Subgroups — C₂×#22549639 #30638444

Order 16256.

512 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₂₇	C₂	1	2	yes	maximal
2	#22549639	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₂₇	—	2	2	no	—
5	C₂×C₄⋊Dic₁₂₇	—	2	4	no	—
6	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
7	C₂×C₄⋊Dic₁₂₇	C₄	1	8	yes	complemented
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
9	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₂₇	—	2	4	no	—
11	C₂²×C₈×C₁₂₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×Dic₁₂₇	—	2	8	no	—
14	C₂²×Dic₁₂₇	—	2	4	no	—
15	C₂²×Dic₁₂₇	—	4	4	no	—
16	C₂²×Dic₁₂₇	—	4	4	no	—
17	C₄⋊Dic₁₂₇	—	2	8	no	—
18	C₄⋊Dic₁₂₇	—	2	4	no	—
19	C₄⋊Dic₁₂₇	C₂×C₄	1	8	yes	complemented
20	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
24	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	—	2	8	no	—
28	C₂×C₈×C₁₂₇	—	2	4	no	—
29	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
30	C₂×Dic₁₂₇	—	2	8	no	—
31	C₂×Dic₁₂₇	—	4	16	no	—
32	C₂×Dic₁₂₇	—	4	4	no	—
33	C₂×Dic₁₂₇	—	4	4	no	—
34	C₂³×C₁₂₇	C₂²⋊C₄	1	4	yes	—
35	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
36	C₂³×C₁₂₇	D₈	1	2	yes	—
37	C₂³×C₁₂₇	SD₁₆	1	4	yes	—
38	C₂³×C₁₂₇	Q₁₆	1	2	yes	—
39	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
41	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
44	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₈×C₁₂₇	—	2	4	no	—
50	C₂²×C₁₂₇	#95	1	4	yes	—
51	C₂²×C₁₂₇	#102	1	8	yes	—
52	C₂²×C₁₂₇	#103	1	8	yes	—
53	C₂²×C₁₂₇	#106	1	4	yes	—
54	C₂²×C₁₂₇	#107	1	4	yes	—
55	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₁₂₇	C₂×SD₁₆	1	2	yes	—
59	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₁₂₇	#95	1	4	yes	—
61	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
64	Dic₁₂₇	—	4	8	no	—
65	C₂×C₁₂₇	#340	1	8	yes	complemented
66	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#102	1	2	yes	—
68	C₂×C₁₂₇	C₂×#103	1	2	yes	—
69	C₂×C₁₂₇	C₂×#106	1	1	yes	characteristic
70	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
71	C₂×#340	—	127	1	no	maximal, Sylow
72	C₁₂₇	C₂×#340	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₈	D₁₂₇	1	1	yes	characteristic
74	C₂²×C₄⋊C₄	—	127	2	no	—
75	#340	—	127	4	no	—
76	C₂³×C₄	—	254	2	no	—
77	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	Dic₁₂₇	1	4	yes	complemented
80	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
81	C₂×C₄⋊C₄	—	127	8	no	—
82	C₂×C₄⋊C₄	—	127	4	no	—
83	C₂×C₄⋊C₄	—	254	4	no	—
84	C₂×C₄⋊C₄	—	254	2	no	—
85	C₂⁴	D₅₀₈	1	1	yes	characteristic
86	C₂²×C₄	—	508	4	no	—
87	C₂²×C₄	—	508	4	no	—
88	C₂²×C₄	—	254	8	no	—
89	C₂²×C₄	—	254	4	no	—
90	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
91	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
92	C₂²×C₄	C₂²⋊D₁₂₇	1	4	yes	—
93	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
94	C₂²×C₄	Dic₂₅₄	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	complemented
99	C₄⋊C₄	—	127	8	no	—
100	C₄⋊C₄	—	254	8	no	—
101	C₄⋊C₄	—	254	4	no	—
102	C₂³	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
103	C₂³	C₄⋊Dic₁₂₇	1	2	yes	—
104	C₂³	D₁₀₁₆	1	2	yes	—
105	C₂³	#14523237	1	4	yes	—
106	C₂³	Dic₅₀₈	1	2	yes	—
107	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
108	C₂×C₄	—	508	4	no	—
109	C₂×C₄	—	508	4	no	—
110	C₂×C₄	—	508	16	no	—
111	C₂×C₄	—	254	8	no	—
112	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
113	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
114	C₂×C₄	C₂²⋊Dic₁₂₇	1	4	yes	—
115	C₂×C₄	Dic₁₂₇⋊C₄	1	8	yes	—
116	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
118	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
120	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	C₁₂₇⋊#95	1	4	yes	—
123	C₂²	#16861131	1	8	yes	—
124	C₂²	#16861134	1	8	yes	—
125	C₂²	#16861141	1	4	yes	—
126	C₂²	#16861143	1	4	yes	—
127	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	2	yes	—
128	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
129	C₂²	C₂×D₁₀₁₆	1	1	yes	characteristic
130	C₂²	C₂×#14523237	1	2	yes	—
131	C₂²	C₂×Dic₅₀₈	1	1	yes	characteristic
132	C₄	—	508	8	no	—
133	C₄	C₁₂₇⋊#95	1	4	yes	—
134	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
135	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
136	C₄	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
137	C₂	#22549639	1	8	yes	direct product factor
138	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
139	C₂	C₂×#16861131	1	2	yes	—
140	C₂	C₂×#16861134	1	2	yes	—
141	C₂	C₂×#16861141	1	1	yes	characteristic
142	C₂	C₂×#16861143	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₀₈
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂³×C₈×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂³×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#340
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #22549639	C₂	direct (N × H)
7. C₂×C₄⋊Dic₁₂₇	C₄	semidirect
11. C₂²×C₈×C₁₂₇	C₄	semidirect
19. C₄⋊Dic₁₂₇	C₂×C₄	semidirect
29. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
65. C₂×C₁₂₇	#340	semidirect
72. C₁₂₇	C₂×#340	semidirect
79. C₂²×C₈	Dic₁₂₇	semidirect
98. C₂×C₈	C₂×Dic₁₂₇	semidirect
137. C₂	#22549639	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₂₇	2
#22549639	2
C₂³×C₈×C₁₂₇	2
C₂×#340	127

Subgroups — C2×#22549639 #30638444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549639 #30638444