Subgroups — #14312611

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#90935	C₂	1	2	yes	maximal
3	#11746282	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64681	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃×#672	C₂	1	2	yes	maximal, complemented
6	C₄×#3944	—	3	1	no	maximal, Hall
7	C₂×C₅×D₃×C₈	C₄	1	1	yes	characteristic
8	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₅×D₃	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊#672	C₂²	1	2	yes	—
12	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
13	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
14	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₃×#672	C₂²	1	2	yes	—
16	#90935	C₄	1	8	yes	complemented
17	#69071	—	5	1	no	maximal, Hall
18	C₂×C₄×C₅×C₈	—	3	1	no	—
19	C₂×#3944	—	3	2	no	—
20	#64681	—	3	1	no	—
21	#64681	D₃	1	1	yes	characteristic, complemented
22	C₄×#672	—	3	2	no	—
23	C₂×C₄×C₅×D₃	C₈	1	2	yes	—
24	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×Dic₃	C₈	1	2	yes	—
29	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₅×D₃×C₈	C₈	1	4	yes	—
33	C₅×D₃×C₈	C₂×C₄	1	4	yes	—
34	C₃⋊#672	C₂×C₄	1	4	yes	complemented
35	C₃×#672	C₂×C₄	1	4	yes	complemented
36	C₂×#798	—	5	2	no	—
37	C₄×D₃×C₈	D₅	1	1	yes	characteristic
38	C₃×#346	—	5	1	no	—
39	Dic₃×C₁₆	—	5	2	no	—
40	#4461	—	5	1	no	—
41	C₂²×C₅×C₈	—	3	1	no	—
42	C₂²×C₅×C₈	—	3	1	no	—
43	C₂×C₄²×C₅	—	3	1	no	—
44	C₄×C₅×C₈	—	3	1	no	—
45	C₄×C₅×C₈	—	3	1	no	—
46	C₄×C₅×C₈	—	3	1	no	—
47	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
48	#3944	—	3	8	no	—
49	C₂×#672	—	3	2	no	—
50	C₂×#672	C₂×D₃	1	2	yes	—
51	C₄×M₃₂	—	15	1	no	Sylow
52	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
57	C₄×C₅×D₃	—	2	4	no	—
58	C₄×C₅×D₃	C₄²	1	2	yes	—
59	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
60	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
61	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
62	C₅×C₃⋊C₈	C₄²	1	2	yes	—
63	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
64	C₂×D₃×C₈	Dic₅	1	1	yes	characteristic
65	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
66	C₄²×D₃	Dic₅	1	1	yes	characteristic
67	C₂×C₃×C₁₆	—	5	2	no	—
68	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
69	C₈×Dic₃	Dic₅	1	1	yes	characteristic
70	C₈×Dic₃	C₂×D₅	1	1	yes	characteristic
71	C₂×C₃⋊C₁₆	—	5	2	no	—
72	C₄×C₃⋊C₈	Dic₅	1	1	yes	characteristic
73	#798	—	5	8	no	—
74	C₂²×C₄×C₅	—	3	1	no	—
75	C₂²×C₄×C₅	—	3	2	no	—
76	C₂×C₅×C₈	—	3	1	no	—
77	C₂×C₅×C₈	—	3	1	no	—
78	C₂×C₅×C₈	—	3	4	no	—
79	C₂×C₅×C₈	—	3	4	no	—
80	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₄²×C₅	—	3	2	no	—
83	C₄²×C₅	—	3	1	no	—
84	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
85	#672	—	3	4	no	—
86	#672	C₄×D₃	1	4	yes	complemented
87	C₂×M₃₂	—	15	2	no	—
88	C₂×C₄×C₈	—	3	1	no	—
89	#346	—	5	1	no	—
90	#346	—	15	1	no	—
91	C₄×C₁₆	—	15	2	no	—
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₅×D₃	—	2	2	no	—
94	C₂×C₅×D₃	C₄×C₈	1	2	yes	—
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₅×Dic₃	C₄×C₈	1	2	yes	—
99	C₅×Dic₃	M₃₂	1	4	yes	complemented
100	C₂×C₄×D₃	#168	1	2	yes	—
101	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
104	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
105	C₄×Dic₃	#168	1	2	yes	—
106	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
107	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
108	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
109	D₃×C₈	#168	1	4	yes	—
110	D₃×C₈	C₄×D₅	1	4	yes	—
111	C₃×C₁₆	—	5	4	no	—
112	C₃⋊C₁₆	—	5	4	no	—
113	C₂³×C₅	—	3	1	no	—
114	C₂×C₄×C₅	—	3	2	no	—
115	C₂×C₄×C₅	—	3	1	no	—
116	C₂×C₄×C₅	—	3	2	no	—
117	C₂×C₄×C₅	—	3	2	no	—
118	C₂×C₄×C₅	—	6	4	no	—
119	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
121	C₅×C₈	—	3	2	no	—
122	C₅×C₈	—	3	2	no	—
123	C₅×C₈	D₃×C₈	1	2	yes	—
124	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
125	C₂²×C₈	—	3	1	no	—
126	C₂²×C₈	—	3	1	no	—
127	C₂×C₄²	—	3	1	no	—
128	M₃₂	—	15	8	no	—
129	C₂×C₁₆	—	5	2	no	—
130	C₂×C₁₆	—	15	2	no	—
131	C₄×C₈	—	3	1	no	—
132	C₄×C₈	—	3	1	no	—
133	C₄×C₈	—	3	1	no	—
134	C₄×C₈	D₃×D₅	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
137	C₅×D₃	—	2	2	no	—
138	C₂²×D₃	C₂×#168	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×#168	1	2	yes	—
140	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
141	C₂×Dic₃	C₂×#168	1	1	yes	characteristic
142	C₂×Dic₃	C₂×#168	1	2	yes	—
143	C₄×D₃	—	2	4	no	—
144	C₄×D₃	C₂×#168	1	2	yes	—
145	C₄×D₃	C₄×Dic₅	1	2	yes	—
146	C₃×C₈	C₂×#168	1	2	yes	—
147	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
148	C₃⋊C₈	C₂×#168	1	2	yes	—
149	C₃⋊C₈	C₄×Dic₅	1	2	yes	—
150	C₂²×C₅	—	3	2	no	—
151	C₂²×C₅	—	6	2	no	—
152	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	3	4	no	—
155	C₄×C₅	—	3	2	no	—
156	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
157	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
158	C₂²×C₄	—	3	1	no	—
159	C₂²×C₄	—	3	2	no	—
160	C₂×C₈	—	3	1	no	—
161	C₂×C₈	—	3	1	no	—
162	C₂×C₈	—	3	4	no	—
163	C₂×C₈	—	3	4	no	—
164	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
165	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
166	C₄²	—	3	1	no	—
167	C₄²	—	3	2	no	—
168	C₄²	D₃×Dic₅	1	1	yes	characteristic
169	C₁₆	—	5	4	no	—
170	C₁₆	—	15	4	no	—
171	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
173	C₂×D₃	—	2	2	no	—
174	C₂×D₃	C₄×#168	1	2	yes	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	C₂²×#168	1	1	yes	characteristic
177	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
178	Dic₃	C₄×#168	1	2	yes	—
179	Dic₃	#3944	1	4	yes	complemented
180	C₂×C₅	—	6	2	no	—
181	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
182	C₂×C₅	D₃×M₃₂	1	2	yes	—
183	C₂³	—	3	1	no	—
184	C₂×C₄	—	3	1	no	—
185	C₂×C₄	—	3	2	no	—
186	C₂×C₄	—	3	2	no	—
187	C₂×C₄	—	3	2	no	—
188	C₂×C₄	—	6	4	no	—
189	C₂×C₄	D₃×#168	1	2	yes	—
190	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
191	C₈	—	3	2	no	—
192	C₈	—	3	2	no	—
193	C₈	D₃×#168	1	2	yes	—
194	C₈	C₄×D₃×D₅	1	2	yes	—
195	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
196	C₂×C₃	C₂×#3944	1	2	yes	—
197	D₃	—	2	2	no	—
198	C₅	#69071	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	3	2	no	—
200	C₂²	—	6	2	no	—
201	C₂²	C₂×D₃×#168	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	3	2	no	—
204	C₄	—	3	4	no	—
205	C₄	C₂×D₃×#168	1	1	yes	characteristic
206	C₄	C₄×D₃×Dic₅	1	1	yes	characteristic
207	C₃	C₄×#3944	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	6	2	no	—
209	C₂	C₄×D₃×#168	1	1	yes	characteristic
210	C₂	D₃×#3944	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₃	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	C₄×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11746282	C₂	semidirect
4. C₃×#64681	C₂	semidirect
5. Dic₃×#672	C₂	semidirect
16. #90935	C₄	semidirect
21. #64681	D₃	semidirect
34. C₃⋊#672	C₂×C₄	semidirect
35. C₃×#672	C₂×C₄	semidirect
86. #672	C₄×D₃	semidirect
99. C₅×Dic₃	M₃₂	semidirect
171. C₃×C₅	C₄×M₃₂	semidirect
179. Dic₃	#3944	semidirect
198. C₅	#69071	semidirect
207. C₃	C₄×#3944	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₃×C₈	2
C₂×#90935	2
#11746282	2
C₃×#64681	2
Dic₃×#672	2
C₄×#3944	3
#69071	5