Subgroups — C₂×#11751174 #14438771

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×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#6715	C₂	1	4	yes	maximal, complemented
3	C₂×#91763	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751174	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#65362	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×D₆₀	—	2	2	no	—
10	C₂²×D₆₀	C₂²	1	2	yes	—
11	C₃×C₅×#127	—	2	2	no	—
12	C₃×C₅×#127	C₂²	1	4	yes	complemented
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×#6715	—	2	16	no	—
15	C₂×#6715	C₂²	1	16	yes	complemented
16	C₂²×C₃⋊#168	C₂²	1	2	yes	—
17	#91763	—	2	2	no	—
18	#91763	C₂²	1	4	yes	complemented
19	C₄×C₃⋊#168	C₂²	1	4	yes	complemented
20	C₂×#5073	—	5	1	no	maximal, Hall
21	C₂×C₅×#127	D₃	1	1	yes	characteristic, complemented
22	C₂²×#686	—	3	4	no	—
23	C₂×#4020	—	3	1	no	—
24	#65362	—	3	8	no	—
25	C₂×C₄×#168	—	3	1	no	—
26	C₂⁴×C₃×C₅	—	2	2	no	—
27	C₂³×D₁₅	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	2	8	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	4	8	no	—
32	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×D₆₀	—	2	8	no	—
36	C₂×D₆₀	—	2	2	no	—
37	C₂×D₆₀	—	2	8	no	—
38	C₂×D₆₀	—	4	8	no	—
39	C₂×D₆₀	C₂³	1	4	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	2	yes	—
42	#6715	—	2	32	no	—
43	C₂×C₃⋊#168	D₄	1	8	yes	complemented
44	C₂×C₃⋊#168	C₂³	1	4	yes	—
45	C₂×C₃×#127	D₅	1	1	yes	characteristic, complemented
46	C₂²×#213	—	5	4	no	—
47	C₂×#874	—	5	1	no	—
48	#5073	—	5	8	no	—
49	C₂×C₄×C₃⋊C₈	—	5	1	no	—
50	C₂²×C₅×D₄	—	2	2	no	—
51	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
52	C₂²×D₂₀	—	3	2	no	—
53	C₂²×D₂₀	—	6	2	no	—
54	C₅×#127	—	2	2	no	—
55	C₅×#127	C₂×D₃	1	4	yes	complemented
56	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂×#686	—	3	16	no	—
58	C₂×#686	—	6	16	no	—
59	C₂²×#168	—	3	2	no	—
60	#4020	—	3	4	no	—
61	#4020	—	6	2	no	—
62	C₄×#168	—	3	4	no	—
63	C₂×#493	—	15	1	no	Sylow
64	C₂³×C₃×C₅	—	2	4	no	—
65	C₂³×C₃×C₅	—	4	4	no	—
66	C₂³×C₃×C₅	—	4	8	no	—
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×D₁₅	—	2	4	no	—
69	C₂²×D₁₅	—	4	4	no	—
70	C₂²×D₁₅	—	4	8	no	—
71	C₃×C₅×D₄	—	2	8	no	—
72	C₃×C₅×D₄	—	4	4	no	—
73	C₃×C₅×D₄	—	4	16	no	—
74	C₂×C₃×C₄×C₅	D₈	1	8	yes	complemented
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	D₆₀	—	2	8	no	—
80	D₆₀	—	4	4	no	—
81	D₆₀	—	4	16	no	—
82	C₃⋊#168	C₂×D₄	1	8	yes	complemented
83	C₂²×C₃×D₄	—	2	2	no	—
84	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
85	C₂²×D₁₂	—	5	2	no	—
86	C₂²×D₁₂	—	10	2	no	—
87	C₃×#127	—	2	2	no	—
88	C₃×#127	C₂×D₅	1	4	yes	complemented
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	—	5	2	no	—
93	#874	—	5	4	no	—
94	#874	—	10	2	no	—
95	C₄×C₃⋊C₈	—	5	4	no	—
96	C₂⁴×C₅	—	2	2	no	—
97	C₂³×D₅	—	6	2	no	—
98	C₂×C₅×D₄	—	2	8	no	—
99	C₂×C₅×D₄	—	2	2	no	—
100	C₂×C₅×D₄	—	2	8	no	—
101	C₂×C₅×D₄	—	4	8	no	—
102	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
103	C₂²×C₄×C₅	#67	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₂×D₂₀	—	3	4	no	—
106	C₂×D₂₀	—	6	8	no	—
107	C₂×D₂₀	—	6	2	no	—
108	C₂×D₂₀	—	6	8	no	—
109	C₂×D₂₀	—	12	8	no	—
110	C₄²×C₅	#67	1	2	yes	—
111	C₄²×C₅	C₂²×D₃	1	2	yes	—
112	#686	—	6	32	no	—
113	C₂×#168	—	3	8	no	—
114	C₂×#168	—	3	4	no	—
115	C₂²×D₈	—	15	4	no	—
116	C₂×#127	—	15	1	no	—
117	C₂×#127	D₁₅	1	1	yes	characteristic, complemented
118	#493	—	15	8	no	—
119	C₂×C₄×C₈	—	15	1	no	—
120	C₂²×C₃×C₅	—	4	4	no	—
121	C₂²×C₃×C₅	—	4	8	no	—
122	C₂²×C₃×C₅	—	8	8	no	—
123	C₂²×C₃×C₅	#127	1	2	yes	—
124	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
125	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×D₁₅	—	4	4	no	—
127	C₂×D₁₅	—	4	8	no	—
128	C₂×D₁₅	—	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	—	2	2	no	—
133	C₂³×D₃	—	10	2	no	—
134	C₂×C₃×D₄	—	2	8	no	—
135	C₂×C₃×D₄	—	2	2	no	—
136	C₂×C₃×D₄	—	2	8	no	—
137	C₂×C₃×D₄	—	4	8	no	—
138	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
139	C₂²×C₃×C₄	C₂²⋊D₅	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	C₂²⋊D₅	1	2	yes	—
147	C₃×C₄²	C₂²×D₅	1	2	yes	—
148	#213	—	10	32	no	—
149	C₂×C₃⋊C₈	—	5	8	no	—
150	C₂×C₃⋊C₈	—	5	4	no	—
151	C₂³×C₅	—	2	4	no	—
152	C₂³×C₅	—	4	8	no	—
153	C₂³×C₅	—	4	4	no	—
154	C₂³×C₅	C₂×#67	1	1	yes	characteristic
155	C₂²×D₅	—	6	4	no	—
156	C₂²×D₅	—	12	8	no	—
157	C₂²×D₅	—	12	4	no	—
158	C₅×D₄	—	2	8	no	—
159	C₅×D₄	—	4	16	no	—
160	C₅×D₄	—	4	4	no	—
161	C₂×C₄×C₅	#213	1	8	yes	complemented
162	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
163	C₂×C₄×C₅	C₂×#67	1	4	yes	—
164	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
166	D₂₀	—	6	8	no	—
167	D₂₀	—	12	16	no	—
168	D₂₀	—	12	4	no	—
169	#168	—	3	8	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	15	2	no	—
172	C₂²×D₄	—	30	2	no	—
173	C₂²×D₄	C₂×D₁₅	1	2	yes	—
174	C₂×D₈	—	15	16	no	—
175	C₂×D₈	—	30	16	no	—
176	C₂²×C₈	—	15	2	no	—
177	#127	—	2	2	no	—
178	#127	—	15	4	no	—
179	#127	—	30	2	no	—
180	#127	C₂×D₁₅	1	4	yes	complemented
181	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
182	C₄×C₈	—	15	4	no	—
183	C₂×C₃×C₅	—	8	4	no	—
184	C₂×C₃×C₅	#493	1	4	yes	complemented
185	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
187	D₁₅	—	8	4	no	—
188	C₂³×C₃	—	2	4	no	—
189	C₂³×C₃	—	4	8	no	—
190	C₂³×C₃	—	4	4	no	—
191	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
192	C₂²×D₃	—	10	4	no	—
193	C₂²×D₃	—	20	8	no	—
194	C₂²×D₃	—	20	4	no	—
195	C₃×D₄	—	2	8	no	—
196	C₃×D₄	—	4	16	no	—
197	C₃×D₄	—	4	4	no	—
198	C₂×C₃×C₄	#686	1	8	yes	complemented
199	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
200	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
201	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
203	D₁₂	—	10	8	no	—
204	D₁₂	—	20	16	no	—
205	D₁₂	—	20	4	no	—
206	C₃⋊C₈	—	5	8	no	—
207	C₂²×C₅	—	4	8	no	—
208	C₂²×C₅	—	4	4	no	—
209	C₂²×C₅	—	8	8	no	—
210	C₂²×C₅	C₂×#213	1	4	yes	—
211	C₂²×C₅	#940	1	2	yes	—
212	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
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	—	2	2	no	—
220	C₂⁴	—	30	2	no	—
221	C₂×D₄	—	2	8	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	4	8	no	—
225	C₂×D₄	—	15	4	no	—
226	C₂×D₄	—	30	8	no	—
227	C₂×D₄	—	30	8	no	—
228	C₂×D₄	—	30	2	no	—
229	C₂×D₄	—	60	8	no	—
230	C₂×D₄	C₂²×D₁₅	1	4	yes	—
231	C₂²×C₄	#1220	1	2	yes	—
232	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
233	D₈	—	30	32	no	—
234	C₂×C₈	—	15	8	no	—
235	C₂×C₈	—	15	4	no	—
236	C₄²	#1220	1	2	yes	—
237	C₄²	C₂²×D₁₅	1	2	yes	—
238	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	4	8	no	—
240	C₂²×C₃	—	4	4	no	—
241	C₂²×C₃	—	8	8	no	—
242	C₂²×C₃	C₂×#686	1	4	yes	—
243	C₂²×C₃	#4086	1	2	yes	—
244	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
245	C₂×D₃	—	20	8	no	—
246	C₂×D₃	—	20	4	no	—
247	C₂×D₃	—	40	8	no	—
248	C₃×C₄	C₂×#686	1	8	yes	complemented
249	C₃×C₄	#4086	1	2	yes	—
250	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
251	C₂×C₅	—	8	4	no	—
252	C₂×C₅	#5073	1	4	yes	complemented
253	C₂×C₅	C₂²×#213	1	2	yes	—
254	C₂×C₅	C₂×#940	1	1	yes	characteristic
255	D₅	—	24	4	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	4	8	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	30	4	no	—
260	C₂³	—	60	8	no	—
261	C₂³	—	60	4	no	—
262	C₂³	C₂×#1220	1	1	yes	characteristic
263	D₄	—	2	8	no	—
264	D₄	—	4	16	no	—
265	D₄	—	4	4	no	—
266	D₄	—	30	8	no	—
267	D₄	—	60	16	no	—
268	D₄	—	60	4	no	—
269	C₂×C₄	#6715	1	8	yes	complemented
270	C₂×C₄	D₄×D₁₅	1	4	yes	—
271	C₂×C₄	C₂×#1220	1	4	yes	—
272	C₂×C₄	C₂×#1220	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
274	C₈	—	15	8	no	—
275	C₂×C₃	—	8	4	no	—
276	C₂×C₃	#65362	1	4	yes	complemented
277	C₂×C₃	C₂²×#686	1	2	yes	—
278	C₂×C₃	C₂×#4086	1	1	yes	characteristic
279	D₃	—	40	4	no	—
280	C₅	C₂×#5073	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	8	8	no	—
284	C₂²	—	60	8	no	—
285	C₂²	—	60	4	no	—
286	C₂²	—	120	8	no	—
287	C₂²	C₂×#6715	1	4	yes	—
288	C₂²	#91829	1	2	yes	—
289	C₂²	C₂²×#1220	1	1	yes	characteristic
290	C₄	C₂×#6715	1	8	yes	complemented
291	C₄	#91829	1	2	yes	—
292	C₄	C₂×D₄×D₁₅	1	2	yes	—
293	C₃	C₂×#65362	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	8	4	no	—
295	C₂	—	120	4	no	—
296	C₂	#11751174	1	4	yes	direct product factor
297	C₂	C₂²×#6715	1	2	yes	—
298	C₂	C₂×#91829	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×#127	D₁₅
hypocenter	C₃×C₅	C₂×#493
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#127	C₂	semidirect
2. C₂²×#6715	C₂	semidirect
3. C₂×#91763	C₂	semidirect
4. #11751174	C₂	direct (N × H)
5. C₂×C₄×C₃⋊#168	C₂	semidirect
12. C₃×C₅×#127	C₂²	semidirect
15. C₂×#6715	C₂²	semidirect
18. #91763	C₂²	semidirect
19. C₄×C₃⋊#168	C₂²	semidirect
21. C₂×C₅×#127	D₃	semidirect
43. C₂×C₃⋊#168	D₄	semidirect
45. C₂×C₃×#127	D₅	semidirect
55. C₅×#127	C₂×D₃	semidirect
74. C₂×C₃×C₄×C₅	D₈	semidirect
82. C₃⋊#168	C₂×D₄	semidirect
88. C₃×#127	C₂×D₅	semidirect
117. C₂×#127	D₁₅	semidirect
130. C₃×C₄×C₅	C₂×D₈	semidirect
161. C₂×C₄×C₅	#213	semidirect
180. #127	C₂×D₁₅	semidirect
184. C₂×C₃×C₅	#493	semidirect
198. C₂×C₃×C₄	#686	semidirect
216. C₄×C₅	C₂×#213	semidirect
238. C₃×C₅	C₂×#493	semidirect
248. C₃×C₄	C₂×#686	semidirect
252. C₂×C₅	#5073	semidirect
269. C₂×C₄	#6715	semidirect
276. C₂×C₃	#65362	semidirect
280. C₅	C₂×#5073	semidirect
290. C₄	C₂×#6715	semidirect
293. C₃	C₂×#65362	semidirect
296. C₂	#11751174	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×C₃×C₅×#127	2
C₂²×#6715	2
C₂×#91763	2
#11751174	2
C₂×C₄×C₃⋊#168	2
C₂×#65362	3
C₂×#5073	5

Subgroups — C2×#11751174 #14438771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751174 #14438771