Subgroups — C₄×#90952 #14294221

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#90952	C₂	1	4	yes	maximal
5	C₄×#3947	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊#168	C₂²	1	2	yes	—
8	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Dic₅	C₄	1	4	yes	—
10	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
11	C₄×C₃⋊#168	C₄	1	4	yes	complemented
12	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
13	#90952	C₄	1	16	yes	direct product factor
14	C₄×#803	—	5	1	no	maximal, Hall
15	C₂×C₄×C₅×C₈	—	3	1	no	—
16	C₂×#3947	—	3	4	no	—
17	C₄²×Dic₅	D₃	1	1	yes	characteristic
18	C₂×C₄×#168	—	3	1	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	C₂×C₄	1	4	yes	—
22	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₂×C₃⋊#168	C₂×C₄	1	8	yes	—
25	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
26	C₂×C₅×C₃⋊C₈	C₂×C₄	1	8	yes	—
27	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₃×C₄×Dic₅	—	2	8	no	—
29	C₃×C₄×Dic₅	C₂×C₄	1	8	yes	—
30	C₃×C₄³	—	5	1	no	—
31	C₂×#803	—	5	4	no	—
32	C₂×C₄×C₃⋊C₈	—	5	1	no	—
33	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic
34	C₂²×C₅×C₈	—	3	2	no	—
35	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
36	C₂×C₄×Dic₅	Dic₃	1	4	yes	—
37	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
38	C₂²×#168	—	3	2	no	—
39	C₄×C₅×C₈	—	3	4	no	—
40	#3947	—	3	16	no	—
41	C₄×#168	—	3	4	no	—
42	C₄×#97	—	15	1	no	Sylow
43	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
45	C₂×C₃×C₄×C₅	M₁₆	1	8	yes	—
46	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
47	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×Dic₅	—	2	8	no	—
50	C₂×C₃×Dic₅	C₄²	1	8	yes	—
51	C₃⋊#168	C₄²	1	8	yes	complemented
52	C₅×C₃⋊C₈	C₄²	1	8	yes	complemented
53	C₂×C₃×C₄²	—	5	2	no	—
54	C₂×C₃×C₄²	—	5	4	no	—
55	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
56	C₂²×C₃⋊C₈	—	5	2	no	—
57	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
58	#803	—	5	16	no	—
59	C₄×C₃⋊C₈	—	5	4	no	—
60	C₄×C₃⋊C₈	Dic₅	1	4	yes	complemented
61	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
63	C₂²×Dic₅	C₂×Dic₃	1	4	yes	—
64	C₂×C₅×C₈	—	3	8	no	—
65	C₂×C₅×C₈	—	3	4	no	—
66	C₄²×C₅	C₄×D₃	1	2	yes	—
67	C₄²×C₅	C₄×D₃	1	2	yes	—
68	C₄×Dic₅	—	2	8	no	—
69	C₄×Dic₅	C₄×D₃	1	8	yes	—
70	C₂×#168	—	3	4	no	—
71	C₂×#168	—	3	8	no	—
72	C₂×#97	—	15	4	no	—
73	C₂×C₄×C₈	—	3	1	no	—
74	C₂×C₄×C₈	—	15	1	no	—
75	C₄³	—	5	1	no	—
76	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
77	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
78	C₂²×C₃×C₅	C₂×M₁₆	1	4	yes	—
79	C₃×C₄×C₅	#97	1	8	yes	complemented
80	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
81	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
82	C₃×Dic₅	—	2	8	no	—
83	C₂²×C₃×C₄	—	5	4	no	—
84	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
85	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
86	C₃×C₄²	—	5	8	no	—
87	C₃×C₄²	—	10	8	no	—
88	C₃×C₄²	C₄×D₅	1	2	yes	—
89	C₃×C₄²	C₂×Dic₅	1	2	yes	—
90	C₂×C₃⋊C₈	—	5	4	no	—
91	C₂×C₃⋊C₈	—	5	8	no	—
92	C₂×C₃⋊C₈	C₄×D₅	1	8	yes	—
93	C₂×C₃⋊C₈	C₂×Dic₅	1	4	yes	—
94	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
95	C₂×C₄×C₅	#208	1	8	yes	—
96	C₂×C₄×C₅	C₄×Dic₃	1	4	yes	—
97	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
98	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
100	C₂×Dic₅	—	2	8	no	—
101	C₂×Dic₅	C₄×Dic₃	1	8	yes	—
102	C₅×C₈	—	3	8	no	—
103	#168	—	3	8	no	—
104	C₂²×C₈	—	3	2	no	—
105	C₂²×C₈	—	15	2	no	—
106	C₂×C₄²	—	5	2	no	—
107	C₂×C₄²	—	5	4	no	—
108	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
109	#97	—	15	16	no	—
110	C₄×C₈	—	3	4	no	—
111	C₄×C₈	—	15	4	no	—
112	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
113	C₂×C₃×C₅	C₂×#97	1	2	yes	—
114	C₂×C₃×C₅	C₄×M₁₆	1	4	yes	—
115	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
116	C₂×C₃×C₄	—	5	8	no	—
117	C₂×C₃×C₄	—	10	8	no	—
118	C₂×C₃×C₄	#676	1	8	yes	—
119	C₂×C₃×C₄	C₄×Dic₅	1	4	yes	—
120	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
121	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
123	C₃⋊C₈	—	5	8	no	—
124	C₃⋊C₈	C₄×Dic₅	1	8	yes	complemented
125	C₂²×C₅	C₂×#208	1	4	yes	—
126	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
127	C₂²×C₅	C₂×C₄×Dic₃	1	2	yes	—
128	C₄×C₅	#803	1	8	yes	complemented
129	C₄×C₅	C₄²×D₃	1	2	yes	—
130	C₄×C₅	C₄²×D₃	1	2	yes	—
131	Dic₅	—	2	8	no	—
132	C₂²×C₄	—	5	4	no	—
133	C₂²×C₄	D₅×Dic₃	1	2	yes	—
134	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
135	C₂×C₈	—	3	8	no	—
136	C₂×C₈	—	3	4	no	—
137	C₂×C₈	—	15	8	no	—
138	C₂×C₈	—	15	4	no	—
139	C₄²	—	5	8	no	—
140	C₄²	—	10	8	no	—
141	C₄²	D₃×Dic₅	1	2	yes	—
142	C₄²	#1200	1	2	yes	—
143	C₃×C₅	C₄×#97	1	1	yes	characteristic, complemented, Hall
144	C₂²×C₃	C₄²×D₅	1	2	yes	—
145	C₂²×C₃	C₂×#676	1	4	yes	—
146	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
147	C₃×C₄	—	10	8	no	—
148	C₃×C₄	#3947	1	8	yes	complemented
149	C₃×C₄	C₄²×D₅	1	2	yes	—
150	C₃×C₄	C₂×C₄×Dic₅	1	2	yes	—
151	C₂×C₅	C₂×#803	1	2	yes	—
152	C₂×C₅	C₄×#208	1	4	yes	—
153	C₂×C₅	C₄²×Dic₃	1	1	yes	characteristic
154	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
155	C₂×C₄	—	5	8	no	—
156	C₂×C₄	—	10	8	no	—
157	C₂×C₄	#6649	1	8	yes	—
158	C₂×C₄	Dic₃×Dic₅	1	4	yes	—
159	C₂×C₄	C₄×D₃×D₅	1	4	yes	—
160	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
161	C₂×C₄	C₂×#1200	1	1	yes	characteristic
162	C₈	—	3	8	no	—
163	C₈	—	15	8	no	—
164	C₂×C₃	C₄×#676	1	4	yes	—
165	C₂×C₃	C₄²×Dic₅	1	1	yes	characteristic
166	C₂×C₃	C₂×#3947	1	2	yes	—
167	C₅	C₄×#803	1	1	yes	characteristic, complemented, Sylow
168	C₂²	C₂×#6649	1	4	yes	—
169	C₂²	C₄×D₅×Dic₃	1	2	yes	—
170	C₂²	C₂×Dic₃×Dic₅	1	1	yes	characteristic
171	C₄	—	10	8	no	—
172	C₄	#90952	1	8	yes	direct product factor
173	C₄	C₄×D₃×Dic₅	1	2	yes	—
174	C₄	C₄×#1200	1	2	yes	—
175	C₃	C₄×#3947	1	1	yes	characteristic, complemented, Sylow
176	C₂	C₄×#6649	1	4	yes	—
177	C₂	C₂×#90952	1	2	yes	—
178	C₂	C₄×Dic₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₃×D₅
commutator	C₂×C₃×C₅	C₄³
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypocenter	C₃×C₅	C₄×#97
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. C₄×C₃⋊#168	C₄	semidirect
12. C₄×C₅×C₃⋊C₈	C₄	semidirect
13. #90952	C₄	direct (N × H)
51. C₃⋊#168	C₄²	semidirect
52. C₅×C₃⋊C₈	C₄²	semidirect
60. C₄×C₃⋊C₈	Dic₅	semidirect
79. C₃×C₄×C₅	#97	semidirect
124. C₃⋊C₈	C₄×Dic₅	semidirect
128. C₄×C₅	#803	semidirect
143. C₃×C₅	C₄×#97	semidirect
148. C₃×C₄	#3947	semidirect
167. C₅	C₄×#803	semidirect
172. C₄	#90952	direct (N × H)
175. C₃	C₄×#3947	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊#168	2
C₂×C₄×C₅×C₃⋊C₈	2
C₃×C₄²×Dic₅	2
C₂×#90952	2
C₄×#3947	3
C₄×#803	5

Subgroups — C4×#90952 #14294221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#90952 #14294221