Subgroups — C₄×D₄×D₂₅ #13098869

Order 1600.

639 conjugacy classes of subgroups.

The subgroup lattice has 264 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	C₂²×C₄×D₂₅	C₂	1	2	yes	maximal, complemented
3	C₄×D₄×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×Dic₂₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊D₂₅	C₂	1	2	yes	maximal, complemented
6	#11716455	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×D₂₅	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	C₄²×D₂₅	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	direct product factor
18	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₂₅	—	2	8	no	—
20	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₅	C₂²	1	2	yes	—
22	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₂₅	C₂²	1	2	yes	complemented
25	C₂×C₄×D₂₅	C₂²	1	2	yes	complemented
26	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
27	C₂²⋊C₄×C₂₅	C₂²	1	2	yes	complemented
28	C₄²×C₂₅	C₂²	1	1	yes	characteristic, complemented
29	C₂²⋊Dic₂₅	C₂²	1	2	yes	complemented
30	#87720	C₂²	1	2	yes	complemented
31	#87720	C₂²	1	2	yes	complemented
32	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic, complemented
33	#87718	C₂²	1	2	yes	complemented
34	C₄×Dic₂₅	C₂²	1	1	yes	characteristic, complemented
35	C₄×Dic₂₅	C₂²	1	1	yes	characteristic, complemented
36	C₄×Dic₂₅	C₂²	1	1	yes	characteristic, complemented
37	C₄×D₄×D₅	—	5	1	no	maximal
38	C₂³×C₂₅	C₂³	1	2	yes	—
39	C₂²×D₂₅	—	2	4	no	—
40	C₂²×D₂₅	—	2	4	no	—
41	C₂²×D₂₅	C₂×C₄	1	8	yes	complemented
42	C₂²×D₂₅	C₂³	1	2	yes	—
43	C₂²×D₂₅	C₂³	1	1	yes	characteristic
44	D₄×C₂₅	C₂×C₄	1	4	yes	complemented
45	C₂×C₄×C₂₅	—	2	4	no	—
46	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₅	C₂³	1	2	yes	—
48	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
50	C₂²⋊D₂₅	C₂×C₄	1	8	yes	complemented
51	C₂×Dic₂₅	—	2	4	no	—
52	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₅	C₂³	1	2	yes	—
54	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
56	D₁₀₀	C₂×C₄	1	4	yes	complemented
57	C₄×D₂₅	—	2	4	no	—
58	C₄×D₂₅	—	2	2	no	—
59	C₄×D₂₅	—	2	4	no	—
60	C₄×D₂₅	—	2	4	no	—
61	C₄×D₂₅	C₂×C₄	1	4	yes	complemented
62	C₄×D₂₅	D₄	1	4	yes	direct product factor
63	C₂×D₄×D₅	—	5	1	no	—
64	C₂²×C₄×D₅	—	5	2	no	—
65	C₄×C₅×D₄	D₅	1	1	yes	characteristic
66	D₄×Dic₅	—	5	1	no	—
67	C₄×C₂²⋊D₅	—	5	2	no	—
68	#4039	—	5	1	no	—
69	D₅×C₄⋊C₄	—	5	1	no	—
70	#4027	—	5	2	no	—
71	D₅×C₂²⋊C₄	—	5	2	no	—
72	C₄×D₂₀	—	5	1	no	—
73	C₄²×D₅	—	5	1	no	—
74	C₂²×C₂₅	—	2	2	no	—
75	C₂²×C₂₅	—	2	2	no	—
76	C₂²×C₂₅	C₂²×C₄	1	4	yes	complemented
77	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
78	C₂×D₂₅	—	2	2	no	—
79	C₂×D₂₅	—	2	2	no	—
80	C₂×D₂₅	—	2	16	no	—
81	C₂×D₂₅	C₂²×C₄	1	2	yes	—
82	C₂×D₂₅	C₂²×C₄	1	4	yes	complemented
83	C₂×D₂₅	C₂×D₄	1	2	yes	—
84	C₂×D₂₅	#41	1	2	yes	—
85	C₄×C₂₅	—	2	2	no	—
86	C₄×C₂₅	—	2	1	no	—
87	C₄×C₂₅	C₂²×C₄	1	2	yes	complemented
88	C₄×C₂₅	C₂×D₄	1	2	yes	complemented
89	Dic₂₅	—	2	2	no	—
90	Dic₂₅	—	2	1	no	—
91	Dic₂₅	C₂²×C₄	1	2	yes	complemented
92	Dic₂₅	C₂×D₄	1	2	yes	complemented
93	C₂³×D₅	—	5	2	no	—
94	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
96	C₂×C₂²⋊D₅	—	5	2	no	—
97	C₂²×Dic₅	—	5	2	no	—
98	D₄×D₅	—	5	8	no	—
99	C₂×D₂₀	—	5	1	no	—
100	C₂×C₄×D₅	—	5	1	no	—
101	C₂×C₄×D₅	—	5	1	no	—
102	C₂×C₄×D₅	—	5	2	no	—
103	C₂×C₄×D₅	—	5	2	no	—
104	C₂×C₄×D₅	—	5	2	no	—
105	C₂×C₄×D₅	—	5	1	no	—
106	C₂×C₄×D₅	—	10	8	no	—
107	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
108	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	—
109	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
110	C₂²⋊Dic₅	—	5	2	no	—
111	#685	—	5	2	no	—
112	#685	—	5	2	no	—
113	C₄⋊Dic₅	—	5	1	no	—
114	#683	—	5	2	no	—
115	C₄×Dic₅	—	5	1	no	—
116	C₄×Dic₅	—	5	1	no	—
117	C₄×Dic₅	—	5	1	no	—
118	C₂×C₄×D₄	—	25	1	no	Sylow
119	C₂×C₂₅	—	2	4	no	—
120	C₂×C₂₅	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₂₅	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
123	D₂₅	—	2	4	no	—
124	D₂₅	C₄×D₄	1	4	yes	direct product factor
125	C₂³×C₅	C₂²×D₅	1	2	yes	—
126	C₂²×D₅	—	5	1	no	—
127	C₂²×D₅	—	5	8	no	—
128	C₂²×D₅	—	5	2	no	—
129	C₂²×D₅	—	10	4	no	—
130	C₂²×D₅	—	10	4	no	—
131	C₅×D₄	C₄×D₅	1	4	yes	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
135	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
137	C₂²⋊D₅	—	5	8	no	—
138	C₂×Dic₅	—	5	1	no	—
139	C₂×Dic₅	—	5	1	no	—
140	C₂×Dic₅	—	5	1	no	—
141	C₂×Dic₅	—	5	2	no	—
142	C₂×Dic₅	—	10	4	no	—
143	D₂₀	—	5	4	no	—
144	C₄×D₅	—	5	4	no	—
145	C₄×D₅	—	5	4	no	—
146	C₄×D₅	—	10	4	no	—
147	C₄×D₅	—	10	2	no	—
148	C₄×D₅	—	10	4	no	—
149	C₄×D₅	—	10	4	no	—
150	C₂²×D₄	—	25	1	no	—
151	C₂³×C₄	—	25	2	no	—
152	C₄×D₄	—	25	1	no	—
153	C₄×D₄	—	25	2	no	—
154	C₄×D₄	—	25	1	no	—
155	C₄×D₄	—	25	2	no	—
156	C₄×D₄	—	25	1	no	—
157	C₄×D₄	D₂₅	1	1	yes	characteristic, direct product factor
158	C₂×C₄⋊C₄	—	25	1	no	—
159	C₂×C₂²⋊C₄	—	25	2	no	—
160	C₂×C₄²	—	25	1	no	—
161	C₂₅	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	C₂×C₄×D₅	1	4	yes	—
165	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
166	C₂×D₅	—	5	2	no	—
167	C₂×D₅	—	5	2	no	—
168	C₂×D₅	—	5	2	no	—
169	C₂×D₅	—	5	4	no	—
170	C₂×D₅	—	10	16	no	—
171	C₂×D₅	—	10	2	no	—
172	C₂×D₅	—	10	2	no	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
176	C₄×C₅	D₄×D₅	1	2	yes	—
177	Dic₅	—	5	2	no	—
178	Dic₅	—	5	2	no	—
179	Dic₅	—	10	2	no	—
180	Dic₅	—	10	1	no	—
181	C₂⁴	—	25	2	no	—
182	C₂×D₄	—	25	2	no	—
183	C₂×D₄	—	25	8	no	—
184	C₂×D₄	—	25	1	no	—
185	C₂×D₄	C₂×D₂₅	1	1	yes	characteristic
186	C₂²×C₄	—	25	1	no	—
187	C₂²×C₄	—	25	1	no	—
188	C₂²×C₄	—	25	2	no	—
189	C₂²×C₄	—	25	1	no	—
190	C₂²×C₄	—	25	2	no	—
191	C₂²×C₄	—	25	2	no	—
192	C₂²×C₄	—	25	2	no	—
193	C₂²×C₄	—	50	8	no	—
194	C₂²×C₄	C₂×D₂₅	1	2	yes	complemented
195	C₄⋊C₄	—	25	2	no	—
196	C₄⋊C₄	—	25	1	no	—
197	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
198	C₂²⋊C₄	—	25	2	no	—
199	C₂²⋊C₄	—	25	2	no	—
200	C₂²⋊C₄	—	25	2	no	—
201	C₂²⋊C₄	C₂×D₂₅	1	2	yes	complemented
202	C₄²	—	25	1	no	—
203	C₄²	—	25	1	no	—
204	C₄²	—	25	1	no	—
205	C₄²	C₂×D₂₅	1	1	yes	characteristic, complemented
206	C₂×C₅	—	2	4	no	—
207	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
208	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
209	C₂×C₅	D₅×#41	1	1	yes	characteristic
210	D₅	—	5	4	no	—
211	D₅	—	10	4	no	—
212	C₂³	—	25	1	no	—
213	C₂³	—	25	8	no	—
214	C₂³	—	25	2	no	—
215	C₂³	—	50	4	no	—
216	C₂³	—	50	4	no	—
217	C₂³	C₂²×D₂₅	1	2	yes	—
218	D₄	—	25	8	no	—
219	D₄	—	25	4	no	—
220	D₄	C₄×D₂₅	1	4	yes	direct product factor
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	25	1	no	—
223	C₂×C₄	—	25	1	no	—
224	C₂×C₄	—	25	4	no	—
225	C₂×C₄	—	25	4	no	—
226	C₂×C₄	—	25	1	no	—
227	C₂×C₄	—	25	2	no	—
228	C₂×C₄	—	50	4	no	—
229	C₂×C₄	—	50	4	no	—
230	C₂×C₄	—	50	2	no	—
231	C₂×C₄	—	50	4	no	—
232	C₂×C₄	—	50	4	no	—
233	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₂₅	1	2	yes	—
235	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
237	C₅	C₄×D₄×D₅	1	1	yes	characteristic
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	25	2	no	—
241	C₂²	—	25	2	no	—
242	C₂²	—	25	2	no	—
243	C₂²	—	25	4	no	—
244	C₂²	—	50	16	no	—
245	C₂²	—	50	2	no	—
246	C₂²	—	50	2	no	—
247	C₂²	C₂×C₄×D₂₅	1	4	yes	complemented
248	C₂²	C₂³×D₂₅	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	25	2	no	—
252	C₄	—	25	2	no	—
253	C₄	—	50	2	no	—
254	C₄	—	50	1	no	—
255	C₄	C₂×C₄×D₂₅	1	2	yes	complemented
256	C₄	D₄×D₂₅	1	2	yes	direct product factor
257	C₂	—	2	4	no	—
258	C₂	—	25	4	no	—
259	C₂	—	50	4	no	—
260	C₂	C₂²×C₄×D₂₅	1	1	yes	characteristic
261	C₂	C₂×D₄×D₂₅	1	1	yes	characteristic
262	C₂	#41×D₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×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	C₂×C₄×D₄
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₂₅	C₂	semidirect
3. C₄×D₄×C₂₅	C₂	semidirect
4. D₄×Dic₂₅	C₂	semidirect
5. C₄×C₂²⋊D₂₅	C₂	semidirect
6. #11716455	C₂	semidirect
7. C₄⋊C₄×D₂₅	C₂	semidirect
8. #11716443	C₂	semidirect
9. C₂²⋊C₄×D₂₅	C₂	semidirect
10. C₄×D₁₀₀	C₂	semidirect
11. C₄²×D₂₅	C₂	semidirect
14. C₂²×C₄×C₂₅	C₂²	semidirect
16. C₂²×Dic₂₅	C₂²	semidirect
17. D₄×D₂₅	C₄	direct (N × H)
24. C₂×C₄×D₂₅	C₂²	semidirect
25. C₂×C₄×D₂₅	C₂²	semidirect
26. C₄⋊C₄×C₂₅	C₂²	semidirect
27. C₂²⋊C₄×C₂₅	C₂²	semidirect
28. C₄²×C₂₅	C₂²	semidirect
29. C₂²⋊Dic₂₅	C₂²	semidirect
30. #87720	C₂²	semidirect
31. #87720	C₂²	semidirect
32. C₄⋊Dic₂₅	C₂²	semidirect
33. #87718	C₂²	semidirect
34. C₄×Dic₂₅	C₂²	semidirect
35. C₄×Dic₂₅	C₂²	semidirect
36. C₄×Dic₂₅	C₂²	semidirect
41. C₂²×D₂₅	C₂×C₄	semidirect
44. D₄×C₂₅	C₂×C₄	semidirect
50. C₂²⋊D₂₅	C₂×C₄	semidirect
56. D₁₀₀	C₂×C₄	semidirect
61. C₄×D₂₅	C₂×C₄	semidirect
62. C₄×D₂₅	D₄	direct (N × H)
76. C₂²×C₂₅	C₂²×C₄	semidirect
82. C₂×D₂₅	C₂²×C₄	semidirect
87. C₄×C₂₅	C₂²×C₄	semidirect
88. C₄×C₂₅	C₂×D₄	semidirect
91. Dic₂₅	C₂²×C₄	semidirect
92. Dic₂₅	C₂×D₄	semidirect
124. D₂₅	C₄×D₄	direct (N × H)
157. C₄×D₄	D₂₅	direct (N × H)
161. C₂₅	C₂×C₄×D₄	semidirect
194. C₂²×C₄	C₂×D₂₅	semidirect
197. C₄⋊C₄	C₂×D₂₅	semidirect
201. C₂²⋊C₄	C₂×D₂₅	semidirect
205. C₄²	C₂×D₂₅	semidirect
220. D₄	C₄×D₂₅	direct (N × H)
247. C₂²	C₂×C₄×D₂₅	semidirect
255. C₄	C₂×C₄×D₂₅	semidirect
256. C₄	D₄×D₂₅	direct (N × H)

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
C₂²×C₄×D₂₅	2
C₄×D₄×C₂₅	2
D₄×Dic₂₅	2
C₄×C₂²⋊D₂₅	2
#11716455	2
C₄⋊C₄×D₂₅	2
#11716443	2
C₂²⋊C₄×D₂₅	2
C₄×D₁₀₀	2
C₄²×D₂₅	2
C₄×D₄×D₅	5

Subgroups — C4×D4×D25 #13098869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄×D₂₅ #13098869