Subgroups — #13107229

Order 1600.

768 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#126	C₂	1	1	yes	maximal, characteristic
2	#11717210	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717208	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717207	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717199	C₂	1	1	yes	maximal, characteristic
9	#11717199	C₂	1	1	yes	maximal, characteristic
10	#11717199	C₂	1	1	yes	maximal, characteristic
11	#11717198	C₂	1	1	yes	maximal, characteristic
12	#11717198	C₂	1	1	yes	maximal, characteristic
13	#11717197	C₂	1	1	yes	maximal, characteristic
14	#11717190	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#6045	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#1039	C₂²	1	1	yes	characteristic
17	C₂×#6045	C₂²	1	1	yes	characteristic
18	C₂×#6045	C₂²	1	1	yes	characteristic
19	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₄²×C₅²	C₂²	1	1	yes	characteristic
26	C₂²⋊#1039	C₂²	1	1	yes	characteristic
27	C₂²⋊#1039	C₂²	1	1	yes	characteristic
28	C₂²⋊#1039	C₂²	1	1	yes	characteristic
29	C₄⋊#1039	C₂²	1	1	yes	characteristic
30	C₄⋊#1039	C₂²	1	1	yes	characteristic
31	C₄⋊#1039	C₂²	1	1	yes	characteristic
32	C₄⋊#1039	C₂²	1	1	yes	characteristic
33	C₄⋊#1039	C₂²	1	1	yes	characteristic
34	#87806	C₂²	1	1	yes	characteristic
35	#87806	C₂²	1	1	yes	characteristic
36	#87806	C₂²	1	1	yes	characteristic
37	#87806	C₂²	1	1	yes	characteristic
38	#87806	C₂²	1	1	yes	characteristic
39	#87806	C₂²	1	1	yes	characteristic
40	#87806	C₂²	1	1	yes	characteristic
41	#87806	C₂²	1	1	yes	characteristic
42	#87806	C₂²	1	1	yes	characteristic
43	#87806	C₂²	1	1	yes	characteristic
44	#87806	C₂²	1	1	yes	characteristic
45	#87806	C₂²	1	1	yes	characteristic
46	C₄×#1039	C₂²	1	1	yes	characteristic
47	C₄×#1039	C₂²	1	1	yes	characteristic
48	C₄×#1039	C₂²	1	1	yes	characteristic
49	C₄×#1039	C₂²	1	1	yes	characteristic
50	C₄×#1039	C₂²	1	1	yes	characteristic
51	#66035	—	5	6	no	maximal
52	C₂³×C₅²	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
59	C₂×#1039	—	2	2	no	—
60	C₂×#1039	C₂³	1	1	yes	characteristic
61	C₂×#1039	C₂³	1	1	yes	characteristic
62	C₂×#1039	C₂³	1	1	yes	characteristic
63	C₂×#1039	C₂³	1	1	yes	characteristic
64	C₂×#1039	C₂³	1	1	yes	characteristic
65	C₂×#1039	C₂³	1	1	yes	characteristic
66	C₂×#1039	C₂³	1	1	yes	characteristic
67	C₂×#1039	C₂³	1	1	yes	characteristic
68	#6045	—	2	2	no	—
69	#6045	—	2	2	no	—
70	C₅×#126	D₅	1	6	yes	—
71	#4036	—	5	6	no	—
72	#4035	—	5	6	no	—
73	#4035	—	5	6	no	—
74	#4035	—	5	6	no	—
75	#4034	—	5	6	no	—
76	#4033	—	5	6	no	—
77	#4025	—	5	6	no	—
78	#4025	—	5	6	no	—
79	#4025	—	5	6	no	—
80	#4024	—	5	6	no	—
81	#4024	—	5	6	no	—
82	#4023	—	5	6	no	—
83	#4016	—	5	6	no	—
84	C₄×Dic₁₀	—	5	6	no	—
85	C₂²×C₅²	—	2	1	no	—
86	C₂²×C₅²	—	2	1	no	—
87	C₂²×C₅²	—	2	1	no	—
88	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
89	C₄×C₅²	—	2	1	no	—
90	C₄×C₅²	—	2	1	no	—
91	C₄×C₅²	—	2	1	no	—
92	C₄×C₅²	—	2	1	no	—
93	C₄×C₅²	—	2	1	no	—
94	C₄×C₅²	—	2	1	no	—
95	#1039	—	2	1	no	—
96	#1039	—	2	1	no	—
97	#1039	—	2	1	no	—
98	#1039	—	2	1	no	—
99	#1039	—	2	1	no	—
100	#1039	—	2	1	no	—
101	#1039	—	2	1	no	—
102	#1039	#41	1	2	yes	—
103	C₂²×Dic₅	—	5	6	no	—
104	C₂×Dic₁₀	—	5	6	no	—
105	C₂×Dic₁₀	—	5	6	no	—
106	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
107	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
108	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
109	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
110	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
111	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
112	C₄²×C₅	C₂×D₅	1	6	yes	—
113	C₂²⋊Dic₅	—	5	6	no	—
114	C₂²⋊Dic₅	—	5	6	no	—
115	C₂²⋊Dic₅	—	5	6	no	—
116	C₄⋊Dic₅	—	5	6	no	—
117	C₄⋊Dic₅	—	5	6	no	—
118	C₄⋊Dic₅	—	5	6	no	—
119	C₄⋊Dic₅	—	5	6	no	—
120	C₄⋊Dic₅	—	5	6	no	—
121	#683	—	5	6	no	—
122	#683	—	5	6	no	—
123	#683	—	5	6	no	—
124	#683	—	5	6	no	—
125	#683	—	5	6	no	—
126	#683	—	5	6	no	—
127	#683	—	5	6	no	—
128	#683	—	5	6	no	—
129	#683	—	5	6	no	—
130	#683	—	5	6	no	—
131	#683	—	5	6	no	—
132	#683	—	5	6	no	—
133	C₄×Dic₅	—	5	6	no	—
134	C₄×Dic₅	—	5	6	no	—
135	C₄×Dic₅	—	5	6	no	—
136	C₄×Dic₅	—	5	6	no	—
137	C₄×Dic₅	—	5	6	no	—
138	#541	—	25	1	no	Sylow
139	C₂×C₅²	—	4	1	no	—
140	C₂×C₅²	C₂×#41	1	1	yes	characteristic
141	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂³×C₅	C₂²×D₅	1	6	yes	—
144	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
145	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
146	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
147	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
148	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
149	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
150	C₂×Dic₅	—	5	6	no	—
151	C₂×Dic₅	—	5	6	no	—
152	C₂×Dic₅	—	5	6	no	—
153	C₂×Dic₅	—	5	6	no	—
154	C₂×Dic₅	—	5	6	no	—
155	C₂×Dic₅	—	5	6	no	—
156	C₂×Dic₅	—	5	6	no	—
157	C₂×Dic₅	—	5	6	no	—
158	C₂×Dic₅	—	10	12	no	—
159	Dic₁₀	—	10	12	no	—
160	Dic₁₀	—	10	12	no	—
161	C₄⋊Q₈	—	25	1	no	—
162	#126	—	25	1	no	—
163	#126	—	25	1	no	—
164	#126	—	25	1	no	—
165	#126	C₅⋊D₅	1	1	yes	characteristic
166	#125	—	25	1	no	—
167	#125	—	25	1	no	—
168	#125	—	25	1	no	—
169	#125	—	25	1	no	—
170	#125	—	25	1	no	—
171	C₂²⋊Q₈	—	25	1	no	—
172	C₂²⋊Q₈	—	25	1	no	—
173	C₄×Q₈	—	25	1	no	—
174	C₄×Q₈	—	25	1	no	—
175	#117	—	25	1	no	—
176	C₅²	#541	1	1	yes	characteristic, complemented, Sylow
177	C₂²×C₅	—	2	6	no	—
178	C₂²×C₅	—	2	6	no	—
179	C₂²×C₅	—	2	6	no	—
180	C₂²×C₅	C₂³×D₅	1	6	yes	—
181	C₄×C₅	—	2	6	no	—
182	C₄×C₅	—	2	6	no	—
183	C₄×C₅	—	2	6	no	—
184	C₄×C₅	—	2	6	no	—
185	C₄×C₅	—	2	6	no	—
186	C₄×C₅	—	2	6	no	—
187	Dic₅	—	5	12	no	—
188	Dic₅	—	10	6	no	—
189	Dic₅	—	10	6	no	—
190	Dic₅	—	10	6	no	—
191	Dic₅	—	10	6	no	—
192	Dic₅	—	10	6	no	—
193	Dic₅	—	10	6	no	—
194	Dic₅	—	10	6	no	—
195	C₂×Q₈	—	25	1	no	—
196	C₂×Q₈	—	25	1	no	—
197	C₂²×C₄	—	25	1	no	—
198	C₄⋊C₄	—	25	1	no	—
199	C₄⋊C₄	—	25	1	no	—
200	C₄⋊C₄	—	25	1	no	—
201	C₄⋊C₄	—	25	1	no	—
202	C₄⋊C₄	—	25	1	no	—
203	C₄⋊C₄	—	25	1	no	—
204	C₄⋊C₄	—	25	1	no	—
205	C₄⋊C₄	—	25	1	no	—
206	C₄⋊C₄	—	25	1	no	—
207	C₄⋊C₄	—	25	1	no	—
208	C₄⋊C₄	—	25	1	no	—
209	C₄⋊C₄	—	25	1	no	—
210	C₄⋊C₄	—	25	1	no	—
211	C₄⋊C₄	—	25	1	no	—
212	C₄⋊C₄	—	25	1	no	—
213	C₄⋊C₄	—	25	1	no	—
214	C₄⋊C₄	—	25	1	no	—
215	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
216	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
217	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
218	C₂²⋊C₄	—	25	1	no	—
219	C₂²⋊C₄	—	25	1	no	—
220	C₂²⋊C₄	—	25	1	no	—
221	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
222	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
223	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
224	C₄²	—	25	1	no	—
225	C₄²	—	25	1	no	—
226	C₄²	—	25	1	no	—
227	C₄²	—	25	1	no	—
228	C₄²	—	25	1	no	—
229	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
230	C₂×C₅	—	4	6	no	—
231	C₂×C₅	D₅×#41	1	6	yes	—
232	C₂×C₅	#4150	1	6	yes	—
233	C₂×C₅	#4150	1	6	yes	—
234	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
235	Q₈	—	50	2	no	—
236	Q₈	—	50	2	no	—
237	C₂×C₄	—	25	1	no	—
238	C₂×C₄	—	25	1	no	—
239	C₂×C₄	—	25	1	no	—
240	C₂×C₄	—	25	1	no	—
241	C₂×C₄	—	25	1	no	—
242	C₂×C₄	—	25	1	no	—
243	C₂×C₄	—	25	1	no	—
244	C₂×C₄	—	25	1	no	—
245	C₂×C₄	—	50	2	no	—
246	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
252	C₅	#66035	1	6	yes	complemented
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	25	2	no	—
264	C₄	—	50	1	no	—
265	C₄	—	50	1	no	—
266	C₄	—	50	1	no	—
267	C₄	—	50	1	no	—
268	C₄	—	50	1	no	—
269	C₄	—	50	1	no	—
270	C₄	—	50	1	no	—
271	C₂	—	4	1	no	—
272	C₂	#41×C₅⋊D₅	1	1	yes	characteristic
273	C₂	#11717523	1	1	yes	characteristic
274	C₂	#11717523	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#126	C₂
prime radical	C₂³×C₅²	C₂³
hypercenter	#126	C₅⋊D₅
hypocenter	C₅²	#541
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
2. #11717210	C₂	semidirect
3. #11717209	C₂	semidirect
4. #11717209	C₂	semidirect
5. #11717209	C₂	semidirect
6. #11717208	C₂	semidirect
7. #11717207	C₂	semidirect
14. #11717190	C₂	semidirect
15. C₄×#6045	C₂	semidirect
176. C₅²	#541	semidirect
252. C₅	#66035	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#126	2
#11717210	2
#11717209	2
#11717208	2
#11717207	2
#11717199	2
#11717198	2
#11717197	2
#11717190	2
C₄×#6045	2
#66035	5