Subgroups — C₅×#13244 #11935964

Order 1280.

402 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#362	C₂	1	2	yes	maximal
4	C₅×#2108	C₂	1	1	yes	maximal, characteristic
5	C₅×#1979	C₂	1	2	yes	maximal
6	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₅×Q₃₂	—	2	1	no	—
8	C₂×C₅×Q₃₂	—	2	1	no	—
9	C₂×C₅×Q₃₂	—	2	2	no	—
10	C₂×C₅×Q₃₂	C₄	1	4	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	C₅×#419	—	2	2	no	—
15	C₂×C₅×#101	—	2	2	no	—
16	C₅×#405	C₂²	1	1	yes	characteristic
17	C₅×#368	—	2	2	no	—
18	C₅×#362	—	2	4	no	—
19	C₅×#346	—	2	1	no	—
20	C₄×C₅×C₁₆	—	2	1	no	—
21	#13244	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	—	2	2	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	—	2	2	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	2	no	—
28	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₅×#117	D₄	1	1	yes	characteristic
33	C₅×Q₃₂	—	2	4	no	—
34	C₅×Q₃₂	—	2	4	no	—
35	C₅×M₃₂	—	2	1	no	—
36	C₅×M₃₂	D₄	1	2	yes	—
37	C₂×C₅×C₁₆	—	2	1	no	—
38	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₅×C₁₆	D₄	1	2	yes	—
42	C₅×#108	—	2	2	no	—
43	C₅×#108	—	2	2	no	—
44	C₅×#103	—	4	4	no	—
45	C₅×#101	—	4	4	no	—
46	C₅×#97	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic
49	C₂×#368	C₂×C₅	1	1	yes	characteristic
50	C₂×#362	C₂×C₅	1	2	yes	—
51	#2108	C₂×C₅	1	1	yes	characteristic
52	#1979	C₂×C₅	1	2	yes	—
53	C₂×C₅×Q₈	—	2	4	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	C₂×C₅×Q₈	—	4	2	no	—
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₅×Q₁₆	—	2	4	no	—
62	C₅×Q₁₆	—	2	4	no	—
63	C₅×Q₁₆	—	4	2	no	—
64	C₅×Q₁₆	—	4	2	no	—
65	C₅×M₁₆	—	2	1	no	—
66	C₅×M₁₆	—	4	2	no	—
67	C₅×M₁₆	—	4	2	no	—
68	C₅×M₁₆	D₈	1	2	yes	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	#41	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	1	yes	characteristic
77	C₅×C₄⋊C₄	D₈	1	2	yes	—
78	C₅×C₂²⋊C₄	—	2	1	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₅×C₁₆	—	2	2	no	—
81	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
82	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
83	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	—	2	1	no	—
86	C₂×Q₃₂	—	2	2	no	—
87	C₂×Q₃₂	C₄×C₅	1	4	yes	complemented
88	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
90	C₂×#108	C₂²×C₅	1	2	yes	—
91	#419	—	2	2	no	—
92	C₂×#101	—	2	2	no	—
93	#405	C₂²×C₅	1	1	yes	characteristic
94	#368	—	2	2	no	—
95	#362	—	2	4	no	—
96	#346	—	2	1	no	—
97	C₄×C₁₆	—	2	1	no	—
98	C₂³×C₅	#135	1	1	yes	characteristic
99	C₅×Q₈	—	4	4	no	—
100	C₅×Q₈	—	4	4	no	—
101	C₅×Q₈	—	8	2	no	—
102	C₅×Q₈	—	8	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	8	2	no	—
106	C₂×C₄×C₅	—	8	2	no	—
107	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅	#121	1	1	yes	characteristic
109	C₂×C₄×C₅	#127	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅	#135	1	1	yes	characteristic
113	C₅×C₈	—	2	2	no	—
114	C₅×C₈	—	8	2	no	—
115	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₅×C₈	C₄×D₄	1	1	yes	characteristic
117	C₅×C₈	#121	1	1	yes	characteristic
118	C₅×C₈	#124	1	1	yes	characteristic
119	C₂²×Q₈	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	2	2	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	—	2	2	no	—
125	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
129	#117	C₅×D₄	1	1	yes	characteristic
130	Q₃₂	—	2	4	no	—
131	Q₃₂	—	2	4	no	—
132	M₃₂	—	2	1	no	—
133	M₃₂	C₅×D₄	1	2	yes	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
136	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
137	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
138	C₂×C₁₆	C₅×D₄	1	2	yes	—
139	#108	—	2	2	no	—
140	#108	—	2	2	no	—
141	#103	—	4	4	no	—
142	#101	—	4	4	no	—
143	#97	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
147	C₂²×C₅	#466	1	1	yes	characteristic
148	C₂²×C₅	#495	1	1	yes	characteristic
149	C₄×C₅	—	4	2	no	—
150	C₄×C₅	—	8	4	no	—
151	C₄×C₅	#390	1	1	yes	characteristic
152	C₄×C₅	C₄×D₈	1	1	yes	characteristic
153	C₄×C₅	#466	1	1	yes	characteristic
154	C₄×C₅	#493	1	1	yes	characteristic
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	Q₁₆	—	2	4	no	—
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	4	2	no	—
166	Q₁₆	—	4	2	no	—
167	M₁₆	—	2	1	no	—
168	M₁₆	—	4	2	no	—
169	M₁₆	—	4	2	no	—
170	M₁₆	C₅×D₈	1	2	yes	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
174	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
176	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
177	C₂×C₈	C₅×#41	1	1	yes	characteristic
178	C₂×C₈	C₅×#41	1	1	yes	characteristic
179	C₄⋊C₄	C₅×D₈	1	2	yes	—
180	C₂²⋊C₄	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₁₆	—	2	2	no	—
183	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
184	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
185	C₂×C₅	—	2	1	no	—
186	C₂×C₅	—	2	1	no	—
187	C₂×C₅	#1970	1	1	yes	characteristic
188	C₂×C₅	#2222	1	2	yes	—
189	C₂³	C₅×#135	1	1	yes	characteristic
190	Q₈	—	4	4	no	—
191	Q₈	—	4	4	no	—
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
199	C₂×C₄	C₅×#121	1	1	yes	characteristic
200	C₂×C₄	C₅×#127	1	1	yes	characteristic
201	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
202	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
203	C₂×C₄	C₅×#135	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	8	2	no	—
206	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₈	C₄×C₅×D₄	1	1	yes	characteristic
208	C₈	C₅×#121	1	1	yes	characteristic
209	C₈	C₅×#124	1	1	yes	characteristic
210	C₅	#13244	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
213	C₂²	C₅×#466	1	1	yes	characteristic
214	C₂²	C₅×#495	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	8	4	no	—
217	C₄	C₅×#390	1	1	yes	characteristic
218	C₄	C₄×C₅×D₈	1	1	yes	characteristic
219	C₄	C₅×#466	1	1	yes	characteristic
220	C₄	C₅×#493	1	1	yes	characteristic
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	C₅×#1970	1	1	yes	characteristic
224	C₂	C₅×#2222	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×C₅×Q₃₂	C₄	semidirect
21. #13244	C₅	direct (N × H)
87. C₂×Q₃₂	C₄×C₅	semidirect
210. C₅	#13244	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×C₅×Q₃₂	2
C₂×C₅×#368	2
C₂×C₅×#362	2
C₅×#2108	2
C₅×#1979	2
#13244	5

Subgroups — C5×#13244 #11935964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13244 #11935964