Subgroups — C₃×#65979 #11755656

Order 960.

436 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4092	C₂	1	2	yes	maximal, complemented
3	C₃×#4076	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#683	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4044	C₂	1	2	yes	maximal, complemented
6	C₃×#4040	C₂	1	2	yes	maximal, complemented
7	C₃×#4036	C₂	1	2	yes	maximal, complemented
8	C₃×#4032	C₂	1	2	yes	maximal, complemented
9	C₃×#4027	C₂	1	2	yes	maximal, complemented
10	#65979	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×#685	C₂²	1	2	yes	—
22	C₃×#685	C₂²	1	2	yes	—
23	C₃×#685	C₂²	1	2	yes	—
24	C₃×#685	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×#683	C₂²	1	2	yes	—
31	C₃×C₄×Dic₅	C₂²	1	2	yes	—
32	C₃×#539	—	5	1	no	maximal, Hall
33	C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
34	#4092	C₂×C₃	1	2	yes	complemented
35	#4076	C₂×C₃	1	1	yes	characteristic
36	C₂×#683	C₂×C₃	1	1	yes	characteristic, complemented
37	#4044	C₂×C₃	1	2	yes	complemented
38	#4040	C₂×C₃	1	2	yes	complemented
39	#4036	C₂×C₃	1	2	yes	complemented
40	#4032	C₂×C₃	1	2	yes	complemented
41	#4027	C₂×C₃	1	2	yes	complemented
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂²×C₃×D₅	C₂³	1	2	yes	—
44	C₃×C₅×Q₈	—	4	1	no	—
45	C₂×C₃×C₄×C₅	—	4	1	no	—
46	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₃×C₂²⋊D₅	—	2	4	no	—
51	C₂×C₃×Dic₅	—	2	2	no	—
52	C₂×C₃×Dic₅	—	4	1	no	—
53	C₂×C₃×Dic₅	C₂³	1	2	yes	—
54	C₂×C₃×Dic₅	C₂³	1	2	yes	—
55	C₂×C₃×Dic₅	C₂³	1	2	yes	—
56	C₃×D₂₀	—	4	1	no	—
57	C₃×C₄×D₅	—	4	2	no	—
58	C₃×#126	—	5	2	no	—
59	C₃×#125	—	5	2	no	—
60	C₃×#123	—	5	2	no	—
61	C₃×#123	—	5	2	no	—
62	C₃×C₂²⋊Q₈	—	5	2	no	—
63	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
64	C₃×#121	—	5	1	no	—
65	C₃×C₄×D₄	—	5	2	no	—
66	C₂×C₃×C₄⋊C₄	—	5	1	no	—
67	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
70	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
71	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
72	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
74	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	#685	C₂²×C₃	1	2	yes	—
78	#685	C₂²×C₃	1	2	yes	—
79	#685	C₂²×C₃	1	2	yes	—
80	#685	C₂²×C₃	1	2	yes	—
81	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
82	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
83	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
84	#683	C₂²×C₃	1	2	yes	—
85	#683	C₂²×C₃	1	2	yes	—
86	#683	C₂²×C₃	1	2	yes	—
87	C₄×Dic₅	C₂²×C₃	1	2	yes	—
88	#539	—	5	1	no	Sylow
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	#41	1	2	yes	—
92	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
93	C₂×C₃×D₅	—	2	2	no	—
94	C₂×C₃×D₅	—	2	2	no	—
95	C₂×C₃×D₅	—	2	2	no	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×Dic₅	—	2	2	no	—
101	C₃×Dic₅	—	2	2	no	—
102	C₃×Dic₅	—	2	2	no	—
103	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
104	C₂×C₃×D₄	—	5	2	no	—
105	C₂×C₃×D₄	—	5	1	no	—
106	C₂²×C₃×C₄	—	5	1	no	—
107	C₂²×C₃×C₄	—	5	1	no	—
108	C₂²×C₃×C₄	—	5	2	no	—
109	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	5	2	no	—
111	C₃×C₄⋊C₄	—	5	2	no	—
112	C₃×C₄⋊C₄	—	5	2	no	—
113	C₃×C₄⋊C₄	—	5	1	no	—
114	C₃×C₄⋊C₄	—	5	2	no	—
115	C₃×C₄⋊C₄	—	5	2	no	—
116	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
117	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
118	C₃×C₂²⋊C₄	—	5	2	no	—
119	C₃×C₂²⋊C₄	—	5	2	no	—
120	C₃×C₂²⋊C₄	—	5	2	no	—
121	C₃×C₂²⋊C₄	—	5	2	no	—
122	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
123	C₃×C₄²	—	5	2	no	—
124	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
125	C₂²×D₅	C₂³×C₃	1	2	yes	—
126	C₅×Q₈	—	4	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
129	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
132	C₂²⋊D₅	—	2	4	no	—
133	C₂×Dic₅	—	2	2	no	—
134	C₂×Dic₅	—	4	1	no	—
135	C₂×Dic₅	C₂³×C₃	1	2	yes	—
136	C₂×Dic₅	C₂³×C₃	1	2	yes	—
137	C₂×Dic₅	C₂³×C₃	1	2	yes	—
138	D₂₀	—	4	1	no	—
139	C₄×D₅	—	4	2	no	—
140	#126	—	5	2	no	—
141	#125	—	5	2	no	—
142	#123	—	5	2	no	—
143	#123	—	5	2	no	—
144	C₂²⋊Q₈	—	5	2	no	—
145	C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
146	#121	—	5	1	no	—
147	C₄×D₄	—	5	2	no	—
148	C₂×C₄⋊C₄	—	5	1	no	—
149	C₂×C₃×C₅	—	2	2	no	—
150	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
151	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
153	C₃×D₅	—	4	2	no	—
154	C₂³×C₃	—	5	2	no	—
155	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
156	C₃×Q₈	—	4	1	no	—
157	C₃×D₄	—	10	4	no	—
158	C₃×D₄	—	20	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	5	2	no	—
161	C₂×C₃×C₄	—	5	2	no	—
162	C₂×C₃×C₄	—	5	2	no	—
163	C₂×C₃×C₄	—	10	2	no	—
164	C₂×C₃×C₄	—	20	2	no	—
165	C₂×C₃×C₄	—	20	1	no	—
166	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
167	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
169	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	C₃×#41	1	2	yes	—
173	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×D₅	—	2	2	no	—
175	C₂×D₅	—	2	2	no	—
176	C₂×D₅	—	2	2	no	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	2	no	—
181	Dic₅	—	2	2	no	—
182	Dic₅	—	2	2	no	—
183	Dic₅	—	2	2	no	—
184	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
185	C₂×D₄	—	5	2	no	—
186	C₂×D₄	—	5	1	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	1	no	—
189	C₂²×C₄	—	5	2	no	—
190	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
191	C₄⋊C₄	—	5	2	no	—
192	C₄⋊C₄	—	5	2	no	—
193	C₄⋊C₄	—	5	2	no	—
194	C₄⋊C₄	—	5	1	no	—
195	C₄⋊C₄	—	5	2	no	—
196	C₄⋊C₄	—	5	2	no	—
197	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
198	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
199	C₂²⋊C₄	—	5	2	no	—
200	C₂²⋊C₄	—	5	2	no	—
201	C₂²⋊C₄	—	5	2	no	—
202	C₂²⋊C₄	—	5	2	no	—
203	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
204	C₄²	—	5	2	no	—
205	C₃×C₅	#539	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	10	2	no	—
209	C₂²×C₃	—	10	2	no	—
210	C₂²×C₃	—	10	2	no	—
211	C₂²×C₃	#713	1	2	yes	—
212	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	10	2	no	—
218	C₃×C₄	—	10	2	no	—
219	C₃×C₄	—	10	2	no	—
220	C₂×C₅	—	2	2	no	—
221	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
222	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
223	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
224	D₅	—	4	2	no	—
225	C₂³	—	5	2	no	—
226	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
227	Q₈	—	4	1	no	—
228	D₄	—	10	4	no	—
229	D₄	—	20	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	5	2	no	—
232	C₂×C₄	—	5	2	no	—
233	C₂×C₄	—	5	2	no	—
234	C₂×C₄	—	10	2	no	—
235	C₂×C₄	—	20	2	no	—
236	C₂×C₄	—	20	1	no	—
237	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
238	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
240	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	20	2	no	—
243	C₂×C₃	#4144	1	1	yes	characteristic
244	C₂×C₃	C₂×#713	1	1	yes	characteristic
245	C₂×C₃	#4150	1	1	yes	characteristic
246	C₅	C₃×#539	1	1	yes	characteristic, complemented, Sylow
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	10	2	no	—
250	C₂²	—	10	2	no	—
251	C₂²	—	10	2	no	—
252	C₂²	C₃×#713	1	2	yes	—
253	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	10	2	no	—
259	C₄	—	10	2	no	—
260	C₄	—	10	2	no	—
261	C₃	#65979	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	—	2	2	no	—
263	C₂	—	20	2	no	—
264	C₂	C₃×#4144	1	1	yes	characteristic
265	C₂	C₂×C₃×#713	1	1	yes	characteristic
266	C₂	C₃×#4150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₃×#4076	C₂
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×#539
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
2. C₃×#4092	C₂	semidirect
4. C₂×C₃×#683	C₂	semidirect
5. C₃×#4044	C₂	semidirect
6. C₃×#4040	C₂	semidirect
7. C₃×#4036	C₂	semidirect
8. C₃×#4032	C₂	semidirect
9. C₃×#4027	C₂	semidirect
10. #65979	C₃	direct (N × H)
33. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
34. #4092	C₂×C₃	semidirect
36. C₂×#683	C₂×C₃	semidirect
37. #4044	C₂×C₃	semidirect
38. #4040	C₂×C₃	semidirect
39. #4036	C₂×C₃	semidirect
40. #4032	C₂×C₃	semidirect
41. #4027	C₂×C₃	semidirect
63. C₃×C₂²⋊Q₈	D₅	semidirect
145. C₂²⋊Q₈	C₃×D₅	semidirect
205. C₃×C₅	#539	semidirect
246. C₅	C₃×#539	semidirect
261. C₃	#65979	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₂²⋊Q₈	2
C₃×#4092	2
C₃×#4076	2
C₂×C₃×#683	2
C₃×#4044	2
C₃×#4040	2
C₃×#4036	2
C₃×#4032	2
C₃×#4027	2
#65979	3
C₃×#539	5

Subgroups — C3×#65979 #11755656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65979 #11755656