Subgroups — C₅×#5553 #11755938

Order 960.

624 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#235	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#940	C₂	1	2	yes	maximal, complemented
4	C₅×#930	C₂	1	2	yes	maximal, complemented
5	C₅×#893	C₂	1	2	yes	maximal, complemented
6	C₅×#884	C₂	1	4	yes	maximal, complemented
7	C₅×#874	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#550	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₂×C₅×#67	C₂²	1	4	yes	—
13	C₂×C₅×#67	C₂²	1	4	yes	—
14	C₅×#235	—	2	8	no	—
15	C₂×C₅×D₁₂	C₂²	1	2	yes	—
16	C₂×C₅×D₁₂	C₂²	1	2	yes	—
17	C₂×C₅×D₁₂	C₂²	1	2	yes	—
18	C₂×C₄×C₅×D₃	C₂²	1	4	yes	—
19	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₄×C₅×Dic₃	C₂²	1	2	yes	—
27	#5553	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂×C₅×#41	—	3	2	no	—
29	C₅×#127	—	3	2	no	—
30	C₅×#127	—	3	1	no	—
31	C₅×#121	—	3	2	no	—
32	C₅×#121	—	3	4	no	—
33	C₄×C₅×Q₈	—	3	1	no	—
34	C₄×C₅×D₄	—	3	2	no	—
35	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
36	C₂³×C₃×C₅	C₂³	1	2	yes	—
37	C₂²×C₅×D₃	C₂³	1	4	yes	—
38	C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₄×C₅	—	2	4	no	—
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₅×#67	—	2	8	no	—
45	C₅×#67	—	4	4	no	—
46	C₂×C₅×Dic₃	C₂³	1	2	yes	—
47	C₂×C₅×Dic₃	C₂³	1	2	yes	—
48	C₅×D₁₂	—	2	4	no	—
49	C₅×D₁₂	—	2	4	no	—
50	C₅×D₁₂	—	4	2	no	—
51	C₄×C₅×D₃	—	2	8	no	—
52	C₅×Dic₆	D₄	1	4	yes	complemented
53	C₂×#235	C₂×C₅	1	2	yes	complemented
54	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
55	#940	C₂×C₅	1	2	yes	complemented
56	#930	C₂×C₅	1	2	yes	complemented
57	#893	C₂×C₅	1	2	yes	complemented
58	#884	C₂×C₅	1	4	yes	complemented
59	#874	C₂×C₅	1	1	yes	characteristic, complemented
60	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
61	C₅×#41	—	6	8	no	—
62	C₂×C₅×Q₈	—	3	1	no	—
63	C₂×C₅×D₄	—	3	4	no	—
64	C₂×C₅×D₄	—	3	4	no	—
65	C₂×C₅×D₄	—	3	2	no	—
66	C₂×C₅×D₄	—	3	2	no	—
67	C₂×C₅×D₄	—	3	2	no	—
68	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	3	4	no	—
70	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
71	C₅×C₄⋊C₄	—	3	2	no	—
72	C₅×C₄⋊C₄	—	3	1	no	—
73	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
74	C₅×C₂²⋊C₄	—	3	4	no	—
75	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
76	C₄²×C₅	—	3	2	no	—
77	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
78	#550	—	3	1	no	Sylow
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	—	2	2	no	—
81	C₂²×C₃×C₅	—	2	2	no	—
82	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₅×D₃	—	2	4	no	—
84	C₂×C₅×D₃	—	2	4	no	—
85	C₂×C₅×D₃	—	2	4	no	—
86	C₃×C₄×C₅	—	2	2	no	—
87	C₃×C₄×C₅	—	2	1	no	—
88	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₃×C₄×C₅	#41	1	2	yes	complemented
90	C₅×Dic₃	—	2	2	no	—
91	C₅×Dic₃	C₂×D₄	1	4	yes	—
92	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
94	C₂×#67	C₂²×C₅	1	4	yes	—
95	C₂×#67	C₂²×C₅	1	4	yes	—
96	#235	—	2	8	no	—
97	C₂×D₁₂	C₂²×C₅	1	2	yes	—
98	C₂×D₁₂	C₂²×C₅	1	2	yes	—
99	C₂×D₁₂	C₂²×C₅	1	2	yes	—
100	C₂×C₄×D₃	C₂²×C₅	1	4	yes	—
101	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
102	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
103	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
104	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
105	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
106	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
107	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
108	C₄×Dic₃	C₂²×C₅	1	2	yes	—
109	C₂³×C₅	—	3	4	no	—
110	C₂³×C₅	C₂²×D₃	1	2	yes	—
111	C₅×Q₈	—	3	4	no	—
112	C₅×D₄	—	2	2	no	—
113	C₅×D₄	—	6	8	no	—
114	C₅×D₄	—	6	4	no	—
115	C₅×D₄	—	6	4	no	—
116	C₅×D₄	—	12	2	no	—
117	C₅×D₄	—	12	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	3	2	no	—
120	C₂×C₄×C₅	—	3	2	no	—
121	C₂×C₄×C₅	—	6	8	no	—
122	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
124	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
126	C₂×#41	—	3	2	no	—
127	#127	—	3	2	no	—
128	#127	—	3	1	no	—
129	#121	—	3	2	no	—
130	#121	—	3	4	no	—
131	C₄×Q₈	—	3	1	no	—
132	C₄×D₄	—	3	2	no	—
133	C₄×D₄	C₅×D₃	1	1	yes	characteristic, complemented
134	C₂×C₃×C₅	—	4	2	no	—
135	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
137	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
138	C₅×D₃	—	4	4	no	—
139	C₂³×C₃	C₂³×C₅	1	2	yes	—
140	C₂²×D₃	C₂³×C₅	1	4	yes	—
141	C₃×D₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
145	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
147	#67	—	2	8	no	—
148	#67	—	4	4	no	—
149	C₂×Dic₃	C₂³×C₅	1	2	yes	—
150	C₂×Dic₃	C₂³×C₅	1	2	yes	—
151	D₁₂	—	2	4	no	—
152	D₁₂	—	2	4	no	—
153	D₁₂	—	4	2	no	—
154	C₄×D₃	—	2	8	no	—
155	Dic₆	C₅×D₄	1	4	yes	complemented
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	6	4	no	—
160	C₂²×C₅	—	6	4	no	—
161	C₂²×C₅	—	6	4	no	—
162	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	3	4	no	—
166	C₄×C₅	—	6	2	no	—
167	C₄×C₅	#235	1	2	yes	complemented
168	C₄×C₅	D₃×D₄	1	2	yes	—
169	#41	—	6	8	no	—
170	C₂×Q₈	—	3	1	no	—
171	C₂×D₄	—	3	4	no	—
172	C₂×D₄	—	3	4	no	—
173	C₂×D₄	—	3	2	no	—
174	C₂×D₄	—	3	2	no	—
175	C₂×D₄	—	3	2	no	—
176	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
177	C₂²×C₄	—	3	4	no	—
178	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
179	C₄⋊C₄	—	3	2	no	—
180	C₄⋊C₄	—	3	1	no	—
181	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
182	C₂²⋊C₄	—	3	4	no	—
183	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
184	C₄²	—	3	2	no	—
185	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
186	C₃×C₅	#550	1	1	yes	characteristic, complemented, Hall
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
191	C₂×D₃	—	2	4	no	—
192	C₂×D₃	—	2	4	no	—
193	C₂×D₃	—	2	4	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
197	C₃×C₄	C₅×#41	1	2	yes	complemented
198	Dic₃	—	2	2	no	—
199	Dic₃	C₂×C₅×D₄	1	4	yes	—
200	C₂×C₅	—	4	2	no	—
201	C₂×C₅	—	12	4	no	—
202	C₂×C₅	C₂×#235	1	1	yes	characteristic
203	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
204	C₂×C₅	#1009	1	1	yes	characteristic
205	C₂³	—	3	4	no	—
206	C₂³	C₂²×C₅×D₃	1	2	yes	—
207	Q₈	—	3	4	no	—
208	D₄	—	2	2	no	—
209	D₄	—	6	8	no	—
210	D₄	—	6	4	no	—
211	D₄	—	6	4	no	—
212	D₄	—	12	2	no	—
213	D₄	—	12	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	3	2	no	—
216	C₂×C₄	—	3	2	no	—
217	C₂×C₄	—	6	8	no	—
218	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
220	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
222	C₂×C₃	—	4	2	no	—
223	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
224	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
225	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
226	D₃	—	4	4	no	—
227	C₅	#5553	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	6	4	no	—
232	C₂²	—	6	4	no	—
233	C₂²	—	6	4	no	—
234	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	3	4	no	—
238	C₄	—	6	2	no	—
239	C₄	C₅×#235	1	2	yes	complemented
240	C₄	C₅×D₃×D₄	1	2	yes	—
241	C₃	C₅×#550	1	1	yes	characteristic, complemented, Sylow
242	C₂	—	4	2	no	—
243	C₂	—	12	4	no	—
244	C₂	C₂×C₅×#235	1	1	yes	characteristic
245	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
246	C₂	C₅×#1009	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×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#550
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#235	C₂	semidirect
2. C₃×C₄×C₅×D₄	C₂	semidirect
3. C₅×#940	C₂	semidirect
4. C₅×#930	C₂	semidirect
5. C₅×#893	C₂	semidirect
6. C₅×#884	C₂	semidirect
7. C₅×#874	C₂	semidirect
8. C₄×C₅×Dic₆	C₂	semidirect
27. #5553	C₅	direct (N × H)
35. C₄×C₅×D₄	D₃	semidirect
52. C₅×Dic₆	D₄	semidirect
53. C₂×#235	C₂×C₅	semidirect
54. C₃×C₄×D₄	C₂×C₅	semidirect
55. #940	C₂×C₅	semidirect
56. #930	C₂×C₅	semidirect
57. #893	C₂×C₅	semidirect
58. #884	C₂×C₅	semidirect
59. #874	C₂×C₅	semidirect
60. C₄×Dic₆	C₂×C₅	semidirect
89. C₃×C₄×C₅	#41	semidirect
133. C₄×D₄	C₅×D₃	semidirect
155. Dic₆	C₅×D₄	semidirect
167. C₄×C₅	#235	semidirect
186. C₃×C₅	#550	semidirect
197. C₃×C₄	C₅×#41	semidirect
227. C₅	#5553	direct (N × H)
239. C₄	C₅×#235	semidirect
241. C₃	C₅×#550	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#235	2
C₃×C₄×C₅×D₄	2
C₅×#940	2
C₅×#930	2
C₅×#893	2
C₅×#884	2
C₅×#874	2
C₄×C₅×Dic₆	2
C₅×#550	3
#5553	5

Subgroups — C5×#5553 #11755938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5553 #11755938