Subgroups — C₂×#22549949 #30647411

Order 16256.

564 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic
4	#22549949	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₀₈	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×#14523231	—	2	4	no	—
13	C₂×#14523231	C₂²	1	4	yes	complemented
14	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₂₇	—	2	4	no	—
17	C₂×C₈×D₁₂₇	C₂²	1	4	yes	complemented
18	C₄×D₅₀₈	C₄	1	4	yes	—
19	C₄×D₅₀₈	C₂²	1	4	yes	—
20	C₂×C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
21	C₂×(C₂×D₁₂₇)⋊C₄	C₄	1	1	yes	characteristic
22	C₂×(C₂×D₁₂₇)⋊C₄	C₄	1	1	yes	characteristic
23	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	complemented
24	(C₂×D₁₂₇)⋊C₈	C₂²	1	4	yes	complemented
25	(C₂×D₁₂₇)⋊C₈	C₂²	1	4	yes	complemented
26	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
29	#97×C₁₂₇	C₂²	1	4	yes	complemented
30	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂×D₅₀₈	—	2	4	no	—
33	C₂×D₅₀₈	C₂×C₄	1	4	yes	—
34	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₂₇	—	2	2	no	—
37	C₂×C₄×D₁₂₇	—	2	2	no	—
38	C₂×C₄×D₁₂₇	—	2	2	no	—
39	C₂×C₄×D₁₂₇	—	2	2	no	—
40	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
43	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
44	#14523231	—	2	8	no	—
45	C₂×C₁₂₇⋊C₈	—	2	2	no	—
46	C₂×C₁₂₇⋊C₈	—	2	2	no	—
47	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
49	C₈×D₁₂₇	—	2	8	no	—
50	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
51	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
52	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	—	2	2	no	—
57	C₂×C₈×C₁₂₇	D₄	1	4	yes	complemented
58	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
59	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
60	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
61	C₄²×C₁₂₇	C₂³	1	2	yes	—
62	C₂²×D₁₂₇	—	2	2	no	—
63	C₂²×D₁₂₇	—	2	2	no	—
64	C₂²×D₁₂₇	—	2	2	no	—
65	C₂²×D₁₂₇	—	2	2	no	—
66	C₂²×D₁₂₇	—	2	2	no	—
67	C₂²×D₁₂₇	M₁₆	1	4	yes	complemented
68	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
70	D₅₀₈	—	2	8	no	—
71	C₂×Dic₁₂₇	—	2	2	no	—
72	C₂×Dic₁₂₇	—	2	2	no	—
73	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
75	C₄×D₁₂₇	—	2	4	no	—
76	C₄×D₁₂₇	—	2	4	no	—
77	C₄×D₁₂₇	—	2	2	no	—
78	C₄×D₁₂₇	—	2	2	no	—
79	C₄×D₁₂₇	—	2	2	no	—
80	C₄×D₁₂₇	—	2	2	no	—
81	C₁₂₇⋊C₈	—	2	2	no	—
82	C₁₂₇⋊C₈	—	2	2	no	—
83	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₂₇	#41	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₂₇	—	2	2	no	—
93	C₈×C₁₂₇	C₂×D₄	1	4	yes	complemented
94	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
95	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
96	C₂²×C₁₂₇	#131	1	2	yes	—
97	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₂₇	—	2	2	no	—
99	C₂×D₁₂₇	—	2	2	no	—
100	C₂×D₁₂₇	—	2	8	no	—
101	C₂×D₁₂₇	—	2	2	no	—
102	C₂×D₁₂₇	—	2	2	no	—
103	C₂×D₁₂₇	—	2	2	no	—
104	C₂×D₁₂₇	—	4	4	no	—
105	C₂×D₁₂₇	C₂×M₁₆	1	4	yes	complemented
106	C₄×C₁₂₇	—	2	2	no	—
107	C₄×C₁₂₇	—	2	2	no	—
108	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
109	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
111	Dic₁₂₇	—	2	2	no	—
112	Dic₁₂₇	—	2	2	no	—
113	C₂×C₁₂₇	#435	1	4	yes	complemented
114	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
116	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
117	D₁₂₇	—	2	4	no	—
118	D₁₂₇	—	4	2	no	—
119	C₂×#435	—	127	1	no	maximal, Sylow
120	C₁₂₇	C₂×#435	1	1	yes	characteristic, complemented, Sylow
121	C₂²×M₁₆	—	127	1	no	—
122	C₂³×C₈	—	127	1	no	—
123	C₂×C₄×D₄	—	127	1	no	—
124	#435	—	127	8	no	—
125	C₂×C₄⋊C₈	—	127	1	no	—
126	C₂×C₂²⋊C₈	—	127	1	no	—
127	C₂×C₂²⋊C₈	—	127	1	no	—
128	C₂×#97	D₁₂₇	1	1	yes	characteristic, complemented
129	C₂²×D₄	—	127	1	no	—
130	C₂³×C₄	—	127	1	no	—
131	C₂³×C₄	—	127	1	no	—
132	C₂×M₁₆	—	127	4	no	—
133	C₂×M₁₆	—	254	4	no	—
134	C₂²×C₈	—	127	1	no	—
135	C₂²×C₈	—	127	1	no	—
136	C₂²×C₈	—	127	4	no	—
137	C₂²×C₈	—	254	4	no	—
138	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
139	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
140	C₄×D₄	—	127	4	no	—
141	C₄×D₄	—	127	4	no	—
142	C₂×C₄⋊C₄	—	127	1	no	—
143	C₂×C₂²⋊C₄	—	127	1	no	—
144	C₂×C₂²⋊C₄	—	127	1	no	—
145	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
146	C₄⋊C₈	—	127	4	no	—
147	C₂²⋊C₈	—	127	4	no	—
148	C₂²⋊C₈	—	127	4	no	—
149	#97	C₂×D₁₂₇	1	4	yes	complemented
150	C₂⁴	—	127	1	no	—
151	C₂⁴	—	127	1	no	—
152	C₂×D₄	—	127	4	no	—
153	C₂×D₄	—	254	4	no	—
154	C₂²×C₄	—	127	1	no	—
155	C₂²×C₄	—	127	1	no	—
156	C₂²×C₄	—	127	2	no	—
157	C₂²×C₄	—	127	2	no	—
158	C₂²×C₄	—	127	2	no	—
159	C₂²×C₄	—	127	2	no	—
160	C₂²×C₄	—	254	2	no	—
161	C₂²×C₄	—	254	2	no	—
162	C₂²×C₄	—	254	2	no	—
163	C₂²×C₄	—	254	2	no	—
164	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
165	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
167	M₁₆	—	254	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	127	2	no	—
170	C₂×C₈	—	127	2	no	—
171	C₂×C₈	—	254	2	no	—
172	C₂×C₈	—	254	2	no	—
173	C₂×C₈	—	254	8	no	—
174	C₂×C₈	D₅₀₈	1	4	yes	complemented
175	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
176	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
177	C₄⋊C₄	—	127	4	no	—
178	C₂²⋊C₄	—	127	4	no	—
179	C₂²⋊C₄	—	127	4	no	—
180	C₄²	C₄×D₁₂₇	1	2	yes	—
181	C₄²	C₂²×D₁₂₇	1	2	yes	—
182	C₂³	—	127	4	no	—
183	C₂³	—	127	2	no	—
184	C₂³	—	127	2	no	—
185	C₂³	—	254	2	no	—
186	C₂³	—	254	2	no	—
187	C₂³	—	254	2	no	—
188	C₂³	—	254	2	no	—
189	C₂³	—	254	2	no	—
190	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
191	D₄	—	254	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	127	2	no	—
195	C₂×C₄	—	127	2	no	—
196	C₂×C₄	—	254	2	no	—
197	C₂×C₄	—	254	2	no	—
198	C₂×C₄	—	254	2	no	—
199	C₂×C₄	—	254	2	no	—
200	C₂×C₄	—	254	2	no	—
201	C₂×C₄	—	254	2	no	—
202	C₂×C₄	—	254	4	no	—
203	C₂×C₄	—	254	4	no	—
204	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
207	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
208	C₂×C₄	#14523248	1	2	yes	—
209	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	254	2	no	—
212	C₈	—	254	2	no	—
213	C₈	C₂×D₅₀₈	1	4	yes	complemented
214	C₂²	—	127	4	no	—
215	C₂²	—	508	4	no	—
216	C₂²	—	254	2	no	—
217	C₂²	—	254	2	no	—
218	C₂²	—	254	2	no	—
219	C₂²	—	254	2	no	—
220	C₂²	—	254	2	no	—
221	C₂²	—	254	8	no	—
222	C₂²	C₄×D₅₀₈	1	2	yes	—
223	C₂²	M₁₆×D₁₂₇	1	2	yes	—
224	C₂²	#16861217	1	2	yes	—
225	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	254	2	no	—
229	C₄	—	254	2	no	—
230	C₄	C₄×D₅₀₈	1	2	yes	—
231	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
232	C₄	C₂×#14523248	1	1	yes	characteristic
233	C₂	—	508	2	no	—
234	C₂	—	254	4	no	—
235	C₂	#22549949	1	4	yes	direct product factor
236	C₂	C₂×C₄×D₅₀₈	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₁₂₇	1	1	yes	characteristic
238	C₂	C₂×#16861217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#97×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂²×D₅₀₈	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	C₂×#435
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523231	C₂	semidirect
2. C₂²×C₈×D₁₂₇	C₂	semidirect
4. #22549949	C₂	direct (N × H)
5. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
6. C₂×(C₂×D₁₂₇)⋊C₈	C₂	semidirect
7. C₂×(C₂×D₁₂₇)⋊C₈	C₂	semidirect
8. C₂×#97×C₁₂₇	C₂	semidirect
13. C₂×#14523231	C₂²	semidirect
17. C₂×C₈×D₁₂₇	C₂²	semidirect
23. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
24. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
25. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
29. #97×C₁₂₇	C₂²	semidirect
57. C₂×C₈×C₁₂₇	D₄	semidirect
67. C₂²×D₁₂₇	M₁₆	semidirect
93. C₈×C₁₂₇	C₂×D₄	semidirect
105. C₂×D₁₂₇	C₂×M₁₆	semidirect
113. C₂×C₁₂₇	#435	semidirect
120. C₁₂₇	C₂×#435	semidirect
128. C₂×#97	D₁₂₇	semidirect
149. #97	C₂×D₁₂₇	semidirect
174. C₂×C₈	D₅₀₈	semidirect
213. C₈	C₂×D₅₀₈	semidirect
235. C₂	#22549949	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×#14523231	2
C₂²×C₈×D₁₂₇	2
C₂×C₄×D₅₀₈	2
#22549949	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×(C₂×D₁₂₇)⋊C₈	2
C₂×#97×C₁₂₇	2
C₂×#435	127

Subgroups — C2×#22549949 #30647411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549949 #30647411