Subgroups — D₈×C₃⋊#168 #14286074

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	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	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×#168	—	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	D₈×C₃⋊C₈	—	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	#64678	—	3	1	no	—
22	C₈×#168	—	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	1	no	—
32	C₂×C₃⋊#168	—	2	2	no	—
33	C₂×C₃⋊#168	—	4	4	no	—
34	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
35	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
36	C₃×C₄×D₈	D₅	1	1	yes	characteristic
37	D₄×C₃⋊C₈	—	5	2	no	—
38	D₄⋊(C₃⋊C₈)	—	5	2	no	—
39	#4458	—	5	1	no	—
40	C₈×C₃⋊C₈	—	5	1	no	—
41	C₂×C₅×D₈	Dic₃	1	1	yes	characteristic
42	C₄×C₅×D₄	C₂×D₃	1	2	yes	—
43	C₂²×#168	—	6	2	no	—
44	C₅×#107	Dic₃	1	1	yes	characteristic
45	C₅×#102	Dic₃	1	2	yes	—
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	—	3	1	no	—
50	C₄×#168	—	3	1	no	—
51	C₈×D₈	—	15	1	no	Sylow
52	C₂³×C₃×C₅	—	2	2	no	—
53	C₃×C₅×D₄	—	2	2	no	—
54	C₃×C₅×D₄	C₂×C₈	1	4	yes	complemented
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
61	C₃×C₅×C₈	—	2	1	no	—
62	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
63	C₃⋊#168	—	2	2	no	—
64	C₃⋊#168	—	2	1	no	—
65	C₃⋊#168	—	4	2	no	—
66	C₃⋊#168	D₈	1	2	yes	direct product factor
67	C₂×C₃×D₈	Dic₅	1	1	yes	characteristic
68	C₃×C₄×D₄	C₂×D₅	1	2	yes	—
69	C₂²×C₃⋊C₈	—	10	2	no	—
70	C₃×#107	Dic₅	1	1	yes	characteristic
71	C₃×#102	Dic₅	1	2	yes	—
72	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#830	—	10	2	no	—
74	C₄⋊(C₃⋊C₈)	—	5	2	no	—
75	C₄×C₃⋊C₈	—	5	1	no	—
76	C₄×C₃⋊C₈	—	5	1	no	—
77	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₅×D₈	C₃⋊C₈	1	4	yes	complemented
80	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
82	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
83	C₅×C₂²⋊C₄	—	2	2	no	—
84	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₂×#168	—	3	1	no	—
86	C₂×#168	—	3	1	no	—
87	C₂×#168	—	6	1	no	—
88	C₂×#168	—	6	2	no	—
89	C₂×#168	—	12	4	no	—
90	C₄×D₈	D₁₅	1	1	yes	characteristic
91	C₈×D₄	—	15	2	no	—
92	#335	—	15	1	no	—
93	D₄⋊C₈	—	15	2	no	—
94	C₈²	—	15	1	no	—
95	C₂²×C₃×C₅	—	2	4	no	—
96	C₂²×C₃×C₅	—	4	2	no	—
97	C₂²×C₃×C₅	—	4	2	no	—
98	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	4	2	no	—
101	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
102	C₃×C₄×C₅	#131	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
104	C₃×C₄×C₅	#135	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₃×D₈	#168	1	4	yes	complemented
108	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
111	C₃×C₂²⋊C₄	—	2	2	no	—
112	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
113	C₂×C₃⋊C₈	—	5	1	no	—
114	C₂×C₃⋊C₈	—	5	1	no	—
115	C₂×C₃⋊C₈	—	10	1	no	—
116	C₂×C₃⋊C₈	—	10	2	no	—
117	C₂×C₃⋊C₈	—	20	4	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₅×D₄	—	2	2	no	—
120	C₅×D₄	C₂×C₃⋊C₈	1	4	yes	complemented
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#237	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	C₂×C₃⋊C₈	1	2	yes	complemented
129	#168	—	3	2	no	—
130	#168	—	6	2	no	—
131	#168	—	6	1	no	—
132	#168	—	12	2	no	—
133	C₂×D₈	Dic₁₅	1	1	yes	characteristic
134	C₂²×C₈	—	30	2	no	—
135	C₄×D₄	C₂×D₁₅	1	2	yes	—
136	#107	Dic₁₅	1	1	yes	characteristic
137	C₄⋊C₈	—	15	2	no	—
138	#102	Dic₁₅	1	2	yes	—
139	C₂²⋊C₈	—	30	2	no	—
140	C₄×C₈	—	15	1	no	—
141	C₄×C₈	—	15	1	no	—
142	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
143	C₂×C₃×C₅	—	4	4	no	—
144	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
145	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
146	C₂×C₃×C₅	#443	1	1	yes	characteristic
147	C₂³×C₃	—	2	2	no	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	C₂×#168	1	4	yes	complemented
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
154	C₂×C₃×C₄	#711	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	C₂×#168	1	2	yes	complemented
158	C₃⋊C₈	—	5	2	no	—
159	C₃⋊C₈	—	10	2	no	—
160	C₃⋊C₈	—	10	1	no	—
161	C₃⋊C₈	—	20	2	no	—
162	C₂²×C₅	—	2	4	no	—
163	C₂²×C₅	—	4	2	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	D₃×D₈	1	1	yes	characteristic
169	C₄×C₅	#912	1	1	yes	characteristic
170	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
171	C₄×C₅	#948	1	1	yes	characteristic
172	C₂×D₄	C₂×Dic₁₅	1	2	yes	—
173	C₂²×C₄	—	2	2	no	—
174	D₈	C₃⋊#168	1	4	yes	direct product factor
175	C₂×C₈	—	15	1	no	—
176	C₂×C₈	—	15	1	no	—
177	C₂×C₈	—	30	1	no	—
178	C₂×C₈	—	30	2	no	—
179	C₂×C₈	—	60	4	no	—
180	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
181	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
182	C₄⋊C₄	C₂×Dic₁₅	1	2	yes	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	C₂²×D₁₅	1	1	yes	characteristic
185	C₃×C₅	C₈×D₈	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	—	2	4	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	—	4	2	no	—
189	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	4	2	no	—
192	C₃×C₄	D₅×D₈	1	1	yes	characteristic
193	C₃×C₄	#4058	1	1	yes	characteristic
194	C₃×C₄	C₂²×#168	1	1	yes	characteristic
195	C₃×C₄	#4094	1	1	yes	characteristic
196	C₂×C₅	—	4	4	no	—
197	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
198	C₂×C₅	Dic₃×D₈	1	1	yes	characteristic
199	C₂×C₅	#5196	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	C₂×C₃⋊#168	1	4	yes	complemented
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
207	C₂×C₄	#6819	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	15	2	no	—
211	C₈	—	30	2	no	—
212	C₈	—	30	1	no	—
213	C₈	—	60	2	no	—
214	C₈	C₂×C₃⋊#168	1	2	yes	complemented
215	C₂×C₃	—	4	4	no	—
216	C₂×C₃	D₄×#168	1	1	yes	characteristic
217	C₂×C₃	D₈×Dic₅	1	1	yes	characteristic
218	C₂×C₃	#65485	1	1	yes	characteristic
219	C₅	D₈×C₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	4	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	D₄×Dic₁₅	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	D₈×D₁₅	1	1	yes	characteristic
227	C₄	#91801	1	1	yes	characteristic
228	C₄	C₂²×C₃⋊#168	1	1	yes	characteristic
229	C₄	#91837	1	1	yes	characteristic
230	C₃	D₈×#168	1	1	yes	characteristic, complemented, Sylow
231	C₂	—	4	4	no	—
232	C₂	D₄×C₃⋊#168	1	1	yes	characteristic
233	C₂	D₈×Dic₁₅	1	1	yes	characteristic
234	C₂	#11751297	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	C₈×D₈
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. #11746279	C₂	semidirect
5. C₈×C₃⋊#168	C₂	semidirect
25. C₃×C₅×D₈	C₈	semidirect
54. C₃×C₅×D₄	C₂×C₈	semidirect
62. C₃×C₅×C₈	C₂×C₈	semidirect
66. C₃⋊#168	D₈	direct (N × H)
79. C₅×D₈	C₃⋊C₈	semidirect
107. C₃×D₈	#168	semidirect
120. C₅×D₄	C₂×C₃⋊C₈	semidirect
128. C₅×C₈	C₂×C₃⋊C₈	semidirect
149. C₃×D₄	C₂×#168	semidirect
157. C₃×C₈	C₂×#168	semidirect
174. D₈	C₃⋊#168	direct (N × H)
185. C₃×C₅	C₈×D₈	semidirect
202. D₄	C₂×C₃⋊#168	semidirect
214. C₈	C₂×C₃⋊#168	semidirect
219. C₅	D₈×C₃⋊C₈	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
C₃×C₄×C₅×D₈	2
D₄×C₃⋊#168	2
D₄⋊(C₃⋊#168)	2
#11746279	2
C₈×C₃⋊#168	2
D₈×#168	3
D₈×C₃⋊C₈	5

Subgroups — D8×C3⋊#168 #14286074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×C₃⋊#168 #14286074