₁₃₁	1	3	yes	complemented
99	#41	—	262	3	no	—
100	#41	—	262	3	no	—
101	#41	—	524	1	no	—
102	#41	—	524	3	no	—
103	#41	D₅₂₄	1	1	yes	characteristic
104	C₂×Q₈	—	262	3	no	—
105	C₂×Q₈	—	524	1	no	—
106	C₂×D₄	—	262	3	no	—
107	C₂×D₄	—	524	3	no	—
108	Q₁₆	—	262	3	no	—
109	Q₁₆	—	131	3	no	—
110	Q₁₆	—	131	1	no	—
111	SD₁₆	—	262	3	no	—
112	SD₁₆	—	262	3	no	—
113	D₈	—	262	3	no	—
114	D₈	—	131	1	no	—
115	D₈	—	131	3	no	—
116	M₁₆	D₅₂₄	1	3	yes	—
117	C₂×C₈	C₂²×D₁₃₁	1	3	yes	—
118	C₁₆	C₂²×D₁₃₁	1	1	yes	characteristic
119	C₁₆	C₂²×D₁₃₁	1	3	yes	—
120	C₂³	—	524	3	no	—
121	Q₈	—	262	1	no	—
122	Q₈	—	262	3	no	—
123	Q₈	—	524	3	no	—
124	Q₈	D₁₀₄₈	1	1	yes	characteristic
125	D₄	—	262	3	no	—
126	D₄	—	262	1	no	—
127	D₄	—	524	3	no	—
128	D₄	—	524	3	no	—
129	D₄	—	524	3	no	—
130	D₄	D₁₀₄₈	1	3	yes	—
131	C₂×C₄	—	524	3	no	—
132	C₂×C₄	—	524	3	no	—
133	C₂×C₄	—	524	3	no	—
134	C₂×C₄	C₂×D₅₂₄	1	3	yes	—
135	C₈	C₂×D₅₂₄	1	3	yes	—
136	C₈	C₂³×D₁₃₁	1	1	yes	characteristic
137	C₂²	—	524	1	no	—
138	C₂²	—	524	3	no	—
139	C₂²	—	2096	3	no	—
140	C₂²	C₂×D₁₀₄₈	1	3	yes	—
141	C₄	—	524	1	no	—
142	C₄	—	524	3	no	—
143	C₄	C₂×D₁₀₄₈	1	3	yes	—
144	C₄	C₂²×D₅₂₄	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	1048	3	no	—
147	C₂	—	1048	1	no	—
148	C₂	C₂²×D₁₀₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₀₄₈
commutator, cosocle	C₈×C₁₃₁	C₂⁴
Frattini	C₈	C₂³×D₁₃₁
Fitting	#504×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₂²×D₈
hypercenter	#504	D₁₃₁
hypocenter	C₁₃₁	#3416
LCS	G ▷ C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #22554129	C₂	semidirect
2. #22554117	C₂	semidirect
3. #22553755	C₂	semidirect
4. #22553750	C₂	semidirect
5. #22553744	C₂	semidirect
6. C₂×#16862191	C₂	semidirect
7. #504×C₁₃₁	C₂	semidirect
13. #16862268	C₂²	semidirect
14. C₂×Dic₅₂₄	C₂²	semidirect
16. Dic₁₀₄₈	C₂²	semidirect
17. #16862191	C₂²	semidirect
18. #16862191	C₂²	semidirect
19. #16862191	C₂²	semidirect
20. D₂₀₉₆	C₂²	semidirect
22. M₃₂×C₁₃₁	C₂²	semidirect
23. C₂×C₁₆×C₁₃₁	C₂²	semidirect
74. C₁₃₁	#3416	semidirect
82. #504	D₁₃₁	semidirect
97. M₃₂	C₂×D₁₃₁	semidirect
98. C₂×C₁₆	C₂×D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22554129	2
#22554117	2
#22553755	2
#22553750	2
#22553744	2
C₂×#16862191	2
#504×C₁₃₁	2
#3416	131