Subgroups — C₅×#13212 #11935932

Order 1280.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#362	C₂	1	4	yes	maximal
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₅×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×Q₁₆	—	2	1	no	—
6	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₅×#494	—	2	1	no	—
10	C₅×#494	—	2	1	no	—
11	C₅×#494	C₄	1	2	yes	—
12	C₅×#494	C₄	1	2	yes	—
13	C₂×C₅×#108	—	2	1	no	—
14	C₂×C₅×#108	C₂²	1	2	yes	—
15	C₅×#423	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#362	—	2	8	no	—
18	C₅×#346	—	2	1	no	—
19	C₄×C₅×C₁₆	—	2	1	no	—
20	#13212	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	2	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	—	2	4	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	—	2	4	no	—
27	C₂×C₅×Q₁₆	—	4	2	no	—
28	C₂×C₅×Q₁₆	—	4	2	no	—
29	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×C₄⋊Q₈	—	2	2	no	—
34	C₅×C₄⋊Q₈	—	2	2	no	—
35	C₅×C₄⋊Q₈	—	4	2	no	—
36	C₂×C₅×C₄⋊C₄	—	4	2	no	—
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₅×M₃₂	—	2	2	no	—
39	C₅×M₃₂	D₄	1	4	yes	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₅×#108	—	2	4	no	—
42	C₅×#108	—	4	2	no	—
43	C₅×C₄⋊C₈	—	4	2	no	—
44	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂×#494	C₂×C₅	1	1	yes	characteristic
49	C₂×#362	C₂×C₅	1	4	yes	—
50	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
51	#1848	C₂×C₅	1	1	yes	characteristic
52	C₂×C₅×Q₈	—	2	4	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂×C₅×Q₈	—	4	2	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₅×Q₁₆	—	2	8	no	—
62	C₅×Q₁₆	—	4	4	no	—
63	C₅×Q₁₆	—	4	4	no	—
64	C₅×Q₁₆	—	4	4	no	—
65	C₅×M₁₆	—	4	2	no	—
66	C₅×M₁₆	—	4	2	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	D₈	1	2	yes	—
71	C₂×C₅×C₈	SD₁₆	1	2	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	#41	1	2	yes	—
76	C₅×C₄⋊C₄	—	4	4	no	—
77	C₅×C₄⋊C₄	—	4	4	no	—
78	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅	D₈	1	2	yes	—
81	C₅×C₁₆	—	2	4	no	—
82	C₂²×Q₁₆	—	2	1	no	—
83	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
84	C₂×C₄⋊Q₈	—	2	1	no	—
85	C₂×M₃₂	C₂²×C₅	1	2	yes	—
86	#494	—	2	1	no	—
87	#494	—	2	1	no	—
88	#494	C₄×C₅	1	2	yes	—
89	#494	C₄×C₅	1	2	yes	—
90	C₂×#108	—	2	1	no	—
91	C₂×#108	C₂²×C₅	1	2	yes	—
92	#423	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	#362	—	2	8	no	—
95	#346	—	2	1	no	—
96	C₄×C₁₆	—	2	1	no	—
97	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
98	C₅×Q₈	—	4	4	no	—
99	C₅×Q₈	—	4	4	no	—
100	C₅×Q₈	—	8	2	no	—
101	C₅×Q₈	—	8	2	no	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	4	4	no	—
104	C₂×C₄×C₅	—	8	2	no	—
105	C₂×C₄×C₅	—	8	2	no	—
106	C₂×C₄×C₅	#102	1	2	yes	—
107	C₂×C₄×C₅	#102	1	2	yes	—
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	#124	1	1	yes	characteristic
110	C₂×C₄×C₅	#127	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
114	C₅×C₈	—	8	2	no	—
115	C₅×C₈	#102	1	2	yes	—
116	C₅×C₈	#102	1	2	yes	—
117	C₅×C₈	C₄×D₄	1	2	yes	—
118	C₅×C₈	#121	1	2	yes	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	—	2	4	no	—
125	C₂×Q₁₆	—	4	2	no	—
126	C₂×Q₁₆	—	4	2	no	—
127	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	2	no	—
134	C₂×C₄⋊C₄	—	4	2	no	—
135	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
136	M₃₂	—	2	2	no	—
137	M₃₂	C₅×D₄	1	4	yes	—
138	C₂×C₁₆	—	2	2	no	—
139	#108	—	2	4	no	—
140	#108	—	4	2	no	—
141	C₄⋊C₈	—	4	2	no	—
142	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
143	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
144	C₄×C₈	C₅×D₄	1	1	yes	characteristic
145	C₄×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	C₂×#102	1	1	yes	characteristic
148	C₂²×C₅	#486	1	1	yes	characteristic
149	C₂²×C₅	#492	1	1	yes	characteristic
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	8	4	no	—
152	C₄×C₅	#362	1	4	yes	complemented
153	C₄×C₅	#390	1	1	yes	characteristic
154	C₄×C₅	C₂×#102	1	1	yes	characteristic
155	C₄×C₅	#486	1	1	yes	characteristic
156	C₄×C₅	#493	1	1	yes	characteristic
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
165	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	Q₁₆	—	2	8	no	—
167	Q₁₆	—	4	4	no	—
168	Q₁₆	—	4	4	no	—
169	Q₁₆	—	4	4	no	—
170	M₁₆	—	4	2	no	—
171	M₁₆	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₅×D₈	1	2	yes	—
176	C₂×C₈	C₅×SD₁₆	1	2	yes	—
177	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
178	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
179	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₈	C₅×#41	1	2	yes	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₄²	C₅×D₈	1	2	yes	—
186	C₁₆	—	2	4	no	—
187	C₂×C₅	—	2	1	no	—
188	C₂×C₅	—	2	1	no	—
189	C₂×C₅	#1957	1	1	yes	characteristic
190	C₂×C₅	C₂×#362	1	2	yes	—
191	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
192	Q₈	—	4	4	no	—
193	Q₈	—	4	4	no	—
194	Q₈	—	8	2	no	—
195	Q₈	—	8	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	C₅×#102	1	2	yes	—
201	C₂×C₄	C₅×#102	1	2	yes	—
202	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₅×#124	1	1	yes	characteristic
204	C₂×C₄	C₅×#127	1	1	yes	characteristic
205	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
206	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
207	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
208	C₈	—	8	2	no	—
209	C₈	C₅×#102	1	2	yes	—
210	C₈	C₅×#102	1	2	yes	—
211	C₈	C₄×C₅×D₄	1	2	yes	—
212	C₈	C₅×#121	1	2	yes	—
213	C₅	#13212	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	C₂×C₅×#102	1	1	yes	characteristic
216	C₂²	C₅×#486	1	1	yes	characteristic
217	C₂²	C₅×#492	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	4	no	—
220	C₄	C₅×#362	1	4	yes	complemented
221	C₄	C₅×#390	1	1	yes	characteristic
222	C₄	C₂×C₅×#102	1	1	yes	characteristic
223	C₄	C₅×#486	1	1	yes	characteristic
224	C₄	C₅×#493	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	C₅×#1957	1	1	yes	characteristic
228	C₂	C₂×C₅×#362	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
20. #13212	C₅	direct (N × H)
152. C₄×C₅	#362	semidirect
213. C₅	#13212	direct (N × H)
220. C₄	C₅×#362	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#494	2
C₂×C₅×#362	2
C₄×C₅×M₃₂	2
C₅×#1848	2
#13212	5

Subgroups — C5×#13212 #11935932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13212 #11935932