Subgroups — C₄×C₅×#362 #11935365

Order 1280.

434 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	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#362	C₂	1	2	yes	maximal
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
5	C₅×#1387	C₂	1	2	yes	maximal
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₄×C₅×Q₁₆	—	2	1	no	—
10	C₄×C₅×Q₁₆	—	2	1	no	—
11	C₄×C₅×Q₁₆	C₄	1	2	yes	—
12	C₄×C₅×Q₁₆	C₄	1	2	yes	—
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	C₂×C₅×#107	C₂²	1	1	yes	characteristic
15	C₂×C₅×#103	—	2	1	no	—
16	C₄×C₅×M₁₆	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#362	C₄	1	8	yes	direct product factor
19	C₅×#346	—	2	1	no	—
20	C₄×C₅×C₁₆	—	2	1	no	—
21	C₅×#341	—	2	1	no	—
22	C₄×#362	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	—	2	1	no	—
24	C₂×C₅×Q₁₆	—	2	1	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₄×C₅×Q₈	—	2	2	no	—
32	C₄×C₅×Q₈	—	2	2	no	—
33	C₄×C₅×Q₈	—	4	1	no	—
34	C₄×C₅×Q₈	—	4	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₅	—	4	1	no	—
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₅×M₃₂	—	2	2	no	—
40	C₅×M₃₂	C₂×C₄	1	4	yes	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₅×#108	—	2	2	no	—
43	C₅×#108	C₂×C₄	1	4	yes	—
44	C₅×#107	—	2	1	no	—
45	C₅×#107	C₂×C₄	1	2	yes	—
46	C₅×#103	—	2	2	no	—
47	C₅×#103	—	2	2	no	—
48	C₅×#97	—	4	1	no	—
49	C₄×C₅×C₈	—	4	1	no	—
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
55	C₂×#362	C₂×C₅	1	2	yes	—
56	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
57	C₄×#108	C₂×C₅	1	1	yes	characteristic
58	#1387	C₂×C₅	1	2	yes	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×Q₈	—	2	4	no	—
61	C₂×C₅×Q₈	—	4	1	no	—
62	C₂×C₅×Q₈	—	4	2	no	—
63	C₂²×C₄×C₅	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₅×Q₁₆	—	2	2	no	—
69	C₅×Q₁₆	—	2	4	no	—
70	C₅×Q₁₆	C₄²	1	4	yes	—
71	C₅×M₁₆	—	2	4	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	#41	1	2	yes	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₄²×C₅	—	4	2	no	—
86	C₄²×C₅	—	4	2	no	—
87	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₅	D₈	1	1	yes	characteristic
90	C₄²×C₅	D₈	1	1	yes	characteristic
91	C₅×C₁₆	—	2	4	no	—
92	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄×Q₈	—	2	1	no	—
94	C₂×M₃₂	C₂²×C₅	1	2	yes	—
95	C₄×Q₁₆	—	2	1	no	—
96	C₄×Q₁₆	—	2	1	no	—
97	C₄×Q₁₆	C₄×C₅	1	2	yes	—
98	C₄×Q₁₆	C₄×C₅	1	2	yes	—
99	C₂×#108	C₂²×C₅	1	2	yes	—
100	C₂×#107	C₂²×C₅	1	1	yes	characteristic
101	C₂×#103	—	2	1	no	—
102	C₄×M₁₆	—	2	1	no	—
103	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	#362	C₄×C₅	1	8	yes	direct product factor
105	#346	—	2	1	no	—
106	C₄×C₁₆	—	2	1	no	—
107	#341	—	2	1	no	—
108	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
109	C₅×Q₈	—	2	4	no	—
110	C₅×Q₈	—	4	2	no	—
111	C₅×Q₈	—	4	4	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	4	1	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	4	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	8	2	no	—
118	C₂×C₄×C₅	#102	1	1	yes	characteristic
119	C₂×C₄×C₅	#102	1	1	yes	characteristic
120	C₂×C₄×C₅	#102	1	1	yes	characteristic
121	C₂×C₄×C₅	#102	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅	#135	1	2	yes	—
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	—	4	4	no	—
129	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
130	C₅×C₈	#117	1	1	yes	characteristic
131	C₅×C₈	C₄×D₄	1	2	yes	—
132	C₂²×Q₈	—	2	1	no	—
133	C₂×Q₁₆	—	2	1	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
136	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
140	C₄×Q₈	—	2	2	no	—
141	C₄×Q₈	—	2	2	no	—
142	C₄×Q₈	—	4	1	no	—
143	C₄×Q₈	—	4	1	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄²	—	4	1	no	—
147	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	C₂×C₄×C₅	1	4	yes	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	2	2	no	—
152	#108	C₂×C₄×C₅	1	4	yes	—
153	#107	—	2	1	no	—
154	#107	C₂×C₄×C₅	1	2	yes	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	#97	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₄×C₈	C₅×D₄	1	1	yes	characteristic
162	C₄×C₈	C₅×D₄	1	1	yes	characteristic
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	C₂×#102	1	1	yes	characteristic
166	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	4	4	no	—
171	C₄×C₅	—	8	2	no	—
172	C₄×C₅	#362	1	2	yes	direct product factor
173	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
174	C₄×C₅	#416	1	1	yes	characteristic
175	C₄×C₅	C₄×D₈	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	Q₁₆	—	2	2	no	—
186	Q₁₆	—	2	4	no	—
187	Q₁₆	C₄²×C₅	1	4	yes	—
188	M₁₆	—	2	4	no	—
189	M₁₆	—	4	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
196	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂×C₈	C₅×#41	1	2	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	1	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄²	—	4	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₄²	C₅×D₈	1	1	yes	characteristic
207	C₄²	C₅×D₈	1	1	yes	characteristic
208	C₁₆	—	2	4	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	C₄×#102	1	1	yes	characteristic
211	C₂×C₅	C₂×#362	1	1	yes	characteristic
212	C₂×C₅	#2148	1	1	yes	characteristic
213	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
214	Q₈	—	2	4	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	C₅×#102	1	1	yes	characteristic
224	C₂×C₄	C₅×#102	1	1	yes	characteristic
225	C₂×C₄	C₅×#102	1	1	yes	characteristic
226	C₂×C₄	C₅×#102	1	1	yes	characteristic
227	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
229	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
230	C₂×C₄	C₅×#135	1	2	yes	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	4	4	no	—
234	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
235	C₈	C₅×#117	1	1	yes	characteristic
236	C₈	C₄×C₅×D₄	1	2	yes	—
237	C₅	C₄×#362	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×C₅×#102	1	1	yes	characteristic
241	C₂²	C₄×C₅×SD₁₆	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	4	no	—
246	C₄	—	8	2	no	—
247	C₄	C₅×#362	1	2	yes	direct product factor
248	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
249	C₄	C₅×#416	1	1	yes	characteristic
250	C₄	C₄×C₅×D₈	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×C₅×#102	1	1	yes	characteristic
253	C₂	C₂×C₅×#362	1	1	yes	characteristic
254	C₂	C₅×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
18. C₅×#362	C₄	direct (N × H)
22. C₄×#362	C₅	direct (N × H)
104. #362	C₄×C₅	direct (N × H)
172. C₄×C₅	#362	direct (N × H)
237. C₅	C₄×#362	direct (N × H)
247. C₄	C₅×#362	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Q₁₆	2
C₂×C₅×#362	2
C₄×C₅×M₃₂	2
C₄×C₅×#108	2
C₅×#1387	2
C₄×#362	5

Subgroups — C4×C5×#362 #11935365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#362 #11935365