Finite Group Databasebeta

Subgroups — #30640501

Order 16256.

144 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₃₂×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#108×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549911	C₂	1	1	yes	maximal, characteristic
4	#22549700	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549699	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549698	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×D₁₂₇	—	2	1	no	—
9	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
10	#16861167	—	2	1	no	—
11	#16861149	C₂²	1	1	yes	characteristic
12	#16861148	C₄	1	2	yes	complemented
13	C₁₆×D₁₂₇	C₄	1	2	yes	complemented
14	#16861145	C₂²	1	1	yes	characteristic
15	#16861144	—	2	1	no	—
16	#16861142	—	2	1	no	—
17	#16861141	C₂²	1	1	yes	characteristic
18	M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
19	#108×C₁₂₇	C₂²	1	1	yes	characteristic
20	#106×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
22	#14523232	—	2	1	no	—
23	#14523231	—	4	1	no	—
24	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
25	C₈×D₁₂₇	—	4	1	no	—
26	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₈×D₁₂₇	Q₈	1	1	yes	characteristic
29	C₈×D₁₂₇	Q₈	1	1	yes	characteristic
30	C₄⋊Dic₁₂₇	—	2	1	no	—
31	(C₂×D₁₂₇)⋊C₄	—	4	1	no	—
32	C₄×Dic₁₂₇	—	4	1	no	—
33	C₁₂₇⋊C₁₆	C₂×C₄	1	2	yes	—
34	M₁₆×C₁₂₇	—	2	1	no	—
35	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₂₇	—	2	1	no	—
37	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²×D₁₂₇	SD₁₆	1	1	yes	characteristic
39	C₂×Dic₁₂₇	—	4	1	no	—
40	C₂×Dic₁₂₇	SD₁₆	1	1	yes	characteristic
41	C₄×D₁₂₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄×D₁₂₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄×D₁₂₇	SD₁₆	1	1	yes	characteristic
44	C₄×D₁₂₇	SD₁₆	1	1	yes	characteristic
45	C₁₂₇⋊C₈	—	4	1	no	—
46	C₁₂₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₁₂₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	—	4	1	no	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₂₇	—	4	1	no	—
51	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×D₁₂₇	—	4	1	no	—
55	C₂×D₁₂₇	#106	1	1	yes	characteristic
56	C₂×D₁₂₇	#106	1	1	yes	characteristic
57	C₄×C₁₂₇	—	8	1	no	—
58	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₁₂₇	—	8	1	no	—
61	Dic₁₂₇	#106	1	1	yes	characteristic
62	Dic₁₂₇	#106	1	1	yes	characteristic
63	C₂×C₁₂₇	—	2	1	no	—
64	C₂×C₁₂₇	C₂×#106	1	1	yes	characteristic
65	D₁₂₇	—	2	1	no	—
66	D₁₂₇	—	2	1	no	—
67	#2155	—	127	1	no	maximal, Sylow
68	C₁₂₇	#2155	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₃₂	—	127	1	no	—
70	C₂×#108	—	127	1	no	—
71	#427	—	127	1	no	—
72	#365	—	127	1	no	—
73	#365	—	127	1	no	—
74	#365	—	127	1	no	—
75	#365	D₁₂₇	1	1	yes	characteristic, complemented
76	C₂×M₁₆	—	254	1	no	—
77	C₂²×C₈	—	127	1	no	—
78	#117	—	254	1	no	—
79	M₃₂	—	127	1	no	—
80	M₃₂	—	127	2	no	—
81	M₃₂	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂×C₁₆	—	127	2	no	—
83	#108	—	127	1	no	—
84	#108	—	254	1	no	—
85	#108	C₂×D₁₂₇	1	1	yes	characteristic
86	#107	—	254	1	no	—
87	#106	—	127	1	no	—
88	#106	C₂×D₁₂₇	1	1	yes	characteristic
89	C₂²×C₄	—	127	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	508	1	no	—
92	M₁₆	—	254	1	no	—
93	C₂×C₈	—	127	1	no	—
94	C₂×C₈	—	127	1	no	—
95	C₂×C₈	—	127	1	no	—
96	C₂×C₈	—	127	1	no	—
97	C₂×C₈	—	127	1	no	—
98	C₂×C₈	—	508	1	no	—
99	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	254	1	no	—
102	C₂²⋊C₄	—	508	1	no	—
103	C₄²	—	508	1	no	—
104	C₁₆	—	127	2	no	—
105	C₁₆	C₄×D₁₂₇	1	2	yes	—
106	C₂³	—	127	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	127	1	no	—
109	C₂×C₄	—	127	1	no	—
110	C₂×C₄	—	127	1	no	—
111	C₂×C₄	—	127	1	no	—
112	C₂×C₄	—	127	1	no	—
113	C₂×C₄	—	508	1	no	—
114	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
115	C₈	—	4	1	no	—
116	C₈	—	127	1	no	—
117	C₈	—	127	1	no	—
118	C₈	—	508	1	no	—
119	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
120	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
121	C₂²	—	127	1	no	—
122	C₂²	—	127	1	no	—
123	C₂²	—	508	1	no	—
124	C₂²	SD₁₆×D₁₂₇	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	127	1	no	—
127	C₄	—	127	1	no	—
128	C₄	—	1016	1	no	—
129	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
130	C₄	SD₁₆×D₁₂₇	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	254	1	no	—
133	C₂	—	254	1	no	—
134	C₂	#106×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#106×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#365×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×#106
prime radical	C₂²×D₁₂₇	SD₁₆
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#365	D₁₂₇
hypocenter	C₁₂₇	#2155
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

Complements

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

N	H ≅ G/N	type
4. #22549700	C₂	semidirect
5. #22549699	C₂	semidirect
6. #22549698	C₂	semidirect
7. #365×C₁₂₇	C₂	semidirect
12. #16861148	C₄	semidirect
13. C₁₆×D₁₂₇	C₄	semidirect
68. C₁₂₇	#2155	semidirect
75. #365	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
M₃₂×D₁₂₇	2
#108×D₁₂₇	2
#22549911	2
#22549700	2
#22549699	2
#22549698	2
#365×C₁₂₇	2
#2155	127