Subgroups — C₂₅×#388 #13098537

Order 1600.

285 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×C₂₅	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₅	—	2	2	no	—
16	C₂²⋊C₄×C₂₅	—	2	2	no	—
17	C₂²⋊C₄×C₂₅	—	2	2	no	—
18	C₂²⋊C₄×C₂₅	C₄	1	4	yes	complemented
19	C₄²×C₂₅	—	2	2	no	—
20	C₅×#388	C₅	1	1	yes	maximal, characteristic
21	C₂³×C₂₅	—	2	1	no	—
22	C₂³×C₂₅	—	2	1	no	—
23	C₂³×C₂₅	—	2	1	no	—
24	C₂³×C₂₅	—	2	1	no	—
25	C₂³×C₂₅	—	2	1	no	—
26	C₂³×C₂₅	—	2	1	no	—
27	C₂³×C₂₅	C₂×C₄	1	2	yes	—
28	C₂³×C₂₅	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₅	—	2	1	no	—
30	C₂×C₄×C₂₅	—	2	1	no	—
31	C₂×C₄×C₂₅	—	2	1	no	—
32	C₂×C₄×C₂₅	—	2	1	no	—
33	C₂×C₄×C₂₅	—	2	1	no	—
34	C₂×C₄×C₂₅	—	2	1	no	—
35	C₂×C₄×C₂₅	—	2	1	no	—
36	C₂×C₄×C₂₅	—	2	1	no	—
37	C₂×C₄×C₂₅	—	2	1	no	—
38	C₂×C₄×C₂₅	—	2	1	no	—
39	C₂×C₄×C₂₅	—	2	1	no	—
40	C₂×C₄×C₂₅	—	2	1	no	—
41	C₂×C₄×C₂₅	—	2	1	no	—
42	C₂×C₄×C₂₅	—	2	1	no	—
43	C₂×C₄×C₂₅	C₂×C₄	1	2	yes	—
44	C₂×C₄×C₂₅	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₂₅	D₄	1	2	yes	—
46	C₂×C₄×C₂₅	D₄	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
48	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
49	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
50	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic
52	C₅×#95	C₂×C₅	1	1	yes	characteristic
53	C₅×#95	C₂×C₅	1	1	yes	characteristic
54	C₂²×C₂₅	—	2	2	no	—
55	C₂²×C₂₅	—	2	2	no	—
56	C₂²×C₂₅	—	2	2	no	—
57	C₂²×C₂₅	—	4	1	no	—
58	C₂²×C₂₅	—	4	1	no	—
59	C₂²×C₂₅	—	4	1	no	—
60	C₂²×C₂₅	—	4	1	no	—
61	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₂₅	#41	1	1	yes	characteristic
65	C₂²×C₂₅	#41	1	1	yes	characteristic
66	C₂²×C₂₅	#41	1	1	yes	characteristic
67	C₂²×C₂₅	#41	1	1	yes	characteristic
68	C₄×C₂₅	—	2	2	no	—
69	C₄×C₂₅	—	2	2	no	—
70	C₄×C₂₅	—	2	2	no	—
71	C₄×C₂₅	—	2	2	no	—
72	C₄×C₂₅	—	4	1	no	—
73	C₄×C₂₅	—	4	1	no	—
74	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₂²⋊C₄	—	2	2	no	—
83	C₅×C₂²⋊C₄	—	2	2	no	—
84	C₅×C₂²⋊C₄	C₄×C₅	1	4	yes	—
85	C₄²×C₅	—	2	2	no	—
86	#388	C₂₅	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×C₂₅	—	4	2	no	—
88	C₂×C₂₅	#117	1	1	yes	characteristic
89	C₂×C₂₅	C₄×D₄	1	1	yes	characteristic
90	C₂×C₂₅	C₄×D₄	1	1	yes	characteristic
91	C₂×C₂₅	#121	1	1	yes	characteristic
92	C₂×C₂₅	#123	1	1	yes	characteristic
93	C₂×C₂₅	#124	1	1	yes	characteristic
94	C₂×C₂₅	#126	1	1	yes	characteristic
95	C₂³×C₅	—	2	1	no	—
96	C₂³×C₅	—	2	1	no	—
97	C₂³×C₅	—	2	1	no	—
98	C₂³×C₅	—	2	1	no	—
99	C₂³×C₅	—	2	1	no	—
100	C₂³×C₅	—	2	1	no	—
101	C₂³×C₅	C₂×C₄×C₅	1	2	yes	—
102	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
118	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
119	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
120	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×C₂₅	1	1	yes	characteristic, complemented
122	C₂×C₂²⋊C₄	C₂×C₂₅	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	C₂×C₂₅	1	1	yes	characteristic
124	C₂×C₂²⋊C₄	C₂×C₂₅	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₂₅	1	1	yes	characteristic, complemented
126	#95	C₂×C₂₅	1	1	yes	characteristic, complemented
127	#95	C₂×C₂₅	1	1	yes	characteristic, complemented
128	C₂₅	#388	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	2	2	no	—
132	C₂²×C₅	—	4	1	no	—
133	C₂²×C₅	—	4	1	no	—
134	C₂²×C₅	—	4	1	no	—
135	C₂²×C₅	—	4	1	no	—
136	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
137	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
138	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
139	C₂²×C₅	C₅×#41	1	1	yes	characteristic
140	C₂²×C₅	C₅×#41	1	1	yes	characteristic
141	C₂²×C₅	C₅×#41	1	1	yes	characteristic
142	C₂²×C₅	C₅×#41	1	1	yes	characteristic
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	4	1	no	—
148	C₄×C₅	—	4	1	no	—
149	C₂⁴	C₂²×C₂₅	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	C₄×C₂₅	1	4	yes	complemented
160	C₄²	—	2	2	no	—
161	C₂×C₅	—	4	2	no	—
162	C₂×C₅	C₅×#117	1	1	yes	characteristic
163	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
164	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
165	C₂×C₅	C₅×#121	1	1	yes	characteristic
166	C₂×C₅	C₅×#123	1	1	yes	characteristic
167	C₂×C₅	C₅×#124	1	1	yes	characteristic
168	C₂×C₅	C₅×#126	1	1	yes	characteristic
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	1	no	—
172	C₂³	—	2	1	no	—
173	C₂³	—	2	1	no	—
174	C₂³	—	2	1	no	—
175	C₂³	C₂×C₄×C₂₅	1	2	yes	—
176	C₂³	C₂³×C₂₅	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	C₂×C₄×C₂₅	1	2	yes	—
192	C₂×C₄	C₂×C₄×C₂₅	1	2	yes	—
193	C₂×C₄	D₄×C₂₅	1	2	yes	—
194	C₂×C₄	D₄×C₂₅	1	2	yes	—
195	C₅	C₅×#388	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	4	1	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	1	no	—
202	C₂²	—	4	1	no	—
203	C₂²	C₂²×C₄×C₂₅	1	1	yes	characteristic
204	C₂²	C₂×D₄×C₂₅	1	1	yes	characteristic
205	C₂²	C₂×D₄×C₂₅	1	1	yes	characteristic
206	C₂²	#41×C₂₅	1	1	yes	characteristic
207	C₂²	#41×C₂₅	1	1	yes	characteristic
208	C₂²	#41×C₂₅	1	1	yes	characteristic
209	C₂²	#41×C₂₅	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	1	no	—
216	C₂	—	4	2	no	—
217	C₂	C₂₅×#117	1	1	yes	characteristic
218	C₂	C₄×D₄×C₂₅	1	1	yes	characteristic
219	C₂	C₄×D₄×C₂₅	1	1	yes	characteristic
220	C₂	C₂₅×#121	1	1	yes	characteristic
221	C₂	C₂₅×#123	1	1	yes	characteristic
222	C₂	C₂₅×#124	1	1	yes	characteristic
223	C₂	C₂₅×#126	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₂₅	C₂³
commutator	C₂²	C₂²×C₄×C₂₅
Frattini, socle, cosocle	C₂³×C₅	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊C₄×C₂₅	C₂	semidirect
5. C₂×C₄²×C₂₅	C₂	semidirect
6. C₂₅×#95	C₂	semidirect
7. C₂₅×#95	C₂	semidirect
18. C₂²⋊C₄×C₂₅	C₄	semidirect
86. #388	C₂₅	direct (N × H)
121. C₂×C₄⋊C₄	C₂×C₂₅	semidirect
125. C₂×C₄²	C₂×C₂₅	semidirect
126. #95	C₂×C₂₅	semidirect
127. #95	C₂×C₂₅	semidirect
128. C₂₅	#388	direct (N × H)
159. C₂²⋊C₄	C₄×C₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₂₅	2
C₂×C₂²⋊C₄×C₂₅	2
C₂×C₄²×C₂₅	2
C₂₅×#95	2
C₅×#388	5

Subgroups — C25×#388 #13098537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₅×#388 #13098537