Finite Group Databasebeta

Subgroups — #30651878

Order 16256.

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	#22550693	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550681	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550319	C₂	1	3	yes	maximal, complemented
4	#22550314	C₂	1	3	yes	maximal, complemented
5	#22550308	C₂	1	3	yes	maximal, complemented
6	C₂×#16861154	C₂	1	3	yes	maximal, complemented
7	#504×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16861256	—	2	1	no	—
9	#16861255	—	2	1	no	—
10	#16861242	—	2	3	no	—
11	#16861237	—	2	3	no	—
12	#16861231	C₂²	1	3	yes	—
13	#16861231	C₂²	1	3	yes	complemented
14	C₂×Dic₅₀₈	C₂²	1	3	yes	complemented
15	C₂×D₁₀₁₆	C₂²	1	3	yes	—
16	Dic₁₀₁₆	C₂²	1	3	yes	complemented
17	#16861154	C₂²	1	3	yes	complemented
18	#16861154	C₂²	1	6	yes	complemented
19	#16861154	C₂²	1	1	yes	characteristic, complemented
20	D₂₀₃₂	C₂²	1	3	yes	complemented
21	#131×C₁₂₇	C₂²	1	1	yes	characteristic
22	M₃₂×C₁₂₇	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₁₂₇	C₂²	1	3	yes	complemented
24	#14523249	—	4	1	no	—
25	#14523248	—	2	3	no	—
26	#14523248	—	2	3	no	—
27	#14523247	—	4	3	no	—
28	C₂×Dic₂₅₄	—	2	3	no	—
29	Q₈×D₁₂₇	—	4	1	no	—
30	C₂×D₅₀₈	—	2	3	no	—
31	D₄×D₁₂₇	—	4	3	no	—
32	Dic₅₀₈	—	2	3	no	—
33	Dic₅₀₈	C₂³	1	3	yes	—
34	Dic₅₀₈	C₂³	1	1	yes	characteristic
35	#14523237	—	2	3	no	—
36	#14523237	—	2	3	no	—
37	D₁₀₁₆	—	2	3	no	—
38	D₁₀₁₆	C₂³	1	3	yes	—
39	D₁₀₁₆	C₂³	1	1	yes	characteristic
40	#41×C₁₂₇	D₄	1	1	yes	characteristic
41	M₁₆×C₁₂₇	D₄	1	3	yes	—
42	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
43	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₁₆×C₁₂₇	C₂³	1	3	yes	—
45	C₂²×D₁₂₇	—	4	3	no	—
46	Dic₂₅₄	—	2	1	no	—
47	Dic₂₅₄	—	2	3	no	—
48	Dic₂₅₄	—	4	3	no	—
49	D₅₀₈	—	2	3	no	—
50	D₅₀₈	—	2	1	no	—
51	D₅₀₈	—	4	3	no	—
52	C₂²⋊D₁₂₇	—	4	3	no	—
53	C₂²⋊D₁₂₇	—	4	3	no	—
54	C₂×Dic₁₂₇	—	4	3	no	—
55	C₄×D₁₂₇	—	4	3	no	—
56	C₄×D₁₂₇	—	4	3	no	—
57	Q₈×C₁₂₇	D₈	1	1	yes	characteristic
58	D₄×C₁₂₇	D₈	1	3	yes	—
59	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
60	C₈×C₁₂₇	C₂×D₄	1	3	yes	—
61	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₂₇	C₂×D₈	1	3	yes	—
63	C₂×D₁₂₇	—	4	1	no	—
64	C₂×D₁₂₇	—	4	3	no	—
65	C₂×D₁₂₇	—	16	3	no	—
66	C₄×C₁₂₇	C₂×D₈	1	3	yes	—
67	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₂₇	—	4	1	no	—
69	Dic₁₂₇	—	4	3	no	—
70	C₂×C₁₂₇	—	2	3	no	—
71	C₂×C₁₂₇	C₂²×D₈	1	1	yes	characteristic
72	D₁₂₇	—	8	1	no	—
73	D₁₂₇	—	8	3	no	—
74	#3416	—	127	1	no	maximal, Sylow
75	C₁₂₇	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	127	1	no	—
77	#576	—	127	1	no	—
78	#510	—	127	3	no	—
79	#509	—	127	3	no	—
80	#508	—	127	3	no	—
81	C₂×SD₃₂	—	127	3	no	—
82	#504	D₁₂₇	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	254	1	no	—
84	H(5,𝔽₂)	—	254	1	no	—
85	#137	—	254	3	no	—
86	AGL(1,ℤ/8)	—	254	3	no	—
87	#135	—	127	3	no	—
88	#135	—	127	3	no	—
89	C₂×Q₁₆	—	127	3	no	—
90	C₂×D₈	—	127	3	no	—
91	#131	C₂×D₁₂₇	1	1	yes	characteristic
92	Q₃₂	—	127	3	no	—
93	SD₃₂	—	127	3	no	—
94	SD₃₂	—	127	6	no	—
95	SD₃₂	—	127	1	no	—
96	D₁₆	—	127	3	no	—
97	M₃₂	C₂×D₁₂₇	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₁₂₇	1	3	yes	complemented
99	#41	—	508	1	no	—
100	#41	—	508	3	no	—
101	#41	—	254	3	no	—
102	#41	—	254	3	no	—
103	#41	D₅₀₈	1	1	yes	characteristic
104	C₂×Q₈	—	508	1	no	—
105	C₂×Q₈	—	254	3	no	—
106	C₂×D₄	—	508	3	no	—
107	C₂×D₄	—	254	3	no	—
108	Q₁₆	—	127	3	no	—
109	Q₁₆	—	127	1	no	—
110	Q₁₆	—	254	3	no	—
111	SD₁₆	—	254	3	no	—
112	SD₁₆	—	254	3	no	—
113	D₈	—	127	1	no	—
114	D₈	—	127	3	no	—
115	D₈	—	254	3	no	—
116	M₁₆	D₅₀₈	1	3	yes	—
117	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
118	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
119	C₁₆	C₂²×D₁₂₇	1	3	yes	—
120	C₂³	—	508	3	no	—
121	Q₈	—	508	3	no	—
122	Q₈	—	254	1	no	—
123	Q₈	—	254	3	no	—
124	Q₈	D₁₀₁₆	1	1	yes	characteristic
125	D₄	—	508	3	no	—
126	D₄	—	508	3	no	—
127	D₄	—	508	3	no	—
128	D₄	—	254	3	no	—
129	D₄	—	254	1	no	—
130	D₄	D₁₀₁₆	1	3	yes	—
131	C₂×C₄	—	508	3	no	—
132	C₂×C₄	—	508	3	no	—
133	C₂×C₄	—	508	3	no	—
134	C₂×C₄	C₂×D₅₀₈	1	3	yes	—
135	C₈	C₂×D₅₀₈	1	3	yes	—
136	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
137	C₂²	—	2032	3	no	—
138	C₂²	—	508	1	no	—
139	C₂²	—	508	3	no	—
140	C₂²	C₂×D₁₀₁₆	1	3	yes	—
141	C₄	—	508	1	no	—
142	C₄	—	508	3	no	—
143	C₄	C₂×D₁₀₁₆	1	3	yes	—
144	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	1016	3	no	—
147	C₂	—	1016	1	no	—
148	C₂	C₂²×D₁₀₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₀₁₆
commutator, cosocle	C₈×C₁₂₇	C₂⁴
Frattini	C₈	C₂³×D₁₂₇
Fitting	#504×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²×D₈
hypercenter	#504	D₁₂₇
hypocenter	C₁₂₇	#3416
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #22550693	C₂	semidirect
2. #22550681	C₂	semidirect
3. #22550319	C₂	semidirect
4. #22550314	C₂	semidirect
5. #22550308	C₂	semidirect
6. C₂×#16861154	C₂	semidirect
7. #504×C₁₂₇	C₂	semidirect
13. #16861231	C₂²	semidirect
14. C₂×Dic₅₀₈	C₂²	semidirect
16. Dic₁₀₁₆	C₂²	semidirect
17. #16861154	C₂²	semidirect
18. #16861154	C₂²	semidirect
19. #16861154	C₂²	semidirect
20. D₂₀₃₂	C₂²	semidirect
22. M₃₂×C₁₂₇	C₂²	semidirect
23. C₂×C₁₆×C₁₂₇	C₂²	semidirect
75. C₁₂₇	#3416	semidirect
82. #504	D₁₂₇	semidirect
97. M₃₂	C₂×D₁₂₇	semidirect
98. C₂×C₁₆	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550693	2
#22550681	2
#22550319	2
#22550314	2
#22550308	2
C₂×#16861154	2
#504×C₁₂₇	2
#3416	127