Subgroups — C₅×#33392 #11956112

Order 1280.

1154 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	C₂×C₅×#576	C₂	1	1	yes	maximal, characteristic
2	C₅×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2148	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#361	C₂	1	6	yes	maximal, complemented
6	C₂×C₅×H(5,𝔽₂)	—	2	1	no	—
7	C₅×#576	—	2	4	no	—
8	C₅×#576	C₄	1	8	yes	—
9	C₂×C₅×AGL(1,ℤ/8)	—	2	3	no	—
10	C₂×C₅×#135	C₂²	1	3	yes	—
11	C₂²×C₅×D₈	C₂²	1	3	yes	—
12	C₅×#568	—	2	1	no	—
13	C₂×C₅×#131	C₂²	1	1	yes	characteristic
14	C₅×#504	—	2	4	no	—
15	C₂×C₅×M₃₂	C₂²	1	6	yes	complemented
16	C₅×#430	—	2	3	no	—
17	C₂×C₅×#108	C₂²	1	6	yes	complemented
18	C₅×#362	C₂²	1	4	yes	complemented
19	C₅×#361	C₂²	1	12	yes	complemented
20	#33392	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₅×H(5,𝔽₂)	—	2	8	no	—
22	C₅×H(5,𝔽₂)	—	4	4	no	—
23	C₂×C₅×#41	—	2	3	no	—
24	C₂×C₅×#41	—	4	1	no	—
25	C₂×C₅×#41	D₄	1	1	yes	characteristic
26	C₂²×C₅×D₄	—	2	3	no	—
27	C₂²×C₅×D₄	—	4	3	no	—
28	C₅×AGL(1,ℤ/8)	—	2	24	no	—
29	C₅×#135	—	2	6	no	—
30	C₅×#135	C₂×C₄	1	12	yes	—
31	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	3	no	—
33	C₂×C₅×D₈	—	2	3	no	—
34	C₂×C₅×D₈	—	2	6	no	—
35	C₂×C₅×D₈	C₂×C₄	1	12	yes	—
36	C₂×C₅×D₈	C₂³	1	3	yes	—
37	C₅×#131	—	4	4	no	—
38	C₅×#131	C₂×C₄	1	4	yes	—
39	C₅×#131	D₄	1	4	yes	—
40	C₂×C₅×M₁₆	—	2	6	no	—
41	C₂×C₅×M₁₆	D₄	1	3	yes	—
42	C₂²×C₅×C₈	C₂³	1	3	yes	—
43	C₅×M₃₂	—	2	6	no	—
44	C₅×M₃₂	C₂³	1	4	yes	—
45	C₂×C₅×C₁₆	—	2	6	no	—
46	C₅×#108	—	2	6	no	—
47	C₅×#108	C₂³	1	4	yes	—
48	C₂×#576	C₂×C₅	1	1	yes	characteristic
49	#3407	C₂×C₅	1	1	yes	characteristic, complemented
50	#2916	C₂×C₅	1	1	yes	characteristic, complemented
51	#2148	C₂×C₅	1	6	yes	complemented
52	C₂×#361	C₂×C₅	1	6	yes	complemented
53	C₂⁴×C₅	—	4	3	no	—
54	C₅×#41	—	2	12	no	—
55	C₅×#41	—	4	8	no	—
56	C₅×#41	—	4	6	no	—
57	C₅×#41	C₂²⋊C₄	1	4	yes	—
58	C₅×#41	D₈	1	4	yes	—
59	C₂×C₅×Q₈	—	2	1	no	—
60	C₂×C₅×Q₈	SD₁₆	1	1	yes	characteristic
61	C₂×C₅×D₄	—	2	12	no	—
62	C₂×C₅×D₄	—	2	3	no	—
63	C₂×C₅×D₄	—	4	3	no	—
64	C₂×C₅×D₄	—	4	24	no	—
65	C₂×C₅×D₄	—	4	6	no	—
66	C₂×C₅×D₄	—	4	3	no	—
67	C₂×C₅×D₄	SD₁₆	1	3	yes	—
68	C₂²×C₄×C₅	—	4	3	no	—
69	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
70	C₅×Q₁₆	—	2	1	no	—
71	C₅×Q₁₆	C₂²×C₄	1	2	yes	—
72	C₅×SD₁₆	—	2	12	no	—
73	C₅×D₈	—	2	12	no	—
74	C₅×D₈	—	2	3	no	—
75	C₅×D₈	C₂²×C₄	1	6	yes	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	4	6	no	—
78	C₅×M₁₆	C₂²⋊C₄	1	12	yes	—
79	C₂×C₅×C₈	—	4	6	no	—
80	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
83	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
84	C₅×C₁₆	—	2	4	no	—
85	C₂×H(5,𝔽₂)	—	2	1	no	—
86	#576	—	2	4	no	—
87	#576	C₄×C₅	1	8	yes	—
88	C₂×AGL(1,ℤ/8)	—	2	3	no	—
89	C₂×#135	C₂²×C₅	1	3	yes	—
90	C₂²×D₈	C₂²×C₅	1	3	yes	—
91	#568	—	2	1	no	—
92	C₂×#131	C₂²×C₅	1	1	yes	characteristic
93	#504	—	2	4	no	—
94	C₂×M₃₂	C₂²×C₅	1	6	yes	complemented
95	#430	—	2	3	no	—
96	C₂×#108	C₂²×C₅	1	6	yes	complemented
97	#362	C₂²×C₅	1	4	yes	complemented
98	#361	C₂²×C₅	1	12	yes	complemented
99	C₂³×C₅	—	4	3	no	—
100	C₂³×C₅	—	4	12	no	—
101	C₂³×C₅	—	16	6	no	—
102	C₂³×C₅	C₂×SD₁₆	1	3	yes	—
103	C₅×Q₈	—	2	2	no	—
104	C₅×Q₈	—	4	1	no	—
105	C₅×Q₈	#102	1	4	yes	—
106	C₅×D₄	—	2	6	no	—
107	C₅×D₄	—	4	12	no	—
108	C₅×D₄	—	4	3	no	—
109	C₅×D₄	—	4	12	no	—
110	C₅×D₄	#102	1	12	yes	—
111	C₂×C₄×C₅	—	4	1	no	—
112	C₂×C₄×C₅	—	4	12	no	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
114	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
115	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
116	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
117	C₅×C₈	—	4	4	no	—
118	C₅×C₈	C₂×C₂²⋊C₄	1	6	yes	—
119	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
120	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
121	H(5,𝔽₂)	—	2	8	no	—
122	H(5,𝔽₂)	—	4	4	no	—
123	C₂×#41	—	2	3	no	—
124	C₂×#41	—	4	1	no	—
125	C₂×#41	C₅×D₄	1	1	yes	characteristic
126	C₂²×D₄	—	2	3	no	—
127	C₂²×D₄	—	4	3	no	—
128	AGL(1,ℤ/8)	—	2	24	no	—
129	#135	—	2	6	no	—
130	#135	C₂×C₄×C₅	1	12	yes	—
131	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
132	C₂×SD₁₆	—	2	3	no	—
133	C₂×D₈	—	2	3	no	—
134	C₂×D₈	—	2	6	no	—
135	C₂×D₈	C₂×C₄×C₅	1	12	yes	—
136	C₂×D₈	C₂³×C₅	1	3	yes	—
137	#131	—	4	4	no	—
138	#131	C₂×C₄×C₅	1	4	yes	—
139	#131	C₅×D₄	1	4	yes	—
140	C₂×M₁₆	—	2	6	no	—
141	C₂×M₁₆	C₅×D₄	1	3	yes	—
142	C₂²×C₈	C₂³×C₅	1	3	yes	—
143	M₃₂	—	2	6	no	—
144	M₃₂	C₂³×C₅	1	4	yes	—
145	C₂×C₁₆	—	2	6	no	—
146	#108	—	2	6	no	—
147	#108	C₂³×C₅	1	4	yes	—
148	C₂²×C₅	—	4	3	no	—
149	C₂²×C₅	—	4	6	no	—
150	C₂²×C₅	—	8	6	no	—
151	C₂²×C₅	—	16	12	no	—
152	C₂²×C₅	C₂×#102	1	6	yes	—
153	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
154	C₄×C₅	—	4	2	no	—
155	C₄×C₅	C₂×#102	1	6	yes	—
156	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
158	C₂⁴	—	4	3	no	—
159	#41	—	2	12	no	—
160	#41	—	4	8	no	—
161	#41	—	4	6	no	—
162	#41	C₅×C₂²⋊C₄	1	4	yes	—
163	#41	C₅×D₈	1	4	yes	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₅×SD₁₆	1	1	yes	characteristic
166	C₂×D₄	—	2	12	no	—
167	C₂×D₄	—	2	3	no	—
168	C₂×D₄	—	4	3	no	—
169	C₂×D₄	—	4	24	no	—
170	C₂×D₄	—	4	6	no	—
171	C₂×D₄	—	4	3	no	—
172	C₂×D₄	C₅×SD₁₆	1	3	yes	—
173	C₂²×C₄	—	4	3	no	—
174	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
175	Q₁₆	—	2	1	no	—
176	Q₁₆	C₂²×C₄×C₅	1	2	yes	—
177	SD₁₆	—	2	12	no	—
178	D₈	—	2	12	no	—
179	D₈	—	2	3	no	—
180	D₈	C₂²×C₄×C₅	1	6	yes	—
181	M₁₆	—	2	4	no	—
182	M₁₆	—	4	6	no	—
183	M₁₆	C₅×C₂²⋊C₄	1	12	yes	—
184	C₂×C₈	—	4	6	no	—
185	C₂×C₈	C₂²×C₄×C₅	1	6	yes	—
186	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
187	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
188	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
189	C₁₆	—	2	4	no	—
190	C₂×C₅	—	2	1	no	—
191	C₂×C₅	—	2	6	no	—
192	C₂×C₅	—	8	6	no	—
193	C₂×C₅	C₂²×#102	1	1	yes	characteristic
194	C₂³	—	4	3	no	—
195	C₂³	—	4	12	no	—
196	C₂³	—	16	6	no	—
197	C₂³	C₂×C₅×SD₁₆	1	3	yes	—
198	Q₈	—	2	2	no	—
199	Q₈	—	4	1	no	—
200	Q₈	C₅×#102	1	4	yes	—
201	D₄	—	2	6	no	—
202	D₄	—	4	12	no	—
203	D₄	—	4	12	no	—
204	D₄	—	4	3	no	—
205	D₄	C₅×#102	1	12	yes	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	12	no	—
208	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
209	C₂×C₄	C₂×C₅×D₈	1	6	yes	—
210	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
211	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
212	C₈	—	4	4	no	—
213	C₈	C₂×C₅×C₂²⋊C₄	1	6	yes	—
214	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
215	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
216	C₅	#33392	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	4	3	no	—
218	C₂²	—	4	6	no	—
219	C₂²	—	8	6	no	—
220	C₂²	—	16	12	no	—
221	C₂²	C₂×C₅×#102	1	6	yes	—
222	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
223	C₄	—	4	2	no	—
224	C₄	C₂×C₅×#102	1	6	yes	—
225	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	6	no	—
229	C₂	—	8	6	no	—
230	C₂	C₂²×C₅×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	C₂²×#102
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#576	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3407	C₂	semidirect
3. C₅×#2916	C₂	semidirect
4. C₅×#2148	C₂	semidirect
5. C₂×C₅×#361	C₂	semidirect
15. C₂×C₅×M₃₂	C₂²	semidirect
17. C₂×C₅×#108	C₂²	semidirect
18. C₅×#362	C₂²	semidirect
19. C₅×#361	C₂²	semidirect
20. #33392	C₅	direct (N × H)
49. #3407	C₂×C₅	semidirect
50. #2916	C₂×C₅	semidirect
51. #2148	C₂×C₅	semidirect
52. C₂×#361	C₂×C₅	semidirect
94. C₂×M₃₂	C₂²×C₅	semidirect
96. C₂×#108	C₂²×C₅	semidirect
97. #362	C₂²×C₅	semidirect
98. #361	C₂²×C₅	semidirect
216. C₅	#33392	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×C₅×#576	2
C₅×#3407	2
C₅×#2916	2
C₅×#2148	2
C₂×C₅×#361	2
#33392	5

Subgroups — C5×#33392 #11956112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33392 #11956112