Subgroups — C₅×#29812 #11952532

Order 1280.

824 conjugacy classes of subgroups.

The subgroup lattice has 242 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, complemented
2	C₂×C₅×#433	C₂	1	1	yes	maximal, characteristic
3	C₅×#2621	C₂	1	2	yes	maximal
4	C₅×#2056	C₂	1	4	yes	maximal, complemented
5	C₅×#2001	C₂	1	4	yes	maximal
6	C₅×#1960	C₂	1	2	yes	maximal, complemented
7	C₅×#1835	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
11	C₂×C₅×#117	C₂²	1	1	yes	characteristic
12	C₅×#494	—	2	4	no	—
13	C₅×#433	—	2	4	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×#103	C₂²	1	8	yes	—
16	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#398	—	2	4	no	—
20	C₅×#397	—	2	4	no	—
21	C₅×#391	C₂²	1	4	yes	—
22	C₅×#382	C₂²	1	2	yes	—
23	C₅×#340	C₂²	1	4	yes	—
24	#29812	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	C₂³	1	4	yes	—
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₅×Q₁₆	—	2	8	no	—
28	C₂×C₅×Q₁₆	—	4	8	no	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₅×C₄⋊Q₈	—	2	4	no	—
32	C₅×C₄⋊Q₈	—	2	4	no	—
33	C₅×C₂²⋊Q₈	—	4	8	no	—
34	C₅×#117	—	2	2	no	—
35	C₅×#117	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	4	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
41	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×C₄⋊C₈	—	2	4	no	—
45	C₅×#103	—	4	8	no	—
46	C₅×C₂²⋊C₈	—	2	4	no	—
47	C₅×#97	—	2	2	no	—
48	C₄×C₅×C₈	D₄	1	4	yes	—
49	C₅×#95	—	2	4	no	—
50	C₅×#95	—	2	8	no	—
51	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#433	C₂×C₅	1	1	yes	characteristic
53	#2621	C₂×C₅	1	2	yes	—
54	#2056	C₂×C₅	1	4	yes	complemented
55	#2001	C₂×C₅	1	4	yes	—
56	#1960	C₂×C₅	1	2	yes	complemented
57	#1835	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×Q₈	—	2	4	no	—
60	C₂×C₅×Q₈	—	2	4	no	—
61	C₂×C₅×Q₈	—	2	8	no	—
62	C₂×C₅×Q₈	—	2	8	no	—
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
72	C₅×Q₁₆	—	4	16	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
77	C₅×C₄⋊C₄	—	2	4	no	—
78	C₅×C₄⋊C₄	—	2	8	no	—
79	C₅×C₄⋊C₄	—	4	8	no	—
80	C₅×C₄⋊C₄	—	4	8	no	—
81	C₅×C₂²⋊C₄	—	2	4	no	—
82	C₅×C₂²⋊C₄	—	4	8	no	—
83	C₄²×C₅	—	2	1	no	—
84	C₄²×C₅	—	2	1	no	—
85	C₄²×C₅	C₂×D₄	1	2	yes	—
86	C₄²×C₅	#41	1	2	yes	—
87	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
88	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
89	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
90	C₂×#117	C₂²×C₅	1	1	yes	characteristic
91	#494	—	2	4	no	—
92	#433	—	2	4	no	—
93	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
94	C₂×#103	C₂²×C₅	1	8	yes	—
95	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
96	C₂×#97	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	#398	—	2	4	no	—
99	#397	—	2	4	no	—
100	#391	C₂²×C₅	1	4	yes	—
101	#382	C₂²×C₅	1	2	yes	—
102	#340	C₂²×C₅	1	4	yes	—
103	C₂³×C₅	—	2	1	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	—	2	4	no	—
106	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
107	C₅×Q₈	—	4	8	no	—
108	C₅×Q₈	—	4	8	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	4	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	—	4	4	no	—
119	C₂×C₄×C₅	—	4	4	no	—
120	C₂×C₄×C₅	—	4	8	no	—
121	C₂×C₄×C₅	#121	1	4	yes	—
122	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
125	C₅×C₈	—	2	4	no	—
126	C₅×C₈	—	4	2	no	—
127	C₂²×Q₈	C₂³×C₅	1	4	yes	—
128	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
129	C₂×Q₁₆	—	2	8	no	—
130	C₂×Q₁₆	—	4	8	no	—
131	C₂²×C₈	C₂³×C₅	1	2	yes	—
132	C₂²×C₈	C₂³×C₅	1	2	yes	—
133	C₄⋊Q₈	—	2	4	no	—
134	C₄⋊Q₈	—	2	4	no	—
135	C₂²⋊Q₈	—	4	8	no	—
136	#117	—	2	2	no	—
137	#117	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	2	4	no	—
140	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
141	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
142	C₂×C₂²⋊C₄	—	2	4	no	—
143	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
144	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
145	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
146	C₄⋊C₈	—	2	4	no	—
147	#103	—	4	8	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	#97	—	2	2	no	—
150	C₄×C₈	C₅×D₄	1	4	yes	—
151	#95	—	2	4	no	—
152	#95	—	2	8	no	—
153	C₂²×C₅	—	4	1	no	—
154	C₂²×C₅	—	4	4	no	—
155	C₂²×C₅	—	4	2	no	—
156	C₂²×C₅	C₂×#121	1	1	yes	characteristic
157	C₂²×C₅	#535	1	2	yes	—
158	C₂²×C₅	#578	1	4	yes	—
159	C₄×C₅	—	2	4	no	—
160	C₄×C₅	—	4	1	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	—	8	4	no	—
163	C₄×C₅	—	8	4	no	—
164	C₄×C₅	#535	1	2	yes	—
165	C₄×C₅	#540	1	2	yes	—
166	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	2	8	no	—
170	C₂×Q₈	—	2	8	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
177	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
179	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
180	Q₁₆	—	4	16	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	4	8	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	4	8	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	C₂×C₅×D₄	1	2	yes	—
194	C₄²	C₅×#41	1	2	yes	—
195	C₂×C₅	—	8	1	no	—
196	C₂×C₅	#2619	1	1	yes	characteristic
197	C₂×C₅	#3059	1	2	yes	—
198	C₂×C₅	#3223	1	4	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	4	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	C₅×#121	1	4	yes	—
218	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
220	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₅	#29812	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	4	2	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	4	no	—
227	C₂²	C₂×C₅×#121	1	1	yes	characteristic
228	C₂²	C₅×#535	1	2	yes	—
229	C₂²	C₅×#578	1	4	yes	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	8	4	no	—
234	C₄	—	8	4	no	—
235	C₄	C₅×#535	1	2	yes	—
236	C₄	C₅×#540	1	2	yes	—
237	C₂	—	8	1	no	—
238	C₂	C₅×#2619	1	1	yes	characteristic
239	C₂	C₅×#3059	1	2	yes	—
240	C₂	C₅×#3223	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#494	C₂	semidirect
4. C₅×#2056	C₂	semidirect
6. C₅×#1960	C₂	semidirect
7. C₅×#1835	C₂	semidirect
24. #29812	C₅	direct (N × H)
51. C₂×#494	C₂×C₅	semidirect
54. #2056	C₂×C₅	semidirect
56. #1960	C₂×C₅	semidirect
57. #1835	C₂×C₅	semidirect
223. C₅	#29812	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×#494	2
C₂×C₅×#433	2
C₅×#2621	2
C₅×#2056	2
C₅×#2001	2
C₅×#1960	2
C₅×#1835	2
#29812	5

Subgroups — C5×#29812 #11952532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#29812 #11952532