Finite Group Databasebeta

Subgroups — #22549891

Order 8128.

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

normal subgroups only

This group has 154 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	#41×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861166	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861137	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861136	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861135	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523249	C₄	1	1	yes	characteristic, complemented
9	#14523248	—	2	1	no	—
10	#14523248	C₂²	1	1	yes	characteristic
11	#14523247	—	2	1	no	—
12	#14523247	C₄	1	1	yes	characteristic, complemented
13	Q₈×D₁₂₇	C₄	1	1	yes	characteristic, complemented
14	D₄×D₁₂₇	—	2	1	no	—
15	D₄×D₁₂₇	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×D₁₂₇	—	2	1	no	—
17	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
18	#14523232	C₂²	1	1	yes	characteristic, complemented
19	#14523231	—	2	1	no	—
20	C₈×D₁₂₇	—	2	1	no	—
21	Dic₁₂₇⋊C₄	—	2	1	no	—
22	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
23	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
24	#41×C₁₂₇	C₂²	1	1	yes	characteristic
25	M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
26	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₁₂₇	—	2	1	no	—
28	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
29	Dic₂₅₄	—	2	1	no	—
30	Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
31	D₅₀₈	—	2	1	no	—
32	D₅₀₈	C₂×C₄	1	1	yes	characteristic
33	C₂²⋊D₁₂₇	—	2	1	no	—
34	C₂²⋊D₁₂₇	—	2	1	no	—
35	C₂²⋊D₁₂₇	—	2	1	no	—
36	C₂×Dic₁₂₇	—	2	1	no	—
37	C₂×Dic₁₂₇	—	2	1	no	—
38	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
39	C₄×D₁₂₇	—	2	1	no	—
40	C₄×D₁₂₇	—	2	1	no	—
41	C₄×D₁₂₇	—	2	1	no	—
42	C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₄×D₁₂₇	D₄	1	1	yes	characteristic
45	C₄×D₁₂₇	D₄	1	1	yes	characteristic
46	C₁₂₇⋊C₈	—	2	1	no	—
47	Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	D₄×C₁₂₇	—	2	1	no	—
49	D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	—	2	1	no	—
51	C₂×C₄×C₁₂₇	—	2	1	no	—
52	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₈×C₁₂₇	—	2	1	no	—
54	C₂²×C₁₂₇	—	2	1	no	—
55	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂×D₁₂₇	—	2	1	no	—
57	C₂×D₁₂₇	—	4	1	no	—
58	C₂×D₁₂₇	—	4	2	no	—
59	C₂×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₂₇	—	2	1	no	—
62	C₄×C₁₂₇	—	4	1	no	—
63	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	Dic₁₂₇	—	2	1	no	—
66	Dic₁₂₇	—	4	1	no	—
67	Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
68	Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₁₂₇	—	2	1	no	—
70	C₂×C₁₂₇	—	4	1	no	—
71	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	D₁₂₇	—	2	1	no	—
73	D₁₂₇	—	2	1	no	—
74	D₁₂₇	—	4	1	no	—
75	C₁₂₇	#421	1	1	yes	characteristic, complemented, Sylow
76	#421	—	127	1	no	maximal, Sylow
77	C₂×#41	—	127	1	no	—
78	C₂×M₁₆	—	127	1	no	—
79	#117	—	127	1	no	—
80	#104	—	127	1	no	—
81	#104	—	127	1	no	—
82	#104	—	127	1	no	—
83	#104	D₁₂₇	1	1	yes	characteristic, complemented
84	#41	—	127	1	no	—
85	#41	—	127	1	no	—
86	#41	—	127	1	no	—
87	#41	—	254	1	no	—
88	#41	—	254	1	no	—
89	#41	C₂×D₁₂₇	1	1	yes	characteristic
90	C₂×Q₈	—	127	1	no	—
91	C₂×D₄	—	127	1	no	—
92	C₂×D₄	—	254	1	no	—
93	C₂²×C₄	—	127	1	no	—
94	C₂²×C₄	—	254	1	no	—
95	M₁₆	—	127	1	no	—
96	M₁₆	—	254	1	no	—
97	M₁₆	C₂×D₁₂₇	1	1	yes	characteristic, complemented
98	C₂×C₈	—	254	1	no	—
99	C₄⋊C₄	—	254	1	no	—
100	C₂²⋊C₄	—	254	1	no	—
101	C₄²	—	127	1	no	—
102	C₄²	C₂×D₁₂₇	1	1	yes	characteristic, complemented
103	C₂³	—	127	1	no	—
104	C₂³	—	254	1	no	—
105	Q₈	—	127	1	no	—
106	Q₈	—	254	1	no	—
107	Q₈	C₄×D₁₂₇	1	1	yes	characteristic
108	D₄	—	2	1	no	—
109	D₄	—	127	1	no	—
110	D₄	—	254	1	no	—
111	D₄	—	254	1	no	—
112	D₄	—	254	1	no	—
113	D₄	—	254	1	no	—
114	D₄	C₄×D₁₂₇	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	127	1	no	—
118	C₂×C₄	—	127	1	no	—
119	C₂×C₄	—	127	1	no	—
120	C₂×C₄	—	127	1	no	—
121	C₂×C₄	—	127	1	no	—
122	C₂×C₄	—	254	1	no	—
123	C₂×C₄	—	254	1	no	—
124	C₂×C₄	—	254	1	no	—
125	C₂×C₄	—	254	1	no	—
126	C₂×C₄	—	254	1	no	—
127	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	254	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	127	1	no	—
132	C₂²	—	127	1	no	—
133	C₂²	—	508	1	no	—
134	C₂²	—	508	2	no	—
135	C₂²	—	254	1	no	—
136	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	127	1	no	—
140	C₄	—	127	1	no	—
141	C₄	—	508	1	no	—
142	C₄	—	254	1	no	—
143	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
144	C₄	D₄×D₁₂₇	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	508	1	no	—
148	C₂	—	254	1	no	—
149	C₂	—	254	1	no	—
150	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#104×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₂²⋊C₄
hypercenter	#104	D₁₂₇
hypocenter	C₁₂₇	#421
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₁₂₇	C₂	semidirect
3. #16861166	C₂	semidirect
4. #16861137	C₂	semidirect
5. #16861136	C₂	semidirect
6. #16861135	C₂	semidirect
7. #104×C₁₂₇	C₂	semidirect
8. #14523249	C₄	semidirect
12. #14523247	C₄	semidirect
13. Q₈×D₁₂₇	C₄	semidirect
15. D₄×D₁₂₇	C₄	semidirect
18. #14523232	C₂²	semidirect
23. C₄×Dic₁₂₇	C₂²	semidirect
25. M₁₆×C₁₂₇	C₂²	semidirect
26. C₄²×C₁₂₇	C₂²	semidirect
75. C₁₂₇	#421	semidirect
83. #104	D₁₂₇	semidirect
97. M₁₆	C₂×D₁₂₇	semidirect
102. C₄²	C₂×D₁₂₇	semidirect