Subgroups — C₂×#11751296 #14478382

Order 1920.

720 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×C₃×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₆₀	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751296	C₂	1	8	yes	maximal, direct product factor
4	C₂×#91113	C₂	1	2	yes	maximal, complemented
5	C₂×#91111	C₂	1	2	yes	maximal, complemented
6	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65484	—	3	1	no	maximal, Hall
8	C₂×C₃×C₅×#41	—	2	1	no	—
9	C₂²×Dic₃₀	—	2	1	no	—
10	C₃×C₅×#135	—	2	2	no	—
11	C₃×C₅×#135	C₂²	1	4	yes	—
12	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
14	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₀	—	2	2	no	—
17	C₂×Dic₆₀	—	2	1	no	—
18	C₂×Dic₆₀	C₂²	1	2	yes	—
19	C₂×Dic₆₀	C₂²	1	4	yes	complemented
20	#91113	C₂²	1	8	yes	complemented
21	#91111	C₂²	1	8	yes	complemented
22	#91096	C₂²	1	4	yes	complemented
23	C₂×C₃⋊#672	C₂²	1	2	yes	—
24	C₂×#5195	—	5	1	no	maximal, Hall
25	C₂×C₅×#135	D₃	1	1	yes	characteristic
26	C₂²×Dic₂₀	—	3	1	no	—
27	#65484	—	3	8	no	—
28	C₂×#3961	—	3	2	no	—
29	C₂×#3959	—	3	2	no	—
30	C₂×#3944	—	3	1	no	—
31	C₃×C₅×#41	—	2	4	no	—
32	C₃×C₅×#41	—	4	2	no	—
33	C₂×C₃×C₅×Q₈	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	4	1	no	—
36	C₂²×C₃×C₄×C₅	—	4	1	no	—
37	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
38	C₂²×Dic₁₅	—	4	1	no	—
39	C₂×Dic₃₀	—	2	2	no	—
40	C₂×Dic₃₀	—	2	4	no	—
41	C₂×Dic₃₀	—	4	1	no	—
42	C₂×Dic₃₀	—	4	2	no	—
43	C₃×C₅×Q₁₆	—	2	1	no	—
44	C₃×C₅×Q₁₆	C₂³	1	2	yes	—
45	C₃×C₅×SD₁₆	—	2	4	no	—
46	C₃×C₅×D₈	—	2	1	no	—
47	C₃×C₅×D₈	C₂³	1	2	yes	—
48	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
49	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
52	Dic₆₀	—	2	2	no	—
53	Dic₆₀	—	2	4	no	—
54	Dic₆₀	C₂³	1	4	yes	—
55	C₃⋊#672	C₂³	1	4	yes	—
56	C₂×C₃×#135	D₅	1	1	yes	characteristic
57	C₂²×Dic₁₂	—	5	1	no	—
58	#5195	—	5	8	no	—
59	C₂×#829	—	5	2	no	—
60	C₂×#827	—	5	2	no	—
61	C₂×#812	—	5	1	no	—
62	C₂×C₅×#41	—	2	1	no	—
63	C₂²×Dic₁₀	—	6	1	no	—
64	C₅×#135	—	2	2	no	—
65	C₅×#135	C₂×D₃	1	4	yes	—
66	C₂×C₅×Q₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂×C₅×SD₁₆	—	2	1	no	—
68	C₂×C₅×D₈	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
70	C₂×Dic₂₀	—	3	2	no	—
71	C₂×Dic₂₀	—	3	4	no	—
72	C₂×Dic₂₀	—	6	1	no	—
73	C₂×Dic₂₀	—	6	2	no	—
74	#3961	—	3	8	no	—
75	#3959	—	3	8	no	—
76	#3944	—	3	4	no	—
77	C₂×#672	—	3	2	no	—
78	C₂×#510	—	15	1	no	Sylow
79	C₂³×C₃×C₅	—	4	1	no	—
80	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
81	C₃×C₅×Q₈	—	2	2	no	—
82	C₃×C₅×Q₈	—	4	1	no	—
83	C₃×C₅×D₄	—	2	2	no	—
84	C₃×C₅×D₄	—	4	4	no	—
85	C₃×C₅×D₄	—	4	1	no	—
86	C₂×C₃×C₄×C₅	—	4	1	no	—
87	C₂×C₃×C₄×C₅	—	4	4	no	—
88	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×Dic₁₅	—	4	4	no	—
93	C₂×Dic₁₅	—	4	2	no	—
94	Dic₃₀	—	2	4	no	—
95	Dic₃₀	—	4	4	no	—
96	Dic₃₀	—	4	2	no	—
97	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×#41	—	2	1	no	—
101	C₂²×Dic₆	—	10	1	no	—
102	C₃×#135	—	2	2	no	—
103	C₃×#135	C₂×D₅	1	4	yes	—
104	C₂×C₃×Q₁₆	C₂×D₅	1	1	yes	characteristic
105	C₂×C₃×SD₁₆	—	2	1	no	—
106	C₂×C₃×D₈	C₂×D₅	1	1	yes	characteristic
107	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×Dic₁₂	—	5	2	no	—
109	C₂×Dic₁₂	—	5	4	no	—
110	C₂×Dic₁₂	—	10	1	no	—
111	C₂×Dic₁₂	—	10	2	no	—
112	#829	—	5	8	no	—
113	#827	—	5	8	no	—
114	#812	—	5	4	no	—
115	C₂×C₃⋊C₁₆	—	5	2	no	—
116	C₅×#41	—	2	4	no	—
117	C₅×#41	—	4	2	no	—
118	C₂×C₅×Q₈	—	2	1	no	—
119	C₂×C₅×D₄	—	2	1	no	—
120	C₂×C₅×D₄	—	4	1	no	—
121	C₂²×C₄×C₅	—	4	1	no	—
122	C₂²×C₄×C₅	#67	1	1	yes	characteristic
123	C₂²×Dic₅	—	12	1	no	—
124	C₂×Dic₁₀	—	6	2	no	—
125	C₂×Dic₁₀	—	6	4	no	—
126	C₂×Dic₁₀	—	12	1	no	—
127	C₂×Dic₁₀	—	12	2	no	—
128	C₅×Q₁₆	—	2	1	no	—
129	C₅×Q₁₆	C₂²×D₃	1	2	yes	—
130	C₅×SD₁₆	—	2	4	no	—
131	C₅×D₈	—	2	1	no	—
132	C₅×D₈	C₂²×D₃	1	2	yes	—
133	C₂×C₅×C₈	#67	1	1	yes	characteristic
134	C₂×C₅×C₈	#67	1	2	yes	—
135	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
137	Dic₂₀	—	3	4	no	—
138	Dic₂₀	—	6	2	no	—
139	Dic₂₀	—	6	4	no	—
140	#672	—	3	4	no	—
141	C₂×#135	D₁₅	1	1	yes	characteristic
142	C₂²×Q₁₆	—	15	1	no	—
143	#510	—	15	8	no	—
144	C₂×Q₃₂	—	15	2	no	—
145	C₂×SD₃₂	—	15	2	no	—
146	C₂×M₃₂	—	15	1	no	—
147	C₂²×C₃×C₅	—	2	2	no	—
148	C₂²×C₃×C₅	—	4	2	no	—
149	C₂²×C₃×C₅	—	8	2	no	—
150	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
151	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
152	C₃×C₄×C₅	—	4	2	no	—
153	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
154	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
155	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
156	Dic₁₅	—	4	4	no	—
157	C₃×#41	—	2	4	no	—
158	C₃×#41	—	4	2	no	—
159	C₂×C₃×Q₈	—	2	1	no	—
160	C₂×C₃×D₄	—	2	1	no	—
161	C₂×C₃×D₄	—	4	1	no	—
162	C₂²×C₃×C₄	—	4	1	no	—
163	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
164	C₂²×Dic₃	—	20	1	no	—
165	C₂×Dic₆	—	10	2	no	—
166	C₂×Dic₆	—	10	4	no	—
167	C₂×Dic₆	—	20	1	no	—
168	C₂×Dic₆	—	20	2	no	—
169	C₃×Q₁₆	—	2	1	no	—
170	C₃×Q₁₆	C₂²×D₅	1	2	yes	—
171	C₃×SD₁₆	—	2	4	no	—
172	C₃×D₈	—	2	1	no	—
173	C₃×D₈	C₂²×D₅	1	2	yes	—
174	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
175	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
176	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
178	Dic₁₂	—	5	4	no	—
179	Dic₁₂	—	10	2	no	—
180	Dic₁₂	—	10	4	no	—
181	C₃⋊C₁₆	—	5	4	no	—
182	C₂³×C₅	—	4	1	no	—
183	C₂³×C₅	#213	1	1	yes	characteristic
184	C₅×Q₈	—	2	2	no	—
185	C₅×Q₈	—	4	1	no	—
186	C₅×D₄	—	2	2	no	—
187	C₅×D₄	—	4	4	no	—
188	C₅×D₄	—	4	1	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	—	4	4	no	—
191	C₂×C₄×C₅	#213	1	1	yes	characteristic
192	C₂×C₄×C₅	#213	1	2	yes	—
193	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×#67	1	2	yes	—
195	C₂×Dic₅	—	12	4	no	—
196	C₂×Dic₅	—	12	2	no	—
197	Dic₁₀	—	6	4	no	—
198	Dic₁₀	—	12	4	no	—
199	Dic₁₀	—	12	2	no	—
200	C₅×C₈	C₂×#67	1	1	yes	characteristic
201	C₅×C₈	C₂×#67	1	2	yes	—
202	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
203	C₂×#41	—	2	1	no	—
204	C₂²×Q₈	—	30	1	no	—
205	#135	—	2	2	no	—
206	#135	C₂×D₁₅	1	4	yes	—
207	C₂×Q₁₆	—	15	2	no	—
208	C₂×Q₁₆	—	15	4	no	—
209	C₂×Q₁₆	—	30	1	no	—
210	C₂×Q₁₆	—	30	2	no	—
211	C₂×Q₁₆	C₂×D₁₅	1	1	yes	characteristic
212	C₂×SD₁₆	—	2	1	no	—
213	C₂×D₈	C₂×D₁₅	1	1	yes	characteristic
214	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
215	Q₃₂	—	15	8	no	—
216	SD₃₂	—	15	8	no	—
217	M₃₂	—	15	4	no	—
218	C₂×C₁₆	—	15	2	no	—
219	C₂×C₃×C₅	—	2	2	no	—
220	C₂×C₃×C₅	—	8	2	no	—
221	C₂×C₃×C₅	#510	1	2	yes	complemented
222	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
223	C₂³×C₃	—	4	1	no	—
224	C₂³×C₃	#686	1	1	yes	characteristic
225	C₃×Q₈	—	2	2	no	—
226	C₃×Q₈	—	4	1	no	—
227	C₃×D₄	—	2	2	no	—
228	C₃×D₄	—	4	4	no	—
229	C₃×D₄	—	4	1	no	—
230	C₂×C₃×C₄	—	4	1	no	—
231	C₂×C₃×C₄	—	4	4	no	—
232	C₂×C₃×C₄	#686	1	1	yes	characteristic
233	C₂×C₃×C₄	#686	1	2	yes	—
234	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
236	C₂×Dic₃	—	20	4	no	—
237	C₂×Dic₃	—	20	2	no	—
238	Dic₆	—	10	4	no	—
239	Dic₆	—	20	4	no	—
240	Dic₆	—	20	2	no	—
241	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	C₃×C₈	C₂×C₂²⋊D₅	1	2	yes	—
243	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
244	C₂²×C₅	—	2	2	no	—
245	C₂²×C₅	—	4	2	no	—
246	C₂²×C₅	—	8	2	no	—
247	C₂²×C₅	C₂×#213	1	2	yes	—
248	C₂²×C₅	C₂×#213	1	1	yes	characteristic
249	C₄×C₅	—	4	2	no	—
250	C₄×C₅	C₂×#213	1	2	yes	—
251	C₄×C₅	C₂×#213	1	1	yes	characteristic
252	C₄×C₅	C₂²×#67	1	1	yes	characteristic
253	Dic₅	—	12	4	no	—
254	#41	—	2	4	no	—
255	#41	—	4	2	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	30	4	no	—
258	C₂×Q₈	—	30	2	no	—
259	C₂×Q₈	—	60	2	no	—
260	C₂×Q₈	—	60	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	60	1	no	—
265	C₂²×C₄	#1220	1	1	yes	characteristic
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	15	4	no	—
268	Q₁₆	—	30	2	no	—
269	Q₁₆	—	30	4	no	—
270	Q₁₆	C₂²×D₁₅	1	2	yes	—
271	SD₁₆	—	2	4	no	—
272	D₈	—	2	1	no	—
273	D₈	C₂²×D₁₅	1	2	yes	—
274	C₂×C₈	#1220	1	1	yes	characteristic
275	C₂×C₈	#1220	1	2	yes	—
276	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
277	C₂×C₈	C₂²×D₁₅	1	2	yes	—
278	C₁₆	—	15	4	no	—
279	C₃×C₅	C₂×#510	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	—	8	2	no	—
283	C₂²×C₃	C₂×#686	1	2	yes	—
284	C₂²×C₃	C₂×#686	1	1	yes	characteristic
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	C₂×#686	1	2	yes	—
287	C₃×C₄	C₂×#686	1	1	yes	characteristic
288	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
289	Dic₃	—	20	4	no	—
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	—	8	2	no	—
292	C₂×C₅	#5195	1	2	yes	complemented
293	C₂×C₅	C₂²×#213	1	1	yes	characteristic
294	C₂³	—	4	1	no	—
295	C₂³	#6715	1	1	yes	characteristic
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	30	4	no	—
299	Q₈	—	60	4	no	—
300	Q₈	—	60	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	60	4	no	—
307	C₂×C₄	—	60	2	no	—
308	C₂×C₄	#6715	1	2	yes	—
309	C₂×C₄	#6715	1	1	yes	characteristic
310	C₂×C₄	C₂×#1220	1	2	yes	—
311	C₂×C₄	C₂×#1220	1	1	yes	characteristic
312	C₈	C₂×#1220	1	1	yes	characteristic
313	C₈	C₂×#1220	1	2	yes	—
314	C₈	C₂³×D₁₅	1	1	yes	characteristic
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	8	2	no	—
317	C₂×C₃	#65484	1	2	yes	complemented
318	C₂×C₃	C₂²×#686	1	1	yes	characteristic
319	C₅	C₂×#5195	1	1	yes	characteristic, complemented, Sylow
320	C₂²	—	2	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₂×#6715	1	2	yes	—
324	C₂²	C₂×#6715	1	1	yes	characteristic
325	C₄	—	4	2	no	—
326	C₄	—	60	4	no	—
327	C₄	C₂×#6715	1	2	yes	—
328	C₄	C₂×#6715	1	1	yes	characteristic
329	C₄	C₂²×#1220	1	1	yes	characteristic
330	C₃	C₂×#65484	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	8	2	no	—
333	C₂	#11751296	1	2	yes	direct product factor
334	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, prime radical	C₂×C₃×C₅×#135	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×#135	D₁₅
hypocenter	C₃×C₅	C₂×#510
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₆₀	C₂	semidirect
3. #11751296	C₂	direct (N × H)
4. C₂×#91113	C₂	semidirect
5. C₂×#91111	C₂	semidirect
6. C₂×#91096	C₂	semidirect
19. C₂×Dic₆₀	C₂²	semidirect
20. #91113	C₂²	semidirect
21. #91111	C₂²	semidirect
22. #91096	C₂²	semidirect
221. C₂×C₃×C₅	#510	semidirect
279. C₃×C₅	C₂×#510	semidirect
292. C₂×C₅	#5195	semidirect
317. C₂×C₃	#65484	semidirect
319. C₅	C₂×#5195	semidirect
330. C₃	C₂×#65484	semidirect
333. C₂	#11751296	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×#135	2
C₂²×Dic₆₀	2
#11751296	2
C₂×#91113	2
C₂×#91111	2
C₂×#91096	2
C₂×#65484	3
C₂×#5195	5

Subgroups — C2×#11751296 #14478382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751296 #14478382