Subgroups — C₅×#5830 #11756215

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#947	C₂	1	2	yes	maximal, complemented
4	C₅×#940	C₂	1	2	yes	maximal, complemented
5	C₅×#937	C₂	1	4	yes	maximal, complemented
6	C₅×#930	C₂	1	4	yes	maximal, complemented
7	C₅×#926	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#535	—	3	1	no	maximal, Hall
9	C₂³×C₅×D₃	C₂²	1	2	yes	—
10	C₃×C₅×#41	—	2	4	no	—
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	complemented
16	C₂×C₅×#67	C₂²	1	8	yes	complemented
17	C₂×C₅×D₁₂	—	2	4	no	—
18	C₂×C₅×D₁₂	C₂²	1	2	yes	—
19	C₂×C₅×D₁₂	C₂²	1	2	yes	—
20	C₅×#217	C₂²	1	2	yes	—
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	8	yes	complemented
22	C₅×C₄⋊Dic₃	C₂²	1	2	yes	complemented
23	C₄×C₅×Dic₃	C₂²	1	2	yes	complemented
24	#5830	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#41	D₃	1	1	yes	characteristic, complemented
26	C₂²×C₅×D₄	—	3	1	no	—
27	C₅×#127	—	3	2	no	—
28	C₅×#124	—	3	2	no	—
29	C₅×#121	—	3	4	no	—
30	C₅×#120	—	3	4	no	—
31	C₅×#117	—	3	1	no	—
32	C₂³×C₃×C₅	C₂³	1	2	yes	—
33	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
34	C₂²×C₅×D₃	—	2	4	no	—
35	C₂²×C₅×D₃	—	4	4	no	—
36	C₂²×C₅×D₃	C₂³	1	4	yes	—
37	C₃×C₅×Q₈	—	2	2	no	—
38	C₃×C₅×D₄	—	2	4	no	—
39	C₃×C₅×D₄	—	2	2	no	—
40	C₂×C₃×C₄×C₅	—	2	4	no	—
41	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
42	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₅×#67	—	4	8	no	—
46	C₂×C₅×Dic₃	C₂³	1	4	yes	—
47	C₅×D₁₂	—	2	4	no	—
48	C₅×D₁₂	—	2	4	no	—
49	C₂×C₃×#41	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
51	#947	C₂×C₅	1	2	yes	complemented
52	#940	C₂×C₅	1	2	yes	complemented
53	#937	C₂×C₅	1	4	yes	complemented
54	#930	C₂×C₅	1	4	yes	complemented
55	#926	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂⁴×C₅	—	3	2	no	—
57	C₅×#41	—	2	4	no	—
58	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
59	C₂×C₅×D₄	—	3	8	no	—
60	C₂×C₅×D₄	—	3	2	no	—
61	C₂×C₅×D₄	—	3	2	no	—
62	C₂×C₅×D₄	—	6	4	no	—
63	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
64	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic, complemented
65	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₃	1	2	yes	complemented
67	C₅×C₄⋊C₄	—	3	2	no	—
68	C₅×C₂²⋊C₄	—	3	2	no	—
69	C₅×C₂²⋊C₄	—	3	8	no	—
70	C₄²×C₅	—	3	2	no	—
71	#535	—	3	1	no	Sylow
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	2	no	—
74	C₂²×C₃×C₅	—	2	4	no	—
75	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
76	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
77	C₂×C₅×D₃	—	2	4	no	—
78	C₂×C₅×D₃	—	2	8	no	—
79	C₂×C₅×D₃	—	4	8	no	—
80	C₃×C₄×C₅	—	2	2	no	—
81	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₅×Dic₃	—	2	4	no	—
84	C₂³×D₃	C₂²×C₅	1	2	yes	—
85	C₃×#41	—	2	4	no	—
86	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
87	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
88	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
89	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	complemented
91	C₂×#67	C₂²×C₅	1	8	yes	complemented
92	C₂×D₁₂	—	2	4	no	—
93	C₂×D₁₂	C₂²×C₅	1	2	yes	—
94	C₂×D₁₂	C₂²×C₅	1	2	yes	—
95	#217	C₂²×C₅	1	2	yes	—
96	(C₂×D₃)⋊C₄	C₂²×C₅	1	8	yes	complemented
97	C₄⋊Dic₃	C₂²×C₅	1	2	yes	complemented
98	C₄×Dic₃	C₂²×C₅	1	2	yes	complemented
99	C₂³×C₅	—	3	4	no	—
100	C₂³×C₅	—	6	4	no	—
101	C₂³×C₅	—	12	4	no	—
102	C₂³×C₅	C₂²×D₃	1	2	yes	—
103	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
104	C₅×Q₈	—	2	2	no	—
105	C₅×D₄	—	2	4	no	—
106	C₅×D₄	—	2	2	no	—
107	C₅×D₄	—	6	4	no	—
108	C₅×D₄	—	6	4	no	—
109	C₅×D₄	—	12	8	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	3	4	no	—
112	C₂×C₄×C₅	#67	1	4	yes	complemented
113	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
114	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂×#41	C₅×D₃	1	1	yes	characteristic, complemented
117	C₂²×D₄	—	3	1	no	—
118	#127	—	3	2	no	—
119	#124	—	3	2	no	—
120	#121	—	3	4	no	—
121	#120	—	3	4	no	—
122	#117	—	3	1	no	—
123	C₂×C₃×C₅	—	2	2	no	—
124	C₂×C₃×C₅	—	4	2	no	—
125	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
127	C₅×D₃	—	4	4	no	—
128	C₂³×C₃	C₂³×C₅	1	2	yes	—
129	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
130	C₂²×D₃	—	2	4	no	—
131	C₂²×D₃	—	4	4	no	—
132	C₂²×D₃	C₂³×C₅	1	4	yes	—
133	C₃×Q₈	—	2	2	no	—
134	C₃×D₄	—	2	4	no	—
135	C₃×D₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	C₅×D₄	1	4	yes	complemented
138	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
139	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	#67	—	4	8	no	—
142	C₂×Dic₃	C₂³×C₅	1	4	yes	—
143	D₁₂	—	2	4	no	—
144	D₁₂	—	2	4	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	6	8	no	—
149	C₂²×C₅	—	6	4	no	—
150	C₂²×C₅	—	12	8	no	—
151	C₂²×C₅	C₂×#67	1	2	yes	—
152	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	6	4	no	—
155	C₄×C₅	C₂×#67	1	2	yes	—
156	C₄×C₅	C₂×#67	1	2	yes	—
157	C₂⁴	—	3	2	no	—
158	#41	—	2	4	no	—
159	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
160	C₂×D₄	—	3	8	no	—
161	C₂×D₄	—	3	2	no	—
162	C₂×D₄	—	3	2	no	—
163	C₂×D₄	—	6	4	no	—
164	C₂×D₄	C₂×C₅×D₃	1	2	yes	—
165	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
166	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₅×D₃	1	2	yes	complemented
168	C₄⋊C₄	—	3	2	no	—
169	C₂²⋊C₄	—	3	2	no	—
170	C₂²⋊C₄	—	3	8	no	—
171	C₄²	—	3	2	no	—
172	C₃×C₅	#535	1	1	yes	characteristic, complemented, Hall
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	4	no	—
176	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
177	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
178	C₂×D₃	—	2	4	no	—
179	C₂×D₃	—	2	8	no	—
180	C₂×D₃	—	4	8	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
183	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
184	Dic₃	—	2	4	no	—
185	C₂×C₅	—	2	2	no	—
186	C₂×C₅	—	4	2	no	—
187	C₂×C₅	—	12	4	no	—
188	C₂×C₅	#1009	1	2	yes	—
189	C₂×C₅	C₂²×#67	1	1	yes	characteristic
190	C₂³	—	3	4	no	—
191	C₂³	—	6	4	no	—
192	C₂³	—	12	4	no	—
193	C₂³	C₂²×C₅×D₃	1	2	yes	—
194	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
195	Q₈	—	2	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	—	6	4	no	—
199	D₄	—	6	4	no	—
200	D₄	—	12	8	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	3	4	no	—
203	C₂×C₄	C₅×#67	1	4	yes	complemented
204	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
205	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	—	4	2	no	—
209	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
210	C₂×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
211	D₃	—	4	4	no	—
212	C₅	#5830	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	6	8	no	—
217	C₂²	—	6	4	no	—
218	C₂²	—	12	8	no	—
219	C₂²	C₂×C₅×#67	1	2	yes	—
220	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	6	4	no	—
223	C₄	C₂×C₅×#67	1	2	yes	—
224	C₄	C₂×C₅×#67	1	2	yes	—
225	C₃	C₅×#535	1	1	yes	characteristic, complemented, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	12	4	no	—
229	C₂	C₅×#1009	1	2	yes	—
230	C₂	C₂²×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂×C₅×#41	D₃
hypocenter	C₃	C₅×#535
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#41	C₂	semidirect
2. C₂²×C₅×D₁₂	C₂	semidirect
3. C₅×#947	C₂	semidirect
4. C₅×#940	C₂	semidirect
5. C₅×#937	C₂	semidirect
6. C₅×#930	C₂	semidirect
7. C₅×#926	C₂	semidirect
11. C₂×C₃×C₅×Q₈	C₂²	semidirect
13. C₂×C₃×C₅×D₄	C₂²	semidirect
15. C₂²×C₃×C₄×C₅	C₂²	semidirect
16. C₂×C₅×#67	C₂²	semidirect
21. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
22. C₅×C₄⋊Dic₃	C₂²	semidirect
23. C₄×C₅×Dic₃	C₂²	semidirect
24. #5830	C₅	direct (N × H)
25. C₂×C₅×#41	D₃	semidirect
41. C₂×C₃×C₄×C₅	D₄	semidirect
49. C₂×C₃×#41	C₂×C₅	semidirect
50. C₂²×D₁₂	C₂×C₅	semidirect
51. #947	C₂×C₅	semidirect
52. #940	C₂×C₅	semidirect
53. #937	C₂×C₅	semidirect
54. #930	C₂×C₅	semidirect
55. #926	C₂×C₅	semidirect
58. C₂×C₅×Q₈	C₂×D₃	semidirect
64. C₂×C₅×D₄	C₂×D₃	semidirect
66. C₂²×C₄×C₅	C₂×D₃	semidirect
86. C₂×C₃×Q₈	C₂²×C₅	semidirect
88. C₂×C₃×D₄	C₂²×C₅	semidirect
90. C₂²×C₃×C₄	C₂²×C₅	semidirect
91. C₂×#67	C₂²×C₅	semidirect
96. (C₂×D₃)⋊C₄	C₂²×C₅	semidirect
97. C₄⋊Dic₃	C₂²×C₅	semidirect
98. C₄×Dic₃	C₂²×C₅	semidirect
112. C₂×C₄×C₅	#67	semidirect
116. C₂×#41	C₅×D₃	semidirect
137. C₂×C₃×C₄	C₅×D₄	semidirect
159. C₂×Q₈	C₂×C₅×D₃	semidirect
165. C₂×D₄	C₂×C₅×D₃	semidirect
167. C₂²×C₄	C₂×C₅×D₃	semidirect
172. C₃×C₅	#535	semidirect
203. C₂×C₄	C₅×#67	semidirect
212. C₅	#5830	direct (N × H)
225. C₃	C₅×#535	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#41	2
C₂²×C₅×D₁₂	2
C₅×#947	2
C₅×#940	2
C₅×#937	2
C₅×#930	2
C₅×#926	2
C₅×#535	3
#5830	5

Subgroups — C5×#5830 #11756215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5830 #11756215