Finite Group Databasebeta

Subgroups — #30640603

Order 16256.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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×#16861150	C₂	1	1	yes	maximal, characteristic
2	D₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549958	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549703	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549678	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549651	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
9	D₄×Dic₁₂₇	—	2	1	no	—
10	#16861171	—	2	1	no	—
11	#16861150	C₄	1	4	yes	complemented
12	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16861143	C₂²	1	1	yes	characteristic
14	#16861130	—	2	1	no	—
15	#16861129	—	2	1	no	—
16	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
19	#107×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×D₅₀₈	—	2	1	no	—
21	C₂²×Dic₁₂₇	—	4	1	no	—
22	C₂×C₄×D₁₂₇	—	4	1	no	—
23	D₁₀₁₆	—	2	1	no	—
24	D₁₀₁₆	C₂×C₄	1	2	yes	complemented
25	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
26	C₄⋊Dic₁₂₇	—	2	1	no	—
27	C₂²⋊Dic₁₂₇	—	4	1	no	—
28	(C₂×D₁₂₇)⋊C₄	—	4	1	no	—
29	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
30	C₁₂₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×D₄×C₁₂₇	—	2	1	no	—
32	D₈×C₁₂₇	—	2	1	no	—
33	D₈×C₁₂₇	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₂₇	—	2	1	no	—
36	C₁₆×C₁₂₇	—	2	1	no	—
37	C₂²×D₁₂₇	—	4	1	no	—
38	D₅₀₈	—	2	2	no	—
39	D₅₀₈	—	4	1	no	—
40	C₂×Dic₁₂₇	—	4	1	no	—
41	C₂×Dic₁₂₇	—	8	2	no	—
42	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
43	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
44	C₄×D₁₂₇	—	8	2	no	—
45	C₁₂₇⋊C₈	—	2	1	no	—
46	C₂³×C₁₂₇	—	4	1	no	—
47	D₄×C₁₂₇	—	2	2	no	—
48	D₄×C₁₂₇	—	4	1	no	—
49	C₂×C₄×C₁₂₇	—	4	1	no	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₁₂₇	#41	1	1	yes	characteristic
53	C₂²×C₁₂₇	—	4	2	no	—
54	C₂²×C₁₂₇	—	8	1	no	—
55	C₂²×C₁₂₇	—	8	1	no	—
56	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
57	C₂×D₁₂₇	—	4	2	no	—
58	C₂×D₁₂₇	—	8	1	no	—
59	C₂×D₁₂₇	—	8	1	no	—
60	C₄×C₁₂₇	—	8	1	no	—
61	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
62	C₄×C₁₂₇	#135	1	1	yes	characteristic
63	Dic₁₂₇	—	2	1	no	—
64	Dic₁₂₇	—	2	1	no	—
65	Dic₁₂₇	—	8	1	no	—
66	C₂×C₁₂₇	—	8	2	no	—
67	C₂×C₁₂₇	C₄×D₈	1	1	yes	characteristic
68	C₂×C₁₂₇	#509	1	1	yes	characteristic
69	C₂×C₁₂₇	#509	1	1	yes	characteristic
70	D₁₂₇	—	8	2	no	—
71	#2177	—	127	1	no	maximal, Sylow
72	C₁₂₇	#2177	1	1	yes	characteristic, complemented, Sylow
73	C₂×D₁₆	—	127	1	no	—
74	C₄×D₈	—	127	1	no	—
75	C₄×D₈	—	127	1	no	—
76	#367	—	127	1	no	—
77	#357	—	127	1	no	—
78	#357	D₁₂₇	1	1	yes	characteristic, complemented
79	#346	—	127	1	no	—
80	C₂×D₈	—	127	1	no	—
81	C₂×D₈	C₂×D₁₂₇	1	1	yes	characteristic
82	C₄×D₄	—	254	1	no	—
83	C₄×D₄	—	254	1	no	—
84	D₁₆	—	127	4	no	—
85	C₂×C₁₆	—	127	1	no	—
86	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
87	#107	—	127	1	no	—
88	#107	C₂×D₁₂₇	1	1	yes	characteristic
89	#102	—	254	1	no	—
90	#102	—	254	1	no	—
91	C₄×C₈	—	127	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	254	1	no	—
94	C₂²×C₄	—	508	1	no	—
95	C₂²×C₄	—	508	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	127	2	no	—
98	D₈	—	254	1	no	—
99	D₈	C₄×D₁₂₇	1	2	yes	complemented
100	C₂×C₈	—	127	1	no	—
101	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	254	1	no	—
104	C₂²⋊C₄	—	508	1	no	—
105	C₂²⋊C₄	—	508	1	no	—
106	C₄²	—	127	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	127	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	508	1	no	—
111	D₄	—	2	2	no	—
112	D₄	—	4	1	no	—
113	D₄	—	508	1	no	—
114	D₄	—	254	2	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	127	1	no	—
117	C₂×C₄	—	127	1	no	—
118	C₂×C₄	—	1016	2	no	—
119	C₂×C₄	—	1016	2	no	—
120	C₂×C₄	—	508	1	no	—
121	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
122	C₈	—	254	1	no	—
123	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
124	C₈	#14523247	1	1	yes	characteristic
125	C₂²	—	4	2	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	1016	1	no	—
129	C₂²	—	1016	1	no	—
130	C₂²	—	508	2	no	—
131	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
132	C₄	—	8	1	no	—
133	C₄	—	1016	1	no	—
134	C₄	—	254	1	no	—
135	C₄	—	254	1	no	—
136	C₄	(C₂²⋊D₁₂₇)⋊C₄	1	1	yes	characteristic
137	C₄	#16861230	1	1	yes	characteristic
138	C₂	—	8	2	no	—
139	C₂	—	1016	2	no	—
140	C₂	#22549956	1	1	yes	characteristic
141	C₂	#22550310	1	1	yes	characteristic
142	C₂	#22550313	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#357×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂×#16861150	C₂
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#357	D₁₂₇
hypocenter	C₁₂₇	#2177
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
2. D₈×Dic₁₂₇	C₂	semidirect
3. #22549958	C₂	semidirect
4. #22549703	C₂	semidirect
5. #22549678	C₂	semidirect
6. #22549651	C₂	semidirect
7. #357×C₁₂₇	C₂	semidirect
11. #16861150	C₄	semidirect
24. D₁₀₁₆	C₂×C₄	semidirect
30. C₁₂₇⋊C₁₆	C₂×C₄	semidirect
33. D₈×C₁₂₇	C₂×C₄	semidirect
72. C₁₂₇	#2177	semidirect
78. #357	D₁₂₇	semidirect
99. D₈	C₄×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861150	2
D₈×Dic₁₂₇	2
#22549958	2
#22549703	2
#22549678	2
#22549651	2
#357×C₁₂₇	2
#2177	127