Subgroups — C₅×#78442 #14439846

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5470	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#5106	C₂	1	4	yes	maximal
4	C₅×#4683	C₂	1	4	yes	maximal, complemented
5	C₅×#4682	C₂	1	2	yes	maximal, complemented
6	C₅×#4680	C₂	1	2	yes	maximal, complemented
7	C₅×#3160	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#926	C₂²	1	4	yes	—
11	C₅×#924	—	2	4	no	—
12	C₅×#870	C₂²	1	4	yes	—
13	C₅×#869	C₂²	1	4	yes	—
14	C₄×C₅×Dic₆	—	2	4	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₅×#818	C₂²	1	8	yes	—
19	C₅×#817	C₂²	1	8	yes	—
20	#78442	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#533	—	3	2	no	—
22	C₅×#500	—	3	2	no	—
23	C₅×#499	—	3	4	no	—
24	C₅×#498	—	3	2	no	—
25	C₅×#427	—	3	4	no	—
26	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₅×Dic₆	—	2	4	no	—
30	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
31	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₅×#217	—	2	4	no	—
37	C₅×C₄⋊Dic₃	—	2	4	no	—
38	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
39	C₅×Dic₃⋊C₄	—	2	8	no	—
40	C₄×C₅×Dic₃	—	2	4	no	—
41	#5470	C₂×C₅	1	2	yes	—
42	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
43	#5106	C₂×C₅	1	4	yes	—
44	#4683	C₂×C₅	1	4	yes	complemented
45	#4682	C₂×C₅	1	2	yes	complemented
46	#4680	C₂×C₅	1	2	yes	complemented
47	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
48	C₅×C₄⋊Q₈	—	3	4	no	—
49	C₅×#125	—	3	4	no	—
50	C₅×C₂²⋊Q₈	—	6	4	no	—
51	C₄×C₅×Q₈	—	6	4	no	—
52	C₅×#117	—	3	4	no	—
53	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
54	C₅×#107	—	3	8	no	—
55	C₅×#106	—	3	8	no	—
56	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
58	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
59	#3160	—	3	1	no	Sylow
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
66	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
67	C₂×C₅×Dic₃	—	2	8	no	—
68	C₅×Dic₆	—	4	8	no	—
69	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
70	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
71	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
73	#926	C₂²×C₅	1	4	yes	—
74	#924	—	2	4	no	—
75	#870	C₂²×C₅	1	4	yes	—
76	#869	C₂²×C₅	1	4	yes	—
77	C₄×Dic₆	—	2	4	no	—
78	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
79	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
81	#818	C₂²×C₅	1	8	yes	—
82	#817	C₂²×C₅	1	8	yes	—
83	C₂×C₅×Q₈	—	6	4	no	—
84	C₂²×C₄×C₅	D₁₂	1	2	yes	—
85	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₅×C₈	Dic₆	1	8	yes	complemented
87	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
88	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
89	C₅×C₄⋊C₄	—	3	8	no	—
90	C₅×C₄⋊C₄	—	6	8	no	—
91	C₅×C₄⋊C₄	—	6	4	no	—
92	C₅×C₂²⋊C₄	—	6	4	no	—
93	C₄²×C₅	—	6	4	no	—
94	C₄²×C₅	D₁₂	1	2	yes	—
95	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
97	#533	—	3	2	no	—
98	#500	—	3	2	no	—
99	#499	—	3	4	no	—
100	#498	—	3	2	no	—
101	#427	—	3	4	no	—
102	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
105	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
108	C₃×C₄×C₅	#135	1	4	yes	—
109	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
110	C₅×Dic₃	—	4	8	no	—
111	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
112	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂×Dic₆	—	2	4	no	—
114	C₂×C₃×C₈	C₅×Q₈	1	8	yes	complemented
115	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
116	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
117	C₃×C₄²	C₅×D₄	1	2	yes	—
118	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
119	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
120	#217	—	2	4	no	—
121	C₄⋊Dic₃	—	2	4	no	—
122	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
123	Dic₃⋊C₄	—	2	8	no	—
124	C₄×Dic₃	—	2	4	no	—
125	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
126	C₅×Q₈	—	12	8	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	6	8	no	—
129	C₂×C₄×C₅	C₂×Dic₆	1	4	yes	—
130	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
134	C₅×C₈	C₂×Dic₆	1	4	yes	—
135	C₅×C₈	C₂×Dic₆	1	4	yes	—
136	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
137	C₄⋊Q₈	—	3	4	no	—
138	#125	—	3	4	no	—
139	C₂²⋊Q₈	—	6	4	no	—
140	C₄×Q₈	—	6	4	no	—
141	#117	—	3	4	no	—
142	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
143	#107	—	3	8	no	—
144	#106	—	3	8	no	—
145	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
146	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
147	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
148	C₂×C₃×C₅	—	2	2	no	—
149	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×#135	1	2	yes	—
151	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
157	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
158	C₂×Dic₃	—	2	8	no	—
159	Dic₆	—	4	8	no	—
160	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
161	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	#869	1	2	yes	—
164	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	12	8	no	—
167	C₄×C₅	#869	1	2	yes	—
168	C₄×C₅	#904	1	4	yes	—
169	C₄×C₅	C₂²×Dic₆	1	2	yes	—
170	C₂×Q₈	—	6	4	no	—
171	C₂²×C₄	C₅×D₁₂	1	2	yes	—
172	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
173	C₂×C₈	C₅×Dic₆	1	8	yes	complemented
174	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
175	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
176	C₄⋊C₄	—	3	8	no	—
177	C₄⋊C₄	—	6	8	no	—
178	C₄⋊C₄	—	6	4	no	—
179	C₂²⋊C₄	—	6	4	no	—
180	C₄²	—	6	4	no	—
181	C₄²	C₅×D₁₂	1	2	yes	—
182	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
183	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
184	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
187	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
190	C₃×C₄	C₅×#135	1	4	yes	—
191	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
192	Dic₃	—	4	8	no	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	C₂×#869	1	1	yes	characteristic
195	C₂×C₅	C₂×#904	1	2	yes	—
196	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
197	Q₈	—	12	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	6	8	no	—
200	C₂×C₄	C₂×C₅×Dic₆	1	4	yes	—
201	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
202	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
203	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
204	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
205	C₈	C₂×C₅×Dic₆	1	4	yes	—
206	C₈	C₂×C₅×Dic₆	1	4	yes	—
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
209	C₂×C₃	C₂×C₅×#135	1	2	yes	—
210	C₅	#78442	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	C₅×#869	1	2	yes	—
213	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	12	8	no	—
216	C₄	C₅×#869	1	2	yes	—
217	C₄	C₅×#904	1	4	yes	—
218	C₄	C₂²×C₅×Dic₆	1	2	yes	—
219	C₃	C₅×#3160	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	2	no	—
221	C₂	C₂×C₅×#869	1	1	yes	characteristic
222	C₂	C₂×C₅×#904	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
4. C₅×#4683	C₂	semidirect
5. C₅×#4682	C₂	semidirect
6. C₅×#4680	C₂	semidirect
20. #78442	C₅	direct (N × H)
30. C₂×C₃×C₅×C₈	Q₈	semidirect
44. #4683	C₂×C₅	semidirect
45. #4682	C₂×C₅	semidirect
46. #4680	C₂×C₅	semidirect
86. C₂×C₅×C₈	Dic₆	semidirect
114. C₂×C₃×C₈	C₅×Q₈	semidirect
173. C₂×C₈	C₅×Dic₆	semidirect
184. C₃×C₅	#3160	semidirect
210. C₅	#78442	direct (N × H)
219. C₃	C₅×#3160	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5470	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#5106	2
C₅×#4683	2
C₅×#4682	2
C₅×#4680	2
C₅×#3160	3
#78442	5

Subgroups — C5×#78442 #14439846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78442 #14439846