Subgroups — C₂×#11756511 #14509965

Order 1920.

3024 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#92100	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#11756511	C₂	1	16	yes	maximal, direct product factor
4	C₂×#91797	C₂	1	6	yes	maximal, complemented
5	C₂×#91793	C₂	1	3	yes	maximal, complemented
6	C₂²×D₁₂₀	C₂	1	3	yes	maximal, complemented
7	C₂×#66086	—	3	1	no	maximal, Hall
8	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
9	#92100	—	2	8	no	—
10	#92100	C₂²	1	16	yes	complemented
11	C₂×#6821	—	2	2	no	—
12	C₂×D₄×D₁₅	—	2	6	no	—
13	C₂×#6817	C₂²	1	6	yes	complemented
14	C₂²×D₆₀	C₂²	1	6	yes	complemented
15	C₃×C₅×#131	C₂²	1	8	yes	complemented
16	C₂×C₃×C₅×M₁₆	C₂²	1	3	yes	complemented
17	C₂²×C₃×C₅×C₈	C₂²	1	3	yes	complemented
18	#91797	C₂²	1	48	yes	complemented
19	C₂×Dic₆₀	C₂²	1	1	yes	characteristic, complemented
20	#91793	C₂²	1	24	yes	complemented
21	C₂×D₁₂₀	C₂²	1	3	yes	complemented
22	C₂×D₁₂₀	C₂²	1	6	yes	complemented
23	C₂×D₁₂₀	C₂²	1	24	yes	complemented
24	C₂×#6706	C₂²	1	6	yes	complemented
25	C₂×#5752	—	5	1	no	maximal, Hall
26	C₂×#4149	—	3	2	no	—
27	C₂×C₅×#131	D₃	1	1	yes	characteristic, complemented
28	#66086	—	3	16	no	—
29	C₂×#4054	—	3	6	no	—
30	C₂×#4050	—	3	3	no	—
31	C₂²×D₄₀	—	3	3	no	—
32	C₂³×D₁₅	—	2	6	no	—
33	C₃×C₅×#41	D₄	1	4	yes	—
34	C₃×C₅×#41	C₂³	1	4	yes	—
35	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
36	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
38	C₂×#1220	—	2	6	no	—
39	#6821	—	2	16	no	—
40	D₄×D₁₅	—	2	48	no	—
41	#6817	—	2	12	no	—
42	#6817	C₂³	1	24	yes	complemented
43	C₂×D₆₀	—	2	6	no	—
44	C₂×D₆₀	—	2	12	no	—
45	C₂×D₆₀	C₂³	1	6	yes	—
46	C₂×D₆₀	C₂³	1	24	yes	complemented
47	C₂×C₄×D₁₅	—	2	6	no	—
48	C₂×Dic₃₀	C₂³	1	2	yes	—
49	C₃×C₅×M₁₆	C₂³	1	12	yes	complemented
50	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
51	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₅×C₈	C₂³	1	12	yes	complemented
53	Dic₆₀	C₂³	1	4	yes	complemented
54	D₁₂₀	C₂³	1	12	yes	complemented
55	D₁₂₀	C₂³	1	24	yes	complemented
56	#6706	C₂³	1	24	yes	complemented
57	C₂×#1009	—	5	2	no	—
58	C₂×C₃×#131	D₅	1	1	yes	characteristic, complemented
59	#5752	—	5	16	no	—
60	C₂×#908	—	5	6	no	—
61	C₂×#904	—	5	3	no	—
62	C₂²×D₂₄	—	5	3	no	—
63	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
64	#4149	—	3	16	no	—
65	#4149	—	6	8	no	—
66	C₂×#713	—	6	2	no	—
67	C₂×D₄×D₅	—	6	6	no	—
68	C₂×#709	—	3	6	no	—
69	C₂²×D₂₀	—	3	6	no	—
70	C₅×#131	C₂×D₃	1	8	yes	complemented
71	C₂×C₅×M₁₆	C₂×D₃	1	3	yes	complemented
72	C₂²×C₅×C₈	C₂×D₃	1	3	yes	complemented
73	#4054	—	3	48	no	—
74	C₂×Dic₂₀	—	3	1	no	—
75	#4050	—	3	24	no	—
76	C₂×D₄₀	—	3	3	no	—
77	C₂×D₄₀	—	3	6	no	—
78	C₂×D₄₀	—	3	24	no	—
79	C₂×#677	—	3	6	no	—
80	C₂×#576	—	15	1	no	Sylow
81	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
82	C₂²×D₁₅	—	2	6	no	—
83	C₂²×D₁₅	—	2	24	no	—
84	C₂²×D₁₅	—	8	12	no	—
85	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
86	C₃×C₅×D₄	C₂×D₄	1	12	yes	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
89	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
91	#1220	—	2	24	no	—
92	C₂×Dic₁₅	—	2	2	no	—
93	D₆₀	—	2	6	no	—
94	D₆₀	—	2	24	no	—
95	D₆₀	C₂⁴	1	12	yes	—
96	C₄×D₁₅	—	2	24	no	—
97	Dic₃₀	—	2	2	no	—
98	Dic₃₀	C₂⁴	1	4	yes	—
99	C₃×C₅×C₈	C₂⁴	1	6	yes	—
100	C₃×C₅×C₈	C₂⁴	1	2	yes	—
101	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
102	#1009	—	5	16	no	—
103	#1009	—	10	8	no	—
104	C₂×#239	—	10	2	no	—
105	C₂×D₃×D₄	—	10	6	no	—
106	C₂×#235	—	5	6	no	—
107	C₂²×D₁₂	—	5	6	no	—
108	C₃×#131	C₂×D₅	1	8	yes	complemented
109	C₂×C₃×M₁₆	C₂×D₅	1	3	yes	complemented
110	C₂²×C₃×C₈	C₂×D₅	1	3	yes	complemented
111	#908	—	5	48	no	—
112	C₂×Dic₁₂	—	5	1	no	—
113	#904	—	5	24	no	—
114	C₂×D₂₄	—	5	3	no	—
115	C₂×D₂₄	—	5	6	no	—
116	C₂×D₂₄	—	5	24	no	—
117	C₂×#204	—	5	6	no	—
118	C₂³×D₅	—	6	6	no	—
119	C₅×#41	D₁₂	1	4	yes	—
120	C₅×#41	C₂²×D₃	1	4	yes	—
121	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
122	C₂×C₅×D₄	D₁₂	1	3	yes	—
123	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
124	C₂×C₂²⋊D₅	—	6	6	no	—
125	#713	—	6	16	no	—
126	D₄×D₅	—	6	48	no	—
127	#709	—	3	24	no	—
128	#709	—	6	12	no	—
129	C₂×D₂₀	—	3	6	no	—
130	C₂×D₂₀	—	3	24	no	—
131	C₂×D₂₀	—	6	6	no	—
132	C₂×D₂₀	—	6	12	no	—
133	C₂×C₄×D₅	—	6	6	no	—
134	C₂×Dic₁₀	—	3	2	no	—
135	C₅×M₁₆	C₂²×D₃	1	12	yes	complemented
136	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
138	C₂×C₅×C₈	C₂²×D₃	1	12	yes	complemented
139	Dic₂₀	—	3	4	no	—
140	D₄₀	—	3	12	no	—
141	D₄₀	—	3	24	no	—
142	#677	—	3	24	no	—
143	C₂×H(5,𝔽₂)	—	15	2	no	—
144	#576	—	15	16	no	—
145	C₂×AGL(1,ℤ/8)	—	15	6	no	—
146	C₂×#135	—	15	3	no	—
147	C₂²×D₈	—	15	3	no	—
148	C₂×#131	D₁₅	1	1	yes	characteristic, complemented
149	C₂²×C₃×C₅	—	2	6	no	—
150	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
151	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₅	—	2	12	no	—
153	C₂×D₁₅	—	4	12	no	—
154	C₂×D₁₅	—	8	24	no	—
155	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²×D₄	1	6	yes	—
157	C₃×C₄×C₅	C₂⁵	1	1	yes	characteristic
158	Dic₁₅	—	2	4	no	—
159	C₂³×D₃	—	10	6	no	—
160	C₃×#41	D₂₀	1	4	yes	—
161	C₃×#41	C₂²×D₅	1	4	yes	—
162	C₂×C₃×Q₈	D₂₀	1	1	yes	characteristic
163	C₂×C₃×D₄	D₂₀	1	3	yes	—
164	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
165	C₂×#67	—	10	6	no	—
166	#239	—	10	16	no	—
167	D₃×D₄	—	10	48	no	—
168	#235	—	5	24	no	—
169	#235	—	10	12	no	—
170	C₂×D₁₂	—	5	6	no	—
171	C₂×D₁₂	—	5	24	no	—
172	C₂×D₁₂	—	10	6	no	—
173	C₂×D₁₂	—	10	12	no	—
174	C₂×C₄×D₃	—	10	6	no	—
175	C₂×Dic₆	—	5	2	no	—
176	C₃×M₁₆	C₂²×D₅	1	12	yes	complemented
177	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
178	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
179	C₂×C₃×C₈	C₂²×D₅	1	12	yes	complemented
180	Dic₁₂	—	5	4	no	—
181	D₂₄	—	5	12	no	—
182	D₂₄	—	5	24	no	—
183	#204	—	5	24	no	—
184	C₂³×C₅	C₂×D₁₂	1	3	yes	—
185	C₂²×D₅	—	6	6	no	—
186	C₂²×D₅	—	6	24	no	—
187	C₂²×D₅	—	24	12	no	—
188	C₅×Q₈	C₂×D₁₂	1	4	yes	—
189	C₅×D₄	C₂×D₁₂	1	12	yes	—
190	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
191	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
192	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
194	C₂²⋊D₅	—	6	24	no	—
195	C₂×Dic₅	—	6	2	no	—
196	D₂₀	—	3	12	no	—
197	D₂₀	—	6	6	no	—
198	D₂₀	—	6	24	no	—
199	C₄×D₅	—	6	24	no	—
200	Dic₁₀	—	3	4	no	—
201	Dic₁₀	—	6	2	no	—
202	C₅×C₈	C₂³×D₃	1	6	yes	—
203	C₅×C₈	C₂³×D₃	1	2	yes	—
204	H(5,𝔽₂)	—	15	16	no	—
205	H(5,𝔽₂)	—	30	8	no	—
206	C₂×#41	—	15	6	no	—
207	C₂×#41	—	30	2	no	—
208	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
209	C₂²×D₄	—	15	6	no	—
210	C₂²×D₄	—	30	6	no	—
211	AGL(1,ℤ/8)	—	15	48	no	—
212	#135	—	15	24	no	—
213	C₂×Q₁₆	—	15	1	no	—
214	C₂×SD₁₆	—	15	6	no	—
215	C₂×D₈	—	15	3	no	—
216	C₂×D₈	—	15	6	no	—
217	C₂×D₈	—	15	24	no	—
218	#131	C₂×D₁₅	1	8	yes	complemented
219	C₂×M₁₆	C₂×D₁₅	1	3	yes	complemented
220	C₂²×C₈	C₂×D₁₅	1	3	yes	complemented
221	C₂×C₃×C₅	—	2	6	no	—
222	C₂×C₃×C₅	#576	1	2	yes	complemented
223	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
224	D₁₅	—	4	12	no	—
225	C₂³×C₃	C₂×D₂₀	1	3	yes	—
226	C₂²×D₃	—	10	6	no	—
227	C₂²×D₃	—	10	24	no	—
228	C₂²×D₃	—	40	12	no	—
229	C₃×Q₈	C₂×D₂₀	1	4	yes	—
230	C₃×D₄	C₂×D₂₀	1	12	yes	—
231	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
232	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
233	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
235	#67	—	10	24	no	—
236	C₂×Dic₃	—	10	2	no	—
237	D₁₂	—	5	12	no	—
238	D₁₂	—	10	6	no	—
239	D₁₂	—	10	24	no	—
240	C₄×D₃	—	10	24	no	—
241	Dic₆	—	5	4	no	—
242	Dic₆	—	10	2	no	—
243	C₃×C₈	C₂³×D₅	1	6	yes	—
244	C₃×C₈	C₂³×D₅	1	2	yes	—
245	C₂²×C₅	—	2	6	no	—
246	C₂²×C₅	C₂²×D₁₂	1	6	yes	—
247	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
248	C₂×D₅	—	6	12	no	—
249	C₂×D₅	—	12	12	no	—
250	C₂×D₅	—	24	24	no	—
251	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
252	C₄×C₅	C₂²×D₁₂	1	6	yes	—
253	C₄×C₅	C₂⁴×D₃	1	1	yes	characteristic
254	Dic₅	—	6	4	no	—
255	C₂⁴	—	30	6	no	—
256	#41	—	15	24	no	—
257	#41	—	30	16	no	—
258	#41	—	30	12	no	—
259	#41	D₆₀	1	4	yes	—
260	#41	C₂²×D₁₅	1	4	yes	—
261	C₂×Q₈	—	15	2	no	—
262	C₂×Q₈	D₆₀	1	1	yes	characteristic
263	C₂×D₄	—	15	6	no	—
264	C₂×D₄	—	15	24	no	—
265	C₂×D₄	—	30	6	no	—
266	C₂×D₄	—	30	48	no	—
267	C₂×D₄	—	30	6	no	—
268	C₂×D₄	—	30	12	no	—
269	C₂×D₄	D₆₀	1	3	yes	—
270	C₂²×C₄	—	30	6	no	—
271	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
272	Q₁₆	—	15	4	no	—
273	SD₁₆	—	15	24	no	—
274	D₈	—	15	12	no	—
275	D₈	—	15	24	no	—
276	M₁₆	C₂²×D₁₅	1	12	yes	complemented
277	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
278	C₂×C₈	C₂²×D₁₅	1	3	yes	—
279	C₂×C₈	C₂²×D₁₅	1	12	yes	complemented
280	C₃×C₅	C₂×#576	1	1	yes	characteristic, complemented, Hall
281	C₂²×C₃	—	2	6	no	—
282	C₂²×C₃	C₂²×D₂₀	1	6	yes	—
283	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
284	C₂×D₃	—	10	12	no	—
285	C₂×D₃	—	20	12	no	—
286	C₂×D₃	—	40	24	no	—
287	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
288	C₃×C₄	C₂²×D₂₀	1	6	yes	—
289	C₃×C₄	C₂⁴×D₅	1	1	yes	characteristic
290	Dic₃	—	10	4	no	—
291	C₂×C₅	—	2	6	no	—
292	C₂×C₅	#5752	1	2	yes	complemented
293	C₂×C₅	C₂³×D₁₂	1	1	yes	characteristic
294	D₅	—	12	12	no	—
295	C₂³	—	30	6	no	—
296	C₂³	—	30	24	no	—
297	C₂³	—	120	12	no	—
298	C₂³	C₂×D₆₀	1	3	yes	—
299	Q₈	—	15	4	no	—
300	Q₈	—	30	2	no	—
301	Q₈	C₂×D₆₀	1	4	yes	—
302	D₄	—	15	12	no	—
303	D₄	—	30	24	no	—
304	D₄	—	30	24	no	—
305	D₄	—	30	6	no	—
306	D₄	C₂×D₆₀	1	12	yes	—
307	C₂×C₄	—	30	2	no	—
308	C₂×C₄	—	30	24	no	—
309	C₂×C₄	C₂×D₆₀	1	6	yes	—
310	C₂×C₄	C₂×D₆₀	1	3	yes	—
311	C₂×C₄	C₂³×D₁₅	1	6	yes	—
312	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
313	C₈	C₂³×D₁₅	1	2	yes	—
314	C₈	C₂³×D₁₅	1	6	yes	—
315	C₂×C₃	—	2	6	no	—
316	C₂×C₃	#66086	1	2	yes	complemented
317	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
318	D₃	—	20	12	no	—
319	C₅	C₂×#5752	1	1	yes	characteristic, complemented, Sylow
320	C₂²	—	2	6	no	—
321	C₂²	—	30	12	no	—
322	C₂²	—	60	12	no	—
323	C₂²	—	120	24	no	—
324	C₂²	C₂²×D₆₀	1	6	yes	—
325	C₂²	C₂²×D₆₀	1	1	yes	characteristic
326	C₄	—	30	4	no	—
327	C₄	C₂²×D₆₀	1	6	yes	—
328	C₄	C₂²×D₆₀	1	1	yes	characteristic
329	C₄	C₂⁴×D₁₅	1	1	yes	characteristic
330	C₃	C₂×#66086	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	6	no	—
332	C₂	—	60	12	no	—
333	C₂	#11756511	1	2	yes	direct product factor
334	C₂	C₂³×D₆₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₀
commutator, cosocle	C₃×C₄×C₅	C₂⁵
Frattini	C₄	C₂⁴×D₁₅
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×#131	D₁₅
hypocenter	C₃×C₅	C₂×#576
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#92100	C₂	semidirect
2. C₂×C₃×C₅×#131	C₂	semidirect
3. #11756511	C₂	direct (N × H)
4. C₂×#91797	C₂	semidirect
5. C₂×#91793	C₂	semidirect
6. C₂²×D₁₂₀	C₂	semidirect
10. #92100	C₂²	semidirect
13. C₂×#6817	C₂²	semidirect
14. C₂²×D₆₀	C₂²	semidirect
15. C₃×C₅×#131	C₂²	semidirect
16. C₂×C₃×C₅×M₁₆	C₂²	semidirect
17. C₂²×C₃×C₅×C₈	C₂²	semidirect
18. #91797	C₂²	semidirect
19. C₂×Dic₆₀	C₂²	semidirect
20. #91793	C₂²	semidirect
21. C₂×D₁₂₀	C₂²	semidirect
22. C₂×D₁₂₀	C₂²	semidirect
23. C₂×D₁₂₀	C₂²	semidirect
24. C₂×#6706	C₂²	semidirect
27. C₂×C₅×#131	D₃	semidirect
42. #6817	C₂³	semidirect
46. C₂×D₆₀	C₂³	semidirect
49. C₃×C₅×M₁₆	C₂³	semidirect
52. C₂×C₃×C₅×C₈	C₂³	semidirect
53. Dic₆₀	C₂³	semidirect
54. D₁₂₀	C₂³	semidirect
55. D₁₂₀	C₂³	semidirect
56. #6706	C₂³	semidirect
58. C₂×C₃×#131	D₅	semidirect
70. C₅×#131	C₂×D₃	semidirect
71. C₂×C₅×M₁₆	C₂×D₃	semidirect
72. C₂²×C₅×C₈	C₂×D₃	semidirect
108. C₃×#131	C₂×D₅	semidirect
109. C₂×C₃×M₁₆	C₂×D₅	semidirect
110. C₂²×C₃×C₈	C₂×D₅	semidirect
135. C₅×M₁₆	C₂²×D₃	semidirect
138. C₂×C₅×C₈	C₂²×D₃	semidirect
148. C₂×#131	D₁₅	semidirect
176. C₃×M₁₆	C₂²×D₅	semidirect
179. C₂×C₃×C₈	C₂²×D₅	semidirect
218. #131	C₂×D₁₅	semidirect
219. C₂×M₁₆	C₂×D₁₅	semidirect
220. C₂²×C₈	C₂×D₁₅	semidirect
222. C₂×C₃×C₅	#576	semidirect
276. M₁₆	C₂²×D₁₅	semidirect
279. C₂×C₈	C₂²×D₁₅	semidirect
280. C₃×C₅	C₂×#576	semidirect
292. C₂×C₅	#5752	semidirect
316. C₂×C₃	#66086	semidirect
319. C₅	C₂×#5752	semidirect
330. C₃	C₂×#66086	semidirect
333. C₂	#11756511	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×#92100	2
C₂×C₃×C₅×#131	2
#11756511	2
C₂×#91797	2
C₂×#91793	2
C₂²×D₁₂₀	2
C₂×#66086	3
C₂×#5752	5

Subgroups — C2×#11756511 #14509965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11756511 #14509965