Subgroups — C₅×#9598 #11932318

Order 1280.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1939	C₂	1	2	yes	maximal
2	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
5	C₅×#1279	C₂	1	2	yes	maximal
6	C₂×C₅×#107	C₄	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	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₅×#340	C₄	1	1	yes	characteristic
14	C₅×#340	C₄	1	1	yes	characteristic
15	C₅×#336	—	2	2	no	—
16	C₅×#335	—	2	2	no	—
17	C₅×#322	—	2	2	no	—
18	#9598	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅×C₈	—	2	1	no	—
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	—	2	2	no	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×#107	—	2	4	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₄⋊C₈	—	2	2	no	—
32	C₅×C₄⋊C₈	—	4	4	no	—
33	C₄×C₅×C₈	—	2	1	no	—
34	C₄×C₅×C₈	—	2	1	no	—
35	C₄×C₅×C₈	—	2	1	no	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	C₄×C₅×C₈	Q₈	1	2	yes	—
38	C₄×C₅×C₈	Q₈	1	2	yes	—
39	C₅×#95	—	2	2	no	—
40	#1939	C₂×C₅	1	2	yes	—
41	C₄×#107	C₂×C₅	1	1	yes	characteristic
42	C₂×#335	C₂×C₅	1	1	yes	characteristic
43	C₂×#322	C₂×C₅	1	1	yes	characteristic
44	#1279	C₂×C₅	1	2	yes	—
45	C₂²×C₄×C₅	—	2	2	no	—
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	#41	1	1	yes	characteristic
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	2	1	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	1	no	—
55	C₂×C₅×C₈	—	2	1	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	4	2	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	—	4	2	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
63	C₂×C₅×C₈	D₈	1	2	yes	—
64	C₂×C₅×C₈	Q₁₆	1	2	yes	—
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
69	C₄²×C₅	—	4	2	no	—
70	C₄²×C₅	—	4	2	no	—
71	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₅	#41	1	1	yes	characteristic
75	C₂×#107	C₄×C₅	1	2	yes	—
76	C₂×C₄⋊C₈	—	2	2	no	—
77	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
82	#340	C₄×C₅	1	1	yes	characteristic
83	#340	C₄×C₅	1	1	yes	characteristic
84	#336	—	2	2	no	—
85	#335	—	2	2	no	—
86	#322	—	2	2	no	—
87	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	4	2	no	—
91	C₂×C₄×C₅	—	4	2	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	—	4	2	no	—
94	C₂×C₄×C₅	—	4	2	no	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅	#121	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅	#125	1	1	yes	characteristic
100	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₅	#131	1	2	yes	—
103	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅	#135	1	1	yes	characteristic
106	C₂×C₄×C₅	#135	1	1	yes	characteristic
107	C₅×C₈	—	2	2	no	—
108	C₅×C₈	—	2	2	no	—
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	—	2	2	no	—
111	C₅×C₈	—	4	2	no	—
112	C₅×C₈	—	8	2	no	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
117	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
118	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
121	C₂×C₄²	—	2	2	no	—
122	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
123	#107	—	2	4	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	4	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	C₅×D₄	1	2	yes	—
131	C₄×C₈	C₅×Q₈	1	2	yes	—
132	C₄×C₈	C₅×Q₈	1	2	yes	—
133	#95	—	2	2	no	—
134	C₂²×C₅	#389	1	1	yes	characteristic
135	C₂²×C₅	#435	1	2	yes	—
136	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
137	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
138	C₂²×C₅	#444	1	2	yes	—
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	4	4	no	—
142	C₄×C₅	—	8	2	no	—
143	C₄×C₅	#423	1	1	yes	characteristic
144	C₄×C₅	#424	1	1	yes	characteristic
145	C₄×C₅	#446	1	2	yes	—
146	C₄×C₅	#466	1	1	yes	characteristic
147	C₄×C₅	#467	1	1	yes	characteristic
148	C₄×C₅	#499	1	1	yes	characteristic
149	C₄×C₅	#500	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
154	C₂²×C₄	C₅×#41	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
167	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
168	C₂×C₈	C₅×D₈	1	2	yes	—
169	C₂×C₈	C₅×Q₁₆	1	2	yes	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
174	C₄²	—	4	2	no	—
175	C₄²	—	4	2	no	—
176	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
177	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
178	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
179	C₄²	C₅×#41	1	1	yes	characteristic
180	C₂×C₅	#1581	1	2	yes	—
181	C₂×C₅	#1584	1	2	yes	—
182	C₂×C₅	#1940	1	1	yes	characteristic
183	C₂×C₅	#1942	1	1	yes	characteristic
184	C₂×C₅	#1946	1	1	yes	characteristic
185	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
195	C₂×C₄	C₅×#121	1	1	yes	characteristic
196	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
197	C₂×C₄	C₅×#125	1	1	yes	characteristic
198	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
199	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
200	C₂×C₄	C₅×#131	1	2	yes	—
201	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
202	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
203	C₂×C₄	C₅×#135	1	1	yes	characteristic
204	C₂×C₄	C₅×#135	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	—	8	2	no	—
211	C₅	#9598	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	C₅×#389	1	1	yes	characteristic
213	C₂²	C₅×#435	1	2	yes	—
214	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
215	C₂²	C₄×C₅×Q₁₆	1	1	yes	characteristic
216	C₂²	C₅×#444	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	C₅×#423	1	1	yes	characteristic
222	C₄	C₅×#424	1	1	yes	characteristic
223	C₄	C₅×#446	1	2	yes	—
224	C₄	C₅×#466	1	1	yes	characteristic
225	C₄	C₅×#467	1	1	yes	characteristic
226	C₄	C₅×#499	1	1	yes	characteristic
227	C₄	C₅×#500	1	1	yes	characteristic
228	C₂	C₅×#1581	1	2	yes	—
229	C₂	C₅×#1584	1	2	yes	—
230	C₂	C₅×#1940	1	1	yes	characteristic
231	C₂	C₅×#1942	1	1	yes	characteristic
232	C₂	C₅×#1946	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
18. #9598	C₅	direct (N × H)
211. C₅	#9598	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1939	2
C₄×C₅×#107	2
C₂×C₅×#335	2
C₂×C₅×#322	2
C₅×#1279	2
#9598	5

Subgroups — C5×#9598 #11932318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9598 #11932318