Subgroups — C₂×#20245278 #26369710

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689566	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689244	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689229	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245278	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688938	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688932	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688900	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	C₂×#13094032	—	2	1	no	—
12	C₂×#13093989	—	2	1	no	—
13	#15689566	C₄	1	4	yes	complemented
14	#15689566	C₂²	1	4	yes	—
15	C₂×Dic₃₇₆	C₄	1	2	yes	complemented
16	C₂×#13093969	C₄	1	2	yes	complemented
17	C₂×#13093964	C₂²	1	1	yes	characteristic
18	#15689244	—	2	2	no	—
19	#15689244	C₂²	1	4	yes	—
20	#15689229	—	2	2	no	—
21	#15689229	C₂²	1	4	yes	complemented
22	C₂×#13093950	—	2	1	no	—
23	C₂×#13093947	—	2	1	no	—
24	C₂×#13093934	C₂²	1	1	yes	characteristic
25	#15688938	C₂²	1	4	yes	complemented
26	#15688932	C₂²	1	4	yes	—
27	#15688900	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#365	C₂²	1	4	yes	complemented
32	C₂×#10625350	—	2	1	no	—
33	C₂²×Dic₉₄	—	2	1	no	—
34	#13094046	—	2	1	no	—
35	#13094046	—	2	1	no	—
36	#13094046	C₂×C₄	1	2	yes	—
37	#13094046	C₂³	1	2	yes	—
38	C₂×Dic₁₈₈	—	2	1	no	—
39	C₂×Dic₁₈₈	—	2	1	no	—
40	C₂×Dic₁₈₈	—	2	1	no	—
41	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
42	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
44	C₂×Dic₁₈₈	C₂³	1	2	yes	—
45	C₂×#10625325	—	2	1	no	—
46	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
47	#13094032	—	2	4	no	—
48	#13094032	—	4	2	no	—
49	C₂×#10625324	—	4	1	no	—
50	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
51	C₂×C₈×D₄₇	—	4	1	no	—
52	#13093989	—	2	4	no	—
53	#13093989	—	4	2	no	—
54	C₂×Dic₄₇⋊C₄	—	4	1	no	—
55	C₂×C₄×Dic₄₇	—	4	1	no	—
56	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
57	Dic₃₇₆	C₂×C₄	1	8	yes	complemented
58	#13093969	C₂×C₄	1	8	yes	complemented
59	#13093964	—	2	1	no	—
60	#13093964	C₂³	1	2	yes	—
61	C₂×C₄₇⋊C₁₆	—	2	1	no	—
62	#13093950	—	2	4	no	—
63	#13093947	—	2	4	no	—
64	#13093934	D₄	1	2	yes	—
65	#13093934	C₂³	1	2	yes	—
66	C₂×M₁₆×C₄₇	—	2	1	no	—
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
69	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
70	M₃₂×C₄₇	C₂³	1	2	yes	—
71	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
72	#108×C₄₇	—	2	1	no	—
73	#108×C₄₇	C₂³	1	2	yes	—
74	#106×C₄₇	—	2	1	no	—
75	#106×C₄₇	C₂³	1	2	yes	—
76	C₂²×C₄×C₄₇	—	4	1	no	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₄₇	—	4	1	no	—
79	C₂²×Dic₄₇	—	2	1	no	—
80	C₂²×Dic₄₇	—	4	1	no	—
81	#10625350	—	2	2	no	—
82	#10625350	—	2	2	no	—
83	#10625350	—	4	1	no	—
84	#10625350	—	4	1	no	—
85	C₂×D₁₈₈	—	2	1	no	—
86	C₂×C₄×D₄₇	—	4	1	no	—
87	C₂×Dic₉₄	—	2	1	no	—
88	C₂×Dic₉₄	—	2	2	no	—
89	C₂×Dic₉₄	—	2	2	no	—
90	C₂×Dic₉₄	—	2	2	no	—
91	C₂×Dic₉₄	—	4	1	no	—
92	C₂×Dic₉₄	—	4	1	no	—
93	C₂×Dic₉₄	—	4	1	no	—
94	M₁₆×C₄₇	—	2	2	no	—
95	M₁₆×C₄₇	—	4	1	no	—
96	C₂×C₈×C₄₇	—	4	1	no	—
97	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
101	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	4	1	no	—
105	Dic₄₇⋊C₄	—	4	4	no	—
106	C₄×Dic₄₇	—	2	1	no	—
107	C₄×Dic₄₇	—	4	4	no	—
108	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
109	C₂×C₄₇⋊C₈	—	2	1	no	—
110	C₂×C₄₇⋊C₈	—	2	1	no	—
111	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
112	Dic₁₈₈	—	2	2	no	—
113	Dic₁₈₈	—	2	2	no	—
114	Dic₁₈₈	—	2	1	no	—
115	Dic₁₈₈	—	2	2	no	—
116	Dic₁₈₈	C₂²×C₄	1	4	yes	—
117	Dic₁₈₈	C₂²×C₄	1	2	yes	—
118	D₃₇₆	—	2	1	no	—
119	D₃₇₆	C₂²×C₄	1	2	yes	—
120	#10625325	—	2	2	no	—
121	#10625325	—	2	2	no	—
122	#10625324	—	4	4	no	—
123	C₈×D₄₇	—	4	4	no	—
124	C₁₆×C₄₇	C₂²×C₄	1	4	yes	—
125	C₄₇⋊C₁₆	—	2	2	no	—
126	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×D₄₇	—	4	1	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	8	2	no	—
130	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₄₇	—	4	2	no	—
137	C₂²⋊D₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	1	no	—
144	D₁₈₈	—	2	2	no	—
145	D₁₈₈	—	4	1	no	—
146	C₄×D₄₇	—	4	2	no	—
147	C₄×D₄₇	—	4	2	no	—
148	Dic₉₄	—	2	2	no	—
149	Dic₉₄	—	2	4	no	—
150	Dic₉₄	—	4	2	no	—
151	Dic₉₄	—	4	2	no	—
152	Dic₉₄	—	4	2	no	—
153	Dic₉₄	—	4	1	no	—
154	C₈×C₄₇	—	4	2	no	—
155	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
156	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
157	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
159	C₄₇⋊C₈	—	2	2	no	—
160	C₂×#2180	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	—	2	2	no	—
162	C₂²×C₄₇	#442	1	1	yes	characteristic
163	C₂²×C₄₇	#442	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×D₄₇	—	4	2	no	—
166	C₂×D₄₇	—	8	2	no	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	#442	1	1	yes	characteristic
169	C₄×C₄₇	#442	1	1	yes	characteristic
170	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	Dic₄₇	—	4	2	no	—
173	Dic₄₇	—	4	4	no	—
174	Dic₄₇	—	4	2	no	—
175	C₂×#510	—	47	1	no	—
176	C₂×#444	—	47	1	no	—
177	C₂×#441	—	47	1	no	—
178	#2180	—	47	8	no	—
179	C₂×#365	D₄₇	1	1	yes	characteristic, complemented
180	C₂×#362	—	47	1	no	—
181	C₂×#360	—	47	1	no	—
182	C₂×#347	—	47	1	no	—
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	2	1	no	—
185	C₂×C₄₇	#2180	1	2	yes	complemented
186	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
187	D₄₇	—	8	2	no	—
188	C₂×#135	—	47	1	no	—
189	C₂²×Q₁₆	—	47	1	no	—
190	C₂×#131	—	94	1	no	—
191	C₂×C₄×Q₈	—	94	1	no	—
192	#510	—	47	4	no	—
193	#510	—	47	4	no	—
194	C₂×Q₃₂	—	47	2	no	—
195	C₂×SD₃₂	—	47	2	no	—
196	C₂×M₃₂	—	47	1	no	—
197	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
198	#444	—	47	4	no	—
199	#444	—	94	2	no	—
200	#441	—	47	4	no	—
201	#441	—	94	2	no	—
202	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
203	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
204	C₂×#104	—	94	1	no	—
205	C₂×#103	—	94	1	no	—
206	C₂×#97	—	47	1	no	—
207	#365	C₂×D₄₇	1	4	yes	complemented
208	#362	—	47	4	no	—
209	#360	—	47	4	no	—
210	#347	—	47	4	no	—
211	C₄₇	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	94	1	no	—
213	C₂²×Q₈	—	94	1	no	—
214	#135	—	47	2	no	—
215	#135	—	47	2	no	—
216	#135	—	94	1	no	—
217	#135	—	94	1	no	—
218	C₂×Q₁₆	—	47	2	no	—
219	C₂×Q₁₆	—	47	2	no	—
220	C₂×Q₁₆	—	47	1	no	—
221	C₂×Q₁₆	—	47	2	no	—
222	C₂×Q₁₆	—	94	1	no	—
223	C₂×Q₁₆	—	94	1	no	—
224	C₂×Q₁₆	—	94	1	no	—
225	C₂×SD₁₆	—	94	1	no	—
226	C₂×D₈	—	47	1	no	—
227	#131	—	94	4	no	—
228	#131	—	188	2	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂×M₁₆	—	188	1	no	—
231	C₂²×C₈	—	47	1	no	—
232	C₂²×C₈	—	188	1	no	—
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	C₄×Q₈	—	94	4	no	—
235	C₄×Q₈	—	188	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	188	1	no	—
238	C₂×C₄²	—	47	1	no	—
239	C₂×C₄²	—	188	1	no	—
240	Q₃₂	—	47	8	no	—
241	SD₃₂	—	47	8	no	—
242	M₃₂	—	47	2	no	—
243	M₃₂	—	94	1	no	—
244	M₃₂	C₄×D₄₇	1	2	yes	—
245	M₃₂	C₂²×D₄₇	1	2	yes	—
246	C₂×C₁₆	—	94	1	no	—
247	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
248	#108	—	2	1	no	—
249	#108	C₂²×D₄₇	1	2	yes	—
250	#106	—	2	1	no	—
251	#106	C₂²×D₄₇	1	2	yes	—
252	#104	—	94	4	no	—
253	#103	—	94	4	no	—
254	#97	—	47	2	no	—
255	#97	—	47	2	no	—
256	#41	—	94	2	no	—
257	#41	—	94	2	no	—
258	#41	—	188	1	no	—
259	#41	—	188	1	no	—
260	C₂×Q₈	—	94	1	no	—
261	C₂×Q₈	—	94	2	no	—
262	C₂×Q₈	—	94	2	no	—
263	C₂×Q₈	—	94	2	no	—
264	C₂×Q₈	—	188	1	no	—
265	C₂×Q₈	—	188	1	no	—
266	C₂×Q₈	—	188	1	no	—
267	C₂×D₄	—	94	1	no	—
268	C₂×D₄	—	188	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	94	1	no	—
271	C₂²×C₄	—	188	1	no	—
272	C₂²×C₄	—	188	1	no	—
273	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
274	Q₁₆	—	47	4	no	—
275	Q₁₆	—	47	2	no	—
276	Q₁₆	—	94	2	no	—
277	Q₁₆	—	94	2	no	—
278	Q₁₆	—	94	1	no	—
279	Q₁₆	—	94	2	no	—
280	SD₁₆	—	94	2	no	—
281	SD₁₆	—	94	2	no	—
282	D₈	—	47	2	no	—
283	D₈	—	94	1	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	4	1	no	—
286	M₁₆	—	188	4	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	94	1	no	—
290	C₂×C₈	—	94	1	no	—
291	C₂×C₈	—	188	4	no	—
292	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
294	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
295	C₂×C₈	#10625354	1	1	yes	characteristic
296	C₂×C₈	#10625354	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	188	4	no	—
301	C₄²	—	47	2	no	—
302	C₄²	—	94	1	no	—
303	C₄²	—	188	4	no	—
304	C₁₆	—	94	2	no	—
305	C₁₆	C₂×C₄×D₄₇	1	4	yes	—
306	C₂³	—	188	1	no	—
307	C₂³	#13094051	1	1	yes	characteristic
308	Q₈	—	94	2	no	—
309	Q₈	—	94	4	no	—
310	Q₈	—	188	2	no	—
311	Q₈	—	188	2	no	—
312	Q₈	—	188	1	no	—
313	Q₈	—	188	2	no	—
314	D₄	—	94	2	no	—
315	D₄	—	188	2	no	—
316	D₄	—	188	2	no	—
317	D₄	—	188	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	188	2	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	—	188	1	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	#13093986	1	1	yes	characteristic
329	C₂×C₄	#13093986	1	1	yes	characteristic
330	C₂×C₄	#13094051	1	1	yes	characteristic
331	C₂×C₄	#13094051	1	1	yes	characteristic
332	C₂×C₄	#13094051	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	94	2	no	—
336	C₈	#13093986	1	1	yes	characteristic
337	C₈	#13093986	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
339	C₈	C₂×#10625354	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	376	2	no	—
342	C₂²	—	188	2	no	—
343	C₂²	#15689234	1	1	yes	characteristic
344	C₂²	#15689234	1	1	yes	characteristic
345	C₂²	C₂×#13094051	1	1	yes	characteristic
346	C₄	—	8	2	no	—
347	C₄	—	188	2	no	—
348	C₄	—	188	2	no	—
349	C₄	—	188	4	no	—
350	C₄	#15689234	1	1	yes	characteristic
351	C₄	#15689234	1	1	yes	characteristic
352	C₄	C₂×#13093986	1	1	yes	characteristic
353	C₄	C₂×#13094051	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	376	2	no	—
357	C₂	#20245278	1	2	yes	direct product factor
358	C₂	C₂×#15689234	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689234
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#365	C₂
socle	C₂²×C₄₇	#442
prime radical	C₂×#13094046	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#365	D₄₇
hypocenter	C₄₇	C₂×#2180
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
3. C₂×#15689229	C₂	semidirect
4. #20245278	C₂	direct (N × H)
5. C₂×#15688938	C₂	semidirect
7. C₂×#15688900	C₂	semidirect
8. C₂×C₄₇×#365	C₂	semidirect
13. #15689566	C₄	semidirect
15. C₂×Dic₃₇₆	C₄	semidirect
16. C₂×#13093969	C₄	semidirect
21. #15689229	C₂²	semidirect
25. #15688938	C₂²	semidirect
27. #15688900	C₂²	semidirect
31. C₄₇×#365	C₂²	semidirect
57. Dic₃₇₆	C₂×C₄	semidirect
58. #13093969	C₂×C₄	semidirect
179. C₂×#365	D₄₇	semidirect
185. C₂×C₄₇	#2180	semidirect
207. #365	C₂×D₄₇	semidirect
211. C₄₇	C₂×#2180	semidirect
357. C₂	#20245278	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×#15689566	2
C₂×#15689244	2
C₂×#15689229	2
#20245278	2
C₂×#15688938	2
C₂×#15688932	2
C₂×#15688900	2
C₂×C₄₇×#365	2
C₂×#2180	47

Subgroups — C2×#20245278 #26369710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245278 #26369710