Subgroups — C₄×C₅×#235 #11755859

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 228 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	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#67	C₂	1	4	yes	maximal, complemented
4	C₅×#926	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₅×D₁₂	C₂	1	2	yes	maximal, complemented
6	C₅×#872	C₂	1	2	yes	maximal, complemented
7	C₄²×C₅×D₃	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×Dic₆	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×#41	—	3	1	no	maximal, Hall
10	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×#67	C₂²	1	2	yes	—
13	C₅×#235	C₄	1	8	yes	direct product factor
14	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	4	yes	—
16	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₄²×C₅	C₂²	1	2	yes	—
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₅×#217	C₂²	1	2	yes	—
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
25	C₄×C₅×Dic₃	C₂²	1	4	yes	—
26	C₄×C₅×Dic₃	C₂²	1	2	yes	—
27	C₄×#235	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂×C₅×#41	—	3	1	no	—
29	C₄×C₅×Q₈	—	3	2	no	—
30	C₄×C₅×D₄	—	3	4	no	—
31	C₄×C₅×D₄	—	3	2	no	—
32	C₅×#117	—	3	1	no	—
33	C₅×#117	—	3	2	no	—
34	C₂×C₄²×C₅	—	3	2	no	—
35	C₂×C₄²×C₅	D₃	1	1	yes	characteristic, complemented
36	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₅×D₃	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₅	—	2	4	no	—
39	C₂×C₃×C₄×C₅	C₂×C₄	1	4	yes	complemented
40	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
44	C₅×#67	C₂×C₄	1	8	yes	complemented
45	C₂×C₅×Dic₃	C₂³	1	4	yes	—
46	C₂×C₅×Dic₃	C₂³	1	2	yes	—
47	C₅×D₁₂	C₂×C₄	1	4	yes	complemented
48	C₄×C₅×D₃	—	2	8	no	—
49	C₄×C₅×D₃	C₂×C₄	1	8	yes	complemented
50	C₅×Dic₆	C₂×C₄	1	4	yes	complemented
51	C₂×#235	C₂×C₅	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
53	C₄×#67	C₂×C₅	1	4	yes	complemented
54	#926	C₂×C₅	1	1	yes	characteristic, complemented
55	C₄×D₁₂	C₂×C₅	1	2	yes	complemented
56	#872	C₂×C₅	1	2	yes	complemented
57	C₄²×D₃	C₂×C₅	1	2	yes	complemented
58	C₄×Dic₆	C₂×C₅	1	2	yes	complemented
59	C₅×#41	—	3	8	no	—
60	C₂×C₅×Q₈	—	3	1	no	—
61	C₂×C₅×D₄	—	3	2	no	—
62	C₂×C₅×D₄	—	3	1	no	—
63	C₂²×C₄×C₅	—	3	2	no	—
64	C₂²×C₄×C₅	—	3	4	no	—
65	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
67	C₅×C₄⋊C₄	—	3	4	no	—
68	C₅×C₄⋊C₄	—	3	2	no	—
69	C₅×C₂²⋊C₄	—	3	2	no	—
70	C₅×C₂²⋊C₄	—	3	4	no	—
71	C₄²×C₅	—	3	2	no	—
72	C₄²×C₅	—	3	4	no	—
73	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
74	C₄²×C₅	C₂×D₃	1	2	yes	—
75	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
76	C₄×#41	—	3	1	no	Sylow
77	C₂²×C₃×C₅	—	2	2	no	—
78	C₂²×C₃×C₅	C₂²×C₄	1	2	yes	—
79	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×D₃	—	2	4	no	—
81	C₂×C₅×D₃	C₂²×C₄	1	4	yes	—
82	C₃×C₄×C₅	—	2	2	no	—
83	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₃×C₄×C₅	#41	1	4	yes	complemented
86	C₅×Dic₃	—	2	4	no	—
87	C₅×Dic₃	C₂²×C₄	1	4	yes	—
88	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
89	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
90	C₂×#67	C₂²×C₅	1	2	yes	—
91	#235	C₄×C₅	1	8	yes	direct product factor
92	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄×D₃	C₂²×C₅	1	4	yes	—
94	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
95	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄²	C₂²×C₅	1	2	yes	—
97	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
99	#217	C₂²×C₅	1	2	yes	—
100	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
101	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
102	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
103	C₄×Dic₃	C₂²×C₅	1	4	yes	—
104	C₄×Dic₃	C₂²×C₅	1	2	yes	—
105	C₂³×C₅	—	3	2	no	—
106	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₅×Q₈	—	3	4	no	—
108	C₅×D₄	—	3	8	no	—
109	C₅×D₄	—	3	4	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	3	2	no	—
112	C₂×C₄×C₅	—	3	8	no	—
113	C₂×C₄×C₅	—	3	4	no	—
114	C₂×C₄×C₅	—	6	8	no	—
115	C₂×C₄×C₅	C₄×D₃	1	4	yes	complemented
116	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
117	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
120	C₂×#41	—	3	1	no	—
121	C₄×Q₈	—	3	2	no	—
122	C₄×D₄	—	3	4	no	—
123	C₄×D₄	—	3	2	no	—
124	#117	—	3	1	no	—
125	#117	—	3	2	no	—
126	C₂×C₄²	—	3	2	no	—
127	C₂×C₄²	C₅×D₃	1	1	yes	characteristic, complemented
128	C₂×C₃×C₅	—	2	2	no	—
129	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₃×C₅	C₂×#41	1	2	yes	—
131	C₅×D₃	—	2	4	no	—
132	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
133	C₂²×D₃	C₂³×C₅	1	2	yes	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	C₂×C₄×C₅	1	4	yes	complemented
136	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
137	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
140	#67	C₂×C₄×C₅	1	8	yes	complemented
141	C₂×Dic₃	C₂³×C₅	1	4	yes	—
142	C₂×Dic₃	C₂³×C₅	1	2	yes	—
143	D₁₂	C₂×C₄×C₅	1	4	yes	complemented
144	C₄×D₃	—	2	8	no	—
145	C₄×D₃	C₂×C₄×C₅	1	8	yes	complemented
146	Dic₆	C₂×C₄×C₅	1	4	yes	complemented
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	3	4	no	—
149	C₂²×C₅	—	6	4	no	—
150	C₂²×C₅	C₂×C₄×D₃	1	2	yes	—
151	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	3	4	no	—
154	C₄×C₅	—	6	4	no	—
155	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
156	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
157	C₄×C₅	#235	1	4	yes	direct product factor
158	#41	—	3	8	no	—
159	C₂×Q₈	—	3	1	no	—
160	C₂×D₄	—	3	2	no	—
161	C₂×D₄	—	3	1	no	—
162	C₂²×C₄	—	3	2	no	—
163	C₂²×C₄	—	3	4	no	—
164	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
165	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
166	C₄⋊C₄	—	3	4	no	—
167	C₄⋊C₄	—	3	2	no	—
168	C₂²⋊C₄	—	3	2	no	—
169	C₂²⋊C₄	—	3	4	no	—
170	C₄²	—	3	2	no	—
171	C₄²	—	3	4	no	—
172	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
173	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
174	C₄²	C₂×C₅×D₃	1	2	yes	—
175	C₃×C₅	C₄×#41	1	1	yes	characteristic, complemented, Hall
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	C₂²×C₄×C₅	1	2	yes	—
178	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
179	C₂×D₃	—	2	4	no	—
180	C₂×D₃	C₂²×C₄×C₅	1	4	yes	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
183	C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
184	C₃×C₄	C₅×#41	1	4	yes	complemented
185	Dic₃	—	2	4	no	—
186	Dic₃	C₂²×C₄×C₅	1	4	yes	—
187	C₂×C₅	—	2	2	no	—
188	C₂×C₅	—	6	4	no	—
189	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
190	C₂×C₅	C₂×#235	1	2	yes	—
191	C₂³	—	3	2	no	—
192	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
193	Q₈	—	3	4	no	—
194	D₄	—	3	8	no	—
195	D₄	—	3	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	3	2	no	—
198	C₂×C₄	—	3	8	no	—
199	C₂×C₄	—	3	4	no	—
200	C₂×C₄	—	6	8	no	—
201	C₂×C₄	C₄×C₅×D₃	1	4	yes	complemented
202	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
203	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
208	C₂×C₃	C₂×C₅×#41	1	2	yes	—
209	D₃	—	2	4	no	—
210	C₅	C₄×#235	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	—	3	4	no	—
213	C₂²	—	6	4	no	—
214	C₂²	C₂×C₄×C₅×D₃	1	2	yes	—
215	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	3	4	no	—
218	C₄	—	6	4	no	—
219	C₄	C₂×C₄×C₅×D₃	1	2	yes	—
220	C₄	C₂×C₄×C₅×D₃	1	2	yes	—
221	C₄	C₅×#235	1	4	yes	direct product factor
222	C₃	C₄×C₅×#41	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	2	2	no	—
224	C₂	—	6	4	no	—
225	C₂	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
226	C₂	C₂×C₅×#235	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅	C₂×D₃
commutator	C₂×C₃	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄²×C₅	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#235	C₂
cosocle	C₂²×C₃	C₂⁴×C₅
hypercenter	C₂×C₄²×C₅	D₃
hypocenter	C₃	C₄×C₅×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₅	C₂	semidirect
3. C₄×C₅×#67	C₂	semidirect
4. C₅×#926	C₂	semidirect
5. C₄×C₅×D₁₂	C₂	semidirect
6. C₅×#872	C₂	semidirect
7. C₄²×C₅×D₃	C₂	semidirect
8. C₄×C₅×Dic₆	C₂	semidirect
13. C₅×#235	C₄	direct (N × H)
27. C₄×#235	C₅	direct (N × H)
35. C₂×C₄²×C₅	D₃	semidirect
39. C₂×C₃×C₄×C₅	C₂×C₄	semidirect
44. C₅×#67	C₂×C₄	semidirect
47. C₅×D₁₂	C₂×C₄	semidirect
49. C₄×C₅×D₃	C₂×C₄	semidirect
50. C₅×Dic₆	C₂×C₄	semidirect
52. C₂×C₃×C₄²	C₂×C₅	semidirect
53. C₄×#67	C₂×C₅	semidirect
54. #926	C₂×C₅	semidirect
55. C₄×D₁₂	C₂×C₅	semidirect
56. #872	C₂×C₅	semidirect
57. C₄²×D₃	C₂×C₅	semidirect
58. C₄×Dic₆	C₂×C₅	semidirect
85. C₃×C₄×C₅	#41	semidirect
91. #235	C₄×C₅	direct (N × H)
115. C₂×C₄×C₅	C₄×D₃	semidirect
127. C₂×C₄²	C₅×D₃	semidirect
135. C₂×C₃×C₄	C₂×C₄×C₅	semidirect
140. #67	C₂×C₄×C₅	semidirect
143. D₁₂	C₂×C₄×C₅	semidirect
145. C₄×D₃	C₂×C₄×C₅	semidirect
146. Dic₆	C₂×C₄×C₅	semidirect
157. C₄×C₅	#235	direct (N × H)
175. C₃×C₅	C₄×#41	semidirect
184. C₃×C₄	C₅×#41	semidirect
201. C₂×C₄	C₄×C₅×D₃	semidirect
210. C₅	C₄×#235	direct (N × H)
221. C₄	C₅×#235	direct (N × H)
222. C₃	C₄×C₅×#41	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
C₂×C₃×C₄²×C₅	2
C₄×C₅×#67	2
C₅×#926	2
C₄×C₅×D₁₂	2
C₅×#872	2
C₄²×C₅×D₃	2
C₄×C₅×Dic₆	2
C₄×C₅×#41	3
C₄×#235	5

Subgroups — C4×C5×#235 #11755859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#235 #11755859