Subgroups — C₂×#11747067 #14438759

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91763	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6660	C₂	1	4	yes	maximal, complemented
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747067	C₂	1	8	yes	maximal, direct product factor
6	C₂×#65374	—	3	1	no	maximal, Hall
7	C₂²×D₆₀	—	2	2	no	—
8	C₂²×D₆₀	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#91763	—	2	2	no	—
12	#91763	C₂²	1	4	yes	—
13	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
14	C₅×#869	—	2	2	no	—
15	C₅×#869	C₂²	1	4	yes	complemented
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₂×#6660	—	2	16	no	—
18	C₂×#6660	C₂²	1	16	yes	complemented
19	C₃×C₄×#168	C₂²	1	4	yes	complemented
20	C₂×#4693	—	5	1	no	maximal, Hall
21	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
22	C₂²×#688	—	3	4	no	—
23	C₂×#4020	—	3	1	no	—
24	#65374	—	3	8	no	—
25	C₂×C₄×#168	D₃	1	1	yes	characteristic, complemented
26	C₂³×D₁₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×D₆₀	—	2	8	no	—
30	C₂×D₆₀	—	2	2	no	—
31	C₂×D₆₀	—	2	8	no	—
32	C₂×D₆₀	—	4	8	no	—
33	C₂×D₆₀	C₂³	1	4	yes	—
34	C₂²×C₅×Dic₃	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	8	no	—
36	C₂×C₅×Dic₆	—	2	2	no	—
37	C₂×C₅×Dic₆	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₅×C₄⋊Dic₃	—	2	8	no	—
41	C₂×C₃×#168	D₄	1	8	yes	complemented
42	C₂×C₃×#168	C₂³	1	4	yes	—
43	#6660	—	2	32	no	—
44	C₂²×#204	—	5	4	no	—
45	C₂×#874	—	5	1	no	—
46	C₂×#869	D₅	1	1	yes	characteristic, complemented
47	C₂×C₃×C₄×C₈	—	5	1	no	—
48	#4693	—	5	8	no	—
49	C₂²×C₅×Q₈	—	3	2	no	—
50	C₂²×D₂₀	—	3	2	no	—
51	C₂²×D₂₀	—	6	2	no	—
52	C₅×C₄⋊Q₈	—	3	4	no	—
53	C₅×C₄⋊Q₈	—	6	2	no	—
54	C₂×C₅×C₄⋊C₄	—	6	2	no	—
55	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×#688	—	3	16	no	—
57	C₂×#688	—	6	16	no	—
58	C₂²×#168	C₂×D₃	1	2	yes	—
59	#4020	—	3	4	no	—
60	#4020	—	6	2	no	—
61	C₄×#168	C₂×D₃	1	4	yes	complemented
62	C₂×#492	—	15	1	no	Sylow
63	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×D₁₅	—	2	4	no	—
65	C₂²×D₁₅	—	4	4	no	—
66	C₂²×D₁₅	—	4	8	no	—
67	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	complemented
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
72	D₆₀	—	2	8	no	—
73	D₆₀	—	4	4	no	—
74	D₆₀	—	4	16	no	—
75	C₂×C₅×Dic₃	—	2	4	no	—
76	C₂×C₅×Dic₃	—	4	4	no	—
77	C₅×Dic₆	—	2	8	no	—
78	C₅×Dic₆	—	4	4	no	—
79	C₃×#168	C₂×D₄	1	8	yes	complemented
80	C₂²×D₁₂	—	5	2	no	—
81	C₂²×D₁₂	—	10	2	no	—
82	C₂²×Dic₆	C₂×D₅	1	2	yes	—
83	C₂²×C₃×C₈	—	5	2	no	—
84	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
85	C₂×C₄⋊Dic₃	—	2	2	no	—
86	C₂×#204	—	5	16	no	—
87	C₂×#204	—	10	16	no	—
88	#874	—	5	4	no	—
89	#874	—	10	2	no	—
90	#869	—	2	2	no	—
91	#869	C₂×D₅	1	4	yes	complemented
92	C₃×C₄×C₈	—	5	4	no	—
93	C₂³×D₅	—	6	2	no	—
94	C₂×C₅×Q₈	—	3	4	no	—
95	C₂×C₅×Q₈	—	6	2	no	—
96	C₂×C₅×Q₈	—	6	8	no	—
97	C₂²×C₄×C₅	—	6	2	no	—
98	C₂²×C₄×C₅	D₁₂	1	2	yes	—
99	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
100	C₂×D₂₀	—	3	4	no	—
101	C₂×D₂₀	—	6	8	no	—
102	C₂×D₂₀	—	6	2	no	—
103	C₂×D₂₀	—	6	8	no	—
104	C₂×D₂₀	—	12	8	no	—
105	C₅×C₄⋊C₄	—	6	8	no	—
106	C₄²×C₅	D₁₂	1	2	yes	—
107	C₄²×C₅	C₂²×D₃	1	2	yes	—
108	#688	—	6	32	no	—
109	C₂×#168	D₁₂	1	8	yes	complemented
110	C₂×#168	C₂²×D₃	1	4	yes	—
111	C₂²×SD₁₆	—	15	4	no	—
112	C₂×C₄⋊Q₈	—	3	1	no	—
113	C₂×#127	—	15	1	no	—
114	#492	—	15	8	no	—
115	C₂×C₄×C₈	—	5	1	no	—
116	C₂²×C₃×C₅	#127	1	2	yes	—
117	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
118	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₅	—	4	4	no	—
120	C₂×D₁₅	—	4	8	no	—
121	C₂×D₁₅	—	8	8	no	—
122	C₃×C₄×C₅	#127	1	2	yes	—
123	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	complemented
124	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
125	C₅×Dic₃	—	4	4	no	—
126	C₂³×D₃	—	10	2	no	—
127	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
128	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
129	C₂²×Dic₃	—	2	2	no	—
130	C₂×D₁₂	—	5	4	no	—
131	C₂×D₁₂	—	10	8	no	—
132	C₂×D₁₂	—	10	8	no	—
133	C₂×D₁₂	—	10	2	no	—
134	C₂×D₁₂	—	20	8	no	—
135	C₂×Dic₆	—	2	8	no	—
136	C₂×Dic₆	—	2	2	no	—
137	C₂×Dic₆	C₂²×D₅	1	4	yes	—
138	C₂×C₃×C₈	—	5	8	no	—
139	C₂×C₃×C₈	—	5	4	no	—
140	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
141	C₃×C₄²	C₂²×D₅	1	2	yes	—
142	C₄⋊Dic₃	—	2	8	no	—
143	#204	—	10	32	no	—
144	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
145	C₂²×D₅	—	6	4	no	—
146	C₂²×D₅	—	12	4	no	—
147	C₂²×D₅	—	12	8	no	—
148	C₅×Q₈	—	6	8	no	—
149	C₅×Q₈	—	12	4	no	—
150	C₂×C₄×C₅	—	6	4	no	—
151	C₂×C₄×C₅	—	12	4	no	—
152	C₂×C₄×C₅	#204	1	8	yes	complemented
153	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
154	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
156	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
157	D₂₀	—	6	8	no	—
158	D₂₀	—	12	4	no	—
159	D₂₀	—	12	16	no	—
160	#168	C₂×D₁₂	1	8	yes	complemented
161	C₂²×Q₈	—	3	2	no	—
162	C₂²×D₄	—	15	2	no	—
163	C₂²×D₄	—	30	2	no	—
164	C₂×SD₁₆	—	15	16	no	—
165	C₂×SD₁₆	—	30	16	no	—
166	C₂²×C₈	—	5	2	no	—
167	C₄⋊Q₈	—	3	4	no	—
168	C₄⋊Q₈	—	6	2	no	—
169	#127	—	15	4	no	—
170	#127	—	30	2	no	—
171	C₂×C₄⋊C₄	—	6	2	no	—
172	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
173	C₄×C₈	—	5	4	no	—
174	C₂×C₃×C₅	#492	1	4	yes	complemented
175	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
176	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
177	D₁₅	—	8	4	no	—
178	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
179	C₂²×D₃	—	10	4	no	—
180	C₂²×D₃	—	20	4	no	—
181	C₂²×D₃	—	20	8	no	—
182	C₂×C₃×C₄	#688	1	8	yes	complemented
183	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
184	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
186	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
187	C₂×Dic₃	—	2	4	no	—
188	C₂×Dic₃	—	4	4	no	—
189	D₁₂	—	10	8	no	—
190	D₁₂	—	20	4	no	—
191	D₁₂	—	20	16	no	—
192	Dic₆	—	2	8	no	—
193	Dic₆	—	4	4	no	—
194	C₃×C₈	—	5	8	no	—
195	C₂²×C₅	#874	1	2	yes	—
196	C₂²×C₅	C₂×#204	1	4	yes	—
197	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
198	C₂×D₅	—	12	4	no	—
199	C₂×D₅	—	12	8	no	—
200	C₂×D₅	—	24	8	no	—
201	C₄×C₅	—	12	4	no	—
202	C₄×C₅	#874	1	2	yes	—
203	C₄×C₅	C₂×#204	1	8	yes	complemented
204	C₄×C₅	C₂²×D₁₂	1	2	yes	—
205	C₂⁴	—	30	2	no	—
206	C₂×Q₈	—	3	4	no	—
207	C₂×Q₈	—	6	2	no	—
208	C₂×Q₈	—	6	8	no	—
209	C₂×D₄	—	15	4	no	—
210	C₂×D₄	—	30	8	no	—
211	C₂×D₄	—	30	8	no	—
212	C₂×D₄	—	30	2	no	—
213	C₂×D₄	—	60	8	no	—
214	C₂²×C₄	—	6	2	no	—
215	C₂²×C₄	#1203	1	2	yes	—
216	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
217	SD₁₆	—	30	32	no	—
218	C₂×C₈	—	5	8	no	—
219	C₂×C₈	—	5	4	no	—
220	C₄⋊C₄	—	6	8	no	—
221	C₄²	#1203	1	2	yes	—
222	C₄²	C₂×D₃×D₅	1	2	yes	—
223	C₃×C₅	C₂×#492	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	#4086	1	2	yes	—
225	C₂²×C₃	C₂×#688	1	4	yes	—
226	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
227	C₂×D₃	—	20	4	no	—
228	C₂×D₃	—	20	8	no	—
229	C₂×D₃	—	40	8	no	—
230	C₃×C₄	#4086	1	2	yes	—
231	C₃×C₄	C₂×#688	1	8	yes	complemented
232	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
233	Dic₃	—	4	4	no	—
234	C₂×C₅	#4693	1	4	yes	complemented
235	C₂×C₅	C₂×#874	1	1	yes	characteristic
236	C₂×C₅	C₂²×#204	1	2	yes	—
237	D₅	—	24	4	no	—
238	C₂³	—	30	4	no	—
239	C₂³	—	60	4	no	—
240	C₂³	—	60	8	no	—
241	C₂³	C₂×#1203	1	1	yes	characteristic
242	Q₈	—	6	8	no	—
243	Q₈	—	12	4	no	—
244	D₄	—	30	8	no	—
245	D₄	—	60	4	no	—
246	D₄	—	60	16	no	—
247	C₂×C₄	—	6	4	no	—
248	C₂×C₄	—	12	4	no	—
249	C₂×C₄	#6660	1	8	yes	complemented
250	C₂×C₄	D₅×D₁₂	1	4	yes	—
251	C₂×C₄	C₂×#1203	1	1	yes	characteristic
252	C₂×C₄	C₂×#1203	1	4	yes	—
253	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
254	C₈	—	5	8	no	—
255	C₂×C₃	#65374	1	4	yes	complemented
256	C₂×C₃	C₂×#4086	1	1	yes	characteristic
257	C₂×C₃	C₂²×#688	1	2	yes	—
258	D₃	—	40	4	no	—
259	C₅	C₂×#4693	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	60	4	no	—
261	C₂²	—	60	8	no	—
262	C₂²	—	120	8	no	—
263	C₂²	C₂×#6660	1	4	yes	—
264	C₂²	#91451	1	2	yes	—
265	C₂²	C₂²×#1203	1	1	yes	characteristic
266	C₄	—	12	4	no	—
267	C₄	C₂×#6660	1	8	yes	complemented
268	C₄	#91451	1	2	yes	—
269	C₄	C₂×D₅×D₁₂	1	2	yes	—
270	C₃	C₂×#65374	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	120	4	no	—
272	C₂	#11747067	1	4	yes	direct product factor
273	C₂	C₂²×#6660	1	2	yes	—
274	C₂	C₂×#91451	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1203
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×#91763	C₂
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#492
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#869	C₂	semidirect
3. C₂²×#6660	C₂	semidirect
4. C₂×C₃×C₄×#168	C₂	semidirect
5. #11747067	C₂	direct (N × H)
15. C₅×#869	C₂²	semidirect
18. C₂×#6660	C₂²	semidirect
19. C₃×C₄×#168	C₂²	semidirect
25. C₂×C₄×#168	D₃	semidirect
41. C₂×C₃×#168	D₄	semidirect
46. C₂×#869	D₅	semidirect
61. C₄×#168	C₂×D₃	semidirect
67. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₃×#168	C₂×D₄	semidirect
91. #869	C₂×D₅	semidirect
109. C₂×#168	D₁₂	semidirect
123. C₃×C₄×C₅	C₂×SD₁₆	semidirect
152. C₂×C₄×C₅	#204	semidirect
160. #168	C₂×D₁₂	semidirect
174. C₂×C₃×C₅	#492	semidirect
182. C₂×C₃×C₄	#688	semidirect
203. C₄×C₅	C₂×#204	semidirect
223. C₃×C₅	C₂×#492	semidirect
231. C₃×C₄	C₂×#688	semidirect
234. C₂×C₅	#4693	semidirect
249. C₂×C₄	#6660	semidirect
255. C₂×C₃	#65374	semidirect
259. C₅	C₂×#4693	semidirect
267. C₄	C₂×#6660	semidirect
270. C₃	C₂×#65374	semidirect
272. C₂	#11747067	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#91763	2
C₂×C₅×#869	2
C₂²×#6660	2
C₂×C₃×C₄×#168	2
#11747067	2
C₂×#65374	3
C₂×#4693	5

Subgroups — C2×#11747067 #14438759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747067 #14438759