Subgroups — #13098802

Order 1600.

351 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×C₄²×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
2	#11716492	C₂	1	1	yes	maximal, characteristic
3	#11716491	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊D₂₅	C₂	1	2	yes	maximal, complemented
5	#11716486	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716438	C₂	1	2	yes	maximal, complemented
7	#11716437	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₀₀	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716434	C₂	1	2	yes	maximal, complemented
10	#11716432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₅₀	C₂	1	1	yes	maximal, characteristic, complemented
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	C₂×C₂²⋊D₂₅	C₂²	1	2	yes	—
16	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₂₅	C₂²	1	2	yes	—
18	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
19	C₄²×C₂₅	C₂²	1	1	yes	characteristic
20	C₄²×C₂₅	C₂²	1	1	yes	characteristic
21	C₄²×C₂₅	C₂²	1	1	yes	characteristic
22	C₄²×C₂₅	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₂₅	C₂²	1	2	yes	—
24	#87720	C₂²	1	2	yes	—
25	#87720	C₂²	1	2	yes	—
26	#87720	C₂²	1	2	yes	—
27	#87720	C₂²	1	2	yes	—
28	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
30	#87718	C₂²	1	2	yes	—
31	#87718	C₂²	1	2	yes	—
32	#87718	C₂²	1	2	yes	—
33	#87718	C₂²	1	2	yes	—
34	C₄×Dic₂₅	C₂²	1	2	yes	—
35	#65813	—	5	1	no	maximal
36	C₂³×C₂₅	C₂³	1	1	yes	characteristic
37	C₂²×D₂₅	C₂³	1	2	yes	—
38	C₂×C₄×C₂₅	—	2	2	no	—
39	C₂×C₄×C₂₅	—	2	2	no	—
40	C₂×C₄×C₂₅	—	2	2	no	—
41	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
47	C₂²⋊D₂₅	—	2	4	no	—
48	C₂×Dic₂₅	C₂³	1	2	yes	—
49	C₂×Dic₂₅	C₂³	1	2	yes	—
50	C₂×Dic₂₅	C₂³	1	2	yes	—
51	D₁₀₀	—	2	2	no	—
52	C₄×D₂₅	—	2	4	no	—
53	Dic₅₀	—	2	2	no	—
54	C₂×C₄²×C₅	D₅	1	1	yes	characteristic
55	#4076	—	5	1	no	—
56	#4075	—	5	2	no	—
57	C₄×C₂²⋊D₅	—	5	2	no	—
58	#4070	—	5	1	no	—
59	#4022	—	5	2	no	—
60	#4021	—	5	1	no	—
61	C₄×D₂₀	—	5	1	no	—
62	#4018	—	5	2	no	—
63	#4016	—	5	1	no	—
64	C₄×Dic₁₀	—	5	1	no	—
65	C₂²×C₂₅	—	2	1	no	—
66	C₂²×C₂₅	—	2	1	no	—
67	C₂²×C₂₅	#41	1	2	yes	—
68	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
69	C₂×D₂₅	—	2	2	no	—
70	C₂×D₂₅	—	2	2	no	—
71	C₂×D₂₅	—	2	2	no	—
72	C₄×C₂₅	—	2	1	no	—
73	C₄×C₂₅	—	2	1	no	—
74	C₄×C₂₅	—	2	1	no	—
75	C₄×C₂₅	—	2	1	no	—
76	C₄×C₂₅	#41	1	2	yes	—
77	C₄×C₂₅	#41	1	2	yes	—
78	Dic₂₅	—	2	2	no	—
79	Dic₂₅	—	2	2	no	—
80	Dic₂₅	—	2	2	no	—
81	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	—	5	2	no	—
85	C₂×D₂₀	—	5	1	no	—
86	C₂×C₄×D₅	—	5	2	no	—
87	C₂×Dic₁₀	—	5	1	no	—
88	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
91	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
92	C₂²⋊Dic₅	—	5	2	no	—
93	#685	—	5	2	no	—
94	#685	—	5	2	no	—
95	#685	—	5	2	no	—
96	#685	—	5	2	no	—
97	C₄⋊Dic₅	—	5	1	no	—
98	C₄⋊Dic₅	—	5	1	no	—
99	#683	—	5	2	no	—
100	#683	—	5	2	no	—
101	#683	—	5	2	no	—
102	#683	—	5	2	no	—
103	C₄×Dic₅	—	5	2	no	—
104	#529	—	25	1	no	Sylow
105	C₂×C₂₅	—	2	2	no	—
106	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
107	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
108	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
109	D₂₅	—	4	2	no	—
110	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
111	C₂²×D₅	—	5	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
121	C₂²⋊D₅	—	10	4	no	—
122	C₂×Dic₅	—	5	2	no	—
123	C₂×Dic₅	—	5	2	no	—
124	C₂×Dic₅	—	5	2	no	—
125	D₂₀	—	10	2	no	—
126	C₄×D₅	—	10	4	no	—
127	Dic₁₀	—	10	2	no	—
128	#126	—	25	2	no	—
129	#125	—	25	1	no	—
130	#124	—	25	1	no	—
131	#123	—	25	2	no	—
132	C₂²⋊Q₈	—	25	1	no	—
133	#121	—	25	1	no	—
134	C₄×Q₈	—	25	1	no	—
135	C₄×D₄	—	25	2	no	—
136	C₄×D₄	—	25	1	no	—
137	#117	—	25	2	no	—
138	C₂×C₄²	D₂₅	1	1	yes	characteristic, complemented
139	C₂₅	#529	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₅	—	2	1	no	—
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	#709	1	2	yes	—
143	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
144	C₂×D₅	—	10	2	no	—
145	C₂×D₅	—	10	2	no	—
146	C₂×D₅	—	10	2	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	#709	1	2	yes	—
152	C₄×C₅	#709	1	2	yes	—
153	Dic₅	—	10	2	no	—
154	Dic₅	—	10	2	no	—
155	Dic₅	—	10	2	no	—
156	C₂×Q₈	—	25	1	no	—
157	C₂×D₄	—	25	2	no	—
158	C₂×D₄	—	25	1	no	—
159	C₂²×C₄	—	25	2	no	—
160	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
163	C₄⋊C₄	—	25	2	no	—
164	C₄⋊C₄	—	25	2	no	—
165	C₄⋊C₄	—	25	1	no	—
166	C₄⋊C₄	—	25	2	no	—
167	C₄⋊C₄	—	25	1	no	—
168	C₄⋊C₄	—	25	2	no	—
169	C₂²⋊C₄	—	25	2	no	—
170	C₂²⋊C₄	—	25	2	no	—
171	C₂²⋊C₄	—	25	2	no	—
172	C₂²⋊C₄	—	25	2	no	—
173	C₂²⋊C₄	—	25	2	no	—
174	C₄²	—	25	2	no	—
175	C₄²	C₂×D₂₅	1	1	yes	characteristic
176	C₄²	C₂×D₂₅	1	1	yes	characteristic
177	C₄²	C₂×D₂₅	1	1	yes	characteristic
178	C₄²	C₂×D₂₅	1	1	yes	characteristic
179	C₂×C₅	—	2	2	no	—
180	C₂×C₅	C₂×#709	1	1	yes	characteristic
181	C₂×C₅	C₂×#709	1	1	yes	characteristic
182	C₂×C₅	C₂×#709	1	1	yes	characteristic
183	D₅	—	20	2	no	—
184	C₂³	—	25	2	no	—
185	C₂³	C₂²×D₂₅	1	1	yes	characteristic
186	Q₈	—	50	2	no	—
187	D₄	—	50	4	no	—
188	D₄	—	50	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	25	2	no	—
193	C₂×C₄	—	25	2	no	—
194	C₂×C₄	—	25	2	no	—
195	C₂×C₄	—	50	4	no	—
196	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
197	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
198	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
199	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
200	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
201	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
202	C₅	#65813	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	50	2	no	—
206	C₂²	—	50	2	no	—
207	C₂²	—	50	2	no	—
208	C₂²	#87744	1	2	yes	—
209	C₂²	C₂³×D₂₅	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	50	2	no	—
215	C₄	—	50	2	no	—
216	C₄	—	50	2	no	—
217	C₄	#87744	1	2	yes	—
218	C₄	#87744	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	—	100	2	no	—
221	C₂	C₂×#87744	1	1	yes	characteristic
222	C₂	C₂×#87744	1	1	yes	characteristic
223	C₂	C₂×#87744	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂⁴
Frattini, socle	C₂²×C₅	C₂³×D₅
Fitting	C₂×C₄²×C₂₅	C₂
prime radical	#11716492	C₂
hypercenter	C₂×C₄²	D₂₅
hypocenter	C₂₅	#529
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₄²×C₂₅	C₂	semidirect
3. #11716491	C₂	semidirect
4. C₄×C₂²⋊D₂₅	C₂	semidirect
5. #11716486	C₂	semidirect
6. #11716438	C₂	semidirect
7. #11716437	C₂	semidirect
8. C₄×D₁₀₀	C₂	semidirect
9. #11716434	C₂	semidirect
10. #11716432	C₂	semidirect
11. C₄×Dic₅₀	C₂	semidirect
138. C₂×C₄²	D₂₅	semidirect
139. C₂₅	#529	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₂₅	2
#11716492	2
#11716491	2
C₄×C₂²⋊D₂₅	2
#11716486	2
#11716438	2
#11716437	2
C₄×D₁₀₀	2
#11716434	2
#11716432	2
C₄×Dic₅₀	2
#65813	5