Subgroups — C₅×#9637 #11932357

Order 1280.

606 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	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×Q₈⋊C₈	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	—	2	2	no	—
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	—	2	2	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₅×#391	C₄	1	2	yes	—
12	C₅×#389	—	2	2	no	—
13	C₅×#384	C₄	1	1	yes	characteristic
14	C₄×C₅×C₄⋊C₄	—	2	2	no	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₅×Q₈⋊C₈	—	2	8	no	—
17	#9637	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
19	C₂²×C₅×C₈	—	2	2	no	—
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₅×C₄⋊Q₈	—	2	2	no	—
22	C₅×C₄⋊Q₈	C₈	1	4	yes	complemented
23	C₄×C₅×Q₈	—	2	4	no	—
24	C₄×C₅×Q₈	—	2	4	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	—	2	2	no	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×C₄⋊C₈	—	4	4	no	—
37	C₄×C₅×C₈	—	2	8	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₄×C₅×C₈	D₄	1	4	yes	—
40	C₅×#95	—	2	4	no	—
41	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
42	#1842	C₂×C₅	1	1	yes	characteristic
43	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
44	C₂×Q₈⋊C₈	C₂×C₅	1	4	yes	—
45	C₂×C₅×Q₈	—	2	2	no	—
46	C₂×C₅×Q₈	—	2	4	no	—
47	C₂×C₅×Q₈	—	2	4	no	—
48	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	—	2	2	no	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅×C₈	—	2	8	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	—	4	4	no	—
60	C₅×C₄⋊C₄	—	2	8	no	—
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	4	4	no	—
63	C₅×C₄⋊C₄	—	4	4	no	—
64	C₄²×C₅	—	2	4	no	—
65	C₄²×C₅	—	2	2	no	—
66	C₄²×C₅	—	4	2	no	—
67	C₄²×C₅	—	4	2	no	—
68	C₄²×C₅	C₂×C₈	1	2	yes	—
69	C₄²×C₅	M₁₆	1	2	yes	—
70	C₄²×C₅	SD₁₆	1	4	yes	—
71	C₄²×C₅	Q₁₆	1	4	yes	—
72	C₄²×C₅	C₂×D₄	1	2	yes	—
73	C₄²×C₅	#41	1	2	yes	—
74	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
75	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
76	C₂×C₄⋊C₈	—	2	2	no	—
77	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₂×C₄×C₈	—	2	2	no	—
79	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
80	#391	C₄×C₅	1	2	yes	—
81	#389	—	2	2	no	—
82	#384	C₄×C₅	1	1	yes	characteristic
83	C₄×C₄⋊C₄	—	2	2	no	—
84	C₄³	C₂²×C₅	1	1	yes	characteristic
85	Q₈⋊C₈	—	2	8	no	—
86	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₅×Q₈	—	2	8	no	—
88	C₅×Q₈	—	4	4	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	4	2	no	—
94	C₂×C₄×C₅	—	4	2	no	—
95	C₂×C₄×C₅	—	4	2	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄×C₅	#103	1	4	yes	—
99	C₂×C₄×C₅	#103	1	4	yes	—
100	C₂×C₄×C₅	#104	1	4	yes	—
101	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₅	#121	1	2	yes	—
103	C₂×C₄×C₅	#124	1	1	yes	characteristic
104	C₂×C₄×C₅	#127	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	#131	1	2	yes	—
108	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
110	C₅×C₈	—	2	8	no	—
111	C₅×C₈	—	8	2	no	—
112	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	C₅×C₈	1	4	yes	complemented
117	C₄×Q₈	—	2	4	no	—
118	C₄×Q₈	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂×C₄²	C₅×D₄	1	2	yes	—
127	C₂×C₄²	C₅×D₄	1	2	yes	—
128	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
129	C₄⋊C₈	—	2	4	no	—
130	C₄⋊C₈	—	4	4	no	—
131	C₄×C₈	—	2	8	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	C₅×D₄	1	4	yes	—
134	#95	—	2	4	no	—
135	C₂²×C₅	#390	1	1	yes	characteristic
136	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₅	#408	1	1	yes	characteristic
138	C₂²×C₅	C₂×#103	1	2	yes	—
139	C₂²×C₅	C₂×#104	1	2	yes	—
140	C₄×C₅	—	2	2	no	—
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	—	4	4	no	—
143	C₄×C₅	—	8	2	no	—
144	C₄×C₅	Q₈⋊C₈	1	8	yes	complemented
145	C₄×C₅	C₈×D₄	1	2	yes	—
146	C₄×C₅	#436	1	2	yes	—
147	C₄×C₅	#487	1	2	yes	—
148	C₄×C₅	#492	1	1	yes	characteristic
149	C₄×C₅	#494	1	1	yes	characteristic
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	C₂×C₅×C₈	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
157	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	4	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂×C₅×C₈	1	2	yes	—
174	C₄²	C₅×M₁₆	1	2	yes	—
175	C₄²	C₅×SD₁₆	1	4	yes	—
176	C₄²	C₅×Q₁₆	1	4	yes	—
177	C₄²	C₂×C₅×D₄	1	2	yes	—
178	C₄²	C₅×#41	1	2	yes	—
179	C₂×C₅	C₂×Q₈⋊C₈	1	2	yes	—
180	C₂×C₅	#1493	1	2	yes	—
181	C₂×C₅	#1954	1	1	yes	characteristic
182	C₂×C₅	#1956	1	1	yes	characteristic
183	C₂×C₅	#1962	1	1	yes	characteristic
184	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
185	Q₈	—	2	8	no	—
186	Q₈	—	4	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
196	C₂×C₄	C₅×#103	1	4	yes	—
197	C₂×C₄	C₅×#103	1	4	yes	—
198	C₂×C₄	C₅×#104	1	4	yes	—
199	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
200	C₂×C₄	C₅×#121	1	2	yes	—
201	C₂×C₄	C₅×#124	1	1	yes	characteristic
202	C₂×C₄	C₅×#127	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
205	C₂×C₄	C₅×#131	1	2	yes	—
206	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
207	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
208	C₈	—	2	8	no	—
209	C₈	—	8	2	no	—
210	C₅	#9637	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	C₅×#390	1	1	yes	characteristic
212	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
213	C₂²	C₅×#408	1	1	yes	characteristic
214	C₂²	C₂×C₅×#103	1	2	yes	—
215	C₂²	C₂×C₅×#104	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	2	no	—
220	C₄	C₅×Q₈⋊C₈	1	8	yes	complemented
221	C₄	C₅×C₈×D₄	1	2	yes	—
222	C₄	C₅×#436	1	2	yes	—
223	C₄	C₅×#487	1	2	yes	—
224	C₄	C₅×#492	1	1	yes	characteristic
225	C₄	C₅×#494	1	1	yes	characteristic
226	C₂	C₂×C₅×Q₈⋊C₈	1	2	yes	—
227	C₂	C₅×#1493	1	2	yes	—
228	C₂	C₅×#1954	1	1	yes	characteristic
229	C₂	C₅×#1956	1	1	yes	characteristic
230	C₂	C₅×#1962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
17. #9637	C₅	direct (N × H)
22. C₅×C₄⋊Q₈	C₈	semidirect
116. C₄⋊Q₈	C₅×C₈	semidirect
144. C₄×C₅	Q₈⋊C₈	semidirect
210. C₅	#9637	direct (N × H)
220. C₄	C₅×Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1842	2
C₄²×C₅×C₈	2
C₂×C₅×Q₈⋊C₈	2
#9637	5

Subgroups — C5×#9637 #11932357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9637 #11932357