Subgroups — C₂×#22549928 #30647345

Order 16256.

528 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22549928	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₁₂₇)⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₄²×D₁₂₇	C₄	1	4	yes	—
15	C₄²×D₁₂₇	C₂²	1	4	yes	—
16	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	complemented
17	(C₂×D₁₂₇)⋊C₈	C₂²	1	8	yes	—
18	C₈×Dic₁₂₇	C₂²	1	8	yes	complemented
19	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	complemented
22	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₁₂₇	—	2	4	no	—
27	C₂×C₄×D₁₂₇	—	2	2	no	—
28	C₂×C₄×D₁₂₇	—	2	2	no	—
29	C₂×C₄×D₁₂₇	C₈	1	8	yes	complemented
30	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
34	C₂×C₁₂₇⋊C₈	—	2	4	no	—
35	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
36	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
38	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
39	C₄×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	—	2	4	no	—
44	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
45	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₄²×C₁₂₇	C₂³	1	2	yes	—
47	C₂²×D₁₂₇	—	2	2	no	—
48	C₂²×D₁₂₇	—	2	2	no	—
49	C₂²×D₁₂₇	C₂×C₈	1	4	yes	—
50	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
51	C₂×Dic₁₂₇	—	2	2	no	—
52	C₂×Dic₁₂₇	C₂×C₈	1	4	yes	—
53	C₂×Dic₁₂₇	M₁₆	1	4	yes	—
54	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
55	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
56	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
57	C₄×D₁₂₇	—	2	4	no	—
58	C₄×D₁₂₇	—	2	2	no	—
59	C₄×D₁₂₇	—	2	2	no	—
60	C₄×D₁₂₇	—	2	8	no	—
61	C₄×D₁₂₇	C₂×C₈	1	16	yes	complemented
62	C₁₂₇⋊C₈	—	2	4	no	—
63	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₂₇	#41	1	2	yes	—
70	C₂×C₄×C₁₂₇	#41	1	2	yes	—
71	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
72	C₈×C₁₂₇	—	2	4	no	—
73	C₂²×C₁₂₇	#117	1	2	yes	—
74	C₂²×C₁₂₇	C₂²×C₈	1	2	yes	—
75	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
76	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₁₂₇	—	2	2	no	—
78	C₂×D₁₂₇	—	2	2	no	—
79	C₂×D₁₂₇	—	2	8	no	—
80	C₂×D₁₂₇	C₂²×C₈	1	4	yes	—
81	C₄×C₁₂₇	—	2	2	no	—
82	C₄×C₁₂₇	#117	1	2	yes	—
83	C₄×C₁₂₇	C₂²×C₈	1	4	yes	—
84	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
85	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
86	Dic₁₂₇	—	2	2	no	—
87	Dic₁₂₇	C₂²×C₈	1	4	yes	—
88	Dic₁₂₇	C₂×M₁₆	1	4	yes	—
89	C₂×C₁₂₇	#431	1	4	yes	complemented
90	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
91	C₂×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
92	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
93	D₁₂₇	—	2	4	no	—
94	C₂×#431	—	127	1	no	maximal, Sylow
95	C₁₂₇	C₂×#431	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₄²	—	127	1	no	—
97	#431	—	127	8	no	—
98	C₂×C₄⋊C₈	—	127	1	no	—
99	C₂×C₄⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
100	C₂×C₂²⋊C₈	—	127	2	no	—
101	C₂×C₄×C₈	—	127	2	no	—
102	C₂³×C₄	—	127	1	no	—
103	C₂³×C₄	—	127	1	no	—
104	C₂³×C₄	—	127	1	no	—
105	C₂²×C₈	—	127	2	no	—
106	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
107	C₂×C₄²	—	127	1	no	—
108	C₂×C₄²	—	127	1	no	—
109	C₂×C₄²	—	127	1	no	—
110	C₂×C₄²	—	127	4	no	—
111	C₂×C₄²	—	127	4	no	—
112	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
113	C₄⋊C₈	—	127	4	no	—
114	C₄⋊C₈	C₂×D₁₂₇	1	4	yes	complemented
115	C₂²⋊C₈	—	127	8	no	—
116	C₄×C₈	—	127	8	no	—
117	C₂⁴	—	127	1	no	—
118	C₂²×C₄	—	127	1	no	—
119	C₂²×C₄	—	127	1	no	—
120	C₂²×C₄	—	127	8	no	—
121	C₂²×C₄	—	127	4	no	—
122	C₂²×C₄	—	127	1	no	—
123	C₂²×C₄	—	127	4	no	—
124	C₂²×C₄	—	127	2	no	—
125	C₂²×C₄	—	127	2	no	—
126	C₂²×C₄	—	254	4	no	—
127	C₂²×C₄	—	254	2	no	—
128	C₂²×C₄	—	254	2	no	—
129	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
130	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	127	4	no	—
134	C₂×C₈	—	254	4	no	—
135	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
136	C₄²	—	127	2	no	—
137	C₄²	—	127	4	no	—
138	C₄²	—	127	4	no	—
139	C₄²	—	127	2	no	—
140	C₄²	C₄×D₁₂₇	1	2	yes	—
141	C₄²	C₂²×D₁₂₇	1	2	yes	—
142	C₂³	—	127	4	no	—
143	C₂³	—	127	2	no	—
144	C₂³	—	254	2	no	—
145	C₂³	—	254	2	no	—
146	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	127	4	no	—
149	C₂×C₄	—	127	4	no	—
150	C₂×C₄	—	127	2	no	—
151	C₂×C₄	—	127	2	no	—
152	C₂×C₄	—	127	16	no	—
153	C₂×C₄	—	127	2	no	—
154	C₂×C₄	—	254	8	no	—
155	C₂×C₄	—	254	2	no	—
156	C₂×C₄	—	254	2	no	—
157	C₂×C₄	—	254	2	no	—
158	C₂×C₄	—	254	4	no	—
159	C₂×C₄	C₈×D₁₂₇	1	4	yes	—
160	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
163	C₂×C₄	#14523247	1	2	yes	—
164	C₂×C₄	#14523249	1	2	yes	—
165	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	—	254	4	no	—
168	C₂²	—	127	4	no	—
169	C₂²	—	254	2	no	—
170	C₂²	—	254	2	no	—
171	C₂²	—	254	8	no	—
172	C₂²	#16861167	1	2	yes	—
173	C₂²	C₂×C₈×D₁₂₇	1	2	yes	—
174	C₂²	M₁₆×D₁₂₇	1	2	yes	—
175	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	127	4	no	—
178	C₄	—	127	4	no	—
179	C₄	—	254	2	no	—
180	C₄	#16861167	1	2	yes	—
181	C₄	C₂×C₈×D₁₂₇	1	4	yes	—
182	C₄	C₂×#14523247	1	1	yes	characteristic
183	C₄	C₂×#14523249	1	1	yes	characteristic
184	C₂	—	254	4	no	—
185	C₂	#22549928	1	4	yes	direct product factor
186	C₂	C₂×#16861167	1	1	yes	characteristic
187	C₂	C₂²×C₈×D₁₂₇	1	1	yes	characteristic
188	C₂	C₂×M₁₆×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄⋊C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#431
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #22549928	C₂	direct (N × H)
3. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
5. C₂×C₈×Dic₁₂₇	C₂	semidirect
6. C₂×C₄⋊C₈×C₁₂₇	C₂	semidirect
16. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
18. C₈×Dic₁₂₇	C₂²	semidirect
21. C₄⋊C₈×C₁₂₇	C₂²	semidirect
29. C₂×C₄×D₁₂₇	C₈	semidirect
61. C₄×D₁₂₇	C₂×C₈	semidirect
89. C₂×C₁₂₇	#431	semidirect
95. C₁₂₇	C₂×#431	semidirect
99. C₂×C₄⋊C₈	D₁₂₇	semidirect
114. C₄⋊C₈	C₂×D₁₂₇	semidirect
185. C₂	#22549928	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄²×D₁₂₇	2
#22549928	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×(C₂×D₁₂₇)⋊C₈	2
C₂×C₈×Dic₁₂₇	2
C₂×C₄⋊C₈×C₁₂₇	2
C₂×#431	127

Subgroups — C2×#22549928 #30647345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549928 #30647345