Subgroups — C₃×#65957 #11755634

Order 960.

412 conjugacy classes of subgroups.

The subgroup lattice has 252 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
2	C₃×#4090	C₂	1	2	yes	maximal, complemented
3	C₃×#4070	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#683	C₂	1	1	yes	maximal, characteristic
5	C₃×#4035	C₂	1	2	yes	maximal, complemented
6	C₃×#4035	C₂	1	2	yes	maximal, complemented
7	C₃×#4034	C₂	1	2	yes	maximal, complemented
8	C₃×#4024	C₂	1	2	yes	maximal
9	C₃×#4023	C₂	1	2	yes	maximal
10	#65957	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	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
16	C₂×C₃×Dic₁₀	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×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
22	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₃×#683	C₂²	1	2	yes	—
24	C₃×#683	C₂²	1	2	yes	—
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×#683	C₂²	1	2	yes	—
27	C₃×#683	C₂²	1	2	yes	—
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×C₄×Dic₅	C₂²	1	2	yes	—
31	C₃×C₄×Dic₅	C₂²	1	2	yes	—
32	C₃×#544	—	5	1	no	maximal, Hall
33	C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
34	#4090	C₂×C₃	1	2	yes	complemented
35	#4070	C₂×C₃	1	1	yes	characteristic
36	C₂×#683	C₂×C₃	1	1	yes	characteristic
37	#4035	C₂×C₃	1	2	yes	complemented
38	#4035	C₂×C₃	1	2	yes	complemented
39	#4034	C₂×C₃	1	2	yes	complemented
40	#4024	C₂×C₃	1	2	yes	—
41	#4023	C₂×C₃	1	2	yes	—
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₅×Q₈	—	4	1	no	—
44	C₂×C₃×C₄×C₅	—	4	1	no	—
45	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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×Dic₅	—	4	1	no	—
50	C₂×C₃×Dic₅	Q₈	1	4	yes	complemented
51	C₂×C₃×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×Dic₅	C₂³	1	2	yes	—
53	C₂×C₃×Dic₅	C₂³	1	2	yes	—
54	C₂×C₃×Dic₅	C₂³	1	2	yes	—
55	C₃×Dic₁₀	—	4	1	no	—
56	C₃×Dic₁₀	—	4	2	no	—
57	C₃×C₄⋊Q₈	—	5	2	no	—
58	C₃×C₄⋊Q₈	—	5	2	no	—
59	C₃×#125	—	5	2	no	—
60	C₃×#125	—	5	2	no	—
61	C₃×C₂²⋊Q₈	—	5	1	no	—
62	C₃×C₂²⋊Q₈	—	5	2	no	—
63	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic
64	C₃×#117	—	5	2	no	—
65	C₂×C₃×C₄⋊C₄	—	5	1	no	—
66	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
68	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
70	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
71	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
72	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
76	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
77	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
78	#683	C₂²×C₃	1	2	yes	—
79	#683	C₂²×C₃	1	2	yes	—
80	#683	C₂²×C₃	1	2	yes	—
81	#683	C₂²×C₃	1	2	yes	—
82	#683	C₂²×C₃	1	2	yes	—
83	#683	C₂²×C₃	1	2	yes	—
84	#683	C₂²×C₃	1	2	yes	—
85	C₄×Dic₅	C₂²×C₃	1	2	yes	—
86	C₄×Dic₅	C₂²×C₃	1	2	yes	—
87	#544	—	5	1	no	Sylow
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
91	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×Dic₅	—	2	2	no	—
97	C₃×Dic₅	—	2	2	no	—
98	C₃×Dic₅	—	2	2	no	—
99	C₃×Dic₅	C₂×Q₈	1	4	yes	—
100	C₂×C₃×Q₈	—	5	1	no	—
101	C₂×C₃×Q₈	—	5	2	no	—
102	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
103	C₂²×C₃×C₄	—	5	1	no	—
104	C₂²×C₃×C₄	—	5	1	no	—
105	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	5	2	no	—
107	C₃×C₄⋊C₄	—	5	2	no	—
108	C₃×C₄⋊C₄	—	5	2	no	—
109	C₃×C₄⋊C₄	—	5	2	no	—
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	C₂×D₅	1	1	yes	characteristic
116	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
117	C₃×C₂²⋊C₄	—	5	2	no	—
118	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
119	C₃×C₄²	—	5	2	no	—
120	C₃×C₄²	—	5	2	no	—
121	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
122	C₅×Q₈	—	4	1	no	—
123	C₂×C₄×C₅	—	4	1	no	—
124	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
126	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₅	—	4	1	no	—
129	C₂×Dic₅	C₃×Q₈	1	4	yes	complemented
130	C₂×Dic₅	C₂³×C₃	1	2	yes	—
131	C₂×Dic₅	C₂³×C₃	1	2	yes	—
132	C₂×Dic₅	C₂³×C₃	1	2	yes	—
133	C₂×Dic₅	C₂³×C₃	1	2	yes	—
134	Dic₁₀	—	4	1	no	—
135	Dic₁₀	—	4	2	no	—
136	C₄⋊Q₈	—	5	2	no	—
137	C₄⋊Q₈	—	5	2	no	—
138	#125	—	5	2	no	—
139	#125	—	5	2	no	—
140	C₂²⋊Q₈	—	5	1	no	—
141	C₂²⋊Q₈	—	5	2	no	—
142	C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic
143	#117	—	5	2	no	—
144	C₂×C₄⋊C₄	—	5	1	no	—
145	C₂×C₃×C₅	—	2	2	no	—
146	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
149	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
150	C₃×Q₈	—	4	1	no	—
151	C₃×Q₈	—	20	1	no	—
152	C₃×Q₈	—	20	2	no	—
153	C₂×C₃×C₄	—	4	1	no	—
154	C₂×C₃×C₄	—	5	4	no	—
155	C₂×C₃×C₄	—	5	2	no	—
156	C₂×C₃×C₄	—	5	2	no	—
157	C₂×C₃×C₄	—	5	2	no	—
158	C₂×C₃×C₄	—	5	2	no	—
159	C₂×C₃×C₄	—	20	1	no	—
160	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
161	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
162	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	C₂×C₃×Q₈	1	2	yes	—
167	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	2	no	—
172	Dic₅	—	2	2	no	—
173	Dic₅	—	2	2	no	—
174	Dic₅	—	2	2	no	—
175	Dic₅	C₂×C₃×Q₈	1	4	yes	—
176	C₂×Q₈	—	5	1	no	—
177	C₂×Q₈	—	5	2	no	—
178	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
179	C₂²×C₄	—	5	1	no	—
180	C₂²×C₄	—	5	1	no	—
181	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
182	C₄⋊C₄	—	5	2	no	—
183	C₄⋊C₄	—	5	2	no	—
184	C₄⋊C₄	—	5	2	no	—
185	C₄⋊C₄	—	5	2	no	—
186	C₄⋊C₄	—	5	2	no	—
187	C₄⋊C₄	—	5	1	no	—
188	C₄⋊C₄	—	5	2	no	—
189	C₄⋊C₄	—	5	2	no	—
190	C₄⋊C₄	—	5	2	no	—
191	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
192	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
193	C₂²⋊C₄	—	5	2	no	—
194	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
195	C₄²	—	5	2	no	—
196	C₄²	—	5	2	no	—
197	C₃×C₅	#544	1	1	yes	characteristic, complemented, Hall
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	Q₈×D₅	1	2	yes	—
201	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	5	4	no	—
207	C₃×C₄	—	10	2	no	—
208	C₃×C₄	—	10	2	no	—
209	C₃×C₄	—	10	2	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
212	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
215	Q₈	—	4	1	no	—
216	Q₈	—	20	1	no	—
217	Q₈	—	20	2	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	5	4	no	—
220	C₂×C₄	—	5	2	no	—
221	C₂×C₄	—	5	2	no	—
222	C₂×C₄	—	5	2	no	—
223	C₂×C₄	—	5	2	no	—
224	C₂×C₄	—	20	1	no	—
225	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
226	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
227	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	#4144	1	1	yes	characteristic
231	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
232	C₂×C₃	#4150	1	1	yes	characteristic
233	C₅	C₃×#544	1	1	yes	characteristic, complemented, Sylow
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	C₃×Q₈×D₅	1	2	yes	—
237	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	5	4	no	—
243	C₄	—	10	2	no	—
244	C₄	—	10	2	no	—
245	C₄	—	10	2	no	—
246	C₃	#65957	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	2	2	no	—
248	C₂	C₃×#4144	1	1	yes	characteristic
249	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
250	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	C₂³×C₃×C₅	C₂³
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×#544
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
2. C₃×#4090	C₂	semidirect
5. C₃×#4035	C₂	semidirect
6. C₃×#4035	C₂	semidirect
7. C₃×#4034	C₂	semidirect
10. #65957	C₃	direct (N × H)
34. #4090	C₂×C₃	semidirect
37. #4035	C₂×C₃	semidirect
38. #4035	C₂×C₃	semidirect
39. #4034	C₂×C₃	semidirect
50. C₂×C₃×Dic₅	Q₈	semidirect
129. C₂×Dic₅	C₃×Q₈	semidirect
197. C₃×C₅	#544	semidirect
233. C₅	C₃×#544	semidirect
246. C₃	#65957	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×#4090	2
C₃×#4070	2
C₂×C₃×#683	2
C₃×#4035	2
C₃×#4034	2
C₃×#4024	2
C₃×#4023	2
#65957	3
C₃×#544	5

Subgroups — C3×#65957 #11755634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65957 #11755634