Subgroups — #2396×C₂₅₁ #44546983

Order 32128.

528 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#117×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#400×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#399×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#398×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#396×C₂₅₁	C₂	1	2	yes	maximal
7	#395×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#393×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊C₄×C₂₅₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₂₅₁	C₂	1	2	yes	maximal
12	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₅₁	—	2	4	no	—
15	#117×C₂₅₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
31	#95×C₂₅₁	C₂²	1	2	yes	—
32	#95×C₂₅₁	C₂²	1	2	yes	—
33	#95×C₂₅₁	C₂²	1	2	yes	—
34	#95×C₂₅₁	C₂²	1	1	yes	characteristic
35	#95×C₂₅₁	C₂²	1	1	yes	characteristic
36	#95×C₂₅₁	C₂²	1	2	yes	—
37	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₂₅₁	—	2	2	no	—
39	C₂×Q₈×C₂₅₁	—	2	4	no	—
40	C₂²×C₄×C₂₅₁	—	2	2	no	—
41	C₂²×C₄×C₂₅₁	—	2	2	no	—
42	C₂²×C₄×C₂₅₁	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₂₅₁	—	2	2	no	—
54	C₄⋊C₄×C₂₅₁	—	2	2	no	—
55	C₄⋊C₄×C₂₅₁	—	2	4	no	—
56	C₄⋊C₄×C₂₅₁	—	2	2	no	—
57	C₄⋊C₄×C₂₅₁	—	2	4	no	—
58	C₄⋊C₄×C₂₅₁	—	2	4	no	—
59	C₄⋊C₄×C₂₅₁	—	2	4	no	—
60	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
61	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
62	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
63	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
64	C₄²×C₂₅₁	—	2	2	no	—
65	C₄²×C₂₅₁	—	2	4	no	—
66	C₄²×C₂₅₁	—	2	4	no	—
67	C₄²×C₂₅₁	D₄	1	4	yes	—
68	C₂³×C₂₅₁	—	2	1	no	—
69	C₂³×C₂₅₁	—	2	2	no	—
70	C₂³×C₂₅₁	—	2	2	no	—
71	C₂³×C₂₅₁	—	2	1	no	—
72	C₂³×C₂₅₁	C₂×Q₈	1	2	yes	—
73	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
74	Q₈×C₂₅₁	—	2	8	no	—
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	2	1	no	—
77	C₂×C₄×C₂₅₁	—	2	2	no	—
78	C₂×C₄×C₂₅₁	—	2	2	no	—
79	C₂×C₄×C₂₅₁	—	2	2	no	—
80	C₂×C₄×C₂₅₁	—	2	1	no	—
81	C₂×C₄×C₂₅₁	—	2	2	no	—
82	C₂×C₄×C₂₅₁	—	2	2	no	—
83	C₂×C₄×C₂₅₁	—	2	8	no	—
84	C₂×C₄×C₂₅₁	—	2	1	no	—
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	2	no	—
89	C₂×C₄×C₂₅₁	—	2	2	no	—
90	C₂×C₄×C₂₅₁	—	2	2	no	—
91	C₂×C₄×C₂₅₁	—	2	2	no	—
92	C₂×C₄×C₂₅₁	—	2	2	no	—
93	C₂×C₄×C₂₅₁	—	2	2	no	—
94	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₂₅₁	#41	1	4	yes	—
100	C₂×C₄×C₂₅₁	#41	1	4	yes	—
101	C₂×C₄×C₂₅₁	#41	1	2	yes	—
102	C₂²×C₂₅₁	—	2	2	no	—
103	C₂²×C₂₅₁	—	2	4	no	—
104	C₂²×C₂₅₁	—	4	1	no	—
105	C₂²×C₂₅₁	—	4	2	no	—
106	C₂²×C₂₅₁	—	4	1	no	—
107	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
108	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
110	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
111	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
112	C₄×C₂₅₁	—	2	2	no	—
113	C₄×C₂₅₁	—	2	2	no	—
114	C₄×C₂₅₁	—	2	4	no	—
115	C₄×C₂₅₁	—	2	4	no	—
116	C₄×C₂₅₁	—	2	2	no	—
117	C₄×C₂₅₁	—	4	1	no	—
118	C₄×C₂₅₁	—	4	2	no	—
119	C₄×C₂₅₁	—	4	1	no	—
120	C₄×C₂₅₁	—	4	2	no	—
121	C₄×C₂₅₁	C₂²⋊Q₈	1	4	yes	—
122	C₂×C₂₅₁	—	4	2	no	—
123	C₂×C₂₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₂₅₁	#525	1	1	yes	characteristic
125	C₂×C₂₅₁	#529	1	2	yes	—
126	C₂×C₂₅₁	#532	1	1	yes	characteristic
127	C₂×C₂₅₁	#533	1	2	yes	—
128	C₂₅₁	#2396	1	1	yes	characteristic, direct product factor, Sylow
129	#2396	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₄×Q₈	C₂×C₂₅₁	1	1	yes	characteristic, complemented
131	C₂×#117	C₂×C₂₅₁	1	1	yes	characteristic
132	#400	C₂×C₂₅₁	1	1	yes	characteristic, complemented
133	#399	C₂×C₂₅₁	1	1	yes	characteristic
134	#398	C₂×C₂₅₁	1	2	yes	complemented
135	#396	C₂×C₂₅₁	1	2	yes	—
136	#395	C₂×C₂₅₁	1	1	yes	characteristic, complemented
137	#393	C₂×C₂₅₁	1	1	yes	characteristic
138	C₄×C₄⋊C₄	C₂×C₂₅₁	1	2	yes	complemented
139	C₄×C₄⋊C₄	C₂×C₂₅₁	1	1	yes	characteristic, complemented
140	C₄×C₂²⋊C₄	C₂×C₂₅₁	1	2	yes	—
141	C₂²×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
143	C₄×Q₈	—	2	4	no	—
144	#117	—	2	4	no	—
145	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
148	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
156	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
157	C₂×C₄²	C₂²×C₂₅₁	1	2	yes	—
158	C₂×C₄²	C₂²×C₂₅₁	1	2	yes	—
159	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
160	#95	C₂²×C₂₅₁	1	2	yes	—
161	#95	C₂²×C₂₅₁	1	2	yes	—
162	#95	C₂²×C₂₅₁	1	2	yes	—
163	#95	C₂²×C₂₅₁	1	1	yes	characteristic
164	#95	C₂²×C₂₅₁	1	1	yes	characteristic
165	#95	C₂²×C₂₅₁	1	2	yes	—
166	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	Q₈×C₂₅₁	1	4	yes	complemented
172	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
173	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
176	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
181	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	D₄×C₂₅₁	1	4	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₂×Q₈×C₂₅₁	1	2	yes	—
202	C₂³	C₂⁴×C₂₅₁	1	1	yes	characteristic
203	Q₈	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
225	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
226	C₂×C₄	C₂×Q₈×C₂₅₁	1	2	yes	—
227	C₂×C₄	C₂×Q₈×C₂₅₁	1	2	yes	—
228	C₂×C₄	#41×C₂₅₁	1	4	yes	—
229	C₂×C₄	#41×C₂₅₁	1	4	yes	—
230	C₂×C₄	#41×C₂₅₁	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
237	C₂²	C₂²×Q₈×C₂₅₁	1	1	yes	characteristic
238	C₂²	C₂×#41×C₂₅₁	1	1	yes	characteristic
239	C₂²	C₂×#41×C₂₅₁	1	2	yes	—
240	C₂²	C₂×#41×C₂₅₁	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂²⋊Q₈×C₂₅₁	1	4	yes	—
251	C₂	—	4	2	no	—
252	C₂	C₂×C₂²⋊Q₈×C₂₅₁	1	1	yes	characteristic
253	C₂	#525×C₂₅₁	1	1	yes	characteristic
254	C₂	#529×C₂₅₁	1	2	yes	—
255	C₂	#532×C₂₅₁	1	1	yes	characteristic
256	C₂	#533×C₂₅₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₅₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₅₁
socle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂⁴×C₂₅₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₂₅₁	C₂	semidirect
3. #400×C₂₅₁	C₂	semidirect
5. #398×C₂₅₁	C₂	semidirect
7. #395×C₂₅₁	C₂	semidirect
9. C₄×C₄⋊C₄×C₂₅₁	C₂	semidirect
10. C₄×C₄⋊C₄×C₂₅₁	C₂	semidirect
42. C₂²×C₄×C₂₅₁	Q₈	semidirect
128. C₂₅₁	#2396	direct (N × H)
129. #2396	C₂₅₁	direct (N × H)
130. C₂×C₄×Q₈	C₂×C₂₅₁	semidirect
132. #400	C₂×C₂₅₁	semidirect
134. #398	C₂×C₂₅₁	semidirect
136. #395	C₂×C₂₅₁	semidirect
138. C₄×C₄⋊C₄	C₂×C₂₅₁	semidirect
139. C₄×C₄⋊C₄	C₂×C₂₅₁	semidirect
171. C₂²×C₄	Q₈×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₂₅₁	2
C₂×#117×C₂₅₁	2
#400×C₂₅₁	2
#399×C₂₅₁	2
#398×C₂₅₁	2
#396×C₂₅₁	2
#395×C₂₅₁	2
#393×C₂₅₁	2
C₄×C₄⋊C₄×C₂₅₁	2
C₄×C₂²⋊C₄×C₂₅₁	2
#2396	251

Subgroups — #2396×C251 #44546983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2396×C₂₅₁ #44546983