Subgroups — C₂×#11751741 #14478101

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×#6648	C₂	1	1	yes	maximal, characteristic
2	C₂×#91227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91166	C₂	1	2	yes	maximal, complemented
5	#11751741	C₂	1	8	yes	maximal, direct product factor
6	C₂×#91164	C₂	1	2	yes	maximal, complemented
7	C₂×D₅⋊M₃₂	—	3	1	no	maximal, Hall
8	C₂²×C₄×D₁₅	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
11	C₂×#6648	C₄	1	1	yes	characteristic
12	C₂×#6648	C₄	1	4	yes	—
13	C₂×#6648	C₄	1	1	yes	characteristic
14	C₂×#6648	C₂²	1	1	yes	characteristic
15	C₂×#6648	C₂²	1	4	yes	—
16	C₂×#6648	C₂²	1	1	yes	characteristic
17	#91227	C₂²	1	4	yes	complemented
18	C₂×C₃⋊#674	C₂²	1	2	yes	complemented
19	C₃×#3998	C₂²	1	4	yes	complemented
20	C₂×C₃×#674	C₂²	1	2	yes	complemented
21	#91166	C₂²	1	8	yes	complemented
22	#91164	C₂²	1	8	yes	complemented
23	C₂×D₃×M₃₂	—	5	1	no	maximal, Hall
24	C₂²×C₈×D₅	—	3	1	no	—
25	C₂×#3998	—	3	1	no	—
26	C₂×#3998	D₃	1	1	yes	characteristic, complemented
27	D₅⋊M₃₂	—	3	8	no	—
28	C₂×#3990	—	3	2	no	—
29	C₂×D₅⋊C₁₆	—	3	2	no	—
30	C₂³×D₁₅	C₈	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₁₅	C₈	1	1	yes	characteristic
33	C₂×C₄×D₁₅	C₈	1	1	yes	characteristic
34	C₂×C₄×D₁₅	C₈	1	4	yes	—
35	C₂×C₄×D₁₅	C₈	1	1	yes	characteristic
36	C₂×C₄×D₁₅	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×#168	C₂³	1	1	yes	characteristic
45	C₂×C₃×#168	C₂³	1	2	yes	—
46	#6648	C₂×C₄	1	2	yes	—
47	#6648	C₂×C₄	1	4	yes	—
48	#6648	C₂×C₄	1	4	yes	—
49	#6648	C₂×C₄	1	2	yes	—
50	#6648	C₂³	1	2	yes	—
51	#6648	C₂³	1	2	yes	—
52	C₃⋊#674	C₂³	1	4	yes	complemented
53	C₃×#674	C₂³	1	4	yes	complemented
54	C₂²×D₃×C₈	—	5	1	no	—
55	C₂×C₃×M₃₂	—	5	1	no	—
56	C₂×#812	—	5	1	no	—
57	D₃×M₃₂	—	5	8	no	—
58	C₂×#798	—	5	2	no	—
59	C₂×D₃×C₁₆	—	5	2	no	—
60	C₂²×C₄×D₅	—	3	1	no	—
61	C₂²×C₅×C₈	—	3	1	no	—
62	C₂²×#168	C₂×D₃	1	1	yes	characteristic
63	C₂×C₈×D₅	—	3	4	no	—
64	C₂×C₈×D₅	—	3	1	no	—
65	C₂×C₈×D₅	—	3	1	no	—
66	C₂×C₈×D₅	—	3	1	no	—
67	C₂×C₈×D₅	—	3	1	no	—
68	C₂×C₈×D₅	—	3	4	no	—
69	#3998	—	3	4	no	—
70	#3998	C₂×D₃	1	4	yes	complemented
71	C₂×#674	—	3	2	no	—
72	C₂×#674	C₂×D₃	1	2	yes	complemented
73	#3990	—	3	8	no	—
74	D₅⋊C₁₆	—	3	8	no	—
75	C₂²×M₃₂	—	15	1	no	Sylow
76	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
77	C₂²×D₁₅	—	2	4	no	—
78	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
79	C₂²×D₁₅	C₂×C₈	1	4	yes	—
80	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₅	C₂×C₈	1	4	yes	—
86	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
87	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
88	C₄×D₁₅	C₂×C₈	1	4	yes	—
89	C₄×D₁₅	C₂×C₈	1	2	yes	—
90	C₄×D₁₅	C₂×C₈	1	2	yes	—
91	C₄×D₁₅	C₂×C₈	1	4	yes	—
92	C₄×D₁₅	C₂²×C₄	1	2	yes	—
93	C₄×D₁₅	C₂²×C₄	1	2	yes	—
94	C₃×#168	C₂²×C₄	1	1	yes	characteristic
95	C₃×#168	C₂²×C₄	1	2	yes	—
96	C₃×#168	C₂⁴	1	1	yes	characteristic
97	C₅×C₃⋊C₈	C₂²×C₄	1	2	yes	—
98	C₅×C₃⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×D₃	—	5	1	no	—
100	C₂²×C₃×C₈	—	5	1	no	—
101	C₂²×C₃⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₂×D₃×C₈	—	5	1	no	—
103	C₂×D₃×C₈	—	5	1	no	—
104	C₂×D₃×C₈	—	5	4	no	—
105	C₂×D₃×C₈	—	5	4	no	—
106	C₂×D₃×C₈	—	5	1	no	—
107	C₂×D₃×C₈	—	5	1	no	—
108	C₃×M₃₂	—	5	4	no	—
109	C₂×C₃×C₁₆	—	5	2	no	—
110	#812	—	5	4	no	—
111	C₂×C₃⋊C₁₆	—	5	2	no	—
112	#798	—	5	8	no	—
113	D₃×C₁₆	—	5	8	no	—
114	C₂³×D₅	—	3	1	no	—
115	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
116	C₂²×Dic₅	—	3	1	no	—
117	C₂×C₄×D₅	—	3	1	no	—
118	C₂×C₄×D₅	—	3	4	no	—
119	C₂×C₄×D₅	—	3	1	no	—
120	C₂×C₄×D₅	—	3	4	no	—
121	C₂×C₄×D₅	—	3	1	no	—
122	C₂×C₄×D₅	—	3	1	no	—
123	C₂×C₅×C₈	—	3	4	no	—
124	C₂×C₅×C₈	—	3	1	no	—
125	C₂×C₅×C₈	—	3	1	no	—
126	C₂×#168	C₄×D₃	1	2	yes	—
127	C₂×#168	C₄×D₃	1	1	yes	characteristic
128	C₂×#168	C₂²×D₃	1	1	yes	characteristic
129	C₂×#168	C₂²×D₃	1	2	yes	—
130	C₈×D₅	—	3	2	no	—
131	C₈×D₅	—	3	2	no	—
132	C₈×D₅	—	3	4	no	—
133	C₈×D₅	—	3	4	no	—
134	C₈×D₅	—	3	2	no	—
135	C₈×D₅	—	3	2	no	—
136	#674	—	3	4	no	—
137	#674	C₂²×D₃	1	4	yes	complemented
138	C₂³×C₈	—	15	1	no	—
139	C₂×M₃₂	—	5	1	no	—
140	C₂×M₃₂	—	15	2	no	—
141	C₂×M₃₂	—	15	8	no	—
142	C₂×M₃₂	—	15	1	no	—
143	C₂²×C₁₆	—	15	2	no	—
144	C₂²×C₃×C₅	—	2	2	no	—
145	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
146	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
147	C₂×D₁₅	—	2	2	no	—
148	C₂×D₁₅	—	2	8	no	—
149	C₂×D₁₅	M₃₂	1	4	yes	complemented
150	C₂×D₁₅	C₂²×C₈	1	2	yes	—
151	C₂×D₁₅	C₂²×C₈	1	2	yes	—
152	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
153	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
154	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
155	Dic₁₅	C₂²×C₈	1	2	yes	—
156	Dic₁₅	C₂²×C₈	1	2	yes	—
157	C₂³×D₃	—	5	1	no	—
158	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂²×Dic₃	—	5	1	no	—
160	C₂×C₄×D₃	—	5	1	no	—
161	C₂×C₄×D₃	—	5	4	no	—
162	C₂×C₄×D₃	—	5	1	no	—
163	C₂×C₄×D₃	—	5	4	no	—
164	C₂×C₄×D₃	—	5	1	no	—
165	C₂×C₄×D₃	—	5	1	no	—
166	C₂×C₃×C₈	—	5	2	no	—
167	C₂×C₃×C₈	—	5	1	no	—
168	C₂×C₃×C₈	—	5	1	no	—
169	C₂×C₃×C₈	—	5	2	no	—
170	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
171	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
173	D₃×C₈	—	5	2	no	—
174	D₃×C₈	—	5	2	no	—
175	D₃×C₈	—	5	2	no	—
176	D₃×C₈	—	5	2	no	—
177	D₃×C₈	—	5	4	no	—
178	D₃×C₈	—	5	4	no	—
179	C₃×C₁₆	—	5	4	no	—
180	C₃⋊C₁₆	—	5	4	no	—
181	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
182	C₂²×D₅	—	3	1	no	—
183	C₂²×D₅	—	3	4	no	—
184	C₂²×D₅	—	3	1	no	—
185	C₂²×D₅	—	6	4	no	—
186	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
187	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
189	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×Dic₅	—	3	4	no	—
191	C₂×Dic₅	—	3	1	no	—
192	C₂×Dic₅	—	3	1	no	—
193	C₄×D₅	—	3	4	no	—
194	C₄×D₅	—	3	2	no	—
195	C₄×D₅	—	3	4	no	—
196	C₄×D₅	—	3	2	no	—
197	C₄×D₅	—	3	2	no	—
198	C₄×D₅	—	3	2	no	—
199	C₅×C₈	—	3	2	no	—
200	C₅×C₈	—	3	2	no	—
201	#168	C₂×C₄×D₃	1	1	yes	characteristic
202	#168	C₂×C₄×D₃	1	2	yes	—
203	#168	C₂³×D₃	1	1	yes	characteristic
204	C₂³×C₄	—	15	1	no	—
205	C₂²×C₈	—	3	1	no	—
206	C₂²×C₈	—	5	1	no	—
207	C₂²×C₈	—	15	4	no	—
208	C₂²×C₈	—	15	1	no	—
209	C₂²×C₈	—	15	1	no	—
210	C₂²×C₈	—	15	1	no	—
211	C₂²×C₈	—	15	4	no	—
212	C₂²×C₈	—	15	1	no	—
213	M₃₂	—	5	4	no	—
214	M₃₂	—	15	8	no	—
215	M₃₂	—	15	4	no	—
216	C₂×C₁₆	—	5	2	no	—
217	C₂×C₁₆	—	15	8	no	—
218	C₂×C₁₆	—	15	2	no	—
219	C₂×C₃×C₅	—	2	2	no	—
220	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	complemented
221	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
222	D₁₅	—	2	2	no	—
223	D₁₅	C₂×M₃₂	1	4	yes	complemented
224	C₂³×C₃	D₅⋊C₈	1	1	yes	characteristic
225	C₂²×D₃	—	5	1	no	—
226	C₂²×D₃	—	5	4	no	—
227	C₂²×D₃	—	5	1	no	—
228	C₂²×D₃	—	10	4	no	—
229	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
230	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
232	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₂×Dic₃	—	5	4	no	—
234	C₂×Dic₃	—	5	1	no	—
235	C₂×Dic₃	—	5	1	no	—
236	C₄×D₃	—	5	4	no	—
237	C₄×D₃	—	5	2	no	—
238	C₄×D₃	—	5	4	no	—
239	C₄×D₃	—	5	2	no	—
240	C₄×D₃	—	5	2	no	—
241	C₄×D₃	—	5	2	no	—
242	C₃×C₈	—	5	1	no	—
243	C₃×C₈	—	5	2	no	—
244	C₃×C₈	—	5	1	no	—
245	C₃⋊C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
246	C₃⋊C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
247	C₂²×C₅	—	2	2	no	—
248	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
249	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
250	C₂×D₅	—	3	4	no	—
251	C₂×D₅	—	3	2	no	—
252	C₂×D₅	—	3	2	no	—
253	C₂×D₅	—	6	8	no	—
254	C₂×D₅	—	6	2	no	—
255	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
256	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
257	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
258	Dic₅	—	3	2	no	—
259	Dic₅	—	3	2	no	—
260	C₂⁴	—	15	1	no	—
261	C₂²×C₄	—	15	1	no	—
262	C₂²×C₄	—	15	1	no	—
263	C₂²×C₄	—	15	4	no	—
264	C₂²×C₄	—	15	1	no	—
265	C₂²×C₄	—	15	4	no	—
266	C₂²×C₄	—	15	1	no	—
267	C₂²×C₄	—	15	1	no	—
268	C₂²×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₂×C₈	—	3	4	no	—
270	C₂×C₈	—	3	1	no	—
271	C₂×C₈	—	3	1	no	—
272	C₂×C₈	—	5	2	no	—
273	C₂×C₈	—	5	2	no	—
274	C₂×C₈	—	5	1	no	—
275	C₂×C₈	—	5	1	no	—
276	C₂×C₈	—	15	2	no	—
277	C₂×C₈	—	15	2	no	—
278	C₂×C₈	—	15	2	no	—
279	C₂×C₈	—	15	4	no	—
280	C₂×C₈	—	15	4	no	—
281	C₂×C₈	—	15	2	no	—
282	C₁₆	—	5	4	no	—
283	C₁₆	—	15	4	no	—
284	C₃×C₅	C₂²×M₃₂	1	1	yes	characteristic, complemented, Hall
285	C₂²×C₃	—	2	2	no	—
286	C₂²×C₃	C₂×D₅⋊C₈	1	2	yes	—
287	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
288	C₂×D₃	—	5	4	no	—
289	C₂×D₃	—	5	2	no	—
290	C₂×D₃	—	5	2	no	—
291	C₂×D₃	—	10	8	no	—
292	C₂×D₃	—	10	2	no	—
293	C₃×C₄	C₂×D₅⋊C₈	1	2	yes	—
294	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
295	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
296	Dic₃	—	5	2	no	—
297	Dic₃	—	5	2	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	D₃×M₃₂	1	2	yes	complemented
300	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
301	D₅	—	3	4	no	—
302	D₅	—	6	2	no	—
303	C₂³	—	15	1	no	—
304	C₂³	—	15	4	no	—
305	C₂³	—	15	1	no	—
306	C₂³	—	30	4	no	—
307	C₂³	#6738	1	1	yes	characteristic
308	C₂×C₄	—	15	4	no	—
309	C₂×C₄	—	15	1	no	—
310	C₂×C₄	—	15	1	no	—
311	C₂×C₄	—	15	4	no	—
312	C₂×C₄	—	15	4	no	—
313	C₂×C₄	—	15	2	no	—
314	C₂×C₄	—	15	2	no	—
315	C₂×C₄	—	15	2	no	—
316	C₂×C₄	—	15	2	no	—
317	C₂×C₄	#6738	1	1	yes	characteristic
318	C₂×C₄	#6738	1	2	yes	—
319	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	2	yes	—
320	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
321	C₈	—	3	2	no	—
322	C₈	—	3	2	no	—
323	C₈	—	5	1	no	—
324	C₈	—	5	2	no	—
325	C₈	—	5	1	no	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	D₅⋊M₃₂	1	2	yes	complemented
328	C₂×C₃	C₂²×D₅⋊C₈	1	1	yes	characteristic
329	D₃	—	5	4	no	—
330	D₃	—	10	2	no	—
331	C₅	C₂×D₃×M₃₂	1	1	yes	characteristic, complemented, Sylow
332	C₂²	—	2	2	no	—
333	C₂²	—	15	4	no	—
334	C₂²	—	15	2	no	—
335	C₂²	—	15	2	no	—
336	C₂²	—	30	8	no	—
337	C₂²	—	30	2	no	—
338	C₂²	C₂×#6738	1	2	yes	—
339	C₂²	C₂×#6738	1	1	yes	characteristic
340	C₄	—	15	2	no	—
341	C₄	—	15	2	no	—
342	C₄	C₂×#6738	1	2	yes	—
343	C₄	C₂×#6738	1	1	yes	characteristic
344	C₄	C₂²×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
345	C₃	C₂×D₅⋊M₃₂	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	2	2	no	—
347	C₂	—	15	4	no	—
348	C₂	—	30	2	no	—
349	C₂	#11751741	1	2	yes	direct product factor
350	C₂	C₂²×#6738	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂³×C₈
Frattini	C₄	C₂²×D₃×AGL(1,𝔽₅)
Fitting	C₂²×C₃×C₄×C₅	C₂×C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×D₁₅	C₈
cosocle	C₃×#168	C₂⁴
hypercenter	C₂²×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₂²×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#91227	C₂	semidirect
3. C₂×C₃×#3998	C₂	semidirect
4. C₂×#91166	C₂	semidirect
5. #11751741	C₂	direct (N × H)
6. C₂×#91164	C₂	semidirect
17. #91227	C₂²	semidirect
18. C₂×C₃⋊#674	C₂²	semidirect
19. C₃×#3998	C₂²	semidirect
20. C₂×C₃×#674	C₂²	semidirect
21. #91166	C₂²	semidirect
22. #91164	C₂²	semidirect
26. C₂×#3998	D₃	semidirect
52. C₃⋊#674	C₂³	semidirect
53. C₃×#674	C₂³	semidirect
70. #3998	C₂×D₃	semidirect
72. C₂×#674	C₂×D₃	semidirect
137. #674	C₂²×D₃	semidirect
149. C₂×D₁₅	M₃₂	semidirect
220. C₂×C₃×C₅	C₂×M₃₂	semidirect
223. D₁₅	C₂×M₃₂	semidirect
284. C₃×C₅	C₂²×M₃₂	semidirect
299. C₂×C₅	D₃×M₃₂	semidirect
327. C₂×C₃	D₅⋊M₃₂	semidirect
331. C₅	C₂×D₃×M₃₂	semidirect
345. C₃	C₂×D₅⋊M₃₂	semidirect
349. C₂	#11751741	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×#6648	2
C₂×#91227	2
C₂×C₃×#3998	2
C₂×#91166	2
#11751741	2
C₂×#91164	2
C₂×D₅⋊M₃₂	3
C₂×D₃×M₃₂	5

Subgroups — C2×#11751741 #14478101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751741 #14478101