Subgroups — C₂×#11746336 #14312967

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91790	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746336	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64735	—	3	1	no	maximal, Hall
6	C₂×#6817	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6710	C₄	1	1	yes	characteristic
9	#91790	—	2	1	no	—
10	#91790	—	2	1	no	—
11	#91790	C₄	1	2	yes	—
12	#91790	C₂²	1	2	yes	—
13	C₂×#6705	—	2	1	no	—
14	C₂×C₈×D₁₅	—	2	1	no	—
15	C₃×C₅×M₃₂	—	2	1	no	—
16	C₃×C₅×M₃₂	C₂²	1	2	yes	complemented
17	C₂×C₃×C₅×C₁₆	—	2	1	no	—
18	C₂×C₃⋊#672	—	2	1	no	—
19	C₂×C₃⋊#672	—	2	1	no	—
20	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
21	C₂×#4515	—	5	1	no	maximal, Hall
22	C₂×#4047	—	3	1	no	—
23	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
24	C₂²×#672	—	3	1	no	—
25	#64735	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×#1220	—	2	1	no	—
28	#6817	—	2	1	no	—
29	#6817	—	2	1	no	—
30	#6817	C₈	1	2	yes	—
31	#6817	C₂×C₄	1	2	yes	—
32	C₂×D₆₀	C₈	1	1	yes	characteristic
33	C₂×C₄×D₁₅	—	2	1	no	—
34	C₂×Dic₃₀	C₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	#6710	—	2	1	no	—
42	#6710	C₂×C₄	1	2	yes	—
43	C₂×C₃⋊#168	—	2	1	no	—
44	#6705	—	2	2	no	—
45	#6705	—	2	2	no	—
46	C₈×D₁₅	—	2	2	no	—
47	C₈×D₁₅	—	2	2	no	—
48	C₃×C₅×C₁₆	—	2	2	no	—
49	C₃⋊#672	—	2	2	no	—
50	C₂×#901	—	5	1	no	—
51	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic, complemented
52	C₂²×C₃⋊C₁₆	—	5	1	no	—
53	#4515	—	5	4	no	—
54	C₂×#709	—	3	1	no	—
55	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×#681	—	3	1	no	—
57	#4047	—	3	2	no	—
58	#4047	—	3	2	no	—
59	#4047	—	6	1	no	—
60	#4047	—	6	1	no	—
61	C₂×#676	—	6	1	no	—
62	C₂×C₈×D₅	—	6	1	no	—
63	C₅×M₃₂	—	2	1	no	—
64	C₅×M₃₂	C₂×D₃	1	2	yes	complemented
65	C₂×C₅×C₁₆	—	2	1	no	—
66	C₂×#672	—	3	2	no	—
67	C₂×#672	—	6	1	no	—
68	C₂×#672	—	6	1	no	—
69	C₂×#350	—	15	1	no	Sylow
70	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
71	C₂²×D₁₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	#1220	—	2	2	no	—
79	#1220	—	2	2	no	—
80	C₂×Dic₁₅	—	2	1	no	—
81	D₆₀	—	2	1	no	—
82	D₆₀	C₂×C₈	1	2	yes	—
83	C₄×D₁₅	—	2	2	no	—
84	C₄×D₁₅	—	2	2	no	—
85	Dic₃₀	—	2	1	no	—
86	Dic₃₀	C₂×C₈	1	2	yes	—
87	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₃⋊#168	—	2	2	no	—
92	C₂×#235	—	5	1	no	—
93	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#208	—	5	1	no	—
95	#901	—	5	2	no	—
96	#901	—	5	2	no	—
97	#901	—	10	1	no	—
98	#901	—	10	1	no	—
99	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
100	C₂×D₃×C₈	—	10	1	no	—
101	C₃×M₃₂	—	2	1	no	—
102	C₃×M₃₂	C₂×D₅	1	2	yes	complemented
103	C₂×C₃×C₁₆	—	2	1	no	—
104	C₂×C₃⋊C₁₆	—	5	2	no	—
105	C₂×C₃⋊C₁₆	—	10	1	no	—
106	C₂×C₃⋊C₁₆	—	10	1	no	—
107	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
108	C₂×C₂²⋊D₅	—	6	1	no	—
109	#709	—	3	2	no	—
110	#709	—	3	2	no	—
111	#709	—	6	1	no	—
112	#709	—	6	1	no	—
113	C₂×D₂₀	—	3	1	no	—
114	C₂×C₄×D₅	—	6	1	no	—
115	C₂×Dic₁₀	—	3	1	no	—
116	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
117	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
118	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
119	C₂×C₅×C₈	#67	1	1	yes	characteristic
120	C₂×C₅×C₈	#67	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
122	#681	—	3	2	no	—
123	#681	—	6	1	no	—
124	C₂×#168	—	6	1	no	—
125	#676	—	6	2	no	—
126	#676	—	6	2	no	—
127	C₈×D₅	—	6	2	no	—
128	C₈×D₅	—	6	2	no	—
129	C₅×C₁₆	—	2	2	no	—
130	#672	—	6	2	no	—
131	C₂×#131	—	15	1	no	—
132	C₂×M₃₂	D₁₅	1	1	yes	characteristic, complemented
133	C₂²×C₁₆	—	15	1	no	—
134	#350	—	15	4	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
139	C₂×D₁₅	—	2	2	no	—
140	C₂×D₁₅	—	4	2	no	—
141	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
145	Dic₁₅	—	2	2	no	—
146	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
147	C₂×#67	—	10	1	no	—
148	#235	—	5	2	no	—
149	#235	—	5	2	no	—
150	#235	—	10	1	no	—
151	#235	—	10	1	no	—
152	C₂×D₁₂	—	5	1	no	—
153	C₂×C₄×D₃	—	10	1	no	—
154	C₂×Dic₆	—	5	1	no	—
155	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
156	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
157	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
158	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
159	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
160	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
161	#208	—	5	2	no	—
162	#208	—	10	1	no	—
163	C₂×C₃⋊C₈	—	10	1	no	—
164	ΓL(1,𝔽₂₅)	—	10	2	no	—
165	ΓL(1,𝔽₂₅)	—	10	2	no	—
166	D₃×C₈	—	10	2	no	—
167	D₃×C₈	—	10	2	no	—
168	C₃×C₁₆	—	2	2	no	—
169	C₃⋊C₁₆	—	10	2	no	—
170	C₂³×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
171	C₂²×D₅	—	6	1	no	—
172	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
173	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
174	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
175	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂²⋊D₅	—	6	2	no	—
179	C₂²⋊D₅	—	6	2	no	—
180	C₂×Dic₅	—	6	1	no	—
181	D₂₀	—	3	2	no	—
182	D₂₀	—	6	1	no	—
183	C₄×D₅	—	6	2	no	—
184	C₄×D₅	—	6	2	no	—
185	Dic₁₀	—	3	2	no	—
186	Dic₁₀	—	6	1	no	—
187	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
188	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
189	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
190	C₅×C₈	C₂×#67	1	1	yes	characteristic
191	#168	—	6	2	no	—
192	C₂×#41	—	15	1	no	—
193	#131	—	15	2	no	—
194	#131	—	15	2	no	—
195	#131	—	30	1	no	—
196	#131	—	30	1	no	—
197	C₂×M₁₆	—	15	1	no	—
198	C₂×M₁₆	—	30	1	no	—
199	C₂²×C₈	—	30	1	no	—
200	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
201	M₃₂	—	2	1	no	—
202	M₃₂	C₂×D₁₅	1	2	yes	complemented
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	15	2	no	—
205	C₂×C₁₆	—	30	1	no	—
206	C₂×C₁₆	—	30	1	no	—
207	C₂×C₃×C₅	—	2	1	no	—
208	C₂×C₃×C₅	—	2	1	no	—
209	C₂×C₃×C₅	#350	1	2	yes	complemented
210	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
211	D₁₅	—	4	2	no	—
212	C₂³×C₃	#676	1	1	yes	characteristic
213	C₂²×D₃	—	10	1	no	—
214	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
216	C₂×C₃×C₄	#676	1	1	yes	characteristic
217	C₂×C₃×C₄	#685	1	1	yes	characteristic
218	C₂×C₃×C₄	#685	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
220	#67	—	10	2	no	—
221	#67	—	10	2	no	—
222	C₂×Dic₃	—	10	1	no	—
223	D₁₂	—	5	2	no	—
224	D₁₂	—	10	1	no	—
225	C₄×D₃	—	10	2	no	—
226	C₄×D₃	—	10	2	no	—
227	Dic₆	—	5	2	no	—
228	Dic₆	—	10	1	no	—
229	C₃×C₈	#685	1	1	yes	characteristic
230	C₃×C₈	#685	1	1	yes	characteristic
231	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
232	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
233	C₃⋊C₈	—	10	2	no	—
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
236	C₂²×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
237	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
238	C₂×D₅	—	6	2	no	—
239	C₂×D₅	—	12	2	no	—
240	C₄×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
241	C₄×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
242	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
243	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
244	Dic₅	—	6	2	no	—
245	#41	—	15	2	no	—
246	#41	—	15	2	no	—
247	#41	—	30	1	no	—
248	#41	—	30	1	no	—
249	C₂×Q₈	—	15	1	no	—
250	C₂×D₄	—	15	1	no	—
251	C₂×D₄	—	30	1	no	—
252	C₂²×C₄	—	30	1	no	—
253	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
254	M₁₆	—	15	2	no	—
255	M₁₆	—	30	1	no	—
256	M₁₆	—	30	2	no	—
257	M₁₆	—	30	2	no	—
258	C₂×C₈	—	30	1	no	—
259	C₂×C₈	—	30	2	no	—
260	C₂×C₈	—	30	2	no	—
261	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
262	C₂×C₈	D₆₀	1	1	yes	characteristic
263	C₂×C₈	D₆₀	1	1	yes	characteristic
264	C₂×C₈	#1220	1	1	yes	characteristic
265	C₂×C₈	#1220	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	30	2	no	—
269	C₃×C₅	C₂×#350	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	#3952	1	1	yes	characteristic
272	C₂²×C₃	#3952	1	1	yes	characteristic
273	C₂²×C₃	C₂×#676	1	1	yes	characteristic
274	C₂×D₃	—	10	2	no	—
275	C₂×D₃	—	20	2	no	—
276	C₃×C₄	#3952	1	1	yes	characteristic
277	C₃×C₄	#3952	1	1	yes	characteristic
278	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
279	C₃×C₄	C₂×#685	1	1	yes	characteristic
280	Dic₃	—	10	2	no	—
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	#4515	1	2	yes	complemented
284	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
285	D₅	—	12	2	no	—
286	C₂³	—	30	1	no	—
287	C₂³	#6705	1	1	yes	characteristic
288	Q₈	—	15	2	no	—
289	Q₈	—	30	1	no	—
290	D₄	—	15	2	no	—
291	D₄	—	30	2	no	—
292	D₄	—	30	2	no	—
293	D₄	—	30	1	no	—
294	C₂×C₄	—	30	1	no	—
295	C₂×C₄	—	30	2	no	—
296	C₂×C₄	—	30	2	no	—
297	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
298	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
299	C₂×C₄	#6705	1	1	yes	characteristic
300	C₂×C₄	#6714	1	1	yes	characteristic
301	C₂×C₄	#6714	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
303	C₈	—	30	2	no	—
304	C₈	#6714	1	1	yes	characteristic
305	C₈	#6714	1	1	yes	characteristic
306	C₈	C₂×D₆₀	1	1	yes	characteristic
307	C₈	C₂×#1220	1	1	yes	characteristic
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	#64735	1	2	yes	complemented
311	C₂×C₃	C₂×#3952	1	1	yes	characteristic
312	D₃	—	20	2	no	—
313	C₅	C₂×#4515	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	2	no	—
315	C₂²	—	30	2	no	—
316	C₂²	—	60	2	no	—
317	C₂²	#91104	1	1	yes	characteristic
318	C₂²	#91104	1	1	yes	characteristic
319	C₂²	C₂×#6705	1	1	yes	characteristic
320	C₄	—	30	2	no	—
321	C₄	#91104	1	1	yes	characteristic
322	C₄	#91104	1	1	yes	characteristic
323	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
324	C₄	C₂×#6714	1	1	yes	characteristic
325	C₃	C₂×#64735	1	1	yes	characteristic, complemented, Sylow
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	60	2	no	—
329	C₂	#11746336	1	2	yes	direct product factor
330	C₂	C₂×#91104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#1220
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂²⋊C₈
prime radical	C₂×#6817	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×M₃₂	D₁₅
hypocenter	C₃×C₅	C₂×#350
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂²×C₃⋊#672	C₂	semidirect
4. #11746336	C₂	direct (N × H)
16. C₃×C₅×M₃₂	C₂²	semidirect
20. C₂×C₃⋊#672	C₂²	semidirect
23. C₂×C₅×M₃₂	D₃	semidirect
51. C₂×C₃×M₃₂	D₅	semidirect
64. C₅×M₃₂	C₂×D₃	semidirect
102. C₃×M₃₂	C₂×D₅	semidirect
132. C₂×M₃₂	D₁₅	semidirect
202. M₃₂	C₂×D₁₅	semidirect
209. C₂×C₃×C₅	#350	semidirect
269. C₃×C₅	C₂×#350	semidirect
283. C₂×C₅	#4515	semidirect
310. C₂×C₃	#64735	semidirect
313. C₅	C₂×#4515	semidirect
325. C₃	C₂×#64735	semidirect
329. C₂	#11746336	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×#91790	2
C₂×C₃×C₅×M₃₂	2
C₂²×C₃⋊#672	2
#11746336	2
C₂×#64735	3
C₂×#4515	5

Subgroups — C2×#11746336 #14312967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746336 #14312967