Subgroups — #168×D₂₄ #14286102

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	D₁₂×#168	C₂	1	2	yes	maximal, complemented
3	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745837	C₂	1	2	yes	maximal, complemented
6	D₈×#168	—	3	1	no	maximal, Hall
7	C₂×C₅×D₂₄	C₄	1	1	yes	characteristic
8	C₄×C₅×D₁₂	C₂²	1	2	yes	—
9	C₂×D₃×#168	—	2	2	no	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
12	C₅×#821	C₄	1	2	yes	—
13	C₅×#818	C₄	1	1	yes	characteristic
14	C₃×C₄×#168	C₂²	1	1	yes	characteristic
15	C₃×C₄×#168	C₂²	1	1	yes	characteristic
16	(C₂×D₃)⋊#168	—	2	2	no	—
17	C₈×D₂₄	—	5	1	no	maximal, Hall
18	C₄×C₅×D₈	—	3	1	no	—
19	D₄×#168	—	3	2	no	—
20	D₄⋊#168	—	3	2	no	—
21	#64678	—	3	1	no	—
22	C₈×#168	D₃	1	1	yes	characteristic, complemented
23	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₅×D₃	—	2	2	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃⋊#168	—	2	2	no	—
29	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
30	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
31	C₅×D₂₄	C₈	1	4	yes	complemented
32	C₂×C₃×#168	—	2	1	no	—
33	C₂×C₃×#168	D₄	1	1	yes	characteristic
34	C₂×C₃×#168	D₄	1	1	yes	characteristic
35	D₃×#168	—	4	4	no	—
36	C₄×D₂₄	D₅	1	1	yes	characteristic
37	C₈×D₁₂	—	5	2	no	—
38	C₃×C₈²	—	5	1	no	—
39	#4459	—	5	2	no	—
40	#4458	—	5	1	no	—
41	C₂×C₅×D₈	—	3	1	no	—
42	C₄×C₅×D₄	—	3	2	no	—
43	C₂²×#168	—	6	2	no	—
44	C₅×#107	—	3	1	no	—
45	C₅×#102	—	3	2	no	—
46	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
47	C₂²⋊#168	—	6	2	no	—
48	C₄⋊#168	—	3	2	no	—
49	C₄×#168	C₂×D₃	1	1	yes	characteristic
50	C₄×#168	C₂×D₃	1	1	yes	characteristic
51	C₈×D₈	—	15	1	no	Sylow
52	C₂²×C₅×D₃	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
56	C₂×C₅×Dic₃	—	2	2	no	—
57	C₅×D₁₂	—	2	2	no	—
58	C₅×D₁₂	C₂×C₈	1	4	yes	complemented
59	C₄×C₅×D₃	—	4	2	no	—
60	C₄×C₅×D₃	—	4	2	no	—
61	C₃×C₅×C₈	—	2	1	no	—
62	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
63	C₃⋊#168	—	4	2	no	—
64	C₃×#168	—	2	2	no	—
65	C₃×#168	—	2	1	no	—
66	C₃×#168	D₈	1	2	yes	complemented
67	C₂×D₂₄	Dic₅	1	1	yes	characteristic
68	C₂×D₃×C₈	—	10	2	no	—
69	C₄×D₁₂	C₂×D₅	1	2	yes	—
70	C₃×C₄×C₈	—	5	1	no	—
71	C₃×C₄×C₈	—	5	1	no	—
72	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#821	Dic₅	1	2	yes	—
74	(C₂×D₃)⋊C₈	—	10	2	no	—
75	#818	Dic₅	1	1	yes	characteristic
76	C₄⋊(C₃⋊C₈)	—	5	2	no	—
77	C₂×C₅×D₄	—	3	2	no	—
78	C₂²×C₄×C₅	—	6	2	no	—
79	C₅×D₈	—	3	4	no	—
80	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
81	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₂²⋊C₄	—	6	2	no	—
84	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₂×#168	—	2	1	no	—
86	C₂×#168	—	6	2	no	—
87	C₂×#168	—	12	4	no	—
88	C₂×#168	D₁₂	1	1	yes	characteristic
89	C₂×#168	D₁₂	1	1	yes	characteristic
90	C₄×D₈	—	3	1	no	—
91	C₈×D₄	—	15	2	no	—
92	#335	—	15	1	no	—
93	D₄⋊C₈	—	15	2	no	—
94	C₈²	—	5	1	no	—
95	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
96	C₂×C₅×D₃	—	2	4	no	—
97	C₂×C₅×D₃	—	4	2	no	—
98	C₂×C₅×D₃	—	4	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₃×C₄×C₅	#131	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
103	C₃×C₄×C₅	#135	1	1	yes	characteristic
104	C₅×Dic₃	—	4	2	no	—
105	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
106	C₂×C₄×D₃	—	2	2	no	—
107	C₂×C₃×C₈	—	5	1	no	—
108	C₂×C₃×C₈	—	5	1	no	—
109	C₂×C₃×C₈	—	10	1	no	—
110	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
112	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
113	(C₂×D₃)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
115	C₂×C₃⋊C₈	—	10	2	no	—
116	D₂₄	#168	1	4	yes	direct product factor
117	D₃×C₈	—	20	4	no	—
118	C₂³×C₅	—	6	2	no	—
119	C₅×D₄	—	3	4	no	—
120	C₅×D₄	—	6	2	no	—
121	C₂×C₄×C₅	—	6	2	no	—
122	C₂×C₄×C₅	—	12	2	no	—
123	C₂×C₄×C₅	—	12	2	no	—
124	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
126	C₂×C₄×C₅	#235	1	1	yes	characteristic
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	D₃×C₈	1	2	yes	complemented
129	#168	—	2	2	no	—
130	#168	—	2	1	no	—
131	#168	—	12	2	no	—
132	#168	D₂₄	1	2	yes	direct product factor
133	C₂×D₈	—	3	1	no	—
134	C₂²×C₈	—	30	2	no	—
135	C₄×D₄	—	3	2	no	—
136	#107	—	3	1	no	—
137	C₄⋊C₈	—	15	2	no	—
138	#102	—	3	2	no	—
139	C₂²⋊C₈	—	30	2	no	—
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	D₃×D₅	1	1	yes	characteristic
143	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
144	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
145	C₂×C₃×C₅	#443	1	1	yes	characteristic
146	C₅×D₃	—	4	4	no	—
147	C₂²×D₃	—	2	2	no	—
148	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
149	C₂×C₃×C₄	#711	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
151	C₂×Dic₃	—	2	2	no	—
152	D₁₂	—	2	2	no	—
153	D₁₂	C₂×#168	1	4	yes	complemented
154	C₄×D₃	—	4	2	no	—
155	C₄×D₃	—	4	2	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	5	2	no	—
158	C₃×C₈	—	10	2	no	—
159	C₃×C₈	—	10	1	no	—
160	C₃×C₈	C₂×#168	1	2	yes	complemented
161	C₃⋊C₈	—	20	2	no	—
162	C₂²×C₅	—	6	4	no	—
163	C₂²×C₅	—	12	2	no	—
164	C₂²×C₅	—	12	2	no	—
165	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	12	2	no	—
168	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
169	C₄×C₅	#901	1	1	yes	characteristic
170	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
171	C₄×C₅	#904	1	1	yes	characteristic
172	C₂×D₄	—	3	2	no	—
173	C₂²×C₄	—	6	2	no	—
174	D₈	—	3	4	no	—
175	C₂×C₈	—	5	1	no	—
176	C₂×C₈	—	5	1	no	—
177	C₂×C₈	—	10	1	no	—
178	C₂×C₈	—	30	2	no	—
179	C₂×C₈	—	60	4	no	—
180	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
181	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
182	C₄⋊C₄	—	3	2	no	—
183	C₂²⋊C₄	—	6	2	no	—
184	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
185	C₃×C₅	C₈×D₈	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
187	C₂×D₃	—	2	4	no	—
188	C₂×D₃	—	4	2	no	—
189	C₂×D₃	—	4	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	D₅×D₈	1	1	yes	characteristic
192	C₃×C₄	#4058	1	1	yes	characteristic
193	C₃×C₄	C₂²×#168	1	1	yes	characteristic
194	C₃×C₄	#4094	1	1	yes	characteristic
195	Dic₃	—	4	2	no	—
196	C₂×C₅	—	12	4	no	—
197	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
198	C₂×C₅	C₄×D₂₄	1	1	yes	characteristic
199	C₂×C₅	#4700	1	1	yes	characteristic
200	C₂³	—	6	2	no	—
201	D₄	—	3	4	no	—
202	D₄	—	6	2	no	—
203	C₂×C₄	—	6	2	no	—
204	C₂×C₄	—	12	2	no	—
205	C₂×C₄	—	12	2	no	—
206	C₂×C₄	#6772	1	1	yes	characteristic
207	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
208	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	5	2	no	—
211	C₈	—	10	2	no	—
212	C₈	—	10	1	no	—
213	C₈	—	60	2	no	—
214	C₈	D₃×#168	1	2	yes	complemented
215	C₂×C₃	D₄×#168	1	1	yes	characteristic
216	C₂×C₃	D₈×Dic₅	1	1	yes	characteristic
217	C₂×C₃	#65485	1	1	yes	characteristic
218	D₃	—	4	4	no	—
219	C₅	C₈×D₂₄	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	6	4	no	—
221	C₂²	—	12	2	no	—
222	C₂²	—	12	2	no	—
223	C₂²	Dic₅×D₁₂	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	12	2	no	—
226	C₄	D₅×D₂₄	1	1	yes	characteristic
227	C₄	#91270	1	1	yes	characteristic
228	C₄	C₂×D₃×#168	1	1	yes	characteristic
229	C₄	#91286	1	1	yes	characteristic
230	C₃	D₈×#168	1	1	yes	characteristic, complemented, Sylow
231	C₂	—	12	4	no	—
232	C₂	D₁₂×#168	1	1	yes	characteristic
233	C₂	Dic₅×D₂₄	1	1	yes	characteristic
234	C₂	#11748062	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₁₂
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₂₄	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	C₈×D₈
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. D₁₂×#168	C₂	semidirect
3. #11746279	C₂	semidirect
4. C₃×C₈×#168	C₂	semidirect
5. #11745837	C₂	semidirect
22. C₈×#168	D₃	semidirect
31. C₅×D₂₄	C₈	semidirect
58. C₅×D₁₂	C₂×C₈	semidirect
62. C₃×C₅×C₈	C₂×C₈	semidirect
66. C₃×#168	D₈	semidirect
116. D₂₄	#168	direct (N × H)
128. C₅×C₈	D₃×C₈	semidirect
132. #168	D₂₄	direct (N × H)
153. D₁₂	C₂×#168	semidirect
160. C₃×C₈	C₂×#168	semidirect
185. C₃×C₅	C₈×D₈	semidirect
214. C₈	D₃×#168	semidirect
219. C₅	C₈×D₂₄	semidirect
230. C₃	D₈×#168	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₂₄	2
D₁₂×#168	2
#11746279	2
C₃×C₈×#168	2
#11745837	2
D₈×#168	3
C₈×D₂₄	5

Subgroups — #168×D24 #14286102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #168×D₂₄ #14286102