Subgroups — C₅²×#546 #13107404

Order 1600.

1336 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×#123	C₂	1	2	yes	maximal, complemented
5	C₅²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅²×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅²×#121	C₂	1	2	yes	maximal, complemented
8	C₅²×#120	C₂	1	2	yes	maximal, complemented
9	C₄×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₂⁴×C₅²	C₂²	1	2	yes	—
13	C₅²×#41	—	2	4	no	—
14	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₅²×D₄	—	2	4	no	—
16	C₂×C₅²×D₄	C₂²	1	2	yes	—
17	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
20	C₂×C₅²×D₄	C₂²	1	2	yes	complemented
21	C₂×C₅²×D₄	C₂²	1	2	yes	complemented
22	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅²	C₂²	1	2	yes	complemented
24	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×C₅²	C₂²	1	2	yes	complemented
26	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
27	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₅²×C₄⋊C₄	C₂²	1	2	yes	complemented
29	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₅²×C₂²⋊C₄	C₂²	1	2	yes	complemented
32	C₅²×C₂²⋊C₄	C₂²	1	2	yes	complemented
33	C₅²×C₂²⋊C₄	C₂²	1	2	yes	complemented
34	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
35	C₅²×C₂²⋊C₄	C₂²	1	2	yes	complemented
36	C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
37	C₅×#546	C₅	1	6	yes	maximal, direct product factor
38	C₂³×C₅²	—	2	4	no	—
39	C₂³×C₅²	—	2	4	no	—
40	C₂³×C₅²	—	4	2	no	—
41	C₂³×C₅²	C₂³	1	2	yes	—
42	C₂³×C₅²	C₂³	1	1	yes	characteristic
43	C₂³×C₅²	C₂³	1	2	yes	—
44	C₂³×C₅²	C₂³	1	1	yes	characteristic
45	C₅²×Q₈	—	2	2	no	—
46	C₅²×D₄	—	2	4	no	—
47	C₅²×D₄	—	2	4	no	—
48	C₅²×D₄	—	2	2	no	—
49	C₅²×D₄	—	2	2	no	—
50	C₅²×D₄	—	4	2	no	—
51	C₅²×D₄	D₄	1	4	yes	complemented
52	C₂×C₄×C₅²	—	2	4	no	—
53	C₂×C₄×C₅²	—	2	2	no	—
54	C₂×C₄×C₅²	—	2	2	no	—
55	C₂×C₄×C₅²	—	4	2	no	—
56	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₅²	C₂³	1	2	yes	—
59	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅²	C₂³	1	2	yes	—
61	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
62	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
63	C₂×C₅×#41	C₂×C₅	1	6	yes	complemented
64	C₂²×C₅×D₄	C₂×C₅	1	6	yes	complemented
65	C₅×#124	C₂×C₅	1	6	yes	complemented
66	C₅×#123	C₂×C₅	1	12	yes	complemented
67	C₅×C₂²⋊Q₈	C₂×C₅	1	6	yes	complemented
68	C₅×#121	C₂×C₅	1	6	yes	complemented
69	C₅×#121	C₂×C₅	1	12	yes	complemented
70	C₅×#120	C₂×C₅	1	12	yes	complemented
71	C₄×C₅×D₄	C₂×C₅	1	6	yes	complemented
72	C₄×C₅×D₄	C₂×C₅	1	6	yes	complemented
73	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	12	yes	complemented
74	C₂²×C₅²	—	2	1	no	—
75	C₂²×C₅²	—	2	2	no	—
76	C₂²×C₅²	—	2	2	no	—
77	C₂²×C₅²	—	2	1	no	—
78	C₂²×C₅²	—	2	1	no	—
79	C₂²×C₅²	—	2	2	no	—
80	C₂²×C₅²	—	2	1	no	—
81	C₂²×C₅²	—	2	2	no	—
82	C₂²×C₅²	—	2	1	no	—
83	C₂²×C₅²	—	2	2	no	—
84	C₂²×C₅²	—	4	8	no	—
85	C₂²×C₅²	C₂×D₄	1	4	yes	—
86	C₂²×C₅²	#41	1	2	yes	complemented
87	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
88	C₄×C₅²	—	2	1	no	—
89	C₄×C₅²	—	2	1	no	—
90	C₄×C₅²	—	2	2	no	—
91	C₄×C₅²	—	2	1	no	—
92	C₄×C₅²	—	2	1	no	—
93	C₄×C₅²	—	2	2	no	—
94	C₄×C₅²	C₂×D₄	1	2	yes	—
95	C₂⁴×C₅	C₂²×C₅	1	12	yes	—
96	C₅×#41	—	2	24	no	—
97	C₂×C₅×Q₈	C₂²×C₅	1	6	yes	complemented
98	C₂×C₅×D₄	—	2	24	no	—
99	C₂×C₅×D₄	C₂²×C₅	1	6	yes	—
100	C₂×C₅×D₄	C₂²×C₅	1	12	yes	—
101	C₂×C₅×D₄	C₂²×C₅	1	6	yes	—
102	C₂×C₅×D₄	C₂²×C₅	1	6	yes	—
103	C₂×C₅×D₄	C₂²×C₅	1	12	yes	complemented
104	C₂×C₅×D₄	C₂²×C₅	1	12	yes	complemented
105	C₂²×C₄×C₅	C₂²×C₅	1	6	yes	—
106	C₂²×C₄×C₅	C₂²×C₅	1	12	yes	complemented
107	C₂²×C₄×C₅	C₂²×C₅	1	6	yes	complemented
108	C₂²×C₄×C₅	C₂²×C₅	1	12	yes	complemented
109	C₅×C₄⋊C₄	C₂²×C₅	1	6	yes	complemented
110	C₅×C₄⋊C₄	C₂²×C₅	1	6	yes	complemented
111	C₅×C₄⋊C₄	C₂²×C₅	1	12	yes	complemented
112	C₅×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
113	C₅×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
114	C₅×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
115	C₅×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
116	C₅×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
117	C₅×C₂²⋊C₄	C₂²×C₅	1	6	yes	complemented
118	C₅×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
119	C₄²×C₅	C₂²×C₅	1	6	yes	complemented
120	#546	C₅²	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₅²	—	2	2	no	—
122	C₂×C₅²	—	2	4	no	—
123	C₂×C₅²	—	4	1	no	—
124	C₂×C₅²	—	4	2	no	—
125	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₅²	C₂×#41	1	1	yes	characteristic
127	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂³×C₅	—	2	24	no	—
129	C₂³×C₅	—	2	24	no	—
130	C₂³×C₅	—	4	12	no	—
131	C₂³×C₅	C₂³×C₅	1	12	yes	—
132	C₂³×C₅	C₂³×C₅	1	6	yes	—
133	C₂³×C₅	C₂³×C₅	1	12	yes	—
134	C₂³×C₅	C₂³×C₅	1	6	yes	—
135	C₅×Q₈	—	2	12	no	—
136	C₅×D₄	—	2	24	no	—
137	C₅×D₄	—	2	24	no	—
138	C₅×D₄	—	2	12	no	—
139	C₅×D₄	—	2	12	no	—
140	C₅×D₄	—	4	12	no	—
141	C₅×D₄	C₅×D₄	1	24	yes	complemented
142	C₂×C₄×C₅	—	2	24	no	—
143	C₂×C₄×C₅	—	2	12	no	—
144	C₂×C₄×C₅	—	2	12	no	—
145	C₂×C₄×C₅	—	4	12	no	—
146	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
147	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
148	C₂×C₄×C₅	C₂³×C₅	1	12	yes	—
149	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
150	C₂×C₄×C₅	C₂³×C₅	1	12	yes	—
151	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
152	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
153	C₂×#41	C₂×C₅²	1	1	yes	characteristic, complemented
154	C₂²×D₄	C₂×C₅²	1	1	yes	characteristic, complemented
155	#124	C₂×C₅²	1	1	yes	characteristic, complemented
156	#123	C₂×C₅²	1	2	yes	complemented
157	C₂²⋊Q₈	C₂×C₅²	1	1	yes	characteristic, complemented
158	#121	C₂×C₅²	1	1	yes	characteristic, complemented
159	#121	C₂×C₅²	1	2	yes	complemented
160	#120	C₂×C₅²	1	2	yes	complemented
161	C₄×D₄	C₂×C₅²	1	1	yes	characteristic, complemented
162	C₄×D₄	C₂×C₅²	1	1	yes	characteristic, complemented
163	C₂×C₂²⋊C₄	C₂×C₅²	1	2	yes	complemented
164	C₅²	#546	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×C₅	—	2	6	no	—
166	C₂²×C₅	—	2	12	no	—
167	C₂²×C₅	—	2	12	no	—
168	C₂²×C₅	—	2	6	no	—
169	C₂²×C₅	—	2	6	no	—
170	C₂²×C₅	—	2	12	no	—
171	C₂²×C₅	—	2	6	no	—
172	C₂²×C₅	—	2	12	no	—
173	C₂²×C₅	—	2	6	no	—
174	C₂²×C₅	—	2	12	no	—
175	C₂²×C₅	—	4	48	no	—
176	C₂²×C₅	C₂×C₅×D₄	1	24	yes	—
177	C₂²×C₅	C₅×#41	1	12	yes	complemented
178	C₂²×C₅	C₂⁴×C₅	1	6	yes	—
179	C₄×C₅	—	2	6	no	—
180	C₄×C₅	—	2	6	no	—
181	C₄×C₅	—	2	12	no	—
182	C₄×C₅	—	2	6	no	—
183	C₄×C₅	—	2	6	no	—
184	C₄×C₅	—	2	12	no	—
185	C₄×C₅	C₂×C₅×D₄	1	12	yes	—
186	C₂⁴	C₂²×C₅²	1	2	yes	—
187	#41	—	2	4	no	—
188	C₂×Q₈	C₂²×C₅²	1	1	yes	characteristic, complemented
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	C₂²×C₅²	1	1	yes	characteristic
191	C₂×D₄	C₂²×C₅²	1	2	yes	—
192	C₂×D₄	C₂²×C₅²	1	1	yes	characteristic
193	C₂×D₄	C₂²×C₅²	1	1	yes	characteristic
194	C₂×D₄	C₂²×C₅²	1	2	yes	complemented
195	C₂×D₄	C₂²×C₅²	1	2	yes	complemented
196	C₂²×C₄	C₂²×C₅²	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₅²	1	2	yes	complemented
198	C₂²×C₄	C₂²×C₅²	1	1	yes	characteristic, complemented
199	C₂²×C₄	C₂²×C₅²	1	2	yes	complemented
200	C₄⋊C₄	C₂²×C₅²	1	1	yes	characteristic, complemented
201	C₄⋊C₄	C₂²×C₅²	1	1	yes	characteristic, complemented
202	C₄⋊C₄	C₂²×C₅²	1	2	yes	complemented
203	C₂²⋊C₄	C₂²×C₅²	1	1	yes	characteristic
204	C₂²⋊C₄	C₂²×C₅²	1	2	yes	—
205	C₂²⋊C₄	C₂²×C₅²	1	2	yes	complemented
206	C₂²⋊C₄	C₂²×C₅²	1	2	yes	complemented
207	C₂²⋊C₄	C₂²×C₅²	1	2	yes	complemented
208	C₂²⋊C₄	C₂²×C₅²	1	1	yes	characteristic, complemented
209	C₂²⋊C₄	C₂²×C₅²	1	2	yes	complemented
210	C₄²	C₂²×C₅²	1	1	yes	characteristic, complemented
211	C₂×C₅	—	2	12	no	—
212	C₂×C₅	—	2	24	no	—
213	C₂×C₅	—	4	6	no	—
214	C₂×C₅	—	4	12	no	—
215	C₂×C₅	C₂²×C₅×D₄	1	6	yes	—
216	C₂×C₅	C₂×C₅×#41	1	6	yes	—
217	C₂×C₅	C₅×H(5,𝔽₂)	1	6	yes	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂³×C₅²	1	2	yes	—
222	C₂³	C₂³×C₅²	1	1	yes	characteristic
223	C₂³	C₂³×C₅²	1	2	yes	—
224	C₂³	C₂³×C₅²	1	1	yes	characteristic
225	Q₈	—	2	2	no	—
226	D₄	—	2	4	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	2	2	no	—
230	D₄	—	4	2	no	—
231	D₄	C₅²×D₄	1	4	yes	complemented
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
237	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
238	C₂×C₄	C₂³×C₅²	1	2	yes	—
239	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
240	C₂×C₄	C₂³×C₅²	1	2	yes	—
241	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
242	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
243	C₅	C₅×#546	1	6	yes	direct product factor
244	C₂²	—	2	1	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	8	no	—
255	C₂²	C₂×C₅²×D₄	1	4	yes	—
256	C₂²	C₅²×#41	1	2	yes	complemented
257	C₂²	C₂⁴×C₅²	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	C₂×C₅²×D₄	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	—	2	4	no	—
267	C₂	—	4	2	no	—
268	C₂	—	4	1	no	—
269	C₂	C₂²×C₅²×D₄	1	1	yes	characteristic
270	C₂	C₂×C₅²×#41	1	1	yes	characteristic
271	C₂	C₅²×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	semidirect
2. C₂²×C₅²×D₄	C₂	semidirect
3. C₅²×#124	C₂	semidirect
4. C₅²×#123	C₂	semidirect
5. C₅²×C₂²⋊Q₈	C₂	semidirect
6. C₅²×#121	C₂	semidirect
7. C₅²×#121	C₂	semidirect
8. C₅²×#120	C₂	semidirect
9. C₄×C₅²×D₄	C₂	semidirect
10. C₄×C₅²×D₄	C₂	semidirect
11. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
14. C₂×C₅²×Q₈	C₂²	semidirect
20. C₂×C₅²×D₄	C₂²	semidirect
21. C₂×C₅²×D₄	C₂²	semidirect
23. C₂²×C₄×C₅²	C₂²	semidirect
24. C₂²×C₄×C₅²	C₂²	semidirect
25. C₂²×C₄×C₅²	C₂²	semidirect
26. C₅²×C₄⋊C₄	C₂²	semidirect
27. C₅²×C₄⋊C₄	C₂²	semidirect
28. C₅²×C₄⋊C₄	C₂²	semidirect
31. C₅²×C₂²⋊C₄	C₂²	semidirect
32. C₅²×C₂²⋊C₄	C₂²	semidirect
33. C₅²×C₂²⋊C₄	C₂²	semidirect
34. C₅²×C₂²⋊C₄	C₂²	semidirect
35. C₅²×C₂²⋊C₄	C₂²	semidirect
36. C₄²×C₅²	C₂²	semidirect
37. C₅×#546	C₅	direct (N × H)
51. C₅²×D₄	D₄	semidirect
63. C₂×C₅×#41	C₂×C₅	semidirect
64. C₂²×C₅×D₄	C₂×C₅	semidirect
65. C₅×#124	C₂×C₅	semidirect
66. C₅×#123	C₂×C₅	semidirect
67. C₅×C₂²⋊Q₈	C₂×C₅	semidirect
68. C₅×#121	C₂×C₅	semidirect
69. C₅×#121	C₂×C₅	semidirect
70. C₅×#120	C₂×C₅	semidirect
71. C₄×C₅×D₄	C₂×C₅	semidirect
72. C₄×C₅×D₄	C₂×C₅	semidirect
73. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
86. C₂²×C₅²	#41	semidirect
97. C₂×C₅×Q₈	C₂²×C₅	semidirect
103. C₂×C₅×D₄	C₂²×C₅	semidirect
104. C₂×C₅×D₄	C₂²×C₅	semidirect
106. C₂²×C₄×C₅	C₂²×C₅	semidirect
107. C₂²×C₄×C₅	C₂²×C₅	semidirect
108. C₂²×C₄×C₅	C₂²×C₅	semidirect
109. C₅×C₄⋊C₄	C₂²×C₅	semidirect
110. C₅×C₄⋊C₄	C₂²×C₅	semidirect
111. C₅×C₄⋊C₄	C₂²×C₅	semidirect
114. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
115. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
116. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
117. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
118. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
119. C₄²×C₅	C₂²×C₅	semidirect
120. #546	C₅²	direct (N × H)
141. C₅×D₄	C₅×D₄	semidirect
153. C₂×#41	C₂×C₅²	semidirect
154. C₂²×D₄	C₂×C₅²	semidirect
155. #124	C₂×C₅²	semidirect
156. #123	C₂×C₅²	semidirect
157. C₂²⋊Q₈	C₂×C₅²	semidirect
158. #121	C₂×C₅²	semidirect
159. #121	C₂×C₅²	semidirect
160. #120	C₂×C₅²	semidirect
161. C₄×D₄	C₂×C₅²	semidirect
162. C₄×D₄	C₂×C₅²	semidirect
163. C₂×C₂²⋊C₄	C₂×C₅²	semidirect
164. C₅²	#546	direct (N × H)
177. C₂²×C₅	C₅×#41	semidirect
188. C₂×Q₈	C₂²×C₅²	semidirect
194. C₂×D₄	C₂²×C₅²	semidirect
195. C₂×D₄	C₂²×C₅²	semidirect
197. C₂²×C₄	C₂²×C₅²	semidirect
198. C₂²×C₄	C₂²×C₅²	semidirect
199. C₂²×C₄	C₂²×C₅²	semidirect
200. C₄⋊C₄	C₂²×C₅²	semidirect
201. C₄⋊C₄	C₂²×C₅²	semidirect
202. C₄⋊C₄	C₂²×C₅²	semidirect
205. C₂²⋊C₄	C₂²×C₅²	semidirect
206. C₂²⋊C₄	C₂²×C₅²	semidirect
207. C₂²⋊C₄	C₂²×C₅²	semidirect
208. C₂²⋊C₄	C₂²×C₅²	semidirect
209. C₂²⋊C₄	C₂²×C₅²	semidirect
210. C₄²	C₂²×C₅²	semidirect
231. D₄	C₅²×D₄	semidirect
243. C₅	C₅×#546	direct (N × H)
256. C₂²	C₅²×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₂²×C₅²×D₄	2
C₅²×#124	2
C₅²×#123	2
C₅²×C₂²⋊Q₈	2
C₅²×#121	2
C₅²×#120	2
C₄×C₅²×D₄	2
C₂×C₅²×C₂²⋊C₄	2
C₅×#546	5

Subgroups — C52×#546 #13107404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#546 #13107404