Finite Group Databasebeta

Subgroups — #30638313

Order 16256.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#22550227	C₂	1	1	yes	maximal, characteristic
2	#22550172	C₂	1	1	yes	maximal, characteristic
3	#22550162	C₂	1	1	yes	maximal, characteristic
4	#22550006	C₂	1	1	yes	maximal, characteristic
5	#22549624	C₂	1	1	yes	maximal, characteristic
6	#22549621	C₂	1	1	yes	maximal, characteristic
7	#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#16861209	C₂²	1	1	yes	characteristic
9	#16861209	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	#16861143	—	2	1	no	—
12	#16861141	—	2	1	no	—
13	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₁₂₇⋊C₈	—	2	1	no	—
15	#16861134	—	2	1	no	—
16	#16861134	—	2	1	no	—
17	#16861134	—	2	1	no	—
18	#16861134	—	2	1	no	—
19	#16861119	—	2	1	no	—
20	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₅₄	—	2	1	no	—
24	C₂×Dic₂₅₄	—	2	1	no	—
25	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
26	C₂×C₁₂₇⋊C₈	—	2	1	no	—
27	C₄⋊Dic₁₂₇	—	2	1	no	—
28	C₄⋊Dic₁₂₇	—	2	1	no	—
29	C₄⋊Dic₁₂₇	—	2	1	no	—
30	C₄⋊Dic₁₂₇	—	2	1	no	—
31	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
32	Dic₁₂₇⋊C₄	—	2	1	no	—
33	C₄×Dic₁₂₇	—	2	1	no	—
34	C₂×C₈×C₁₂₇	—	2	1	no	—
35	C₂×C₈×C₁₂₇	—	2	1	no	—
36	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
39	Dic₂₅₄	—	2	1	no	—
40	Dic₂₅₄	—	2	1	no	—
41	Dic₂₅₄	—	4	1	no	—
42	Dic₂₅₄	—	4	1	no	—
43	Dic₂₅₄	—	8	1	no	—
44	C₂×Dic₁₂₇	—	2	1	no	—
45	C₂×Dic₁₂₇	—	2	1	no	—
46	C₂×Dic₁₂₇	—	4	1	no	—
47	C₂×Dic₁₂₇	—	4	1	no	—
48	C₁₂₇⋊C₈	—	4	1	no	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
52	C₈×C₁₂₇	—	2	1	no	—
53	C₈×C₁₂₇	—	2	1	no	—
54	C₈×C₁₂₇	—	4	1	no	—
55	C₈×C₁₂₇	—	4	1	no	—
56	C₂²×C₁₂₇	#121	1	1	yes	characteristic
57	C₄×C₁₂₇	—	2	1	no	—
58	C₄×C₁₂₇	#135	1	1	yes	characteristic
59	C₄×C₁₂₇	#137	1	1	yes	characteristic
60	C₄×C₁₂₇	#137	1	1	yes	characteristic
61	C₄×C₁₂₇	#137	1	1	yes	characteristic
62	Dic₁₂₇	—	4	1	no	—
63	Dic₁₂₇	—	4	1	no	—
64	Dic₁₂₇	—	8	1	no	—
65	Dic₁₂₇	—	8	1	no	—
66	C₂×C₁₂₇	#464	1	1	yes	characteristic
67	C₂×C₁₂₇	#470	1	1	yes	characteristic
68	C₂×C₁₂₇	#471	1	1	yes	characteristic
69	#1704	—	127	1	no	maximal, Sylow
70	C₁₂₇	#1704	1	1	yes	characteristic, complemented, Sylow
71	#487	—	127	1	no	—
72	#477	—	127	1	no	—
73	#475	—	127	1	no	—
74	#446	—	127	1	no	—
75	#334	D₁₂₇	1	1	yes	characteristic
76	#333	—	127	1	no	—
77	#332	—	127	1	no	—
78	C₄⋊Q₈	—	127	1	no	—
79	C₄⋊Q₈	—	127	1	no	—
80	C₄×Q₈	—	127	1	no	—
81	#107	—	254	1	no	—
82	#106	—	254	1	no	—
83	C₄⋊C₈	—	127	1	no	—
84	C₄⋊C₈	—	254	1	no	—
85	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
87	#103	—	254	1	no	—
88	#103	—	254	1	no	—
89	#103	—	254	1	no	—
90	#103	—	254	1	no	—
91	#97	—	254	1	no	—
92	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
93	C₂×Q₈	—	127	1	no	—
94	C₂×Q₈	—	254	1	no	—
95	C₂×Q₈	—	254	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	254	1	no	—
99	C₂×C₈	D₅₀₈	1	1	yes	characteristic
100	C₂×C₈	D₅₀₈	1	1	yes	characteristic
101	C₄⋊C₄	—	127	1	no	—
102	C₄⋊C₄	—	254	1	no	—
103	C₄⋊C₄	—	254	1	no	—
104	C₄⋊C₄	—	254	1	no	—
105	C₄⋊C₄	—	254	1	no	—
106	C₄⋊C₄	—	254	1	no	—
107	C₄²	—	254	1	no	—
108	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
109	Q₈	—	1016	1	no	—
110	Q₈	—	508	1	no	—
111	Q₈	—	508	1	no	—
112	Q₈	—	254	1	no	—
113	Q₈	—	254	1	no	—
114	C₂×C₄	—	508	1	no	—
115	C₂×C₄	—	508	1	no	—
116	C₂×C₄	—	254	1	no	—
117	C₂×C₄	—	254	1	no	—
118	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
120	C₂×C₄	#14523249	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	508	1	no	—
126	C₂²	#16861184	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	1016	1	no	—
129	C₄	—	1016	1	no	—
130	C₄	—	508	1	no	—
131	C₄	—	508	1	no	—
132	C₄	#16861231	1	1	yes	characteristic
133	C₄	#16861238	1	1	yes	characteristic
134	C₄	#16861238	1	1	yes	characteristic
135	C₄	#16861242	1	1	yes	characteristic
136	C₂	#22550099	1	1	yes	characteristic
137	C₂	#22550135	1	1	yes	characteristic
138	C₂	#22550140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861184
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#334×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#121
hypercenter	#334	D₁₂₇
hypocenter	C₁₂₇	#1704
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
70. C₁₂₇	#1704	semidirect

Maximal subgroups

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

subgroup	index
#22550227	2
#22550172	2
#22550162	2
#22550006	2
#22549624	2
#22549621	2
#334×C₁₂₇	2
#1704	127