Subgroups — C₂×#11747035 #14438792

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 300 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, complemented
2	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6653	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747035	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64984	—	3	1	no	maximal, Hall
7	C₂²×D₆₀	—	2	2	no	—
8	C₂²×D₆₀	C₂²	1	2	yes	—
9	C₂²×C₃×D₂₀	—	2	2	no	—
10	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	#91763	—	2	2	no	—
13	#91763	C₂²	1	4	yes	complemented
14	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃×#4020	—	2	2	no	—
16	C₃×#4020	C₂²	1	4	yes	complemented
17	C₂×#6653	—	2	16	no	—
18	C₂×#6653	C₂²	1	16	yes	complemented
19	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
20	C₂×#5073	—	5	1	no	maximal, Hall
21	C₂²×D₄₀	—	3	4	no	—
22	C₂×#4020	—	3	1	no	—
23	C₂×#4020	D₃	1	1	yes	characteristic, complemented
24	C₂×C₄×C₅×C₈	—	3	1	no	—
25	#64984	—	3	8	no	—
26	C₂³×D₁₅	—	2	2	no	—
27	C₂³×C₃×D₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×D₆₀	—	2	8	no	—
31	C₂×D₆₀	—	2	2	no	—
32	C₂×D₆₀	—	2	8	no	—
33	C₂×D₆₀	—	4	8	no	—
34	C₂×D₆₀	C₂³	1	4	yes	—
35	C₂×C₃×D₂₀	—	2	8	no	—
36	C₂×C₃×D₂₀	—	2	8	no	—
37	C₂×C₃×D₂₀	—	2	2	no	—
38	C₂×C₃×D₂₀	—	4	8	no	—
39	C₂×C₃×D₂₀	C₂³	1	4	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	2	yes	—
42	C₂×C₅×C₃⋊C₈	D₄	1	8	yes	complemented
43	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
44	#6653	—	2	32	no	—
45	C₂×C₃×#127	—	5	1	no	—
46	C₂²×#213	—	5	4	no	—
47	C₂×#874	—	5	1	no	—
48	#5073	—	5	8	no	—
49	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
50	C₂²×D₂₀	—	2	2	no	—
51	C₂²×D₂₀	—	3	2	no	—
52	C₂²×D₂₀	—	6	2	no	—
53	C₂²×D₂₀	C₂×D₃	1	2	yes	—
54	C₂²×C₅×C₈	—	3	2	no	—
55	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×D₄₀	—	3	16	no	—
57	C₂×D₄₀	—	6	16	no	—
58	#4020	—	2	2	no	—
59	#4020	—	3	4	no	—
60	#4020	—	6	2	no	—
61	#4020	C₂×D₃	1	4	yes	complemented
62	C₄×C₅×C₈	—	3	4	no	—
63	C₂×#493	—	15	1	no	Sylow
64	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₁₅	—	2	4	no	—
66	C₂²×D₁₅	—	4	4	no	—
67	C₂²×D₁₅	—	4	8	no	—
68	C₂²×C₃×D₅	—	2	4	no	—
69	C₂²×C₃×D₅	—	4	4	no	—
70	C₂²×C₃×D₅	—	4	8	no	—
71	C₂×C₃×C₄×C₅	D₈	1	8	yes	complemented
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
75	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	D₆₀	—	2	8	no	—
77	D₆₀	—	4	4	no	—
78	D₆₀	—	4	16	no	—
79	C₃×D₂₀	—	2	8	no	—
80	C₃×D₂₀	—	4	4	no	—
81	C₃×D₂₀	—	4	16	no	—
82	C₅×C₃⋊C₈	C₂×D₄	1	8	yes	complemented
83	C₂²×C₃×D₄	—	5	2	no	—
84	C₂²×C₃×D₄	—	10	2	no	—
85	C₂²×D₁₂	—	5	2	no	—
86	C₂²×D₁₂	—	10	2	no	—
87	C₃×#127	—	5	4	no	—
88	C₃×#127	—	10	2	no	—
89	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
90	C₂×#213	—	5	16	no	—
91	C₂×#213	—	10	16	no	—
92	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
93	#874	—	5	4	no	—
94	#874	—	10	2	no	—
95	C₄×C₃⋊C₈	C₂×D₅	1	4	yes	complemented
96	C₂³×D₅	—	2	2	no	—
97	C₂³×D₅	—	6	2	no	—
98	C₂²×C₄×C₅	#67	1	2	yes	—
99	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
100	C₂×D₂₀	—	2	8	no	—
101	C₂×D₂₀	—	2	8	no	—
102	C₂×D₂₀	—	2	2	no	—
103	C₂×D₂₀	—	3	4	no	—
104	C₂×D₂₀	—	4	8	no	—
105	C₂×D₂₀	—	6	8	no	—
106	C₂×D₂₀	—	6	2	no	—
107	C₂×D₂₀	—	6	8	no	—
108	C₂×D₂₀	—	12	8	no	—
109	C₂×D₂₀	C₂²×D₃	1	4	yes	—
110	C₂×C₅×C₈	—	3	8	no	—
111	C₂×C₅×C₈	—	3	4	no	—
112	C₄²×C₅	#67	1	2	yes	—
113	C₄²×C₅	C₂²×D₃	1	2	yes	—
114	D₄₀	—	6	32	no	—
115	C₂²×D₈	—	15	4	no	—
116	C₂×#127	—	5	1	no	—
117	C₂×#127	—	15	1	no	—
118	#493	—	15	8	no	—
119	C₂×C₄×C₈	—	3	1	no	—
120	C₂²×C₃×C₅	#127	1	2	yes	—
121	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
122	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₅	—	4	4	no	—
124	C₂×D₁₅	—	4	8	no	—
125	C₂×D₁₅	—	8	8	no	—
126	C₂×C₃×D₅	—	4	4	no	—
127	C₂×C₃×D₅	—	4	8	no	—
128	C₂×C₃×D₅	—	8	8	no	—
129	C₃×C₄×C₅	#127	1	2	yes	—
130	C₃×C₄×C₅	C₂×D₈	1	8	yes	complemented
131	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
132	C₂⁴×C₃	—	10	2	no	—
133	C₂³×D₃	—	10	2	no	—
134	C₂×C₃×D₄	—	5	4	no	—
135	C₂×C₃×D₄	—	10	8	no	—
136	C₂×C₃×D₄	—	10	8	no	—
137	C₂×C₃×D₄	—	10	2	no	—
138	C₂×C₃×D₄	—	20	8	no	—
139	C₂²×C₃×C₄	D₂₀	1	2	yes	—
140	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
141	C₂×D₁₂	—	5	4	no	—
142	C₂×D₁₂	—	10	8	no	—
143	C₂×D₁₂	—	10	8	no	—
144	C₂×D₁₂	—	10	2	no	—
145	C₂×D₁₂	—	20	8	no	—
146	C₃×C₄²	D₂₀	1	2	yes	—
147	C₃×C₄²	C₂²×D₅	1	2	yes	—
148	#213	—	10	32	no	—
149	C₂×C₃⋊C₈	D₂₀	1	8	yes	complemented
150	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
151	C₂³×C₅	C₂×#67	1	1	yes	characteristic
152	C₂²×D₅	—	2	4	no	—
153	C₂²×D₅	—	4	8	no	—
154	C₂²×D₅	—	4	4	no	—
155	C₂²×D₅	—	6	4	no	—
156	C₂²×D₅	—	12	8	no	—
157	C₂²×D₅	—	12	4	no	—
158	C₂×C₄×C₅	#213	1	8	yes	complemented
159	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
160	C₂×C₄×C₅	C₂×#67	1	4	yes	—
161	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
163	D₂₀	—	2	8	no	—
164	D₂₀	—	4	16	no	—
165	D₂₀	—	4	4	no	—
166	D₂₀	—	6	8	no	—
167	D₂₀	—	12	16	no	—
168	D₂₀	—	12	4	no	—
169	C₅×C₈	—	3	8	no	—
170	C₂²×D₄	—	5	2	no	—
171	C₂²×D₄	—	10	2	no	—
172	C₂²×D₄	—	15	2	no	—
173	C₂²×D₄	—	30	2	no	—
174	C₂×D₈	—	15	16	no	—
175	C₂×D₈	—	30	16	no	—
176	C₂²×C₈	—	3	2	no	—
177	#127	—	5	4	no	—
178	#127	—	10	2	no	—
179	#127	—	15	4	no	—
180	#127	—	30	2	no	—
181	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
182	C₄×C₈	—	3	4	no	—
183	C₂×C₃×C₅	#493	1	4	yes	complemented
184	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
186	D₁₅	—	8	4	no	—
187	C₃×D₅	—	8	4	no	—
188	C₂³×C₃	—	10	4	no	—
189	C₂³×C₃	—	20	4	no	—
190	C₂³×C₃	—	20	8	no	—
191	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
192	C₂²×D₃	—	10	4	no	—
193	C₂²×D₃	—	20	4	no	—
194	C₂²×D₃	—	20	8	no	—
195	C₃×D₄	—	10	8	no	—
196	C₃×D₄	—	20	4	no	—
197	C₃×D₄	—	20	16	no	—
198	C₂×C₃×C₄	D₄₀	1	8	yes	complemented
199	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
200	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
202	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
203	D₁₂	—	10	8	no	—
204	D₁₂	—	20	4	no	—
205	D₁₂	—	20	16	no	—
206	C₃⋊C₈	C₂×D₂₀	1	8	yes	complemented
207	C₂²×C₅	C₂×#213	1	4	yes	—
208	C₂²×C₅	#940	1	2	yes	—
209	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
210	C₂×D₅	—	4	8	no	—
211	C₂×D₅	—	4	4	no	—
212	C₂×D₅	—	8	8	no	—
213	C₂×D₅	—	12	8	no	—
214	C₂×D₅	—	12	4	no	—
215	C₂×D₅	—	24	8	no	—
216	C₄×C₅	C₂×#213	1	8	yes	complemented
217	C₄×C₅	#940	1	2	yes	—
218	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
219	C₂⁴	—	10	2	no	—
220	C₂⁴	—	30	2	no	—
221	C₂×D₄	—	5	4	no	—
222	C₂×D₄	—	10	8	no	—
223	C₂×D₄	—	10	8	no	—
224	C₂×D₄	—	10	2	no	—
225	C₂×D₄	—	15	4	no	—
226	C₂×D₄	—	20	8	no	—
227	C₂×D₄	—	30	8	no	—
228	C₂×D₄	—	30	8	no	—
229	C₂×D₄	—	30	2	no	—
230	C₂×D₄	—	60	8	no	—
231	C₂²×C₄	#1202	1	2	yes	—
232	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
233	D₈	—	30	32	no	—
234	C₂×C₈	—	3	8	no	—
235	C₂×C₈	—	3	4	no	—
236	C₄²	#1202	1	2	yes	—
237	C₄²	C₂×D₃×D₅	1	2	yes	—
238	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	20	4	no	—
240	C₂²×C₃	—	20	8	no	—
241	C₂²×C₃	—	40	8	no	—
242	C₂²×C₃	#4020	1	2	yes	—
243	C₂²×C₃	C₂×D₄₀	1	4	yes	—
244	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
245	C₂×D₃	—	20	4	no	—
246	C₂×D₃	—	20	8	no	—
247	C₂×D₃	—	40	8	no	—
248	C₃×C₄	#4020	1	2	yes	—
249	C₃×C₄	C₂×D₄₀	1	8	yes	complemented
250	C₃×C₄	C₂²×D₂₀	1	2	yes	—
251	C₂×C₅	#5073	1	4	yes	complemented
252	C₂×C₅	C₂²×#213	1	2	yes	—
253	C₂×C₅	C₂×#940	1	1	yes	characteristic
254	D₅	—	8	4	no	—
255	D₅	—	24	4	no	—
256	C₂³	—	10	4	no	—
257	C₂³	—	20	4	no	—
258	C₂³	—	20	8	no	—
259	C₂³	—	30	4	no	—
260	C₂³	—	60	4	no	—
261	C₂³	—	60	8	no	—
262	C₂³	C₂×#1202	1	1	yes	characteristic
263	D₄	—	10	8	no	—
264	D₄	—	20	4	no	—
265	D₄	—	20	16	no	—
266	D₄	—	30	8	no	—
267	D₄	—	60	4	no	—
268	D₄	—	60	16	no	—
269	C₂×C₄	#6653	1	8	yes	complemented
270	C₂×C₄	D₃×D₂₀	1	4	yes	—
271	C₂×C₄	C₂×#1202	1	1	yes	characteristic
272	C₂×C₄	C₂×#1202	1	4	yes	—
273	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
274	C₈	—	3	8	no	—
275	C₂×C₃	—	40	4	no	—
276	C₂×C₃	#64984	1	4	yes	complemented
277	C₂×C₃	C₂×#4020	1	1	yes	characteristic
278	C₂×C₃	C₂²×D₄₀	1	2	yes	—
279	D₃	—	40	4	no	—
280	C₅	C₂×#5073	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	20	4	no	—
282	C₂²	—	20	8	no	—
283	C₂²	—	40	8	no	—
284	C₂²	—	60	4	no	—
285	C₂²	—	60	8	no	—
286	C₂²	—	120	8	no	—
287	C₂²	C₂×#6653	1	4	yes	—
288	C₂²	#91458	1	2	yes	—
289	C₂²	C₂²×#1202	1	1	yes	characteristic
290	C₄	C₂×#6653	1	8	yes	complemented
291	C₄	#91458	1	2	yes	—
292	C₄	C₂×D₃×D₂₀	1	2	yes	—
293	C₃	C₂×#64984	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	40	4	no	—
295	C₂	—	120	4	no	—
296	C₂	#11747035	1	4	yes	direct product factor
297	C₂	C₂²×#6653	1	2	yes	—
298	C₂	C₂×#91458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1202
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
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#493
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
1. C₂×#91763	C₂	semidirect
2. C₂×C₃×#4020	C₂	semidirect
3. C₂²×#6653	C₂	semidirect
4. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
5. #11747035	C₂	direct (N × H)
13. #91763	C₂²	semidirect
16. C₃×#4020	C₂²	semidirect
18. C₂×#6653	C₂²	semidirect
19. C₄×C₅×C₃⋊C₈	C₂²	semidirect
23. C₂×#4020	D₃	semidirect
42. C₂×C₅×C₃⋊C₈	D₄	semidirect
49. C₂×C₄×C₃⋊C₈	D₅	semidirect
61. #4020	C₂×D₃	semidirect
71. C₂×C₃×C₄×C₅	D₈	semidirect
82. C₅×C₃⋊C₈	C₂×D₄	semidirect
95. C₄×C₃⋊C₈	C₂×D₅	semidirect
130. C₃×C₄×C₅	C₂×D₈	semidirect
149. C₂×C₃⋊C₈	D₂₀	semidirect
158. C₂×C₄×C₅	#213	semidirect
183. C₂×C₃×C₅	#493	semidirect
198. C₂×C₃×C₄	D₄₀	semidirect
206. C₃⋊C₈	C₂×D₂₀	semidirect
216. C₄×C₅	C₂×#213	semidirect
238. C₃×C₅	C₂×#493	semidirect
249. C₃×C₄	C₂×D₄₀	semidirect
251. C₂×C₅	#5073	semidirect
269. C₂×C₄	#6653	semidirect
276. C₂×C₃	#64984	semidirect
280. C₅	C₂×#5073	semidirect
290. C₄	C₂×#6653	semidirect
293. C₃	C₂×#64984	semidirect
296. C₂	#11747035	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×C₃×#4020	2
C₂²×#6653	2
C₂×C₄×C₅×C₃⋊C₈	2
#11747035	2
C₂×#64984	3
C₂×#5073	5

Subgroups — C2×#11747035 #14438792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747035 #14438792