Subgroups — C₂²×#90935 #14478099

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90935	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3944	—	3	1	no	maximal, Hall
6	C₂²×C₄×C₅×D₃	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₅×D₃×C₈	C₄	1	6	yes	—
10	C₂×C₅×D₃×C₈	C₂²	1	6	yes	—
11	C₂×C₃⋊#672	C₂²	1	6	yes	complemented
12	C₂×C₃×#672	C₂²	1	6	yes	complemented
13	#90935	C₂²	1	16	yes	direct product factor
14	C₂²×#798	—	5	1	no	maximal, Hall
15	C₂³×C₅×C₈	—	3	1	no	—
16	C₂×#3944	—	3	12	no	—
17	C₂²×#672	—	3	1	no	—
18	C₂²×#672	D₃	1	1	yes	characteristic, complemented
19	C₂³×C₅×D₃	C₈	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	C₈	1	1	yes	characteristic
22	C₂×C₄×C₅×D₃	C₈	1	6	yes	—
23	C₂×C₄×C₅×D₃	C₂×C₄	1	6	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	3	yes	—
25	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
26	C₂×C₅×C₃⋊C₈	C₂×C₄	1	6	yes	—
27	C₅×D₃×C₈	C₂×C₄	1	12	yes	—
28	C₅×D₃×C₈	C₂³	1	4	yes	—
29	C₃⋊#672	C₂³	1	4	yes	complemented
30	C₃×#672	C₂³	1	4	yes	complemented
31	C₂²×D₃×C₈	D₅	1	1	yes	characteristic
32	C₂²×C₃×C₁₆	—	5	1	no	—
33	C₂²×C₃⋊C₁₆	—	5	1	no	—
34	C₂×#798	—	5	12	no	—
35	C₂³×C₄×C₅	—	3	1	no	—
36	C₂²×C₅×C₈	—	3	1	no	—
37	C₂²×C₅×C₈	—	3	6	no	—
38	C₂²×C₅×C₈	—	3	6	no	—
39	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
40	#3944	—	3	16	no	—
41	C₂×#672	—	3	6	no	—
42	C₂×#672	C₂×D₃	1	6	yes	complemented
43	C₂²×M₃₂	—	15	1	no	Sylow
44	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₅×D₃	—	2	4	no	—
46	C₂²×C₅×D₃	C₂×C₈	1	6	yes	—
47	C₂×C₃×C₄×C₅	C₂×C₈	1	3	yes	—
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
49	C₂×C₅×Dic₃	C₂×C₈	1	6	yes	—
50	C₄×C₅×D₃	C₂×C₈	1	12	yes	—
51	C₄×C₅×D₃	C₂²×C₄	1	4	yes	—
52	C₃×C₅×C₈	C₂²×C₄	1	3	yes	—
53	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
54	C₅×C₃⋊C₈	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×D₃	Dic₅	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
57	C₂²×C₃⋊C₈	Dic₅	1	1	yes	characteristic
58	C₂×D₃×C₈	Dic₅	1	6	yes	—
59	C₂×D₃×C₈	C₂×D₅	1	6	yes	—
60	C₂×C₃×C₁₆	—	5	6	no	—
61	C₂×C₃⋊C₁₆	—	5	6	no	—
62	#798	—	5	16	no	—
63	C₂⁴×C₅	—	3	1	no	—
64	C₂²×C₄×C₅	—	3	1	no	—
65	C₂²×C₄×C₅	—	3	6	no	—
66	C₂²×C₄×C₅	—	3	6	no	—
67	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
68	C₂×C₅×C₈	—	3	6	no	—
69	C₂×C₅×C₈	—	3	12	no	—
70	C₂×C₅×C₈	—	3	4	no	—
71	C₂×C₅×C₈	C₄×D₃	1	3	yes	—
72	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
73	#672	—	3	4	no	—
74	#672	C₂²×D₃	1	4	yes	complemented
75	C₂³×C₈	—	3	1	no	—
76	C₂×M₃₂	—	15	12	no	—
77	C₂²×C₁₆	—	5	1	no	—
78	C₂²×C₁₆	—	15	1	no	—
79	C₂²×C₃×C₅	M₃₂	1	4	yes	complemented
80	C₂²×C₃×C₅	C₂²×C₈	1	3	yes	—
81	C₂×C₅×D₃	—	2	12	no	—
82	C₂×C₅×D₃	C₂²×C₈	1	4	yes	—
83	C₃×C₄×C₅	C₂²×C₈	1	3	yes	—
84	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
85	C₅×Dic₃	C₂²×C₈	1	4	yes	—
86	C₂³×D₃	#168	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
88	C₂²×Dic₃	#168	1	1	yes	characteristic
89	C₂×C₄×D₃	#168	1	6	yes	—
90	C₂×C₄×D₃	C₂×Dic₅	1	6	yes	—
91	C₂×C₃×C₈	C₂×Dic₅	1	3	yes	—
92	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
93	C₂×C₃⋊C₈	C₂×Dic₅	1	6	yes	—
94	D₃×C₈	C₂×Dic₅	1	12	yes	—
95	D₃×C₈	C₂²×D₅	1	4	yes	—
96	C₃×C₁₆	—	5	4	no	—
97	C₃⋊C₁₆	—	5	4	no	—
98	C₂³×C₅	—	3	6	no	—
99	C₂³×C₅	—	6	4	no	—
100	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	—	3	6	no	—
102	C₂×C₄×C₅	—	3	12	no	—
103	C₂×C₄×C₅	—	3	4	no	—
104	C₂×C₄×C₅	D₃×C₈	1	3	yes	—
105	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
106	C₅×C₈	—	3	4	no	—
107	C₅×C₈	C₂×C₄×D₃	1	3	yes	—
108	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
109	C₂³×C₄	—	3	1	no	—
110	C₂²×C₈	—	3	1	no	—
111	C₂²×C₈	—	3	6	no	—
112	C₂²×C₈	—	3	6	no	—
113	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
114	M₃₂	—	15	16	no	—
115	C₂×C₁₆	—	5	6	no	—
116	C₂×C₁₆	—	15	6	no	—
117	C₂×C₃×C₅	C₂×M₃₂	1	6	yes	complemented
118	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
119	C₅×D₃	—	2	4	no	—
120	C₂³×C₃	C₂×#168	1	1	yes	characteristic
121	C₂²×D₃	—	2	4	no	—
122	C₂²×D₃	C₂×#168	1	6	yes	—
123	C₂×C₃×C₄	C₂×#168	1	3	yes	—
124	C₂×C₃×C₄	C₂²×Dic₅	1	3	yes	—
125	C₂×Dic₃	C₂×#168	1	6	yes	—
126	C₄×D₃	C₂×#168	1	12	yes	—
127	C₄×D₃	C₂²×Dic₅	1	4	yes	—
128	C₃×C₈	C₂²×Dic₅	1	3	yes	—
129	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
130	C₃⋊C₈	C₂²×Dic₅	1	4	yes	—
131	C₂²×C₅	—	3	4	no	—
132	C₂²×C₅	—	6	12	no	—
133	C₂²×C₅	#798	1	4	yes	complemented
134	C₂²×C₅	C₂×D₃×C₈	1	3	yes	—
135	C₄×C₅	—	3	4	no	—
136	C₄×C₅	C₂×D₃×C₈	1	3	yes	—
137	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
138	C₂⁴	—	3	1	no	—
139	C₂²×C₄	—	3	1	no	—
140	C₂²×C₄	—	3	6	no	—
141	C₂²×C₄	—	3	6	no	—
142	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
143	C₂×C₈	—	3	6	no	—
144	C₂×C₈	—	3	12	no	—
145	C₂×C₈	—	3	4	no	—
146	C₂×C₈	D₃×Dic₅	1	3	yes	—
147	C₂×C₈	C₂×D₃×D₅	1	3	yes	—
148	C₁₆	—	5	4	no	—
149	C₁₆	—	15	4	no	—
150	C₃×C₅	C₂²×M₃₂	1	1	yes	characteristic, complemented, Hall
151	C₂²×C₃	#3944	1	4	yes	complemented
152	C₂²×C₃	C₂²×#168	1	3	yes	—
153	C₂×D₃	—	2	12	no	—
154	C₂×D₃	C₂²×#168	1	4	yes	—
155	C₃×C₄	C₂²×#168	1	3	yes	—
156	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
157	Dic₃	C₂²×#168	1	4	yes	—
158	C₂×C₅	—	6	4	no	—
159	C₂×C₅	C₂×#798	1	6	yes	complemented
160	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
161	C₂³	—	3	6	no	—
162	C₂³	—	6	4	no	—
163	C₂³	D₃×#168	1	1	yes	characteristic
164	C₂×C₄	—	3	6	no	—
165	C₂×C₄	—	3	4	no	—
166	C₂×C₄	—	3	12	no	—
167	C₂×C₄	D₃×#168	1	3	yes	—
168	C₂×C₄	C₂×D₃×Dic₅	1	3	yes	—
169	C₈	—	3	4	no	—
170	C₈	C₂×D₃×Dic₅	1	3	yes	—
171	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
172	C₂×C₃	C₂×#3944	1	6	yes	complemented
173	C₂×C₃	C₂³×#168	1	1	yes	characteristic
174	D₃	—	2	4	no	—
175	C₅	C₂²×#798	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	3	4	no	—
177	C₂²	—	6	12	no	—
178	C₂²	#90935	1	4	yes	direct product factor
179	C₂²	C₂×D₃×#168	1	3	yes	—
180	C₄	—	3	4	no	—
181	C₄	C₂×D₃×#168	1	3	yes	—
182	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
183	C₃	C₂²×#3944	1	1	yes	characteristic, complemented, Sylow
184	C₂	—	6	4	no	—
185	C₂	C₂×#90935	1	6	yes	direct product factor
186	C₂	C₂²×D₃×#168	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₈
Frattini	C₈	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂³×C₃×C₅	C₂×C₈
prime radical	C₂³×C₅×D₃	C₈
cosocle	C₃×C₅×C₈	C₂⁴
hypocenter	C₃×C₅	C₂²×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊#672	C₂	semidirect
3. C₂²×C₃×#672	C₂	semidirect
4. C₂×#90935	C₂	direct (N × H)
11. C₂×C₃⋊#672	C₂²	semidirect
12. C₂×C₃×#672	C₂²	semidirect
13. #90935	C₂²	direct (N × H)
18. C₂²×#672	D₃	semidirect
29. C₃⋊#672	C₂³	semidirect
30. C₃×#672	C₂³	semidirect
42. C₂×#672	C₂×D₃	semidirect
74. #672	C₂²×D₃	semidirect
79. C₂²×C₃×C₅	M₃₂	semidirect
117. C₂×C₃×C₅	C₂×M₃₂	semidirect
133. C₂²×C₅	#798	semidirect
150. C₃×C₅	C₂²×M₃₂	semidirect
151. C₂²×C₃	#3944	semidirect
159. C₂×C₅	C₂×#798	semidirect
172. C₂×C₃	C₂×#3944	semidirect
175. C₅	C₂²×#798	semidirect
178. C₂²	#90935	direct (N × H)
183. C₃	C₂²×#3944	semidirect
185. C₂	C₂×#90935	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×D₃×C₈	2
C₂²×C₃⋊#672	2
C₂²×C₃×#672	2
C₂×#90935	2
C₂²×#3944	3
C₂²×#798	5

Subgroups — C22×#90935 #14478099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#90935 #14478099