Subgroups — C₂×#22550431 #30652019

Order 16256.

1168 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550431	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16861209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16861200	C₂	1	2	yes	maximal, complemented
4	C₂×#16861189	C₂	1	2	yes	maximal
5	C₂×#16861185	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×#16861174	C₂	1	2	yes	maximal, complemented
8	C₂×#16861167	C₂	1	2	yes	maximal
9	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
12	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	#16861209	C₂²	1	8	yes	complemented
16	#16861200	C₂²	1	16	yes	complemented
17	#16861189	C₂²	1	16	yes	—
18	#16861185	C₂²	1	16	yes	—
19	Q₈×Dic₁₂₇	C₂²	1	16	yes	complemented
20	#16861174	C₂²	1	16	yes	complemented
21	#16861167	C₂²	1	16	yes	—
22	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
25	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
26	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
31	C₄²×D₁₂₇	C₂²	1	8	yes	—
32	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₁₂₇	C₂²	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₂³×D₁₂₇	C₂³	1	1	yes	characteristic
38	C₂×Dic₂₅₄	—	2	8	no	—
39	C₂×Dic₂₅₄	C₂³	1	8	yes	—
40	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
43	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₂₇	—	2	4	no	—
45	C₂×C₄×D₁₂₇	—	2	4	no	—
46	C₂×C₄×D₁₂₇	Q₈	1	8	yes	complemented
47	C₂×C₄×D₁₂₇	C₂³	1	4	yes	—
48	C₂×C₄×D₁₂₇	C₂³	1	4	yes	—
49	C₂×C₄×D₁₂₇	C₂³	1	4	yes	—
50	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
51	C₄⋊Dic₁₂₇	C₂³	1	16	yes	—
52	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
53	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
54	(C₂×D₁₂₇)⋊C₄	C₂³	1	16	yes	—
55	C₄×Dic₁₂₇	C₂³	1	4	yes	—
56	C₄×Dic₁₂₇	C₂³	1	4	yes	—
57	C₄×Dic₁₂₇	C₂³	1	4	yes	—
58	C₄×Dic₁₂₇	C₂³	1	16	yes	—
59	C₂×Q₈×C₁₂₇	—	2	8	no	—
60	C₂×Q₈×C₁₂₇	C₂³	1	8	yes	—
61	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
66	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
67	C₄²×C₁₂₇	C₂³	1	4	yes	—
68	C₂²×D₁₂₇	—	2	2	no	—
69	C₂²×D₁₂₇	—	2	2	no	—
70	C₂²×D₁₂₇	C₂×Q₈	1	4	yes	—
71	C₂²×D₁₂₇	C₂⁴	1	2	yes	—
72	Dic₂₅₄	—	2	16	no	—
73	C₂×Dic₁₂₇	—	2	2	no	—
74	C₂×Dic₁₂₇	—	2	2	no	—
75	C₂×Dic₁₂₇	—	2	8	no	—
76	C₂×Dic₁₂₇	C₂×Q₈	1	4	yes	—
77	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
78	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
79	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
80	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
81	C₄×D₁₂₇	—	2	8	no	—
82	C₄×D₁₂₇	—	2	8	no	—
83	C₄×D₁₂₇	C₂×Q₈	1	16	yes	complemented
84	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
85	Q₈×C₁₂₇	—	2	16	no	—
86	C₂×C₄×C₁₂₇	—	2	8	no	—
87	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
88	C₂×C₄×C₁₂₇	#41	1	4	yes	—
89	C₂×C₄×C₁₂₇	#41	1	4	yes	—
90	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
91	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
92	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
93	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
94	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
95	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
96	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
97	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
98	C₂×D₁₂₇	—	2	8	no	—
99	C₂×D₁₂₇	—	2	2	no	—
100	C₂×D₁₂₇	—	2	2	no	—
101	C₂×D₁₂₇	C₂²×Q₈	1	4	yes	—
102	C₄×C₁₂₇	—	2	8	no	—
103	C₄×C₁₂₇	C₂²×Q₈	1	4	yes	—
104	C₄×C₁₂₇	C₂×#41	1	4	yes	—
105	C₄×C₁₂₇	C₂×#41	1	4	yes	—
106	Dic₁₂₇	—	2	2	no	—
107	Dic₁₂₇	—	2	2	no	—
108	Dic₁₂₇	—	2	8	no	—
109	Dic₁₂₇	C₂²×Q₈	1	4	yes	—
110	C₂×C₁₂₇	#533	1	4	yes	complemented
111	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
112	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
113	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
114	D₁₂₇	—	2	4	no	—
115	C₂×#533	—	127	1	no	maximal, Sylow
116	C₁₂₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
117	#533	—	127	16	no	—
118	C₂×C₄⋊Q₈	—	127	1	no	—
119	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
120	C₂×#125	—	127	2	no	—
121	C₂×C₂²⋊Q₈	—	127	2	no	—
122	C₂×C₂²⋊Q₈	—	127	2	no	—
123	C₂×C₄×Q₈	—	127	2	no	—
124	C₂×C₄×Q₈	—	127	2	no	—
125	C₂×#117	—	127	2	no	—
126	C₂²×C₄²	—	127	1	no	—
127	C₂²×Q₈	—	127	2	no	—
128	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
129	C₂³×C₄	—	127	1	no	—
130	C₂³×C₄	—	127	1	no	—
131	C₂³×C₄	—	127	1	no	—
132	C₄⋊Q₈	—	127	8	no	—
133	C₄⋊Q₈	C₂×D₁₂₇	1	8	yes	complemented
134	#125	—	127	16	no	—
135	C₂²⋊Q₈	—	127	16	no	—
136	C₂²⋊Q₈	—	127	16	no	—
137	C₄×Q₈	—	127	16	no	—
138	C₄×Q₈	—	127	16	no	—
139	#117	—	127	16	no	—
140	C₂×C₄⋊C₄	—	127	2	no	—
141	C₂×C₄⋊C₄	—	127	4	no	—
142	C₂×C₄⋊C₄	—	127	4	no	—
143	C₂×C₄⋊C₄	—	127	2	no	—
144	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
145	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
146	C₂×C₂²⋊C₄	—	127	4	no	—
147	C₂×C₄²	—	127	4	no	—
148	C₂×C₄²	—	127	1	no	—
149	C₂×C₄²	—	127	1	no	—
150	C₂×C₄²	—	127	1	no	—
151	C₂×C₄²	—	127	8	no	—
152	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
153	C₂⁴	—	127	1	no	—
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	127	8	no	—
156	C₂×Q₈	—	254	8	no	—
157	C₂×Q₈	C₂²×D₁₂₇	1	8	yes	—
158	C₂²×C₄	—	127	1	no	—
159	C₂²×C₄	—	127	8	no	—
160	C₂²×C₄	—	127	4	no	—
161	C₂²×C₄	—	127	1	no	—
162	C₂²×C₄	—	127	1	no	—
163	C₂²×C₄	—	127	4	no	—
164	C₂²×C₄	—	127	4	no	—
165	C₂²×C₄	—	127	4	no	—
166	C₂²×C₄	—	254	4	no	—
167	C₂²×C₄	—	254	4	no	—
168	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
169	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
172	C₄⋊C₄	—	127	16	no	—
173	C₄⋊C₄	—	127	8	no	—
174	C₄⋊C₄	—	127	16	no	—
175	C₄⋊C₄	—	127	8	no	—
176	C₄⋊C₄	C₂²×D₁₂₇	1	8	yes	—
177	C₄⋊C₄	C₂²×D₁₂₇	1	8	yes	—
178	C₂²⋊C₄	—	127	16	no	—
179	C₄²	—	127	4	no	—
180	C₄²	—	127	4	no	—
181	C₄²	—	127	4	no	—
182	C₄²	—	127	16	no	—
183	C₄²	C₂²×D₁₂₇	1	4	yes	—
184	C₂³	—	127	4	no	—
185	C₂³	—	127	2	no	—
186	C₂³	—	254	2	no	—
187	C₂³	—	254	2	no	—
188	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	254	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	127	4	no	—
193	C₂×C₄	—	127	2	no	—
194	C₂×C₄	—	127	16	no	—
195	C₂×C₄	—	127	2	no	—
196	C₂×C₄	—	127	2	no	—
197	C₂×C₄	—	127	8	no	—
198	C₂×C₄	—	254	2	no	—
199	C₂×C₄	—	254	2	no	—
200	C₂×C₄	—	254	8	no	—
201	C₂×C₄	—	254	8	no	—
202	C₂×C₄	—	254	8	no	—
203	C₂×C₄	Q₈×D₁₂₇	1	4	yes	—
204	C₂×C₄	#14523247	1	4	yes	—
205	C₂×C₄	#14523249	1	4	yes	—
206	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
207	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
208	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
209	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
210	C₂²	—	127	4	no	—
211	C₂²	—	254	8	no	—
212	C₂²	—	254	2	no	—
213	C₂²	—	254	2	no	—
214	C₂²	C₂×Q₈×D₁₂₇	1	2	yes	—
215	C₂²	C₂×#14523247	1	2	yes	—
216	C₂²	C₂×#14523249	1	2	yes	—
217	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
218	C₄	—	2	8	no	—
219	C₄	—	127	4	no	—
220	C₄	—	254	2	no	—
221	C₄	—	254	2	no	—
222	C₄	—	254	8	no	—
223	C₄	C₂×Q₈×D₁₂₇	1	4	yes	—
224	C₄	C₂×#14523247	1	4	yes	—
225	C₄	C₂×#14523249	1	4	yes	—
226	C₂	—	254	4	no	—
227	C₂	#22550431	1	4	yes	direct product factor
228	C₂	C₂²×Q₈×D₁₂₇	1	1	yes	characteristic
229	C₂	C₂²×#14523247	1	1	yes	characteristic
230	C₂	C₂²×#14523249	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂³×D₁₂₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#533
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #22550431	C₂	direct (N × H)
2. C₂×#16861209	C₂	semidirect
3. C₂×#16861200	C₂	semidirect
6. C₂×Q₈×Dic₁₂₇	C₂	semidirect
7. C₂×#16861174	C₂	semidirect
10. C₂×C₄⋊Q₈×C₁₂₇	C₂	semidirect
15. #16861209	C₂²	semidirect
16. #16861200	C₂²	semidirect
19. Q₈×Dic₁₂₇	C₂²	semidirect
20. #16861174	C₂²	semidirect
33. C₄⋊Q₈×C₁₂₇	C₂²	semidirect
46. C₂×C₄×D₁₂₇	Q₈	semidirect
83. C₄×D₁₂₇	C₂×Q₈	semidirect
110. C₂×C₁₂₇	#533	semidirect
116. C₁₂₇	C₂×#533	semidirect
119. C₂×C₄⋊Q₈	D₁₂₇	semidirect
133. C₄⋊Q₈	C₂×D₁₂₇	semidirect
227. C₂	#22550431	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550431	2
C₂×#16861209	2
C₂×#16861200	2
C₂×#16861189	2
C₂×#16861185	2
C₂×Q₈×Dic₁₂₇	2
C₂×#16861174	2
C₂×#16861167	2
C₂×C₄²×D₁₂₇	2
C₂×C₄⋊Q₈×C₁₂₇	2
C₂×#533	127

Subgroups — C2×#22550431 #30652019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550431 #30652019