Subgroups — C₂₅×#435 #13098584

Order 1600.

186 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×C₂₅	C₂	1	1	yes	maximal, characteristic
4	C₂₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×C₂₅	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
11	M₁₆×C₂₅	—	2	2	no	—
12	C₂×C₈×C₂₅	—	2	2	no	—
13	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
15	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₅	C₄	1	1	yes	characteristic
18	C₂²⋊C₄×C₂₅	C₄	1	1	yes	characteristic
19	C₂²⋊C₄×C₂₅	C₄	1	1	yes	characteristic
20	C₄²×C₂₅	C₂²	1	1	yes	characteristic
21	C₅×#435	C₅	1	1	yes	maximal, characteristic
22	C₂³×C₂₅	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₂₅	C₂×C₄	1	1	yes	characteristic
24	D₄×C₂₅	—	2	2	no	—
25	C₂×C₄×C₂₅	—	2	1	no	—
26	C₂×C₄×C₂₅	—	2	1	no	—
27	C₂×C₄×C₂₅	—	2	1	no	—
28	C₂×C₄×C₂₅	—	2	1	no	—
29	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
34	C₈×C₂₅	—	2	1	no	—
35	C₈×C₂₅	—	2	1	no	—
36	C₈×C₂₅	—	2	1	no	—
37	C₈×C₂₅	D₄	1	2	yes	complemented
38	C₂×C₅×M₁₆	C₂×C₅	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂×C₅	1	1	yes	characteristic
40	C₄×C₅×D₄	C₂×C₅	1	1	yes	characteristic
41	C₅×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
42	C₅×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
43	C₅×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
44	C₅×#97	C₂×C₅	1	1	yes	characteristic
45	C₂²×C₂₅	—	2	1	no	—
46	C₂²×C₂₅	—	2	1	no	—
47	C₂²×C₂₅	—	2	1	no	—
48	C₂²×C₂₅	—	2	1	no	—
49	C₂²×C₂₅	—	2	1	no	—
50	C₂²×C₂₅	M₁₆	1	2	yes	complemented
51	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
52	C₄×C₂₅	—	2	1	no	—
53	C₄×C₂₅	—	2	1	no	—
54	C₄×C₂₅	—	2	1	no	—
55	C₄×C₂₅	—	2	1	no	—
56	C₄×C₂₅	C₂×D₄	1	1	yes	characteristic
57	C₄×C₂₅	#41	1	1	yes	characteristic
58	C₂×C₅×D₄	C₄×C₅	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
61	C₅×M₁₆	—	2	2	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
67	C₅×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
68	C₅×C₂²⋊C₄	C₄×C₅	1	1	yes	characteristic
69	C₅×C₂²⋊C₄	C₄×C₅	1	1	yes	characteristic
70	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
71	#435	C₂₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×C₂₅	—	2	2	no	—
73	C₂×C₂₅	—	4	1	no	—
74	C₂×C₂₅	C₄×D₄	1	1	yes	characteristic
75	C₂×C₂₅	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₂₅	#131	1	1	yes	characteristic
77	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
78	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
79	C₅×D₄	—	2	2	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	2	1	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
89	C₅×C₈	—	2	1	no	—
90	C₅×C₈	—	2	1	no	—
91	C₅×C₈	—	2	1	no	—
92	C₅×C₈	C₅×D₄	1	2	yes	—
93	C₂×M₁₆	C₂×C₂₅	1	1	yes	characteristic, complemented
94	C₂²×C₈	C₂×C₂₅	1	1	yes	characteristic, complemented
95	C₄×D₄	C₂×C₂₅	1	1	yes	characteristic
96	C₄⋊C₈	C₂×C₂₅	1	1	yes	characteristic, complemented
97	C₂²⋊C₈	C₂×C₂₅	1	1	yes	characteristic, complemented
98	C₂²⋊C₈	C₂×C₂₅	1	1	yes	characteristic, complemented
99	#97	C₂×C₂₅	1	1	yes	characteristic, complemented
100	C₂₅	#435	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×C₅	—	2	1	no	—
102	C₂²×C₅	—	2	1	no	—
103	C₂²×C₅	—	2	1	no	—
104	C₂²×C₅	—	2	1	no	—
105	C₂²×C₅	—	2	1	no	—
106	C₂²×C₅	C₅×M₁₆	1	2	yes	—
107	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
108	C₄×C₅	—	2	1	no	—
109	C₄×C₅	—	2	1	no	—
110	C₄×C₅	—	2	1	no	—
111	C₄×C₅	—	2	1	no	—
112	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
113	C₄×C₅	C₅×#41	1	1	yes	characteristic
114	C₂×D₄	C₄×C₂₅	1	1	yes	characteristic
115	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
116	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₂²×C₂₅	1	1	yes	characteristic
120	C₂×C₈	C₂²×C₂₅	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₂₅	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₂₅	1	1	yes	characteristic
123	C₄⋊C₄	C₄×C₂₅	1	1	yes	characteristic
124	C₂²⋊C₄	C₄×C₂₅	1	1	yes	characteristic
125	C₂²⋊C₄	C₄×C₂₅	1	1	yes	characteristic
126	C₄²	C₂²×C₂₅	1	1	yes	characteristic
127	C₂×C₅	—	2	2	no	—
128	C₂×C₅	—	4	1	no	—
129	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
130	C₂×C₅	C₂×C₅×M₁₆	1	1	yes	characteristic
131	C₂×C₅	C₅×#131	1	1	yes	characteristic
132	C₂³	C₂×C₄×C₂₅	1	1	yes	characteristic
133	C₂³	C₂×C₄×C₂₅	1	1	yes	characteristic
134	D₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	C₂×C₄×C₂₅	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×C₂₅	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×C₂₅	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×C₂₅	1	1	yes	characteristic
143	C₂×C₄	C₂³×C₂₅	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	D₄×C₂₅	1	2	yes	complemented
148	C₅	C₅×#435	1	1	yes	characteristic
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	M₁₆×C₂₅	1	2	yes	complemented
155	C₂²	C₂²×C₄×C₂₅	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	C₂×D₄×C₂₅	1	1	yes	characteristic
161	C₄	#41×C₂₅	1	1	yes	characteristic
162	C₂	—	2	2	no	—
163	C₂	—	4	1	no	—
164	C₂	C₄×D₄×C₂₅	1	1	yes	characteristic
165	C₂	C₂×M₁₆×C₂₅	1	1	yes	characteristic
166	C₂	C₂₅×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₅	C₂³
commutator	C₂²	C₂²×C₄×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle	C₂²×C₅	C₂²×C₄×C₅
prime radical	C₂×C₅×D₄	C₄×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×C₂₅	C₂	semidirect
2. C₂²×C₈×C₂₅	C₂	semidirect
4. C₂₅×C₄⋊C₈	C₂	semidirect
5. C₂₅×C₂²⋊C₈	C₂	semidirect
6. C₂₅×C₂²⋊C₈	C₂	semidirect
7. C₂₅×#97	C₂	semidirect
37. C₈×C₂₅	D₄	semidirect
50. C₂²×C₂₅	M₁₆	semidirect
71. #435	C₂₅	direct (N × H)
93. C₂×M₁₆	C₂×C₂₅	semidirect
94. C₂²×C₈	C₂×C₂₅	semidirect
96. C₄⋊C₈	C₂×C₂₅	semidirect
97. C₂²⋊C₈	C₂×C₂₅	semidirect
98. C₂²⋊C₈	C₂×C₂₅	semidirect
99. #97	C₂×C₂₅	semidirect
100. C₂₅	#435	direct (N × H)
147. C₈	D₄×C₂₅	semidirect
154. C₂²	M₁₆×C₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₂₅	2
C₂²×C₈×C₂₅	2
C₄×D₄×C₂₅	2
C₂₅×C₄⋊C₈	2
C₂₅×C₂²⋊C₈	2
C₂₅×#97	2
C₅×#435	5

Subgroups — C25×#435 #13098584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₅×#435 #13098584