Subgroups — #14286441

Order 1920.

384 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	D₄×C₃⋊#168	C₂	1	2	yes	maximal, complemented
3	D₄⋊(C₃⋊#168)	C₂	1	2	yes	maximal, complemented
4	#11746278	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746276	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601671	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×D₈	C₄	1	1	yes	characteristic
8	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₃⋊#168	—	2	2	no	—
10	C₃×C₅×#107	C₄	1	1	yes	characteristic
11	C₃×C₅×#102	C₄	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#91114	—	2	2	no	—
14	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
15	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
17	#70878	—	5	1	no	maximal, Hall
18	C₄×C₅×D₈	D₃	1	1	yes	characteristic
19	D₄×#168	—	3	2	no	—
20	D₄⋊#168	—	3	2	no	—
21	#64677	—	3	1	no	—
22	#64675	—	3	1	no	—
23	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₃×C₅×D₈	C₈	1	4	yes	complemented
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃⋊#168	—	2	2	no	—
32	C₂×C₃⋊#168	—	2	1	no	—
33	C₂×C₃⋊#168	—	4	4	no	—
34	C₂×C₃⋊#168	D₄	1	2	yes	—
35	C₃×C₄×D₈	D₅	1	1	yes	characteristic
36	D₄×C₃⋊C₈	—	5	2	no	—
37	D₄⋊(C₃⋊C₈)	—	5	2	no	—
38	#4457	—	5	1	no	—
39	#4455	—	5	1	no	—
40	C₂×C₅×D₈	Dic₃	1	1	yes	characteristic
41	C₄×C₅×D₄	C₂×D₃	1	2	yes	—
42	C₂²×#168	—	6	2	no	—
43	C₅×#107	Dic₃	1	1	yes	characteristic
44	C₅×#102	Dic₃	1	2	yes	—
45	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂²⋊#168	—	6	2	no	—
47	C₄⋊#168	—	3	2	no	—
48	C₄×#168	—	3	1	no	—
49	C₄×#168	—	3	1	no	—
50	#1580	—	15	1	no	Sylow
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₃×C₅×D₄	—	2	2	no	—
53	C₃×C₅×D₄	C₂×C₈	1	4	yes	complemented
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
62	C₃⋊#168	—	2	2	no	—
63	C₃⋊#168	—	4	1	no	—
64	C₃⋊#168	—	4	2	no	—
65	C₂×C₃×D₈	Dic₅	1	1	yes	characteristic
66	C₃×C₄×D₄	C₂×D₅	1	2	yes	—
67	C₂²×C₃⋊C₈	—	10	2	no	—
68	C₃×#107	Dic₅	1	1	yes	characteristic
69	C₃×#102	Dic₅	1	2	yes	—
70	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
71	#830	—	10	2	no	—
72	C₄⋊(C₃⋊C₈)	—	5	2	no	—
73	C₄×C₃⋊C₈	—	5	1	no	—
74	C₄×C₃⋊C₈	—	5	1	no	—
75	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₅×D₈	C₃⋊C₈	1	4	yes	complemented
78	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
81	C₅×C₂²⋊C₄	—	2	2	no	—
82	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
83	C₂×#168	—	3	2	no	—
84	C₂×#168	—	6	2	no	—
85	C₂×#168	—	6	1	no	—
86	C₂×#168	—	12	4	no	—
87	C₄×D₈	D₁₅	1	1	yes	characteristic
88	C₈×D₄	—	15	2	no	—
89	#334	—	15	1	no	—
90	D₄⋊C₈	—	15	2	no	—
91	#322	—	15	1	no	—
92	C₂²×C₃×C₅	—	2	4	no	—
93	C₂²×C₃×C₅	—	4	2	no	—
94	C₂²×C₃×C₅	—	4	2	no	—
95	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	4	2	no	—
98	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₃×C₄×C₅	#131	1	1	yes	characteristic
100	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₃×D₈	#168	1	4	yes	complemented
104	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
106	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
107	C₃×C₂²⋊C₄	—	2	2	no	—
108	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
109	C₂×C₃⋊C₈	—	5	2	no	—
110	C₂×C₃⋊C₈	—	10	2	no	—
111	C₂×C₃⋊C₈	—	10	1	no	—
112	C₂×C₃⋊C₈	—	20	4	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₅×D₄	—	2	2	no	—
115	C₅×D₄	C₂×C₃⋊C₈	1	4	yes	complemented
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#237	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	C₂×C₃⋊C₈	1	2	yes	complemented
124	#168	—	6	2	no	—
125	#168	—	12	1	no	—
126	#168	—	12	2	no	—
127	C₂×D₈	Dic₁₅	1	1	yes	characteristic
128	C₂²×C₈	—	30	2	no	—
129	C₄×D₄	C₂×D₁₅	1	2	yes	—
130	#107	Dic₁₅	1	1	yes	characteristic
131	C₄⋊C₈	—	15	2	no	—
132	#102	Dic₁₅	1	2	yes	—
133	C₂²⋊C₈	—	30	2	no	—
134	C₄×C₈	—	15	1	no	—
135	C₄×C₈	—	15	1	no	—
136	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
137	C₂×C₃×C₅	—	4	4	no	—
138	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
139	C₂×C₃×C₅	#442	1	1	yes	characteristic
140	C₂×C₃×C₅	#444	1	1	yes	characteristic
141	C₂³×C₃	—	2	2	no	—
142	C₃×D₄	—	2	2	no	—
143	C₃×D₄	C₂×#168	1	4	yes	complemented
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
148	C₂×C₃×C₄	#711	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	C₂×#168	1	2	yes	complemented
152	C₃⋊C₈	—	10	2	no	—
153	C₃⋊C₈	—	20	1	no	—
154	C₃⋊C₈	—	20	2	no	—
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	4	2	no	—
157	C₂²×C₅	—	4	2	no	—
158	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	4	2	no	—
161	C₄×C₅	#911	1	2	yes	—
162	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
163	C₄×C₅	#948	1	1	yes	characteristic
164	C₂×D₄	C₂×Dic₁₅	1	2	yes	—
165	C₂²×C₄	—	2	2	no	—
166	D₈	C₃⋊#168	1	4	yes	complemented
167	C₂×C₈	—	15	2	no	—
168	C₂×C₈	—	30	2	no	—
169	C₂×C₈	—	30	1	no	—
170	C₂×C₈	—	60	4	no	—
171	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
172	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
173	C₄⋊C₄	C₂×Dic₁₅	1	2	yes	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₄²	C₂²×D₁₅	1	1	yes	characteristic
176	C₃×C₅	#1580	1	1	yes	characteristic, complemented, Hall
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	#4057	1	2	yes	—
184	C₃×C₄	C₂²×#168	1	1	yes	characteristic
185	C₃×C₄	#4094	1	1	yes	characteristic
186	C₂×C₅	—	4	4	no	—
187	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
188	C₂×C₅	#5152	1	1	yes	characteristic
189	C₂×C₅	#5197	1	1	yes	characteristic
190	C₂³	—	2	2	no	—
191	D₄	—	2	2	no	—
192	D₄	C₂×C₃⋊#168	1	4	yes	complemented
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
197	C₂×C₄	#6819	1	1	yes	characteristic
198	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	30	2	no	—
201	C₈	—	60	1	no	—
202	C₈	—	60	2	no	—
203	C₈	C₂×C₃⋊#168	1	2	yes	complemented
204	C₂×C₃	—	4	4	no	—
205	C₂×C₃	D₄×#168	1	1	yes	characteristic
206	C₂×C₃	#65441	1	1	yes	characteristic
207	C₂×C₃	#65486	1	1	yes	characteristic
208	C₅	#70878	1	1	yes	characteristic, complemented, Sylow
209	C₂²	—	2	4	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	D₄×Dic₁₅	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	#91800	1	2	yes	—
216	C₄	C₂²×C₃⋊#168	1	1	yes	characteristic
217	C₄	#91837	1	1	yes	characteristic
218	C₃	#10601671	1	1	yes	characteristic, complemented, Sylow
219	C₂	—	4	4	no	—
220	C₂	D₄×C₃⋊#168	1	1	yes	characteristic
221	C₂	#11751253	1	1	yes	characteristic
222	C₂	#11751298	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₄×C₅×D₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×C₅×D₈	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×D₈	D₁₅
hypocenter	C₃×C₅	#1580
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×C₃⋊#168	C₂	semidirect
3. D₄⋊(C₃⋊#168)	C₂	semidirect
4. #11746278	C₂	semidirect
5. #11746276	C₂	semidirect
25. C₃×C₅×D₈	C₈	semidirect
53. C₃×C₅×D₄	C₂×C₈	semidirect
61. C₃×C₅×C₈	C₂×C₈	semidirect
77. C₅×D₈	C₃⋊C₈	semidirect
103. C₃×D₈	#168	semidirect
115. C₅×D₄	C₂×C₃⋊C₈	semidirect
123. C₅×C₈	C₂×C₃⋊C₈	semidirect
143. C₃×D₄	C₂×#168	semidirect
151. C₃×C₈	C₂×#168	semidirect
166. D₈	C₃⋊#168	semidirect
176. C₃×C₅	#1580	semidirect
192. D₄	C₂×C₃⋊#168	semidirect
203. C₈	C₂×C₃⋊#168	semidirect
208. C₅	#70878	semidirect
218. C₃	#10601671	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₈	2
D₄×C₃⋊#168	2
D₄⋊(C₃⋊#168)	2
#11746278	2
#11746276	2
#10601671	3
#70878	5