Subgroups — C₂×C₃×C₅×#396 #14275423

Order 1920.

1816 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃×C₅×#396	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#396	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	12	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	12	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	—	2	12	no	—
12	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
13	C₃×C₅×#95	C₂²	1	4	yes	complemented
14	C₂×C₃×#396	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
16	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	—
17	C₅×#396	C₂×C₃	1	8	yes	direct product factor
18	C₂×C₅×#95	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁴×C₃×C₅	—	2	6	no	—
20	C₂⁴×C₃×C₅	—	2	6	no	—
21	C₂⁴×C₃×C₅	Q₈	1	4	yes	—
22	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
23	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	—	2	6	no	—
25	C₂²×C₃×C₄×C₅	—	2	6	no	—
26	C₂²×C₃×C₄×C₅	—	2	6	no	—
27	C₂²×C₃×C₄×C₅	—	2	6	no	—
28	C₂²×C₃×C₄×C₅	—	2	6	no	—
29	C₂²×C₃×C₄×C₅	D₄	1	12	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	24	no	—
33	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
34	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	3	yes	complemented
35	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	3	yes	—
36	C₃×#396	C₂×C₅	1	8	yes	direct product factor
37	C₂×C₃×#95	C₂×C₅	1	1	yes	characteristic, complemented
38	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
40	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
41	C₂×C₅×C₄⋊C₄	—	2	12	no	—
42	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	12	yes	complemented
43	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
44	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
45	C₅×#95	C₂²×C₃	1	4	yes	complemented
46	C₂×#396	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₅	—	2	24	no	—
48	C₂³×C₃×C₅	—	2	6	no	—
49	C₂³×C₃×C₅	—	2	6	no	—
50	C₂³×C₃×C₅	—	4	12	no	—
51	C₂³×C₃×C₅	—	4	4	no	—
52	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
53	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
54	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
55	C₂³×C₃×C₅	#41	1	6	yes	—
56	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	24	no	—
58	C₂×C₃×C₄×C₅	—	2	6	no	—
59	C₂×C₃×C₄×C₅	—	2	6	no	—
60	C₂×C₃×C₄×C₅	—	2	6	no	—
61	C₂×C₃×C₄×C₅	—	2	6	no	—
62	C₂×C₃×C₄×C₅	—	2	6	no	—
63	C₂×C₃×C₄×C₅	—	4	12	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
65	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
66	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
67	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
68	C₂×C₃×C₄⋊C₄	—	2	12	no	—
69	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	12	yes	complemented
70	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
71	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
72	C₃×#95	C₂²×C₅	1	4	yes	complemented
73	C₂⁴×C₅	—	2	6	no	—
74	C₂⁴×C₅	—	2	6	no	—
75	C₂⁴×C₅	C₃×Q₈	1	4	yes	—
76	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
77	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
78	C₂²×C₄×C₅	—	2	6	no	—
79	C₂²×C₄×C₅	—	2	6	no	—
80	C₂²×C₄×C₅	—	2	6	no	—
81	C₂²×C₄×C₅	—	2	6	no	—
82	C₂²×C₄×C₅	—	2	6	no	—
83	C₂²×C₄×C₅	C₃×D₄	1	12	yes	—
84	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
85	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
86	C₅×C₄⋊C₄	—	2	24	no	—
87	C₅×C₂²⋊C₄	—	2	24	no	—
88	C₂²×C₄⋊C₄	C₂×C₃×C₅	1	3	yes	complemented
89	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	—
90	#396	C₂×C₃×C₅	1	8	yes	direct product factor
91	C₂×#95	C₂×C₃×C₅	1	1	yes	characteristic, complemented
92	C₂²×C₃×C₅	—	2	12	no	—
93	C₂²×C₃×C₅	—	4	6	no	—
94	C₂²×C₃×C₅	—	4	2	no	—
95	C₂²×C₃×C₅	—	4	16	no	—
96	C₂²×C₃×C₅	#121	1	12	yes	—
97	C₂²×C₃×C₅	C₂²⋊Q₈	1	12	yes	—
98	C₂²×C₃×C₅	#126	1	4	yes	—
99	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
100	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
102	C₃×C₄×C₅	—	2	12	no	—
103	C₃×C₄×C₅	—	4	6	no	—
104	C₂⁴×C₃	—	2	6	no	—
105	C₂⁴×C₃	—	2	6	no	—
106	C₂⁴×C₃	C₅×Q₈	1	4	yes	—
107	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
108	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	6	no	—
110	C₂²×C₃×C₄	—	2	6	no	—
111	C₂²×C₃×C₄	—	2	6	no	—
112	C₂²×C₃×C₄	—	2	6	no	—
113	C₂²×C₃×C₄	—	2	6	no	—
114	C₂²×C₃×C₄	C₅×D₄	1	12	yes	—
115	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
116	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
117	C₃×C₄⋊C₄	—	2	24	no	—
118	C₃×C₂²⋊C₄	—	2	24	no	—
119	C₂³×C₅	—	2	24	no	—
120	C₂³×C₅	—	2	6	no	—
121	C₂³×C₅	—	2	6	no	—
122	C₂³×C₅	—	4	12	no	—
123	C₂³×C₅	—	4	4	no	—
124	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
125	C₂³×C₅	C₂×C₃×Q₈	1	4	yes	—
126	C₂³×C₅	C₂×C₃×Q₈	1	2	yes	—
127	C₂³×C₅	C₃×#41	1	6	yes	—
128	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
129	C₂×C₄×C₅	—	2	24	no	—
130	C₂×C₄×C₅	—	2	6	no	—
131	C₂×C₄×C₅	—	2	6	no	—
132	C₂×C₄×C₅	—	2	6	no	—
133	C₂×C₄×C₅	—	2	6	no	—
134	C₂×C₄×C₅	—	2	6	no	—
135	C₂×C₄×C₅	—	4	12	no	—
136	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
137	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
138	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
139	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
140	C₂×C₄⋊C₄	—	2	12	no	—
141	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	12	yes	complemented
142	C₂×C₂²⋊C₄	—	2	12	no	—
143	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
144	#95	C₂²×C₃×C₅	1	4	yes	complemented
145	C₂×C₃×C₅	—	4	4	no	—
146	C₂×C₃×C₅	#396	1	8	yes	direct product factor
147	C₂×C₃×C₅	C₂×#121	1	3	yes	—
148	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	3	yes	—
149	C₂×C₃×C₅	C₂×#126	1	1	yes	characteristic
150	C₂³×C₃	—	2	24	no	—
151	C₂³×C₃	—	2	6	no	—
152	C₂³×C₃	—	2	6	no	—
153	C₂³×C₃	—	4	12	no	—
154	C₂³×C₃	—	4	4	no	—
155	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
156	C₂³×C₃	C₂×C₅×Q₈	1	4	yes	—
157	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
158	C₂³×C₃	C₅×#41	1	6	yes	—
159	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	24	no	—
161	C₂×C₃×C₄	—	2	6	no	—
162	C₂×C₃×C₄	—	2	6	no	—
163	C₂×C₃×C₄	—	2	6	no	—
164	C₂×C₃×C₄	—	2	6	no	—
165	C₂×C₃×C₄	—	2	6	no	—
166	C₂×C₃×C₄	—	4	12	no	—
167	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
168	C₂²×C₅	—	2	12	no	—
169	C₂²×C₅	—	4	16	no	—
170	C₂²×C₅	—	4	6	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	C₃×#121	1	12	yes	—
173	C₂²×C₅	C₃×C₂²⋊Q₈	1	12	yes	—
174	C₂²×C₅	C₃×#126	1	4	yes	—
175	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
176	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
177	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
178	C₄×C₅	—	2	12	no	—
179	C₄×C₅	—	4	6	no	—
180	C₂⁴	—	2	6	no	—
181	C₂⁴	—	2	6	no	—
182	C₂⁴	C₃×C₅×Q₈	1	4	yes	—
183	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
184	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
185	C₂²×C₄	—	2	6	no	—
186	C₂²×C₄	—	2	6	no	—
187	C₂²×C₄	—	2	6	no	—
188	C₂²×C₄	—	2	6	no	—
189	C₂²×C₄	—	2	6	no	—
190	C₂²×C₄	C₃×C₅×D₄	1	12	yes	—
191	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
192	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
193	C₄⋊C₄	—	2	24	no	—
194	C₂²⋊C₄	—	2	24	no	—
195	C₃×C₅	C₂×#396	1	1	yes	characteristic, direct product factor, Hall
196	C₂²×C₃	—	2	12	no	—
197	C₂²×C₃	—	4	16	no	—
198	C₂²×C₃	—	4	6	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	C₅×#121	1	12	yes	—
201	C₂²×C₃	C₅×C₂²⋊Q₈	1	12	yes	—
202	C₂²×C₃	C₅×#126	1	4	yes	—
203	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
204	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
205	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
206	C₃×C₄	—	2	12	no	—
207	C₃×C₄	—	4	6	no	—
208	C₂×C₅	—	4	4	no	—
209	C₂×C₅	C₃×#396	1	8	yes	direct product factor
210	C₂×C₅	C₂×C₃×#121	1	3	yes	—
211	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
212	C₂×C₅	C₂×C₃×#126	1	1	yes	characteristic
213	C₂³	—	2	6	no	—
214	C₂³	—	2	24	no	—
215	C₂³	—	2	6	no	—
216	C₂³	—	4	12	no	—
217	C₂³	—	4	4	no	—
218	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
219	C₂³	C₂×C₃×C₅×Q₈	1	4	yes	—
220	C₂³	C₂×C₃×C₅×Q₈	1	2	yes	—
221	C₂³	C₃×C₅×#41	1	6	yes	—
222	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	6	no	—
226	C₂×C₄	—	2	6	no	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	—	4	12	no	—
230	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
231	C₂×C₃	—	4	4	no	—
232	C₂×C₃	C₅×#396	1	8	yes	direct product factor
233	C₂×C₃	C₂×C₅×#121	1	3	yes	—
234	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	3	yes	—
235	C₂×C₃	C₂×C₅×#126	1	1	yes	characteristic
236	C₅	C₂×C₃×#396	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²	—	2	12	no	—
238	C₂²	—	4	6	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	16	no	—
241	C₂²	C₃×C₅×#121	1	12	yes	—
242	C₂²	C₃×C₅×C₂²⋊Q₈	1	12	yes	—
243	C₂²	C₃×C₅×#126	1	4	yes	—
244	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
245	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
246	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
247	C₄	—	2	12	no	—
248	C₄	—	4	6	no	—
249	C₃	C₂×C₅×#396	1	1	yes	characteristic, direct product factor, Sylow
250	C₂	—	4	4	no	—
251	C₂	C₃×C₅×#396	1	8	yes	direct product factor
252	C₂	C₂×C₃×C₅×#121	1	3	yes	—
253	C₂	C₂×C₃×C₅×C₂²⋊Q₈	1	3	yes	—
254	C₂	C₂×C₃×C₅×#126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃×C₅	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁵×C₃×C₅	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×C₄⋊C₄	C₂	semidirect
3. C₃×C₅×#396	C₂	direct (N × H)
4. C₂×C₃×C₅×#95	C₂	semidirect
5. C₂×C₅×#396	C₃	direct (N × H)
10. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
13. C₃×C₅×#95	C₂²	semidirect
14. C₂×C₃×#396	C₅	direct (N × H)
15. C₂²×C₅×C₄⋊C₄	C₂×C₃	semidirect
17. C₅×#396	C₂×C₃	direct (N × H)
18. C₂×C₅×#95	C₂×C₃	semidirect
34. C₂²×C₃×C₄⋊C₄	C₂×C₅	semidirect
36. C₃×#396	C₂×C₅	direct (N × H)
37. C₂×C₃×#95	C₂×C₅	semidirect
42. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
45. C₅×#95	C₂²×C₃	semidirect
46. C₂×#396	C₃×C₅	direct (N × H)
69. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
72. C₃×#95	C₂²×C₅	semidirect
88. C₂²×C₄⋊C₄	C₂×C₃×C₅	semidirect
90. #396	C₂×C₃×C₅	direct (N × H)
91. C₂×#95	C₂×C₃×C₅	semidirect
141. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
144. #95	C₂²×C₃×C₅	semidirect
146. C₂×C₃×C₅	#396	direct (N × H)
195. C₃×C₅	C₂×#396	direct (N × H)
209. C₂×C₅	C₃×#396	direct (N × H)
232. C₂×C₃	C₅×#396	direct (N × H)
236. C₅	C₂×C₃×#396	direct (N × H)
249. C₃	C₂×C₅×#396	direct (N × H)
251. C₂	C₃×C₅×#396	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₄⋊C₄	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#396	2
C₂×C₃×C₅×#95	2
C₂×C₅×#396	3
C₂×C₃×#396	5

Subgroups — C2×C3×C5×#396 #14275423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#396 #14275423