Subgroups — C₂×#11751248 #14478328

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91793	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91111	C₂	1	2	yes	maximal, complemented
4	#11751248	C₂	1	8	yes	maximal, direct product factor
5	C₂×#91110	C₂	1	2	yes	maximal, complemented
6	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65436	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×D₄	—	2	1	no	—
9	C₂×#6817	—	2	1	no	—
10	C₂×C₃×C₅×D₈	—	2	1	no	—
11	C₂×C₃×C₅×D₈	—	2	2	no	—
12	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
13	C₂×C₃×C₅×D₈	C₂²	1	4	yes	complemented
14	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₀	C₂²	1	1	yes	characteristic, complemented
16	#91793	—	2	2	no	—
17	#91793	C₂²	1	4	yes	complemented
18	C₂×D₁₂₀	C₂²	1	1	yes	characteristic, complemented
19	C₂×#6706	—	2	1	no	—
20	#91111	C₂²	1	8	yes	complemented
21	#91110	C₂²	1	8	yes	complemented
22	#91096	C₂²	1	4	yes	complemented
23	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
24	C₂×#5147	—	5	1	no	maximal, Hall
25	C₂²×C₅×D₈	D₃	1	1	yes	characteristic, complemented
26	C₂×#4050	—	3	1	no	—
27	C₂×#3959	—	3	2	no	—
28	#65436	—	3	8	no	—
29	C₂×#3958	—	3	2	no	—
30	C₂×#3944	—	3	1	no	—
31	C₂⁴×C₃×C₅	—	4	1	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	—	2	4	no	—
34	C₂×C₃×C₅×D₄	—	4	1	no	—
35	C₂×C₃×C₅×D₄	—	4	2	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
37	C₂×#1220	—	4	1	no	—
38	#6817	—	2	4	no	—
39	#6817	—	4	2	no	—
40	C₂×D₆₀	—	2	1	no	—
41	C₂×C₄×D₁₅	—	4	1	no	—
42	C₂×Dic₃₀	—	2	1	no	—
43	C₃×C₅×D₈	—	2	2	no	—
44	C₃×C₅×D₈	—	2	4	no	—
45	C₃×C₅×D₈	C₂³	1	4	yes	—
46	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
50	Dic₆₀	—	2	1	no	—
51	Dic₆₀	C₂³	1	2	yes	complemented
52	D₁₂₀	—	2	1	no	—
53	D₁₂₀	C₂³	1	2	yes	complemented
54	#6706	—	2	4	no	—
55	C₃⋊#672	C₂³	1	4	yes	complemented
56	C₂²×C₃×D₈	D₅	1	1	yes	characteristic, complemented
57	C₂×#904	—	5	1	no	—
58	C₂×#827	—	5	2	no	—
59	#5147	—	5	8	no	—
60	C₂×#826	—	5	2	no	—
61	C₂×#812	—	5	1	no	—
62	C₂²×C₅×D₄	—	2	1	no	—
63	C₂×#709	—	6	1	no	—
64	C₂×C₅×D₈	—	2	1	no	—
65	C₂×C₅×D₈	—	2	2	no	—
66	C₂×C₅×D₈	C₂×D₃	1	2	yes	—
67	C₂×C₅×D₈	C₂×D₃	1	4	yes	complemented
68	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
69	C₂×Dic₂₀	—	3	1	no	—
70	#4050	—	3	4	no	—
71	#4050	—	6	2	no	—
72	C₂×D₄₀	—	3	1	no	—
73	C₂×#677	—	6	1	no	—
74	#3959	—	3	8	no	—
75	#3958	—	3	8	no	—
76	#3944	—	3	4	no	—
77	C₂×#672	—	3	2	no	—
78	C₂×#509	—	15	1	no	Sylow
79	C₂³×C₃×C₅	—	4	4	no	—
80	C₂³×C₃×C₅	—	4	2	no	—
81	C₂³×C₃×C₅	—	8	4	no	—
82	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
83	C₂²×D₁₅	—	4	1	no	—
84	C₃×C₅×D₄	—	2	4	no	—
85	C₃×C₅×D₄	—	4	2	no	—
86	C₃×C₅×D₄	—	4	4	no	—
87	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	#1220	—	4	4	no	—
92	C₂×Dic₁₅	—	4	1	no	—
93	D₆₀	—	2	2	no	—
94	D₆₀	—	4	1	no	—
95	C₄×D₁₅	—	4	4	no	—
96	Dic₃₀	—	2	2	no	—
97	Dic₃₀	—	4	1	no	—
98	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₂²×C₃×D₄	—	2	1	no	—
102	C₂×#235	—	10	1	no	—
103	C₂×C₃×D₈	—	2	1	no	—
104	C₂×C₃×D₈	—	2	2	no	—
105	C₂×C₃×D₈	C₂×D₅	1	2	yes	—
106	C₂×C₃×D₈	C₂×D₅	1	4	yes	complemented
107	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×Dic₁₂	—	5	1	no	—
109	#904	—	5	4	no	—
110	#904	—	10	2	no	—
111	C₂×D₂₄	—	5	1	no	—
112	C₂×#204	—	10	1	no	—
113	#827	—	5	8	no	—
114	#826	—	5	8	no	—
115	#812	—	5	4	no	—
116	C₂×C₃⋊C₁₆	—	5	2	no	—
117	C₂⁴×C₅	—	4	1	no	—
118	C₂×C₅×D₄	—	2	2	no	—
119	C₂×C₅×D₄	—	2	4	no	—
120	C₂×C₅×D₄	—	4	1	no	—
121	C₂×C₅×D₄	—	4	2	no	—
122	C₂²×C₄×C₅	#67	1	1	yes	characteristic
123	C₂×C₂²⋊D₅	—	12	1	no	—
124	#709	—	6	4	no	—
125	#709	—	12	2	no	—
126	C₂×D₂₀	—	6	1	no	—
127	C₂×C₄×D₅	—	12	1	no	—
128	C₂×Dic₁₀	—	6	1	no	—
129	C₅×D₈	—	2	2	no	—
130	C₅×D₈	—	2	4	no	—
131	C₅×D₈	C₂²×D₃	1	4	yes	—
132	C₂×C₅×C₈	#67	1	1	yes	characteristic
133	C₂×C₅×C₈	#67	1	2	yes	—
134	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
136	Dic₂₀	—	3	2	no	—
137	Dic₂₀	—	6	1	no	—
138	D₄₀	—	3	2	no	—
139	D₄₀	—	6	1	no	—
140	#677	—	6	4	no	—
141	#672	—	3	4	no	—
142	C₂×#135	—	15	1	no	—
143	C₂²×D₈	D₁₅	1	1	yes	characteristic, complemented
144	#509	—	15	8	no	—
145	C₂×SD₃₂	—	15	2	no	—
146	C₂×D₁₆	—	15	2	no	—
147	C₂×M₃₂	—	15	1	no	—
148	C₂²×C₃×C₅	—	2	2	no	—
149	C₂²×C₃×C₅	—	4	4	no	—
150	C₂²×C₃×C₅	—	8	8	no	—
151	C₂²×C₃×C₅	—	8	4	no	—
152	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
153	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
154	C₂×D₁₅	—	4	2	no	—
155	C₂×D₁₅	—	8	2	no	—
156	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
158	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	Dic₁₅	—	4	2	no	—
160	C₂⁴×C₃	—	4	1	no	—
161	C₂×C₃×D₄	—	2	2	no	—
162	C₂×C₃×D₄	—	2	4	no	—
163	C₂×C₃×D₄	—	4	1	no	—
164	C₂×C₃×D₄	—	4	2	no	—
165	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
166	C₂×#67	—	20	1	no	—
167	#235	—	10	4	no	—
168	#235	—	20	2	no	—
169	C₂×D₁₂	—	10	1	no	—
170	C₂×C₄×D₃	—	20	1	no	—
171	C₂×Dic₆	—	10	1	no	—
172	C₃×D₈	—	2	2	no	—
173	C₃×D₈	—	2	4	no	—
174	C₃×D₈	C₂²×D₅	1	4	yes	—
175	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
176	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
177	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
179	Dic₁₂	—	5	2	no	—
180	Dic₁₂	—	10	1	no	—
181	D₂₄	—	5	2	no	—
182	D₂₄	—	10	1	no	—
183	#204	—	10	4	no	—
184	C₃⋊C₁₆	—	5	4	no	—
185	C₂³×C₅	—	4	4	no	—
186	C₂³×C₅	—	4	2	no	—
187	C₂³×C₅	—	8	4	no	—
188	C₂³×C₅	#213	1	1	yes	characteristic
189	C₂²×D₅	—	12	1	no	—
190	C₅×D₄	—	2	4	no	—
191	C₅×D₄	—	4	2	no	—
192	C₅×D₄	—	4	4	no	—
193	C₂×C₄×C₅	#213	1	1	yes	characteristic
194	C₂×C₄×C₅	#213	1	2	yes	—
195	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂×#67	1	2	yes	—
197	C₂²⋊D₅	—	12	4	no	—
198	C₂×Dic₅	—	12	1	no	—
199	D₂₀	—	6	2	no	—
200	D₂₀	—	12	1	no	—
201	C₄×D₅	—	12	4	no	—
202	Dic₁₀	—	6	2	no	—
203	Dic₁₀	—	12	1	no	—
204	C₅×C₈	C₂×#67	1	2	yes	—
205	C₅×C₈	C₂×#67	1	1	yes	characteristic
206	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
207	C₂×#41	—	30	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	#135	—	15	4	no	—
210	#135	—	30	2	no	—
211	C₂×Q₁₆	—	15	1	no	—
212	C₂×SD₁₆	—	30	1	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	—	15	1	no	—
216	C₂×D₈	C₂×D₁₅	1	2	yes	—
217	C₂×D₈	C₂×D₁₅	1	4	yes	complemented
218	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
219	SD₃₂	—	15	8	no	—
220	D₁₆	—	15	8	no	—
221	M₃₂	—	15	4	no	—
222	C₂×C₁₆	—	15	2	no	—
223	C₂×C₃×C₅	—	2	2	no	—
224	C₂×C₃×C₅	—	8	4	no	—
225	C₂×C₃×C₅	#509	1	2	yes	complemented
226	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
227	D₁₅	—	8	2	no	—
228	C₂³×C₃	—	4	4	no	—
229	C₂³×C₃	—	4	2	no	—
230	C₂³×C₃	—	8	4	no	—
231	C₂³×C₃	#686	1	1	yes	characteristic
232	C₂²×D₃	—	20	1	no	—
233	C₃×D₄	—	2	4	no	—
234	C₃×D₄	—	4	2	no	—
235	C₃×D₄	—	4	4	no	—
236	C₂×C₃×C₄	#686	1	1	yes	characteristic
237	C₂×C₃×C₄	#686	1	2	yes	—
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
240	#67	—	20	4	no	—
241	C₂×Dic₃	—	20	1	no	—
242	D₁₂	—	10	2	no	—
243	D₁₂	—	20	1	no	—
244	C₄×D₃	—	20	4	no	—
245	Dic₆	—	10	2	no	—
246	Dic₆	—	20	1	no	—
247	C₃×C₈	C₂×C₂²⋊D₅	1	2	yes	—
248	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
249	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	—	4	4	no	—
252	C₂²×C₅	—	8	8	no	—
253	C₂²×C₅	—	8	4	no	—
254	C₂²×C₅	C₂×#213	1	2	yes	—
255	C₂²×C₅	C₂×#213	1	1	yes	characteristic
256	C₂×D₅	—	12	2	no	—
257	C₂×D₅	—	24	2	no	—
258	C₄×C₅	C₂×#213	1	1	yes	characteristic
259	C₄×C₅	C₂×#213	1	2	yes	—
260	C₄×C₅	C₂²×#67	1	1	yes	characteristic
261	Dic₅	—	12	2	no	—
262	C₂⁴	—	4	1	no	—
263	#41	—	30	4	no	—
264	#41	—	60	2	no	—
265	C₂×Q₈	—	30	1	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	30	1	no	—
271	C₂×D₄	—	60	1	no	—
272	C₂²×C₄	—	60	1	no	—
273	C₂²×C₄	#1220	1	1	yes	characteristic
274	Q₁₆	—	15	2	no	—
275	Q₁₆	—	30	1	no	—
276	SD₁₆	—	30	4	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	4	no	—
279	D₈	—	15	2	no	—
280	D₈	—	30	1	no	—
281	D₈	C₂²×D₁₅	1	4	yes	—
282	C₂×C₈	#1220	1	1	yes	characteristic
283	C₂×C₈	#1220	1	2	yes	—
284	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
285	C₂×C₈	C₂²×D₁₅	1	2	yes	—
286	C₁₆	—	15	4	no	—
287	C₃×C₅	C₂×#509	1	1	yes	characteristic, complemented, Hall
288	C₂²×C₃	—	2	2	no	—
289	C₂²×C₃	—	4	4	no	—
290	C₂²×C₃	—	8	8	no	—
291	C₂²×C₃	—	8	4	no	—
292	C₂²×C₃	C₂×#686	1	2	yes	—
293	C₂²×C₃	C₂×#686	1	1	yes	characteristic
294	C₂×D₃	—	20	2	no	—
295	C₂×D₃	—	40	2	no	—
296	C₃×C₄	C₂×#686	1	1	yes	characteristic
297	C₃×C₄	C₂×#686	1	2	yes	—
298	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
299	Dic₃	—	20	2	no	—
300	C₂×C₅	—	2	2	no	—
301	C₂×C₅	—	8	4	no	—
302	C₂×C₅	#5147	1	2	yes	complemented
303	C₂×C₅	C₂²×#213	1	1	yes	characteristic
304	D₅	—	24	2	no	—
305	C₂³	—	4	4	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	8	4	no	—
308	C₂³	—	60	1	no	—
309	C₂³	#6715	1	1	yes	characteristic
310	Q₈	—	30	2	no	—
311	Q₈	—	60	1	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	2	no	—
315	D₄	—	30	2	no	—
316	D₄	—	60	4	no	—
317	D₄	—	60	1	no	—
318	C₂×C₄	—	60	1	no	—
319	C₂×C₄	—	60	4	no	—
320	C₂×C₄	#6715	1	2	yes	—
321	C₂×C₄	#6715	1	1	yes	characteristic
322	C₂×C₄	C₂×#1220	1	2	yes	—
323	C₂×C₄	C₂×#1220	1	1	yes	characteristic
324	C₈	C₂×#1220	1	1	yes	characteristic
325	C₈	C₂×#1220	1	2	yes	—
326	C₈	C₂³×D₁₅	1	1	yes	characteristic
327	C₂×C₃	—	2	2	no	—
328	C₂×C₃	—	8	4	no	—
329	C₂×C₃	#65436	1	2	yes	complemented
330	C₂×C₃	C₂²×#686	1	1	yes	characteristic
331	D₃	—	40	2	no	—
332	C₅	C₂×#5147	1	1	yes	characteristic, complemented, Sylow
333	C₂²	—	2	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	8	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	60	2	no	—
338	C₂²	—	120	2	no	—
339	C₂²	C₂×#6715	1	2	yes	—
340	C₂²	C₂×#6715	1	1	yes	characteristic
341	C₄	—	60	2	no	—
342	C₄	C₂×#6715	1	2	yes	—
343	C₄	C₂×#6715	1	1	yes	characteristic
344	C₄	C₂²×#1220	1	1	yes	characteristic
345	C₃	C₂×#65436	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	2	2	no	—
347	C₂	—	8	4	no	—
348	C₂	—	120	2	no	—
349	C₂	#11751248	1	2	yes	direct product factor
350	C₂	C₂²×#6715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6715
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂³×D₁₅
Fitting	C₂²×C₃×C₅×D₈	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂²×D₈	D₁₅
hypocenter	C₃×C₅	C₂×#509
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×D₈	C₂	semidirect
2. C₂×#91793	C₂	semidirect
3. C₂×#91111	C₂	semidirect
4. #11751248	C₂	direct (N × H)
5. C₂×#91110	C₂	semidirect
6. C₂×#91096	C₂	semidirect
13. C₂×C₃×C₅×D₈	C₂²	semidirect
15. C₂×Dic₆₀	C₂²	semidirect
17. #91793	C₂²	semidirect
18. C₂×D₁₂₀	C₂²	semidirect
20. #91111	C₂²	semidirect
21. #91110	C₂²	semidirect
22. #91096	C₂²	semidirect
23. C₂×C₃⋊#672	C₂²	semidirect
25. C₂²×C₅×D₈	D₃	semidirect
51. Dic₆₀	C₂³	semidirect
53. D₁₂₀	C₂³	semidirect
55. C₃⋊#672	C₂³	semidirect
56. C₂²×C₃×D₈	D₅	semidirect
67. C₂×C₅×D₈	C₂×D₃	semidirect
106. C₂×C₃×D₈	C₂×D₅	semidirect
143. C₂²×D₈	D₁₅	semidirect
217. C₂×D₈	C₂×D₁₅	semidirect
225. C₂×C₃×C₅	#509	semidirect
287. C₃×C₅	C₂×#509	semidirect
302. C₂×C₅	#5147	semidirect
329. C₂×C₃	#65436	semidirect
332. C₅	C₂×#5147	semidirect
345. C₃	C₂×#65436	semidirect
349. C₂	#11751248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×D₈	2
C₂×#91793	2
C₂×#91111	2
#11751248	2
C₂×#91110	2
C₂×#91096	2
C₂×#65436	3
C₂×#5147	5

Subgroups — C2×#11751248 #14478328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751248 #14478328