Subgroups — C₂×#30631688 #44558877

Order 32128.

816 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	C₂²×#16860794	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631688	C₂	1	8	yes	maximal, direct product factor
4	C₂×#22548639	C₂	1	2	yes	maximal, complemented
5	C₂×#22548636	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₂₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₅₁	—	2	1	no	—
8	C₂×#16860794	—	2	4	no	—
9	C₂×#16860794	C₂²	1	4	yes	complemented
10	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
12	C₂×C₄⋊Dic₂₅₁	—	2	2	no	—
13	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₂₅₁	C₂²	1	4	yes	complemented
15	#22548639	C₂²	1	8	yes	complemented
16	#22548636	C₂²	1	8	yes	complemented
17	C₂×#41×C₂₅₁	—	2	4	no	—
18	C₂×#41×C₂₅₁	—	2	2	no	—
19	C₂×#41×C₂₅₁	C₄	1	8	yes	complemented
20	C₂×#41×C₂₅₁	C₂²	1	4	yes	—
21	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₂₅₁	—	2	1	no	—
23	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₂₅₁	—	2	1	no	—
25	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₂²×Dic₂₅₁	—	2	2	no	—
27	C₂²×Dic₂₅₁	—	2	4	no	—
28	C₂²×Dic₂₅₁	—	4	4	no	—
29	#16860794	—	2	8	no	—
30	C₂×C₂₅₁⋊C₈	—	2	4	no	—
31	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₂₅₁	—	2	2	no	—
33	C₄⋊Dic₂₅₁	—	2	4	no	—
34	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
35	C₂⁴×C₂₅₁	—	2	1	no	—
36	C₂⁴×C₂₅₁	D₄	1	1	yes	characteristic
37	#41×C₂₅₁	—	2	16	no	—
38	#41×C₂₅₁	—	2	8	no	—
39	#41×C₂₅₁	C₂×C₄	1	16	yes	complemented
40	C₂×Q₈×C₂₅₁	—	2	2	no	—
41	C₂×Q₈×C₂₅₁	—	2	1	no	—
42	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
43	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
44	C₂×D₄×C₂₅₁	—	2	8	no	—
45	C₂×D₄×C₂₅₁	—	2	4	no	—
46	C₂×D₄×C₂₅₁	—	2	1	no	—
47	C₂×D₄×C₂₅₁	—	2	2	no	—
48	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
49	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₅₁	—	2	1	no	—
51	C₂²×C₄×C₂₅₁	—	2	8	no	—
52	C₂²×C₄×C₂₅₁	—	2	4	no	—
53	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
54	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
56	C₂²×C₄×C₂₅₁	D₄	1	4	yes	—
57	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
59	C₂×Dic₂₅₁	—	2	4	no	—
60	C₂×Dic₂₅₁	—	4	8	no	—
61	C₂×Dic₂₅₁	—	4	4	no	—
62	C₂₅₁⋊C₈	—	2	4	no	—
63	C₂³×C₂₅₁	—	2	2	no	—
64	C₂³×C₂₅₁	—	2	2	no	—
65	C₂³×C₂₅₁	—	2	4	no	—
66	C₂³×C₂₅₁	—	2	2	no	—
67	C₂³×C₂₅₁	—	4	2	no	—
68	C₂³×C₂₅₁	—	4	2	no	—
69	C₂³×C₂₅₁	C₂²⋊C₄	1	4	yes	—
70	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
71	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₂₅₁	—	2	4	no	—
73	Q₈×C₂₅₁	—	2	2	no	—
74	Q₈×C₂₅₁	C₂²×C₄	1	4	yes	complemented
75	D₄×C₂₅₁	—	2	16	no	—
76	D₄×C₂₅₁	—	2	4	no	—
77	D₄×C₂₅₁	—	2	2	no	—
78	D₄×C₂₅₁	C₂²×C₄	1	4	yes	complemented
79	C₂×C₄×C₂₅₁	—	2	4	no	—
80	C₂×C₄×C₂₅₁	—	2	2	no	—
81	C₂×C₄×C₂₅₁	—	2	16	no	—
82	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	4	yes	—
83	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	8	yes	—
84	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
91	C₂²×C₂₅₁	—	2	2	no	—
92	C₂²×C₂₅₁	—	2	2	no	—
93	C₂²×C₂₅₁	—	2	8	no	—
94	C₂²×C₂₅₁	—	2	4	no	—
95	C₂²×C₂₅₁	—	4	2	no	—
96	C₂²×C₂₅₁	—	4	2	no	—
97	C₂²×C₂₅₁	—	4	8	no	—
98	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂²×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
101	C₂²×C₂₅₁	#137	1	2	yes	—
102	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₂₅₁	—	2	4	no	—
104	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
105	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
106	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
108	Dic₂₅₁	—	4	4	no	—
109	C₂×C₂₅₁	—	2	4	no	—
110	C₂×C₂₅₁	—	4	4	no	—
111	C₂×C₂₅₁	#417	1	4	yes	complemented
112	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
115	C₂₅₁	C₂×#417	1	1	yes	characteristic, complemented, Sylow
116	C₂×#417	—	251	1	no	maximal, Sylow
117	C₂²×#41	D₂₅₁	1	1	yes	characteristic
118	C₂²×M₁₆	—	251	1	no	—
119	C₂²×C₄⋊C₄	—	251	1	no	—
120	#417	—	251	8	no	—
121	C₂×#103	—	251	2	no	—
122	C₂×#102	—	251	2	no	—
123	C₂×#41	—	2	4	no	—
124	C₂×#41	—	2	2	no	—
125	C₂×#41	Dic₂₅₁	1	8	yes	complemented
126	C₂×#41	C₂×D₂₅₁	1	4	yes	—
127	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
128	C₂²×D₄	—	2	1	no	—
129	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
130	C₂³×C₄	—	2	1	no	—
131	C₂³×C₄	—	502	1	no	—
132	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
133	C₂×M₁₆	—	502	4	no	—
134	C₂×M₁₆	—	251	4	no	—
135	C₂²×C₈	—	251	2	no	—
136	C₂×C₄⋊C₄	—	502	2	no	—
137	C₂×C₄⋊C₄	—	502	1	no	—
138	C₂×C₄⋊C₄	—	251	2	no	—
139	C₂×C₄⋊C₄	—	251	4	no	—
140	#103	—	251	8	no	—
141	#102	—	251	8	no	—
142	C₂⁴	—	2	1	no	—
143	C₂⁴	C₂²⋊D₂₅₁	1	1	yes	characteristic
144	#41	—	2	16	no	—
145	#41	—	2	8	no	—
146	#41	C₂×Dic₂₅₁	1	16	yes	complemented
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	C₂×Dic₂₅₁	1	4	yes	complemented
150	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
151	C₂×D₄	—	2	8	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	1	no	—
155	C₂×D₄	C₂×Dic₂₅₁	1	4	yes	complemented
156	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	1004	4	no	—
161	C₂²×C₄	—	502	4	no	—
162	C₂²×C₄	—	502	2	no	—
163	C₂²×C₄	C₂×Dic₂₅₁	1	4	yes	—
164	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
165	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
166	C₂²×C₄	C₂²⋊D₂₅₁	1	4	yes	—
167	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
168	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
169	M₁₆	—	502	8	no	—
170	C₂×C₈	—	502	4	no	—
171	C₂×C₈	—	251	4	no	—
172	C₄⋊C₄	—	502	4	no	—
173	C₄⋊C₄	—	502	2	no	—
174	C₄⋊C₄	—	251	4	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	4	no	—
179	C₂³	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂²⋊Dic₂₅₁	1	4	yes	—
182	C₂³	C₂×C₂²⋊D₂₅₁	1	2	yes	—
183	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
184	Q₈	—	2	2	no	—
185	Q₈	—	2	4	no	—
186	Q₈	C₂²×Dic₂₅₁	1	4	yes	complemented
187	D₄	—	2	16	no	—
188	D₄	—	2	2	no	—
189	D₄	—	2	4	no	—
190	D₄	C₂²×Dic₂₅₁	1	4	yes	complemented
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	1004	8	no	—
195	C₂×C₄	—	1004	4	no	—
196	C₂×C₄	—	502	4	no	—
197	C₂×C₄	C₂²⋊Dic₂₅₁	1	4	yes	—
198	C₂×C₄	C₂²⋊Dic₂₅₁	1	8	yes	—
199	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
200	C₂×C₄	C₂²×Dic₂₅₁	1	4	yes	—
201	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
202	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	4	yes	—
203	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
204	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
206	C₈	—	502	4	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	8	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₂×C₂²⋊Dic₂₅₁	1	4	yes	—
215	C₂²	C₂×C₂²⋊Dic₂₅₁	1	2	yes	—
216	C₂²	#22548740	1	2	yes	—
217	C₂²	#22548745	1	2	yes	—
218	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	1004	4	no	—
221	C₄	C₂×C₂²⋊Dic₂₅₁	1	4	yes	—
222	C₄	C₂×C₂²⋊Dic₂₅₁	1	2	yes	—
223	C₄	C₂³×Dic₂₅₁	1	1	yes	characteristic
224	C₄	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
225	C₂	—	2	4	no	—
226	C₂	—	4	4	no	—
227	C₂	#30631688	1	4	yes	direct product factor
228	C₂	C₂²×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
229	C₂	C₂×#22548740	1	1	yes	characteristic
230	C₂	C₂×#22548745	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting, prime radical	C₂²×#41×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂×D₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂²×#41	D₂₅₁
hypocenter	C₂₅₁	C₂×#417
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#16860794	C₂	semidirect
2. C₂²×C₄⋊Dic₂₅₁	C₂	semidirect
3. #30631688	C₂	direct (N × H)
4. C₂×#22548639	C₂	semidirect
5. C₂×#22548636	C₂	semidirect
9. C₂×#16860794	C₂²	semidirect
14. C₂×C₄⋊Dic₂₅₁	C₂²	semidirect
15. #22548639	C₂²	semidirect
16. #22548636	C₂²	semidirect
19. C₂×#41×C₂₅₁	C₄	semidirect
39. #41×C₂₅₁	C₂×C₄	semidirect
42. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
48. C₂×D₄×C₂₅₁	C₂×C₄	semidirect
74. Q₈×C₂₅₁	C₂²×C₄	semidirect
78. D₄×C₂₅₁	C₂²×C₄	semidirect
111. C₂×C₂₅₁	#417	semidirect
115. C₂₅₁	C₂×#417	semidirect
125. C₂×#41	Dic₂₅₁	semidirect
146. #41	C₂×Dic₂₅₁	semidirect
149. C₂×Q₈	C₂×Dic₂₅₁	semidirect
155. C₂×D₄	C₂×Dic₂₅₁	semidirect
186. Q₈	C₂²×Dic₂₅₁	semidirect
190. D₄	C₂²×Dic₂₅₁	semidirect
227. C₂	#30631688	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×#16860794	2
C₂²×C₄⋊Dic₂₅₁	2
#30631688	2
C₂×#22548639	2
C₂×#22548636	2
C₂²×#41×C₂₅₁	2
C₂×#417	251

Subgroups — C2×#30631688 #44558877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30631688 #44558877