Subgroups — C₂²×#90934 #14478097

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×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90934	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3930	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₄×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×#168	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₄	1	6	yes	—
10	C₂×C₃×C₈×D₅	C₂²	1	6	yes	—
11	C₂×C₃⋊#672	C₂²	1	6	yes	complemented
12	C₂×C₅×C₃⋊C₁₆	C₂²	1	6	yes	complemented
13	#90934	C₂²	1	16	yes	direct product factor
14	C₂²×#812	—	5	1	no	maximal, Hall
15	C₂²×C₈×D₅	D₃	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	—	3	1	no	—
17	C₂²×#672	—	3	1	no	—
18	C₂×#3930	—	3	12	no	—
19	C₂³×C₃×D₅	C₈	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	C₈	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	C₈	1	6	yes	—
23	C₂×C₃×C₄×D₅	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×C₃×#168	C₂×C₄	1	6	yes	—
27	C₃×C₈×D₅	C₂×C₄	1	12	yes	—
28	C₃×C₈×D₅	C₂³	1	4	yes	—
29	C₃⋊#672	C₂³	1	4	yes	complemented
30	C₅×C₃⋊C₁₆	C₂³	1	4	yes	complemented
31	C₂³×C₃×C₈	—	5	1	no	—
32	C₂×#812	—	5	12	no	—
33	C₂²×C₃⋊C₁₆	—	5	1	no	—
34	C₂²×C₃⋊C₁₆	D₅	1	1	yes	characteristic, complemented
35	C₂²×C₄×D₅	Dic₃	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
37	C₂²×#168	Dic₃	1	1	yes	characteristic
38	C₂×C₈×D₅	Dic₃	1	6	yes	—
39	C₂×C₈×D₅	C₂×D₃	1	6	yes	—
40	C₂×C₅×C₁₆	—	3	6	no	—
41	C₂×#672	—	3	6	no	—
42	#3930	—	3	16	no	—
43	C₂²×M₃₂	—	15	1	no	Sylow
44	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₃×D₅	—	2	4	no	—
46	C₂²×C₃×D₅	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×C₃×Dic₅	C₂×C₈	1	6	yes	—
50	C₃×C₄×D₅	C₂×C₈	1	12	yes	—
51	C₃×C₄×D₅	C₂²×C₄	1	4	yes	—
52	C₃×C₅×C₈	C₂²×C₄	1	3	yes	—
53	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
54	C₃×#168	C₂²×C₄	1	4	yes	—
55	C₂³×C₃×C₄	—	5	1	no	—
56	C₂²×C₃×C₈	—	5	1	no	—
57	C₂²×C₃×C₈	—	5	6	no	—
58	C₂²×C₃×C₈	—	5	6	no	—
59	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
60	#812	—	5	16	no	—
61	C₂×C₃⋊C₁₆	—	5	6	no	—
62	C₂×C₃⋊C₁₆	C₂×D₅	1	6	yes	complemented
63	C₂³×D₅	C₃⋊C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
65	C₂²×Dic₅	C₃⋊C₈	1	1	yes	characteristic
66	C₂×C₄×D₅	C₃⋊C₈	1	6	yes	—
67	C₂×C₄×D₅	C₂×Dic₃	1	6	yes	—
68	C₂×C₅×C₈	C₂×Dic₃	1	3	yes	—
69	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
70	C₂×#168	C₂×Dic₃	1	6	yes	—
71	C₈×D₅	C₂×Dic₃	1	12	yes	—
72	C₈×D₅	C₂²×D₃	1	4	yes	—
73	C₅×C₁₆	—	3	4	no	—
74	#672	—	3	4	no	—
75	C₂³×C₈	—	5	1	no	—
76	C₂×M₃₂	—	15	12	no	—
77	C₂²×C₁₆	—	3	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×C₃×D₅	—	2	12	no	—
82	C₂×C₃×D₅	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	C₃×Dic₅	C₂²×C₈	1	4	yes	—
86	C₂⁴×C₃	—	5	1	no	—
87	C₂²×C₃×C₄	—	5	1	no	—
88	C₂²×C₃×C₄	—	5	6	no	—
89	C₂²×C₃×C₄	—	5	6	no	—
90	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
91	C₂×C₃×C₈	—	5	6	no	—
92	C₂×C₃×C₈	—	5	12	no	—
93	C₂×C₃×C₈	—	5	4	no	—
94	C₂×C₃×C₈	C₄×D₅	1	3	yes	—
95	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
96	C₃⋊C₁₆	—	5	4	no	—
97	C₃⋊C₁₆	C₂²×D₅	1	4	yes	complemented
98	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
99	C₂²×D₅	—	2	4	no	—
100	C₂²×D₅	C₂×C₃⋊C₈	1	6	yes	—
101	C₂×C₄×C₅	C₂×C₃⋊C₈	1	3	yes	—
102	C₂×C₄×C₅	C₂²×Dic₃	1	3	yes	—
103	C₂×Dic₅	C₂×C₃⋊C₈	1	6	yes	—
104	C₄×D₅	C₂×C₃⋊C₈	1	12	yes	—
105	C₄×D₅	C₂²×Dic₃	1	4	yes	—
106	C₅×C₈	C₂²×Dic₃	1	3	yes	—
107	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
108	#168	C₂²×Dic₃	1	4	yes	—
109	C₂³×C₄	—	5	1	no	—
110	C₂²×C₈	—	5	1	no	—
111	C₂²×C₈	—	5	6	no	—
112	C₂²×C₈	—	5	6	no	—
113	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
114	M₃₂	—	15	16	no	—
115	C₂×C₁₆	—	3	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	C₃×D₅	—	2	4	no	—
120	C₂³×C₃	—	5	6	no	—
121	C₂³×C₃	—	10	4	no	—
122	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
123	C₂×C₃×C₄	—	5	6	no	—
124	C₂×C₃×C₄	—	5	4	no	—
125	C₂×C₃×C₄	—	5	12	no	—
126	C₂×C₃×C₄	C₈×D₅	1	3	yes	—
127	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
128	C₃×C₈	—	5	4	no	—
129	C₃×C₈	C₂×C₄×D₅	1	3	yes	—
130	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
131	C₂²×C₅	#812	1	4	yes	complemented
132	C₂²×C₅	C₂²×C₃⋊C₈	1	3	yes	—
133	C₂×D₅	—	2	12	no	—
134	C₂×D₅	C₂²×C₃⋊C₈	1	4	yes	—
135	C₄×C₅	C₂²×C₃⋊C₈	1	3	yes	—
136	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
137	Dic₅	C₂²×C₃⋊C₈	1	4	yes	—
138	C₂⁴	—	5	1	no	—
139	C₂²×C₄	—	5	1	no	—
140	C₂²×C₄	—	5	6	no	—
141	C₂²×C₄	—	5	6	no	—
142	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
143	C₂×C₈	—	5	6	no	—
144	C₂×C₈	—	5	12	no	—
145	C₂×C₈	—	5	4	no	—
146	C₂×C₈	D₅×Dic₃	1	3	yes	—
147	C₂×C₈	C₂×D₃×D₅	1	3	yes	—
148	C₁₆	—	3	4	no	—
149	C₁₆	—	15	4	no	—
150	C₃×C₅	C₂²×M₃₂	1	1	yes	characteristic, complemented, Hall
151	C₂²×C₃	—	5	4	no	—
152	C₂²×C₃	—	10	12	no	—
153	C₂²×C₃	#3930	1	4	yes	complemented
154	C₂²×C₃	C₂×C₈×D₅	1	3	yes	—
155	C₃×C₄	—	5	4	no	—
156	C₃×C₄	C₂×C₈×D₅	1	3	yes	—
157	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
158	C₂×C₅	C₂×#812	1	6	yes	complemented
159	C₂×C₅	C₂³×C₃⋊C₈	1	1	yes	characteristic
160	D₅	—	2	4	no	—
161	C₂³	—	5	6	no	—
162	C₂³	—	10	4	no	—
163	C₂³	D₅×C₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	5	6	no	—
165	C₂×C₄	—	5	4	no	—
166	C₂×C₄	—	5	12	no	—
167	C₂×C₄	D₅×C₃⋊C₈	1	3	yes	—
168	C₂×C₄	C₂×D₅×Dic₃	1	3	yes	—
169	C₈	—	5	4	no	—
170	C₈	C₂×D₅×Dic₃	1	3	yes	—
171	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
172	C₂×C₃	—	10	4	no	—
173	C₂×C₃	C₂×#3930	1	6	yes	complemented
174	C₂×C₃	C₂²×C₈×D₅	1	1	yes	characteristic
175	C₅	C₂²×#812	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	5	4	no	—
177	C₂²	—	10	12	no	—
178	C₂²	#90934	1	4	yes	direct product factor
179	C₂²	C₂×D₅×C₃⋊C₈	1	3	yes	—
180	C₄	—	5	4	no	—
181	C₄	C₂×D₅×C₃⋊C₈	1	3	yes	—
182	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
183	C₃	C₂²×#3930	1	1	yes	characteristic, complemented, Sylow
184	C₂	—	10	4	no	—
185	C₂	C₂×#90934	1	6	yes	direct product factor
186	C₂	C₂²×D₅×C₃⋊C₈	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×C₃×D₅	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×C₅×C₃⋊C₁₆	C₂	semidirect
4. C₂×#90934	C₂	direct (N × H)
11. C₂×C₃⋊#672	C₂²	semidirect
12. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
13. #90934	C₂²	direct (N × H)
29. C₃⋊#672	C₂³	semidirect
30. C₅×C₃⋊C₁₆	C₂³	semidirect
34. C₂²×C₃⋊C₁₆	D₅	semidirect
62. C₂×C₃⋊C₁₆	C₂×D₅	semidirect
79. C₂²×C₃×C₅	M₃₂	semidirect
97. C₃⋊C₁₆	C₂²×D₅	semidirect
117. C₂×C₃×C₅	C₂×M₃₂	semidirect
131. C₂²×C₅	#812	semidirect
150. C₃×C₅	C₂²×M₃₂	semidirect
153. C₂²×C₃	#3930	semidirect
158. C₂×C₅	C₂×#812	semidirect
173. C₂×C₃	C₂×#3930	semidirect
175. C₅	C₂²×#812	semidirect
178. C₂²	#90934	direct (N × H)
183. C₃	C₂²×#3930	semidirect
185. C₂	C₂×#90934	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×C₃×C₈×D₅	2
C₂²×C₃⋊#672	2
C₂²×C₅×C₃⋊C₁₆	2
C₂×#90934	2
C₂²×#3930	3
C₂²×#812	5

Subgroups — C22×#90934 #14478097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#90934 #14478097