Subgroups — #168×#239 #14418940

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#239	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746998	C₂	1	3	yes	maximal, complemented
5	D₁₂×#168	C₂	1	3	yes	maximal, complemented
6	C₄×D₃×#168	C₂	1	3	yes	maximal, complemented
7	#11746853	C₂	1	3	yes	maximal, complemented
8	#41×#168	—	3	1	no	maximal, Hall
9	C₂×C₅×#239	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×Q₈×Dic₃	C₄	1	1	yes	characteristic
12	C₅×#893	C₄	1	3	yes	—
13	C₅×#892	C₄	1	3	yes	—
14	C₄×C₅×D₁₂	C₂²	1	3	yes	—
15	C₄²×C₅×D₃	C₂²	1	3	yes	—
16	C₂×D₃×#168	C₂²	1	3	yes	—
17	C₂×D₃×#168	C₂²	1	3	yes	—
18	C₄⋊(C₃⋊#168)	C₂²	1	3	yes	—
19	C₄×C₃⋊#168	C₂²	1	3	yes	—
20	C₃×C₄⋊#168	C₂²	1	3	yes	—
21	C₃×C₄×#168	C₂²	1	3	yes	—
22	(C₂×D₃)⋊#168	C₂²	1	6	yes	—
23	Dic₃×#168	C₂²	1	3	yes	—
24	Dic₃×#168	C₂²	1	1	yes	characteristic
25	C₈×#239	—	5	1	no	maximal, Hall
26	C₄×C₅×#41	—	3	1	no	—
27	Q₈×#168	—	3	1	no	—
28	Q₈×#168	D₃	1	1	yes	characteristic, complemented
29	D₄×#168	—	3	3	no	—
30	D₄×#168	—	3	3	no	—
31	#65214	—	3	3	no	—
32	C₂×C₄×#168	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₅×#239	C₈	1	8	yes	complemented
35	C₂×C₅×D₁₂	C₂×C₄	1	3	yes	—
36	C₂×C₄×C₅×D₃	C₂×C₄	1	3	yes	—
37	C₂×C₄×C₅×D₃	C₂×C₄	1	3	yes	—
38	C₂×C₄×C₅×D₃	C₂³	1	3	yes	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₃×C₄²×C₅	C₂³	1	3	yes	—
41	C₂×C₃⋊#168	C₂³	1	3	yes	—
42	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
43	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	6	yes	—
44	C₅×C₄⋊Dic₃	C₂×C₄	1	3	yes	—
45	C₄×C₅×Dic₃	C₂×C₄	1	3	yes	—
46	C₄×C₅×Dic₃	C₂×C₄	1	3	yes	—
47	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
48	C₂×C₃×#168	C₂³	1	3	yes	—
49	C₂×C₃×#168	C₂³	1	1	yes	characteristic
50	D₃×#168	—	2	6	no	—
51	D₃×#168	—	2	6	no	—
52	C₄×#239	D₅	1	1	yes	characteristic
53	C₃×C₈×Q₈	—	5	1	no	—
54	Q₈×C₃⋊C₈	—	5	1	no	—
55	#4834	—	5	3	no	—
56	#4833	—	5	3	no	—
57	C₈×D₁₂	—	5	3	no	—
58	C₄×D₃×C₈	—	5	3	no	—
59	C₂×C₅×#41	—	3	1	no	—
60	C₄×C₅×Q₈	—	3	1	no	—
61	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×D₄	—	3	3	no	—
63	C₄×C₅×D₄	—	3	3	no	—
64	C₅×#117	—	3	3	no	—
65	C₂×C₄²×C₅	—	3	3	no	—
66	C₂²×#168	—	3	3	no	—
67	C₂²×#168	—	3	3	no	—
68	C₂²⋊#168	—	3	6	no	—
69	C₄⋊#168	—	3	3	no	—
70	C₄⋊#168	C₂×D₃	1	3	yes	—
71	C₄×#168	—	3	3	no	—
72	C₄×#168	—	3	1	no	—
73	C₄×#168	—	3	3	no	—
74	C₄×#168	C₂×D₃	1	3	yes	—
75	C₈×#41	—	15	1	no	Sylow
76	C₂²×C₅×D₃	C₂²×C₄	1	3	yes	—
77	C₃×C₅×Q₈	C₂×C₈	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₅×Dic₃	C₂²×C₄	1	3	yes	—
82	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
84	C₅×D₁₂	C₂×C₈	1	12	yes	complemented
85	C₄×C₅×D₃	—	2	3	no	—
86	C₄×C₅×D₃	—	2	3	no	—
87	C₄×C₅×D₃	—	2	6	no	—
88	C₄×C₅×D₃	C₂×C₈	1	12	yes	complemented
89	C₃⋊#168	—	2	3	no	—
90	C₃⋊#168	#41	1	2	yes	complemented
91	C₃×#168	—	2	3	no	—
92	C₃×#168	#41	1	2	yes	complemented
93	C₂×#239	Dic₅	1	1	yes	characteristic
94	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
95	Q₈×Dic₃	Dic₅	1	1	yes	characteristic
96	C₂×D₃×C₈	—	5	3	no	—
97	C₂×D₃×C₈	—	5	3	no	—
98	#893	Dic₅	1	3	yes	—
99	#892	Dic₅	1	3	yes	—
100	C₄×D₁₂	C₂×D₅	1	3	yes	—
101	C₄²×D₃	C₂×D₅	1	3	yes	—
102	C₃×C₄⋊C₈	—	5	3	no	—
103	C₃×C₄×C₈	—	5	3	no	—
104	(C₂×D₃)⋊C₈	—	5	6	no	—
105	C₈×Dic₃	—	5	1	no	—
106	C₈×Dic₃	—	5	3	no	—
107	C₄⋊(C₃⋊C₈)	—	5	3	no	—
108	C₄×C₃⋊C₈	—	5	3	no	—
109	C₅×#41	—	3	8	no	—
110	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
111	C₂×C₅×D₄	—	3	3	no	—
112	C₂²×C₄×C₅	—	3	3	no	—
113	C₂²×C₄×C₅	—	3	3	no	—
114	C₂²×C₄×C₅	—	3	3	no	—
115	C₅×C₄⋊C₄	—	3	3	no	—
116	C₅×C₄⋊C₄	C₄×D₃	1	3	yes	—
117	C₅×C₂²⋊C₄	—	3	6	no	—
118	C₄²×C₅	—	3	3	no	—
119	C₄²×C₅	—	3	3	no	—
120	C₄²×C₅	—	3	1	no	—
121	C₄²×C₅	C₂²×D₃	1	3	yes	—
122	C₂×#168	—	3	1	no	—
123	C₂×#168	—	3	3	no	—
124	C₂×#168	—	6	6	no	—
125	C₂×#168	—	6	6	no	—
126	C₂×#168	C₂²×D₃	1	3	yes	—
127	C₂×#168	C₂²×D₃	1	1	yes	characteristic
128	C₄×#41	—	3	1	no	—
129	C₈×Q₈	—	5	1	no	—
130	C₈×Q₈	—	15	1	no	—
131	C₈×D₄	—	15	3	no	—
132	C₈×D₄	—	15	3	no	—
133	#431	—	15	3	no	—
134	C₂×C₄×C₈	—	15	3	no	—
135	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₅×D₃	—	2	3	no	—
137	C₂×C₅×D₃	—	2	3	no	—
138	C₂×C₅×D₃	C₂²×C₈	1	6	yes	—
139	C₃×C₄×C₅	—	2	3	no	—
140	C₃×C₄×C₅	C₂²×C₈	1	6	yes	—
141	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₅×Dic₃	—	2	3	no	—
144	C₅×Dic₃	C₂²×C₈	1	2	yes	—
145	C₅×Dic₃	#131	1	2	yes	—
146	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
147	#239	#168	1	8	yes	direct product factor
148	C₂×D₁₂	C₂×Dic₅	1	3	yes	—
149	C₂×C₄×D₃	C₂×Dic₅	1	3	yes	—
150	C₂×C₄×D₃	C₂×Dic₅	1	3	yes	—
151	C₂×C₄×D₃	C₂²×D₅	1	3	yes	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₂×C₃×C₈	—	5	3	no	—
154	C₃×C₄⋊C₄	C₂×Dic₅	1	3	yes	—
155	C₃×C₄²	C₂²×D₅	1	3	yes	—
156	(C₂×D₃)⋊C₄	C₂×Dic₅	1	6	yes	—
157	C₄⋊Dic₃	C₂×Dic₅	1	3	yes	—
158	C₄×Dic₃	C₂×Dic₅	1	3	yes	—
159	C₄×Dic₃	C₂×Dic₅	1	3	yes	—
160	C₄×Dic₃	C₂²×D₅	1	1	yes	characteristic
161	C₂×C₃⋊C₈	—	5	1	no	—
162	C₂×C₃⋊C₈	—	5	3	no	—
163	D₃×C₈	—	10	6	no	—
164	D₃×C₈	—	10	6	no	—
165	C₂³×C₅	—	3	3	no	—
166	C₅×Q₈	D₃×C₈	1	4	yes	complemented
167	C₅×D₄	—	3	12	no	—
168	C₂×C₄×C₅	—	3	1	no	—
169	C₂×C₄×C₅	—	3	1	no	—
170	C₂×C₄×C₅	—	3	12	no	—
171	C₂×C₄×C₅	—	3	3	no	—
172	C₂×C₄×C₅	—	6	6	no	—
173	C₂×C₄×C₅	—	6	3	no	—
174	C₂×C₄×C₅	—	6	3	no	—
175	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
176	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
177	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
178	#168	—	2	3	no	—
179	#168	—	3	2	no	—
180	#168	—	6	3	no	—
181	#168	#239	1	2	yes	direct product factor
182	C₂×#41	—	3	1	no	—
183	C₂²×C₈	—	15	3	no	—
184	C₂²×C₈	—	15	3	no	—
185	C₄×Q₈	—	3	1	no	—
186	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
187	C₄×D₄	—	3	3	no	—
188	C₄×D₄	—	3	3	no	—
189	#117	—	3	3	no	—
190	C₂×C₄²	—	3	3	no	—
191	C₄⋊C₈	—	5	3	no	—
192	C₄⋊C₈	—	15	3	no	—
193	C₂²⋊C₈	—	15	6	no	—
194	C₄×C₈	—	5	3	no	—
195	C₄×C₈	—	15	1	no	—
196	C₄×C₈	—	15	3	no	—
197	C₄×C₈	—	15	3	no	—
198	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
201	C₅×D₃	—	2	6	no	—
202	C₂²×D₃	C₂²×Dic₅	1	3	yes	—
203	C₃×Q₈	C₂×#168	1	4	yes	complemented
204	C₂×C₃×C₄	C₂²×Dic₅	1	3	yes	—
205	C₂×C₃×C₄	C₂²×Dic₅	1	3	yes	—
206	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
207	C₂×Dic₃	C₂²×Dic₅	1	3	yes	—
208	C₂×Dic₃	C₂²×Dic₅	1	1	yes	characteristic
209	C₂×Dic₃	C₂²×Dic₅	1	1	yes	characteristic
210	D₁₂	C₂×#168	1	12	yes	complemented
211	C₄×D₃	—	2	3	no	—
212	C₄×D₃	—	2	3	no	—
213	C₄×D₃	—	2	6	no	—
214	C₄×D₃	C₂×#168	1	12	yes	complemented
215	C₃×C₈	—	5	2	no	—
216	C₃×C₈	—	10	3	no	—
217	C₃⋊C₈	—	5	2	no	—
218	C₃⋊C₈	—	10	3	no	—
219	C₂²×C₅	—	3	6	no	—
220	C₂²×C₅	—	6	3	no	—
221	C₂²×C₅	—	6	3	no	—
222	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
223	C₄×C₅	—	2	3	no	—
224	C₄×C₅	—	3	2	no	—
225	C₄×C₅	—	3	2	no	—
226	C₄×C₅	—	6	3	no	—
227	C₄×C₅	C₂×D₃×C₈	1	6	yes	—
228	C₄×C₅	C₂×#239	1	1	yes	characteristic
229	C₄×C₅	D₃×#41	1	1	yes	characteristic
230	#41	—	3	8	no	—
231	C₂×Q₈	D₃×Dic₅	1	1	yes	characteristic
232	C₂×D₄	—	3	3	no	—
233	C₂²×C₄	—	3	3	no	—
234	C₂²×C₄	—	3	3	no	—
235	C₂²×C₄	—	3	3	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	3	no	—
238	C₂×C₈	—	15	1	no	—
239	C₂×C₈	—	15	3	no	—
240	C₂×C₈	—	30	6	no	—
241	C₂×C₈	—	30	6	no	—
242	C₄⋊C₄	—	3	3	no	—
243	C₄⋊C₄	D₃×Dic₅	1	3	yes	—
244	C₂²⋊C₄	—	3	6	no	—
245	C₄²	—	3	3	no	—
246	C₄²	—	3	3	no	—
247	C₄²	—	3	1	no	—
248	C₄²	C₂×D₃×D₅	1	3	yes	—
249	C₃×C₅	C₈×#41	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
251	C₂×D₃	—	2	3	no	—
252	C₂×D₃	—	2	3	no	—
253	C₂×D₃	C₂²×#168	1	6	yes	—
254	C₃×C₄	—	2	3	no	—
255	C₃×C₄	C₂²×#168	1	6	yes	—
256	C₃×C₄	D₅×#41	1	1	yes	characteristic
257	C₃×C₄	D₅×#41	1	1	yes	characteristic
258	Dic₃	—	2	3	no	—
259	Dic₃	C₂²×#168	1	2	yes	—
260	Dic₃	#4094	1	2	yes	—
261	C₂×C₅	—	6	6	no	—
262	C₂×C₅	C₄×#239	1	1	yes	characteristic
263	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
264	C₂×C₅	D₃×#131	1	1	yes	characteristic
265	C₂³	—	3	3	no	—
266	Q₈	D₃×#168	1	4	yes	complemented
267	D₄	—	3	12	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	12	no	—
271	C₂×C₄	—	3	3	no	—
272	C₂×C₄	—	6	6	no	—
273	C₂×C₄	—	6	3	no	—
274	C₂×C₄	—	6	3	no	—
275	C₂×C₄	C₂×D₃×Dic₅	1	3	yes	—
276	C₂×C₄	C₂×D₃×Dic₅	1	3	yes	—
277	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
278	C₈	—	5	2	no	—
279	C₈	—	10	3	no	—
280	C₈	—	15	2	no	—
281	C₈	—	30	3	no	—
282	C₂×C₃	C₂³×#168	1	1	yes	characteristic
283	C₂×C₃	C₂×#4094	1	1	yes	characteristic
284	C₂×C₃	#41×Dic₅	1	1	yes	characteristic
285	D₃	—	2	6	no	—
286	C₅	C₈×#239	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	3	6	no	—
288	C₂²	—	6	3	no	—
289	C₂²	—	6	3	no	—
290	C₂²	C₂²×D₃×Dic₅	1	1	yes	characteristic
291	C₄	—	2	3	no	—
292	C₄	—	3	2	no	—
293	C₄	—	3	2	no	—
294	C₄	—	6	3	no	—
295	C₄	C₂×D₃×#168	1	6	yes	—
296	C₄	D₅×#239	1	1	yes	characteristic
297	C₄	#92034	1	1	yes	characteristic
298	C₃	#41×#168	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	6	6	no	—
300	C₂	C₂²×D₃×#168	1	1	yes	characteristic
301	C₂	Dic₅×#239	1	1	yes	characteristic
302	C₂	D₃×#4094	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₈
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×C₅×#239	C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	C₈×#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊#168	C₂	semidirect
3. C₃×Q₈×#168	C₂	semidirect
4. #11746998	C₂	semidirect
5. D₁₂×#168	C₂	semidirect
6. C₄×D₃×#168	C₂	semidirect
7. #11746853	C₂	semidirect
28. Q₈×#168	D₃	semidirect
34. C₅×#239	C₈	semidirect
77. C₃×C₅×Q₈	C₂×C₈	semidirect
84. C₅×D₁₂	C₂×C₈	semidirect
88. C₄×C₅×D₃	C₂×C₈	semidirect
90. C₃⋊#168	#41	semidirect
92. C₃×#168	#41	semidirect
147. #239	#168	direct (N × H)
166. C₅×Q₈	D₃×C₈	semidirect
181. #168	#239	direct (N × H)
203. C₃×Q₈	C₂×#168	semidirect
210. D₁₂	C₂×#168	semidirect
214. C₄×D₃	C₂×#168	semidirect
249. C₃×C₅	C₈×#41	semidirect
266. Q₈	D₃×#168	semidirect
286. C₅	C₈×#239	semidirect
298. C₃	#41×#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×#239	2
Q₈×C₃⋊#168	2
C₃×Q₈×#168	2
#11746998	2
D₁₂×#168	2
C₄×D₃×#168	2
#11746853	2
#41×#168	3
C₈×#239	5