Subgroups — #13098870

Order 1600.

507 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₂₅	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₂₅	C₂	1	2	yes	maximal, complemented
5	C₂×#87720	C₂	1	2	yes	maximal, complemented
6	#11716455	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716454	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716443	C₂	1	2	yes	maximal, complemented
9	C₂²⋊C₄×D₂₅	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₀₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716434	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₂₅	C₂²	1	2	yes	—
13	C₂×D₄×C₂₅	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₂₅	C₂²	1	2	yes	complemented
15	C₂×C₂²⋊D₂₅	C₂²	1	2	yes	—
16	C₂²×Dic₂₅	C₂²	1	2	yes	complemented
17	D₄×D₂₅	C₄	1	8	yes	complemented
18	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₂₅	C₂²	1	2	yes	complemented
21	C₂×C₄×D₂₅	C₂²	1	2	yes	complemented
22	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
23	C₂²⋊C₄×C₂₅	C₂²	1	2	yes	complemented
24	C₄²×C₂₅	C₂²	1	1	yes	characteristic, complemented
25	C₂²⋊Dic₂₅	C₂²	1	2	yes	complemented
26	#87720	C₂²	1	1	yes	characteristic
27	#87720	C₂²	1	1	yes	characteristic
28	#87720	C₂²	1	2	yes	—
29	#87720	C₂²	1	2	yes	complemented
30	#87720	C₂²	1	2	yes	complemented
31	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic, complemented
32	#87718	C₂²	1	1	yes	characteristic, complemented
33	#87718	C₂²	1	1	yes	characteristic, complemented
34	C₄×Dic₂₅	C₂²	1	1	yes	characteristic, complemented
35	C₄×Dic₂₅	C₂²	1	2	yes	complemented
36	#65879	—	5	1	no	maximal
37	C₂³×C₂₅	C₂³	1	2	yes	—
38	C₂²×D₂₅	—	4	2	no	—
39	C₂²×D₂₅	—	4	2	no	—
40	C₂²×D₂₅	C₂×C₄	1	8	yes	complemented
41	C₂²×D₂₅	C₂³	1	1	yes	characteristic
42	C₂²×D₂₅	C₂³	1	2	yes	—
43	D₄×C₂₅	C₂×C₄	1	4	yes	complemented
44	C₂×C₄×C₂₅	—	4	2	no	—
45	C₂×C₄×C₂₅	C₂³	1	2	yes	—
46	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
49	C₂²⋊D₂₅	C₂×C₄	1	8	yes	complemented
50	C₂×Dic₂₅	—	4	2	no	—
51	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₅	C₂³	1	2	yes	—
54	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
55	D₁₀₀	C₂×C₄	1	4	yes	complemented
56	C₄×D₂₅	—	4	2	no	—
57	C₄×D₂₅	—	4	2	no	—
58	C₄×D₂₅	C₂×C₄	1	4	yes	complemented
59	C₂×D₄×D₅	—	5	1	no	—
60	C₄×C₅×D₄	D₅	1	1	yes	characteristic
61	D₄×Dic₅	—	5	1	no	—
62	C₄×C₂²⋊D₅	—	5	2	no	—
63	C₂×#685	—	5	2	no	—
64	#4039	—	5	1	no	—
65	#4038	—	5	1	no	—
66	#4027	—	5	2	no	—
67	D₅×C₂²⋊C₄	—	5	2	no	—
68	C₄×D₂₀	—	5	1	no	—
69	#4018	—	5	1	no	—
70	C₂²×C₂₅	—	2	2	no	—
71	C₂²×C₂₅	—	2	2	no	—
72	C₂²×C₂₅	C₂²×C₄	1	4	yes	—
73	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
74	C₂×D₂₅	—	2	2	no	—
75	C₂×D₂₅	—	2	2	no	—
76	C₂×D₂₅	—	2	1	no	—
77	C₂×D₂₅	—	2	1	no	—
78	C₂×D₂₅	—	4	8	no	—
79	C₂×D₂₅	C₂²×C₄	1	2	yes	—
80	C₂×D₂₅	C₂²×C₄	1	4	yes	—
81	C₄×C₂₅	—	2	1	no	—
82	C₄×C₂₅	—	2	2	no	—
83	C₄×C₂₅	—	2	1	no	—
84	C₄×C₂₅	C₂²×C₄	1	2	yes	—
85	Dic₂₅	—	2	1	no	—
86	Dic₂₅	—	2	2	no	—
87	Dic₂₅	—	2	1	no	—
88	Dic₂₅	C₂²×C₄	1	2	yes	—
89	C₂³×D₅	—	5	2	no	—
90	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
92	C₂×C₂²⋊D₅	—	5	2	no	—
93	C₂²×Dic₅	—	5	2	no	—
94	D₄×D₅	—	5	8	no	—
95	C₂×D₂₀	—	5	1	no	—
96	C₂×C₄×D₅	—	5	1	no	—
97	C₂×C₄×D₅	—	5	2	no	—
98	C₂×C₄×D₅	—	5	2	no	—
99	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
100	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	—
101	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
102	C₂²⋊Dic₅	—	5	2	no	—
103	#685	—	5	1	no	—
104	#685	—	5	1	no	—
105	#685	—	5	2	no	—
106	#685	—	5	2	no	—
107	#685	—	5	2	no	—
108	C₄⋊Dic₅	—	5	1	no	—
109	#683	—	5	1	no	—
110	#683	—	5	1	no	—
111	C₄×Dic₅	—	5	1	no	—
112	C₄×Dic₅	—	5	2	no	—
113	#518	—	25	1	no	Sylow
114	C₂×C₂₅	—	2	4	no	—
115	C₂×C₂₅	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
118	D₂₅	—	2	2	no	—
119	D₂₅	—	2	4	no	—
120	C₂³×C₅	C₂²×D₅	1	2	yes	—
121	C₂²×D₅	—	5	1	no	—
122	C₂²×D₅	—	5	2	no	—
123	C₂²×D₅	—	5	8	no	—
124	C₂²×D₅	—	20	2	no	—
125	C₂²×D₅	—	20	2	no	—
126	C₅×D₄	C₄×D₅	1	4	yes	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
129	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
132	C₂²⋊D₅	—	5	8	no	—
133	C₂×Dic₅	—	5	1	no	—
134	C₂×Dic₅	—	5	1	no	—
135	C₂×Dic₅	—	5	1	no	—
136	C₂×Dic₅	—	5	2	no	—
137	C₂×Dic₅	—	20	2	no	—
138	D₂₀	—	5	4	no	—
139	C₄×D₅	—	5	4	no	—
140	C₄×D₅	—	20	2	no	—
141	C₄×D₅	—	20	2	no	—
142	C₂²×D₄	—	25	1	no	—
143	C₄×D₄	—	25	1	no	—
144	C₄×D₄	—	25	2	no	—
145	C₄×D₄	—	25	1	no	—
146	C₄×D₄	—	25	2	no	—
147	C₄×D₄	—	25	1	no	—
148	C₄×D₄	D₂₅	1	1	yes	characteristic, complemented
149	#117	—	25	1	no	—
150	#117	—	25	1	no	—
151	C₂×C₂²⋊C₄	—	25	2	no	—
152	C₂×C₂²⋊C₄	—	25	2	no	—
153	C₂₅	#518	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	C₂×C₄×D₅	1	4	yes	—
157	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
158	C₂×D₅	—	5	2	no	—
159	C₂×D₅	—	5	4	no	—
160	C₂×D₅	—	10	2	no	—
161	C₂×D₅	—	10	1	no	—
162	C₂×D₅	—	10	2	no	—
163	C₂×D₅	—	10	1	no	—
164	C₂×D₅	—	20	8	no	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
169	Dic₅	—	5	2	no	—
170	Dic₅	—	10	1	no	—
171	Dic₅	—	10	2	no	—
172	Dic₅	—	10	1	no	—
173	C₂⁴	—	25	2	no	—
174	C₂×D₄	—	25	2	no	—
175	C₂×D₄	—	25	8	no	—
176	C₂×D₄	—	25	1	no	—
177	C₂×D₄	C₂×D₂₅	1	1	yes	characteristic
178	C₂²×C₄	—	25	1	no	—
179	C₂²×C₄	—	25	2	no	—
180	C₂²×C₄	—	25	2	no	—
181	C₂²×C₄	—	25	2	no	—
182	C₂²×C₄	C₂×D₂₅	1	2	yes	complemented
183	C₄⋊C₄	—	25	1	no	—
184	C₄⋊C₄	—	25	1	no	—
185	C₄⋊C₄	—	25	1	no	—
186	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
187	C₂²⋊C₄	—	25	2	no	—
188	C₂²⋊C₄	—	25	2	no	—
189	C₂²⋊C₄	—	25	1	no	—
190	C₂²⋊C₄	—	25	1	no	—
191	C₂²⋊C₄	—	25	2	no	—
192	C₂²⋊C₄	—	25	2	no	—
193	C₂²⋊C₄	C₂×D₂₅	1	2	yes	complemented
194	C₄²	—	25	2	no	—
195	C₄²	—	25	1	no	—
196	C₄²	C₂×D₂₅	1	1	yes	characteristic, complemented
197	C₂×C₅	—	2	4	no	—
198	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
199	C₂×C₅	#4144	1	1	yes	characteristic
200	C₂×C₅	#4149	1	1	yes	characteristic
201	D₅	—	10	2	no	—
202	D₅	—	10	4	no	—
203	C₂³	—	25	1	no	—
204	C₂³	—	25	2	no	—
205	C₂³	—	25	8	no	—
206	C₂³	—	100	2	no	—
207	C₂³	—	100	2	no	—
208	C₂³	C₂²×D₂₅	1	2	yes	—
209	D₄	—	25	8	no	—
210	D₄	—	25	4	no	—
211	D₄	C₄×D₂₅	1	4	yes	complemented
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	25	1	no	—
214	C₂×C₄	—	25	4	no	—
215	C₂×C₄	—	25	1	no	—
216	C₂×C₄	—	25	2	no	—
217	C₂×C₄	—	25	1	no	—
218	C₂×C₄	—	100	2	no	—
219	C₂×C₄	—	100	2	no	—
220	C₂×C₄	—	100	2	no	—
221	C₂×C₄	C₂²×D₂₅	1	2	yes	—
222	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
225	C₅	#65879	1	1	yes	characteristic
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	25	2	no	—
229	C₂²	—	25	4	no	—
230	C₂²	—	50	2	no	—
231	C₂²	—	50	1	no	—
232	C₂²	—	50	2	no	—
233	C₂²	—	50	1	no	—
234	C₂²	—	100	8	no	—
235	C₂²	C₂×C₄×D₂₅	1	4	yes	—
236	C₂²	C₂³×D₂₅	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	25	2	no	—
241	C₄	—	50	1	no	—
242	C₄	—	50	2	no	—
243	C₄	—	50	1	no	—
244	C₄	C₂×C₄×D₂₅	1	2	yes	—
245	C₂	—	2	4	no	—
246	C₂	—	50	2	no	—
247	C₂	—	50	4	no	—
248	C₂	C₂²×C₄×D₂₅	1	1	yes	characteristic
249	C₂	#11716563	1	1	yes	characteristic
250	C₂	#11716568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₅
commutator	C₂×C₂₅	C₂³×C₄
Frattini, socle	C₂²×C₅	C₂³×D₅
Fitting	C₄×D₄×C₂₅	C₂
prime radical	C₂×D₄×D₂₅	C₂
cosocle	C₂²×C₂₅	C₂⁴
hypercenter	C₄×D₄	D₂₅
hypocenter	C₂₅	#518
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×C₂₅	C₂	semidirect
3. D₄×Dic₂₅	C₂	semidirect
4. C₄×C₂²⋊D₂₅	C₂	semidirect
5. C₂×#87720	C₂	semidirect
6. #11716455	C₂	semidirect
7. #11716454	C₂	semidirect
8. #11716443	C₂	semidirect
9. C₂²⋊C₄×D₂₅	C₂	semidirect
10. C₄×D₁₀₀	C₂	semidirect
11. #11716434	C₂	semidirect
14. C₂²×C₄×C₂₅	C₂²	semidirect
16. C₂²×Dic₂₅	C₂²	semidirect
17. D₄×D₂₅	C₄	semidirect
20. C₂×C₄×D₂₅	C₂²	semidirect
21. C₂×C₄×D₂₅	C₂²	semidirect
22. C₄⋊C₄×C₂₅	C₂²	semidirect
23. C₂²⋊C₄×C₂₅	C₂²	semidirect
24. C₄²×C₂₅	C₂²	semidirect
25. C₂²⋊Dic₂₅	C₂²	semidirect
29. #87720	C₂²	semidirect
30. #87720	C₂²	semidirect
31. C₄⋊Dic₂₅	C₂²	semidirect
32. #87718	C₂²	semidirect
33. #87718	C₂²	semidirect
34. C₄×Dic₂₅	C₂²	semidirect
35. C₄×Dic₂₅	C₂²	semidirect
40. C₂²×D₂₅	C₂×C₄	semidirect
43. D₄×C₂₅	C₂×C₄	semidirect
49. C₂²⋊D₂₅	C₂×C₄	semidirect
55. D₁₀₀	C₂×C₄	semidirect
58. C₄×D₂₅	C₂×C₄	semidirect
148. C₄×D₄	D₂₅	semidirect
153. C₂₅	#518	semidirect
182. C₂²×C₄	C₂×D₂₅	semidirect
186. C₄⋊C₄	C₂×D₂₅	semidirect
193. C₂²⋊C₄	C₂×D₂₅	semidirect
196. C₄²	C₂×D₂₅	semidirect
211. D₄	C₄×D₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₂₅	2
C₄×D₄×C₂₅	2
D₄×Dic₂₅	2
C₄×C₂²⋊D₂₅	2
C₂×#87720	2
#11716455	2
#11716454	2
#11716443	2
C₂²⋊C₄×D₂₅	2
C₄×D₁₀₀	2
#11716434	2
#65879	5