Subgroups — #2764×C₂₅₁ #44547351

Order 32128.

400 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₂₅₁	C₂	1	2	yes	maximal, complemented
2	#398×C₂₅₁	C₂	1	2	yes	maximal, complemented
3	#397×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#393×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#391×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#388×C₂₅₁	C₂	1	2	yes	maximal
7	#385×C₂₅₁	C₂	1	2	yes	maximal
8	#383×C₂₅₁	C₂	1	2	yes	maximal, complemented
9	#381×C₂₅₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
21	#95×C₂₅₁	C₂²	1	2	yes	—
22	#95×C₂₅₁	C₂²	1	2	yes	—
23	#95×C₂₅₁	C₂²	1	1	yes	characteristic
24	#95×C₂₅₁	C₂²	1	2	yes	—
25	#95×C₂₅₁	C₂²	1	2	yes	—
26	#95×C₂₅₁	C₂²	1	2	yes	—
27	#95×C₂₅₁	C₂²	1	2	yes	—
28	#95×C₂₅₁	C₂²	1	2	yes	—
29	#95×C₂₅₁	C₂²	1	2	yes	—
30	#95×C₂₅₁	C₂²	1	1	yes	characteristic
31	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₂₅₁	—	2	4	no	—
33	C₂×Q₈×C₂₅₁	—	4	1	no	—
34	C₂²×C₄×C₂₅₁	—	2	2	no	—
35	C₂²×C₄×C₂₅₁	—	4	1	no	—
36	C₂²×C₄×C₂₅₁	—	4	1	no	—
37	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
45	C₄⋊C₄×C₂₅₁	—	4	2	no	—
46	C₄⋊C₄×C₂₅₁	—	4	2	no	—
47	C₄⋊C₄×C₂₅₁	—	4	1	no	—
48	C₄⋊C₄×C₂₅₁	—	4	2	no	—
49	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
50	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
51	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
52	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
53	C₄²×C₂₅₁	—	2	4	no	—
54	C₂³×C₂₅₁	—	2	2	no	—
55	C₂³×C₂₅₁	—	2	1	no	—
56	C₂³×C₂₅₁	—	2	1	no	—
57	C₂³×C₂₅₁	—	2	2	no	—
58	C₂³×C₂₅₁	#41	1	2	yes	—
59	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
60	Q₈×C₂₅₁	—	4	4	no	—
61	C₂×C₄×C₂₅₁	—	2	2	no	—
62	C₂×C₄×C₂₅₁	—	2	2	no	—
63	C₂×C₄×C₂₅₁	—	2	2	no	—
64	C₂×C₄×C₂₅₁	—	2	2	no	—
65	C₂×C₄×C₂₅₁	—	2	2	no	—
66	C₂×C₄×C₂₅₁	—	2	2	no	—
67	C₂×C₄×C₂₅₁	—	2	2	no	—
68	C₂×C₄×C₂₅₁	—	2	2	no	—
69	C₂×C₄×C₂₅₁	—	2	2	no	—
70	C₂×C₄×C₂₅₁	—	2	2	no	—
71	C₂×C₄×C₂₅₁	—	2	1	no	—
72	C₂×C₄×C₂₅₁	—	2	1	no	—
73	C₂×C₄×C₂₅₁	—	2	2	no	—
74	C₂×C₄×C₂₅₁	—	2	2	no	—
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	2	2	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	2	no	—
81	C₂×C₄×C₂₅₁	—	2	2	no	—
82	C₂×C₄×C₂₅₁	—	4	4	no	—
83	C₂×C₄×C₂₅₁	#41	1	4	yes	—
84	C₂²×C₂₅₁	—	2	2	no	—
85	C₂²×C₂₅₁	—	2	4	no	—
86	C₂²×C₂₅₁	—	4	2	no	—
87	C₂²×C₂₅₁	—	4	1	no	—
88	C₂²×C₂₅₁	—	4	1	no	—
89	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
90	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
91	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂²×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
94	C₄×C₂₅₁	—	2	4	no	—
95	C₄×C₂₅₁	—	4	2	no	—
96	C₄×C₂₅₁	—	4	2	no	—
97	C₄×C₂₅₁	—	4	2	no	—
98	C₄×C₂₅₁	—	4	2	no	—
99	C₄×C₂₅₁	—	4	1	no	—
100	C₄×C₂₅₁	—	4	2	no	—
101	C₄×C₂₅₁	—	4	1	no	—
102	C₂×C₂₅₁	—	4	2	no	—
103	C₂×C₂₅₁	#538	1	1	yes	characteristic
104	C₂×C₂₅₁	#542	1	2	yes	—
105	C₂×C₂₅₁	#551	1	1	yes	characteristic
106	C₂×C₂₅₁	#552	1	2	yes	—
107	C₂×C₂₅₁	#563	1	1	yes	characteristic
108	C₂₅₁	#2764	1	1	yes	characteristic, direct product factor, Sylow
109	#2764	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	#400	C₂×C₂₅₁	1	2	yes	complemented
111	#398	C₂×C₂₅₁	1	2	yes	complemented
112	#397	C₂×C₂₅₁	1	1	yes	characteristic
113	#393	C₂×C₂₅₁	1	1	yes	characteristic
114	#391	C₂×C₂₅₁	1	2	yes	complemented
115	#388	C₂×C₂₅₁	1	2	yes	—
116	#385	C₂×C₂₅₁	1	2	yes	—
117	#383	C₂×C₂₅₁	1	2	yes	complemented
118	#381	C₂×C₂₅₁	1	1	yes	characteristic
119	C₂²×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
120	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
125	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₂₅₁	1	2	yes	—
130	#95	C₂²×C₂₅₁	1	2	yes	—
131	#95	C₂²×C₂₅₁	1	2	yes	—
132	#95	C₂²×C₂₅₁	1	1	yes	characteristic
133	#95	C₂²×C₂₅₁	1	2	yes	—
134	#95	C₂²×C₂₅₁	1	2	yes	—
135	#95	C₂²×C₂₅₁	1	2	yes	—
136	#95	C₂²×C₂₅₁	1	2	yes	—
137	#95	C₂²×C₂₅₁	1	2	yes	—
138	#95	C₂²×C₂₅₁	1	2	yes	—
139	#95	C₂²×C₂₅₁	1	1	yes	characteristic
140	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
147	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
150	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
151	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
152	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
153	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	1	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	1	no	—
161	C₂²⋊C₄	—	4	1	no	—
162	C₄²	—	2	4	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	1	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	2	2	no	—
167	C₂³	#41×C₂₅₁	1	2	yes	—
168	C₂³	C₂⁴×C₂₅₁	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	#41×C₂₅₁	1	4	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₂×#41×C₂₅₁	1	1	yes	characteristic
199	C₂²	C₂×#41×C₂₅₁	1	2	yes	—
200	C₂²	H(5,𝔽₂)×C₂₅₁	1	1	yes	characteristic
201	C₂²	H(5,𝔽₂)×C₂₅₁	1	1	yes	characteristic
202	C₂²	2₋¹⁺⁴×C₂₅₁	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	1	no	—
211	C₂	—	4	2	no	—
212	C₂	#538×C₂₅₁	1	1	yes	characteristic
213	C₂	#542×C₂₅₁	1	2	yes	—
214	C₂	#551×C₂₅₁	1	1	yes	characteristic
215	C₂	#552×C₂₅₁	1	2	yes	—
216	C₂	#563×C₂₅₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #400×C₂₅₁	C₂	semidirect
2. #398×C₂₅₁	C₂	semidirect
5. #391×C₂₅₁	C₂	semidirect
8. #383×C₂₅₁	C₂	semidirect
108. C₂₅₁	#2764	direct (N × H)
109. #2764	C₂₅₁	direct (N × H)
110. #400	C₂×C₂₅₁	semidirect
111. #398	C₂×C₂₅₁	semidirect
114. #391	C₂×C₂₅₁	semidirect
117. #383	C₂×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#400×C₂₅₁	2
#398×C₂₅₁	2
#397×C₂₅₁	2
#393×C₂₅₁	2
#391×C₂₅₁	2
#388×C₂₅₁	2
#385×C₂₅₁	2
#383×C₂₅₁	2
#381×C₂₅₁	2
#2764	251

Subgroups — #2764×C251 #44547351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2764×C₂₅₁ #44547351