Subgroups — C₂×#11745874 #14312564

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×#672	C₂	1	1	yes	maximal, characteristic
4	#11745874	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64681	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Dic₃	C₄	1	1	yes	characteristic
8	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₃⋊#672	C₄	1	4	yes	complemented
10	C₂×C₃⋊#672	C₂²	1	2	yes	—
11	C₅×C₈×Dic₃	C₄	1	2	yes	—
12	C₅×C₈×Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×#672	C₄	1	4	yes	complemented
14	C₂×C₃×#672	C₂²	1	2	yes	—
15	C₂×#4502	—	5	1	no	maximal, Hall
16	C₂×C₄×C₅×C₈	—	3	1	no	—
17	C₂²×#672	—	3	1	no	—
18	C₂²×#672	D₃	1	1	yes	characteristic
19	#64681	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	C₈	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₄×C₅×Dic₃	C₈	1	2	yes	—
27	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₃⋊C₈	C₈	1	4	yes	—
29	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₃⋊#672	C₂×C₄	1	4	yes	complemented
31	C₃×#672	C₂×C₄	1	4	yes	complemented
32	C₂²×C₃×C₁₆	—	5	1	no	—
33	C₂×C₈×Dic₃	D₅	1	1	yes	characteristic
34	C₂²×C₃⋊C₁₆	—	5	1	no	—
35	#4502	—	5	4	no	—
36	C₂²×C₅×C₈	—	3	1	no	—
37	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂×C₄²×C₅	—	3	1	no	—
39	C₄×C₅×C₈	—	3	2	no	—
40	C₄×C₅×C₈	—	3	2	no	—
41	C₂×#672	—	3	4	no	—
42	C₂×#672	—	3	2	no	—
43	C₂×#672	Dic₃	1	4	yes	complemented
44	C₂×#672	C₂×D₃	1	2	yes	—
45	C₂×#346	—	15	1	no	Sylow
46	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅×Dic₃	—	2	4	no	—
52	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
53	C₃×C₅×C₈	C₄²	1	2	yes	—
54	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₅×C₃⋊C₈	C₂×C₈	1	4	yes	—
57	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
58	C₂×C₄×Dic₃	Dic₅	1	1	yes	characteristic
59	C₂²×C₃⋊C₈	Dic₅	1	1	yes	characteristic
60	C₂×C₃×C₁₆	—	5	4	no	—
61	C₂×C₃×C₁₆	—	5	2	no	—
62	C₈×Dic₃	Dic₅	1	2	yes	—
63	C₈×Dic₃	C₂×D₅	1	2	yes	—
64	C₂×C₃⋊C₁₆	—	5	4	no	—
65	C₂×C₃⋊C₁₆	—	5	2	no	—
66	C₂²×C₄×C₅	—	3	2	no	—
67	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
68	C₂×C₅×C₈	—	3	4	no	—
69	C₂×C₅×C₈	—	3	2	no	—
70	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
71	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
72	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
73	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
74	C₄²×C₅	—	3	2	no	—
75	C₄²×C₅	—	3	2	no	—
76	#672	—	3	4	no	—
77	#672	C₂×Dic₃	1	4	yes	complemented
78	C₂²×C₁₆	—	5	1	no	—
79	C₂²×C₁₆	—	15	1	no	—
80	C₂×C₄×C₈	—	3	1	no	—
81	#346	—	15	4	no	—
82	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
83	C₂²×C₃×C₅	M₃₂	1	4	yes	—
84	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
85	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
86	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
87	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
88	C₅×Dic₃	—	2	4	no	—
89	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
90	C₂²×Dic₃	#168	1	2	yes	—
91	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
92	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
93	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
95	C₄×Dic₃	#168	1	2	yes	—
96	C₄×Dic₃	C₂×Dic₅	1	2	yes	—
97	C₂×C₃⋊C₈	#168	1	4	yes	—
98	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
99	C₃×C₁₆	—	5	4	no	—
100	C₃⋊C₁₆	—	5	4	no	—
101	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅	—	3	4	no	—
103	C₂×C₄×C₅	—	6	4	no	—
104	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
106	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
107	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
108	C₅×C₈	—	3	4	no	—
109	C₅×C₈	C₄×Dic₃	1	2	yes	—
110	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
111	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
112	C₂²×C₈	—	3	1	no	—
113	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
114	C₂×C₄²	—	3	1	no	—
115	C₂×C₁₆	—	5	4	no	—
116	C₂×C₁₆	—	5	2	no	—
117	C₂×C₁₆	—	15	4	no	—
118	C₂×C₁₆	—	15	2	no	—
119	C₄×C₈	—	3	2	no	—
120	C₄×C₈	—	3	2	no	—
121	C₂×C₃×C₅	#346	1	4	yes	complemented
122	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
124	C₂³×C₃	C₂×#168	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×#168	1	2	yes	—
127	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
128	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
129	C₂×Dic₃	—	2	4	no	—
130	C₂×Dic₃	C₂×#168	1	4	yes	—
131	C₃×C₈	C₄×Dic₅	1	2	yes	—
132	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
133	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
134	C₃⋊C₈	C₂×#168	1	4	yes	—
135	C₂²×C₅	#798	1	4	yes	—
136	C₂²×C₅	C₈×Dic₃	1	2	yes	—
137	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
138	C₄×C₅	—	6	4	no	—
139	C₄×C₅	C₈×Dic₃	1	2	yes	—
140	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
141	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
142	C₂²×C₄	—	3	2	no	—
143	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
144	C₂×C₈	—	3	2	no	—
145	C₂×C₈	—	3	4	no	—
146	C₂×C₈	D₅×Dic₃	1	2	yes	—
147	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
148	C₂×C₈	#1200	1	2	yes	—
149	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
150	C₄²	—	3	2	no	—
151	C₄²	—	3	2	no	—
152	C₁₆	—	5	4	no	—
153	C₁₆	—	15	4	no	—
154	C₃×C₅	C₂×#346	1	1	yes	characteristic, complemented, Hall
155	C₂²×C₃	C₄×#168	1	2	yes	—
156	C₂²×C₃	#3944	1	4	yes	—
157	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
158	C₃×C₄	C₄×#168	1	2	yes	—
159	C₃×C₄	C₂²×#168	1	1	yes	characteristic
160	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
161	Dic₃	—	2	4	no	—
162	C₂×C₅	#4502	1	4	yes	complemented
163	C₂×C₅	C₂×#798	1	2	yes	—
164	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
165	C₂³	D₃×#168	1	1	yes	characteristic
166	C₂×C₄	—	3	4	no	—
167	C₂×C₄	—	6	4	no	—
168	C₂×C₄	D₃×#168	1	1	yes	characteristic
169	C₂×C₄	D₃×#168	1	2	yes	—
170	C₂×C₄	Dic₃×Dic₅	1	2	yes	—
171	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
172	C₈	—	3	4	no	—
173	C₈	Dic₃×Dic₅	1	2	yes	—
174	C₈	C₂×D₅×Dic₃	1	1	yes	characteristic
175	C₈	C₂×#1200	1	1	yes	characteristic
176	C₂×C₃	#64681	1	4	yes	complemented
177	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
178	C₂×C₃	C₂×#3944	1	2	yes	—
179	C₅	C₂×#4502	1	1	yes	characteristic, complemented, Sylow
180	C₂²	#90935	1	4	yes	—
181	C₂²	Dic₃×#168	1	2	yes	—
182	C₂²	C₂×D₃×#168	1	1	yes	characteristic
183	C₄	—	6	4	no	—
184	C₄	Dic₃×#168	1	2	yes	—
185	C₄	C₂×D₃×#168	1	1	yes	characteristic
186	C₄	C₂×Dic₃×Dic₅	1	1	yes	characteristic
187	C₃	C₂×#64681	1	1	yes	characteristic, complemented, Sylow
188	C₂	#11745874	1	4	yes	direct product factor
189	C₂	C₂×#90935	1	2	yes	—
190	C₂	C₂×Dic₃×#168	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypocenter	C₃×C₅	C₂×#346
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
4. #11745874	C₂	direct (N × H)
9. C₂×C₃⋊#672	C₄	semidirect
13. C₂×C₃×#672	C₄	semidirect
30. C₃⋊#672	C₂×C₄	semidirect
31. C₃×#672	C₂×C₄	semidirect
43. C₂×#672	Dic₃	semidirect
77. #672	C₂×Dic₃	semidirect
121. C₂×C₃×C₅	#346	semidirect
154. C₃×C₅	C₂×#346	semidirect
162. C₂×C₅	#4502	semidirect
176. C₂×C₃	#64681	semidirect
179. C₅	C₂×#4502	semidirect
187. C₃	C₂×#64681	semidirect
188. C₂	#11745874	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⋊#672	2
C₂×C₅×C₈×Dic₃	2
C₂²×C₃×#672	2
#11745874	2
C₂×#64681	3
C₂×#4502	5

Subgroups — C2×#11745874 #14312564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11745874 #14312564