Subgroups — #3367×C₂₅₁ #44547954

Order 32128.

372 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#489×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#487×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#477×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#474×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#463×C₂₅₁	C₂	1	2	yes	maximal
8	#461×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#446×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#440×C₂₅₁	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₅₁	—	2	1	no	—
13	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂×D₈×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
18	#126×C₂₅₁	—	2	2	no	—
19	#124×C₂₅₁	C₂²	1	2	yes	—
20	C₂²⋊Q₈×C₂₅₁	—	2	1	no	—
21	C₂²⋊Q₈×C₂₅₁	—	2	1	no	—
22	C₂²⋊Q₈×C₂₅₁	—	2	2	no	—
23	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	—
24	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
28	#117×C₂₅₁	—	2	1	no	—
29	#107×C₂₅₁	C₂²	1	1	yes	characteristic
30	#107×C₂₅₁	C₂²	1	2	yes	—
31	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
33	#103×C₂₅₁	C₂²	1	2	yes	—
34	#103×C₂₅₁	C₂²	1	2	yes	—
35	#103×C₂₅₁	C₂²	1	2	yes	—
36	#102×C₂₅₁	C₂²	1	1	yes	characteristic
37	#102×C₂₅₁	C₂²	1	1	yes	characteristic
38	#102×C₂₅₁	C₂²	1	2	yes	—
39	#97×C₂₅₁	C₂²	1	2	yes	—
40	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₅₁	—	2	2	no	—
42	C₂×Q₈×C₂₅₁	—	2	1	no	—
43	C₂×Q₈×C₂₅₁	—	4	4	no	—
44	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
46	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₅₁	—	2	1	no	—
49	C₂²×C₄×C₂₅₁	—	2	2	no	—
50	C₂²×C₄×C₂₅₁	—	2	1	no	—
51	SD₁₆×C₂₅₁	—	2	4	no	—
52	D₈×C₂₅₁	—	2	2	no	—
53	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
56	C₄⋊C₄×C₂₅₁	—	2	1	no	—
57	C₄⋊C₄×C₂₅₁	—	2	2	no	—
58	C₄⋊C₄×C₂₅₁	—	2	2	no	—
59	C₄⋊C₄×C₂₅₁	—	2	1	no	—
60	C₄⋊C₄×C₂₅₁	—	2	1	no	—
61	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
63	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
65	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
67	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
68	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
69	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
70	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
71	C₄²×C₂₅₁	—	2	2	no	—
72	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
73	C₄²×C₂₅₁	C₂³	1	2	yes	—
74	C₂³×C₂₅₁	—	2	1	no	—
75	C₂³×C₂₅₁	—	2	1	no	—
76	Q₈×C₂₅₁	—	2	2	no	—
77	Q₈×C₂₅₁	—	2	2	no	—
78	Q₈×C₂₅₁	—	2	2	no	—
79	Q₈×C₂₅₁	—	4	4	no	—
80	Q₈×C₂₅₁	—	4	2	no	—
81	D₄×C₂₅₁	—	2	1	no	—
82	D₄×C₂₅₁	—	2	1	no	—
83	D₄×C₂₅₁	—	2	1	no	—
84	D₄×C₂₅₁	#41	1	2	yes	—
85	C₂×C₄×C₂₅₁	—	2	1	no	—
86	C₂×C₄×C₂₅₁	—	2	2	no	—
87	C₂×C₄×C₂₅₁	—	2	2	no	—
88	C₂×C₄×C₂₅₁	—	2	1	no	—
89	C₂×C₄×C₂₅₁	—	4	4	no	—
90	C₂×C₄×C₂₅₁	—	4	2	no	—
91	C₂×C₄×C₂₅₁	—	4	2	no	—
92	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₅₁	—	2	1	no	—
98	C₈×C₂₅₁	—	2	2	no	—
99	C₈×C₂₅₁	—	2	1	no	—
100	C₂²×C₂₅₁	—	2	2	no	—
101	C₂²×C₂₅₁	—	4	1	no	—
102	C₂²×C₂₅₁	—	4	1	no	—
103	C₂²×C₂₅₁	—	4	1	no	—
104	C₂²×C₂₅₁	—	4	1	no	—
105	C₂²×C₂₅₁	—	4	1	no	—
106	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₅₁	—	2	2	no	—
108	C₄×C₂₅₁	—	2	2	no	—
109	C₄×C₂₅₁	—	2	1	no	—
110	C₄×C₂₅₁	—	4	2	no	—
111	C₄×C₂₅₁	—	4	1	no	—
112	C₄×C₂₅₁	—	4	2	no	—
113	C₄×C₂₅₁	—	4	1	no	—
114	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
115	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
116	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×C₂₅₁	—	4	2	no	—
118	C₂×C₂₅₁	—	8	1	no	—
119	C₂×C₂₅₁	#548	1	1	yes	characteristic
120	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₂₅₁	#578	1	1	yes	characteristic
122	C₂₅₁	#3367	1	1	yes	characteristic, direct product factor, Sylow
123	#3367	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	#556	C₂×C₂₅₁	1	1	yes	characteristic, complemented
125	D₄×Q₈	C₂×C₂₅₁	1	1	yes	characteristic, complemented
126	#489	C₂×C₂₅₁	1	2	yes	complemented
127	#487	C₂×C₂₅₁	1	1	yes	characteristic, complemented
128	#477	C₂×C₂₅₁	1	2	yes	complemented
129	#474	C₂×C₂₅₁	1	1	yes	characteristic, complemented
130	#463	C₂×C₂₅₁	1	2	yes	—
131	#461	C₂×C₂₅₁	1	1	yes	characteristic, complemented
132	#446	C₂×C₂₅₁	1	1	yes	characteristic, complemented
133	#440	C₂×C₂₅₁	1	2	yes	complemented
134	C₄×D₈	C₂×C₂₅₁	1	1	yes	characteristic
135	C₂²×Q₈	—	2	1	no	—
136	C₂×SD₁₆	C₂²×C₂₅₁	1	2	yes	—
137	C₂×D₈	C₂²×C₂₅₁	1	1	yes	characteristic
138	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
139	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
140	C₄⋊Q₈	C₂²×C₂₅₁	1	2	yes	—
141	#126	—	2	2	no	—
142	#124	C₂²×C₂₅₁	1	2	yes	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	—	2	1	no	—
145	C₂²⋊Q₈	—	2	2	no	—
146	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	—
147	C₄×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
149	C₄×D₄	C₂²×C₂₅₁	1	2	yes	—
150	C₄×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
151	#117	—	2	1	no	—
152	#107	C₂²×C₂₅₁	1	1	yes	characteristic
153	#107	C₂²×C₂₅₁	1	2	yes	—
154	C₄⋊C₈	C₂²×C₂₅₁	1	1	yes	characteristic
155	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	—
156	#103	C₂²×C₂₅₁	1	2	yes	—
157	#103	C₂²×C₂₅₁	1	2	yes	—
158	#103	C₂²×C₂₅₁	1	2	yes	—
159	#102	C₂²×C₂₅₁	1	1	yes	characteristic
160	#102	C₂²×C₂₅₁	1	1	yes	characteristic
161	#102	C₂²×C₂₅₁	1	2	yes	—
162	#97	C₂²×C₂₅₁	1	2	yes	—
163	C₄×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	D₄×C₂₅₁	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₂₅₁	1	2	yes	—
169	C₂×D₄	C₂³×C₂₅₁	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₂₅₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	1	no	—
174	SD₁₆	—	2	4	no	—
175	D₈	—	2	2	no	—
176	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	D₄×C₂₅₁	1	1	yes	characteristic
185	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
186	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
188	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
196	C₄²	C₂³×C₂₅₁	1	2	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	4	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	2	1	no	—
207	D₄	#41×C₂₅₁	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	AGL(1,ℤ/8)×C₂₅₁	1	2	yes	—
238	C₄	C₂×#41×C₂₅₁	1	1	yes	characteristic
239	C₄	2₋¹⁺⁴×C₂₅₁	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	#548×C₂₅₁	1	1	yes	characteristic
243	C₂	C₂×AGL(1,ℤ/8)×C₂₅₁	1	1	yes	characteristic
244	C₂	#578×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
prime radical	C₂×D₈×C₂₅₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ C₄×Q₈×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #556×C₂₅₁	C₂	semidirect
2. D₄×Q₈×C₂₅₁	C₂	semidirect
3. #489×C₂₅₁	C₂	semidirect
4. #487×C₂₅₁	C₂	semidirect
5. #477×C₂₅₁	C₂	semidirect
6. #474×C₂₅₁	C₂	semidirect
8. #461×C₂₅₁	C₂	semidirect
9. #446×C₂₅₁	C₂	semidirect
10. #440×C₂₅₁	C₂	semidirect
122. C₂₅₁	#3367	direct (N × H)
123. #3367	C₂₅₁	direct (N × H)
124. #556	C₂×C₂₅₁	semidirect
125. D₄×Q₈	C₂×C₂₅₁	semidirect
126. #489	C₂×C₂₅₁	semidirect
127. #487	C₂×C₂₅₁	semidirect
128. #477	C₂×C₂₅₁	semidirect
129. #474	C₂×C₂₅₁	semidirect
131. #461	C₂×C₂₅₁	semidirect
132. #446	C₂×C₂₅₁	semidirect
133. #440	C₂×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#556×C₂₅₁	2
D₄×Q₈×C₂₅₁	2
#489×C₂₅₁	2
#487×C₂₅₁	2
#477×C₂₅₁	2
#474×C₂₅₁	2
#463×C₂₅₁	2
#461×C₂₅₁	2
#446×C₂₅₁	2
#440×C₂₅₁	2
C₄×D₈×C₂₅₁	2
#3367	251

Subgroups — #3367×C251 #44547954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3367×C₂₅₁ #44547954