Subgroups — C₅×#5699 #11756084

Order 960.

392 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#898	C₂	1	2	yes	maximal, complemented
3	C₅×#898	C₂	1	2	yes	maximal, complemented
4	C₅×#897	C₂	1	2	yes	maximal, complemented
5	C₅×#896	C₂	1	2	yes	maximal, complemented
6	C₅×#894	C₂	1	2	yes	maximal
7	C₅×#888	C₂	1	2	yes	maximal, complemented
8	C₅×#875	C₂	1	1	yes	maximal, characteristic
9	C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#563	—	3	1	no	maximal, Hall
11	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
13	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
14	C₂×C₅×Dic₆	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₄×C₅×Dic₃	C₂²	1	2	yes	—
32	#5699	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×C₄⋊Q₈	—	3	2	no	—
34	C₅×#126	—	3	2	no	—
35	C₅×#126	—	3	2	no	—
36	C₅×#125	—	3	1	no	—
37	C₅×#125	D₃	1	1	yes	characteristic, complemented
38	C₅×#124	—	3	1	no	—
39	C₅×#123	—	3	2	no	—
40	C₅×C₂²⋊Q₈	—	3	2	no	—
41	C₅×C₂²⋊Q₈	—	3	2	no	—
42	C₂²×C₅×D₃	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
44	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₃	C₂³	1	2	yes	—
49	C₂×C₅×Dic₃	C₂³	1	2	yes	—
50	C₂×C₅×Dic₃	C₂³	1	2	yes	—
51	C₅×D₁₂	—	4	1	no	—
52	C₄×C₅×D₃	—	4	2	no	—
53	C₅×Dic₆	—	4	1	no	—
54	C₅×Dic₆	—	4	2	no	—
55	C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
56	#898	C₂×C₅	1	2	yes	complemented
57	#898	C₂×C₅	1	2	yes	complemented
58	#897	C₂×C₅	1	2	yes	complemented
59	#896	C₂×C₅	1	2	yes	complemented
60	#894	C₂×C₅	1	2	yes	—
61	#888	C₂×C₅	1	2	yes	complemented
62	#875	C₂×C₅	1	1	yes	characteristic
63	#870	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₅×Q₈	—	3	2	no	—
65	C₂×C₅×Q₈	—	3	1	no	—
66	C₂×C₅×D₄	—	3	1	no	—
67	C₂²×C₄×C₅	—	3	2	no	—
68	C₅×C₄⋊C₄	—	3	1	no	—
69	C₅×C₄⋊C₄	—	3	2	no	—
70	C₅×C₄⋊C₄	—	3	2	no	—
71	C₅×C₄⋊C₄	—	3	2	no	—
72	C₅×C₄⋊C₄	—	3	2	no	—
73	C₅×C₄⋊C₄	—	3	1	no	—
74	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
75	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
76	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
77	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
78	C₅×C₂²⋊C₄	—	3	2	no	—
79	C₅×C₂²⋊C₄	—	3	2	no	—
80	C₅×C₂²⋊C₄	—	3	2	no	—
81	C₅×C₂²⋊C₄	—	3	2	no	—
82	C₅×C₂²⋊C₄	—	3	2	no	—
83	C₄²×C₅	—	3	2	no	—
84	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
85	#563	—	3	1	no	Sylow
86	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
87	C₂×C₅×D₃	—	2	2	no	—
88	C₂×C₅×D₃	—	2	2	no	—
89	C₂×C₅×D₃	—	2	2	no	—
90	C₃×C₄×C₅	—	2	2	no	—
91	C₃×C₄×C₅	—	2	1	no	—
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	—	2	1	no	—
95	C₅×Dic₃	—	2	2	no	—
96	C₅×Dic₃	—	2	2	no	—
97	C₅×Dic₃	—	2	2	no	—
98	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
100	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
101	C₂×Dic₆	C₂²×C₅	1	2	yes	—
102	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
103	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
104	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
105	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
106	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
107	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
108	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
109	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
110	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
111	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
112	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
113	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
114	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
115	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
116	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
117	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
118	C₄×Dic₃	C₂²×C₅	1	2	yes	—
119	C₂³×C₅	—	3	2	no	—
120	C₅×Q₈	—	12	2	no	—
121	C₅×Q₈	—	12	1	no	—
122	C₅×D₄	—	12	1	no	—
123	C₂×C₄×C₅	—	3	2	no	—
124	C₂×C₄×C₅	—	3	2	no	—
125	C₂×C₄×C₅	—	3	2	no	—
126	C₂×C₄×C₅	—	12	2	no	—
127	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
128	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
132	C₄⋊Q₈	—	3	2	no	—
133	#126	—	3	2	no	—
134	#126	—	3	2	no	—
135	#125	—	3	1	no	—
136	#125	C₅×D₃	1	1	yes	characteristic, complemented
137	#124	—	3	1	no	—
138	#123	—	3	2	no	—
139	C₂²⋊Q₈	—	3	2	no	—
140	C₂²⋊Q₈	—	3	2	no	—
141	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
144	C₅×D₃	—	4	2	no	—
145	C₂²×D₃	C₂³×C₅	1	2	yes	—
146	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
147	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
150	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
151	C₂×Dic₃	C₂³×C₅	1	2	yes	—
152	C₂×Dic₃	C₂³×C₅	1	2	yes	—
153	C₂×Dic₃	C₂³×C₅	1	2	yes	—
154	D₁₂	—	4	1	no	—
155	C₄×D₃	—	4	2	no	—
156	Dic₆	—	4	1	no	—
157	Dic₆	—	4	2	no	—
158	C₂²×C₅	—	6	2	no	—
159	C₂²×C₅	—	6	2	no	—
160	C₂²×C₅	—	6	2	no	—
161	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	6	2	no	—
168	C₄×C₅	—	6	2	no	—
169	C₄×C₅	—	6	2	no	—
170	C₂×Q₈	—	3	2	no	—
171	C₂×Q₈	—	3	1	no	—
172	C₂×D₄	—	3	1	no	—
173	C₂²×C₄	—	3	2	no	—
174	C₄⋊C₄	—	3	1	no	—
175	C₄⋊C₄	—	3	2	no	—
176	C₄⋊C₄	—	3	2	no	—
177	C₄⋊C₄	—	3	2	no	—
178	C₄⋊C₄	—	3	2	no	—
179	C₄⋊C₄	—	3	1	no	—
180	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
182	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
183	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
184	C₂²⋊C₄	—	3	2	no	—
185	C₂²⋊C₄	—	3	2	no	—
186	C₂²⋊C₄	—	3	2	no	—
187	C₂²⋊C₄	—	3	2	no	—
188	C₂²⋊C₄	—	3	2	no	—
189	C₄²	—	3	2	no	—
190	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
191	C₃×C₅	#563	1	1	yes	characteristic, complemented, Hall
192	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
193	C₂×D₃	—	2	2	no	—
194	C₂×D₃	—	2	2	no	—
195	C₂×D₃	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	1	no	—
201	Dic₃	—	2	2	no	—
202	Dic₃	—	2	2	no	—
203	Dic₃	—	2	2	no	—
204	C₂×C₅	—	12	2	no	—
205	C₂×C₅	#1007	1	1	yes	characteristic
206	C₂×C₅	#1009	1	1	yes	characteristic
207	C₂×C₅	#1010	1	1	yes	characteristic
208	C₂³	—	3	2	no	—
209	Q₈	—	12	2	no	—
210	Q₈	—	12	1	no	—
211	D₄	—	12	1	no	—
212	C₂×C₄	—	3	2	no	—
213	C₂×C₄	—	3	2	no	—
214	C₂×C₄	—	3	2	no	—
215	C₂×C₄	—	12	2	no	—
216	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
217	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
221	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
223	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
224	D₃	—	4	2	no	—
225	C₅	#5699	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²	—	6	2	no	—
227	C₂²	—	6	2	no	—
228	C₂²	—	6	2	no	—
229	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	6	2	no	—
236	C₄	—	6	2	no	—
237	C₄	—	6	2	no	—
238	C₃	C₅×#563	1	1	yes	characteristic, complemented, Sylow
239	C₂	—	12	2	no	—
240	C₂	C₅×#1007	1	1	yes	characteristic
241	C₂	C₅×#1009	1	1	yes	characteristic
242	C₂	C₅×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#125	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×D₁₂	C₂²
hypercenter	C₅×#125	D₃
hypocenter	C₃	C₅×#563
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#125	C₂	semidirect
2. C₅×#898	C₂	semidirect
3. C₅×#898	C₂	semidirect
4. C₅×#897	C₂	semidirect
5. C₅×#896	C₂	semidirect
7. C₅×#888	C₂	semidirect
9. C₅×#870	C₂	semidirect
32. #5699	C₅	direct (N × H)
37. C₅×#125	D₃	semidirect
55. C₃×#125	C₂×C₅	semidirect
56. #898	C₂×C₅	semidirect
57. #898	C₂×C₅	semidirect
58. #897	C₂×C₅	semidirect
59. #896	C₂×C₅	semidirect
61. #888	C₂×C₅	semidirect
63. #870	C₂×C₅	semidirect
136. #125	C₅×D₃	semidirect
191. C₃×C₅	#563	semidirect
225. C₅	#5699	direct (N × H)
238. C₃	C₅×#563	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#125	2
C₅×#898	2
C₅×#897	2
C₅×#896	2
C₅×#894	2
C₅×#888	2
C₅×#875	2
C₅×#870	2
C₅×#563	3
#5699	5

Subgroups — C5×#5699 #11756084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5699 #11756084