Subgroups — C₂×#20245563 #26371013

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689523	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689239	C₂	1	1	yes	maximal, characteristic
3	#20245563	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688947	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688943	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688934	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688932	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₂×#13094032	—	2	1	no	—
11	C₂×#13094015	—	2	1	no	—
12	C₂×#13093964	C₂²	1	1	yes	characteristic
13	#15689523	—	2	2	no	—
14	#15689523	C₂²	1	4	yes	complemented
15	#15689239	—	2	2	no	—
16	#15689239	C₂²	1	4	yes	—
17	C₂×#13093957	—	2	1	no	—
18	C₂×#13093956	C₂²	1	1	yes	characteristic
19	C₂×#13093950	—	2	1	no	—
20	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688947	C₂²	1	4	yes	complemented
22	#15688943	C₂²	1	4	yes	complemented
23	#15688934	C₂²	1	4	yes	—
24	#15688932	C₂²	1	4	yes	—
25	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#365	C₂²	1	4	yes	complemented
29	C₂×#10625350	—	2	1	no	—
30	#13094046	—	2	1	no	—
31	#13094046	—	2	1	no	—
32	#13094046	Q₈	1	2	yes	—
33	#13094046	C₂³	1	2	yes	—
34	C₂×Dic₁₈₈	Q₈	1	1	yes	characteristic
35	C₂×#10625325	—	2	1	no	—
36	C₂×D₃₇₆	Q₈	1	1	yes	characteristic
37	#13094032	—	2	4	no	—
38	#13094032	—	4	2	no	—
39	C₂×#10625324	—	4	1	no	—
40	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₈×D₄₇	—	4	1	no	—
42	#13094015	—	2	4	no	—
43	#13094015	—	4	2	no	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄⋊Dic₄₇	—	4	1	no	—
46	C₂×Dic₄₇⋊C₄	—	4	1	no	—
47	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
48	#13093964	—	2	1	no	—
49	#13093964	C₂³	1	2	yes	—
50	C₂×C₄₇⋊C₁₆	—	2	1	no	—
51	#13093957	—	2	4	no	—
52	#13093956	—	2	1	no	—
53	#13093956	C₂³	1	2	yes	—
54	#13093950	—	2	4	no	—
55	C₈×Dic₄₇	D₄	1	2	yes	—
56	C₈×Dic₄₇	C₂³	1	2	yes	—
57	C₂×M₁₆×C₄₇	—	2	1	no	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
60	M₃₂×C₄₇	—	2	1	no	—
61	M₃₂×C₄₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₄₇	—	2	1	no	—
63	#108×C₄₇	—	2	1	no	—
64	#108×C₄₇	C₂³	1	2	yes	—
65	#106×C₄₇	—	2	1	no	—
66	#106×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₄₇	—	4	1	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	4	1	no	—
70	C₂²×Dic₄₇	—	2	1	no	—
71	C₂²×Dic₄₇	—	4	1	no	—
72	#10625350	—	2	2	no	—
73	#10625350	—	2	2	no	—
74	#10625350	—	4	1	no	—
75	#10625350	—	4	1	no	—
76	C₂×D₁₈₈	—	2	1	no	—
77	C₂×C₄×D₄₇	—	4	1	no	—
78	C₂×Dic₉₄	—	2	1	no	—
79	M₁₆×C₄₇	—	2	2	no	—
80	M₁₆×C₄₇	—	4	1	no	—
81	C₂×C₈×C₄₇	—	4	1	no	—
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
86	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	4	1	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	4	1	no	—
93	Dic₄₇⋊C₄	—	4	4	no	—
94	C₄×Dic₄₇	—	2	1	no	—
95	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	SD₁₆	1	1	yes	characteristic
97	C₂×C₄₇⋊C₈	SD₁₆	1	1	yes	characteristic
98	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
99	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
100	Dic₁₈₈	—	2	1	no	—
101	Dic₁₈₈	C₂×Q₈	1	2	yes	—
102	D₃₇₆	—	2	1	no	—
103	D₃₇₆	C₂×Q₈	1	2	yes	—
104	#10625325	—	2	2	no	—
105	#10625325	—	2	2	no	—
106	#10625324	—	4	4	no	—
107	C₈×D₄₇	—	4	4	no	—
108	C₁₆×C₄₇	—	2	2	no	—
109	C₄₇⋊C₁₆	—	2	2	no	—
110	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂²×D₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	8	2	no	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂²⋊D₄₇	—	4	2	no	—
121	C₂²⋊D₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	1	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	8	2	no	—
127	D₁₈₈	—	2	2	no	—
128	D₁₈₈	—	4	1	no	—
129	C₄×D₄₇	—	4	2	no	—
130	C₄×D₄₇	—	4	2	no	—
131	Dic₉₄	—	2	2	no	—
132	Dic₉₄	—	4	1	no	—
133	C₈×C₄₇	—	4	2	no	—
134	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
137	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
139	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
140	C₂×#2231	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	#476	1	1	yes	characteristic
143	C₂²×C₄₇	#476	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×D₄₇	—	4	2	no	—
146	C₂×D₄₇	—	8	2	no	—
147	C₄×C₄₇	—	8	2	no	—
148	C₄×C₄₇	#476	1	1	yes	characteristic
149	C₄×C₄₇	#476	1	1	yes	characteristic
150	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
152	Dic₄₇	—	4	2	no	—
153	Dic₄₇	—	4	2	no	—
154	Dic₄₇	—	8	2	no	—
155	C₂×#499	—	47	1	no	—
156	C₂×#443	—	47	1	no	—
157	#2231	—	47	8	no	—
158	C₂×#365	—	47	1	no	—
159	C₂×#365	D₄₇	1	1	yes	characteristic, complemented
160	C₂×#364	—	47	1	no	—
161	C₂×#360	—	47	1	no	—
162	C₂×#360	—	47	1	no	—
163	C₂×C₄₇	—	2	1	no	—
164	C₂×C₄₇	—	2	1	no	—
165	C₂×C₄₇	#2231	1	2	yes	complemented
166	C₂×C₄₇	C₂×#476	1	1	yes	characteristic
167	D₄₇	—	8	2	no	—
168	C₂×#135	—	47	1	no	—
169	C₂×#131	—	94	1	no	—
170	C₂×#125	—	94	1	no	—
171	C₂×M₃₂	—	47	1	no	—
172	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
173	#499	—	47	4	no	—
174	#499	—	94	2	no	—
175	#443	—	47	4	no	—
176	#443	—	94	2	no	—
177	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
178	C₂×#107	—	94	1	no	—
179	C₂×#106	—	47	1	no	—
180	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
181	C₂×#104	—	94	1	no	—
182	C₂×C₄×C₈	—	47	1	no	—
183	#365	—	47	4	no	—
184	#365	C₂×D₄₇	1	4	yes	complemented
185	#364	—	47	4	no	—
186	#360	—	47	4	no	—
187	#360	—	47	4	no	—
188	C₄₇	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
189	C₂×#41	—	94	1	no	—
190	#135	—	47	2	no	—
191	#135	—	47	2	no	—
192	#135	—	94	1	no	—
193	#135	—	94	1	no	—
194	C₂×Q₁₆	—	47	1	no	—
195	C₂×SD₁₆	—	94	1	no	—
196	C₂×D₈	—	47	1	no	—
197	#131	—	94	4	no	—
198	#131	—	188	2	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	188	1	no	—
201	C₂²×C₈	—	47	1	no	—
202	C₂²×C₈	—	188	1	no	—
203	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
204	#125	—	94	4	no	—
205	#125	—	188	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	94	1	no	—
208	C₂×C₄⋊C₄	—	188	1	no	—
209	C₂×C₄⋊C₄	—	188	1	no	—
210	C₂×C₄²	—	47	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	47	2	no	—
213	M₃₂	—	94	1	no	—
214	M₃₂	C₂²×D₄₇	1	2	yes	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	94	1	no	—
217	#108	—	2	1	no	—
218	#108	C₂²×D₄₇	1	2	yes	—
219	#107	—	94	4	no	—
220	#106	—	2	1	no	—
221	#106	—	47	2	no	—
222	#106	—	94	1	no	—
223	#106	C₂²×D₄₇	1	2	yes	—
224	#104	—	94	4	no	—
225	C₄×C₈	—	47	2	no	—
226	C₄×C₈	—	47	2	no	—
227	#41	—	94	2	no	—
228	#41	—	94	2	no	—
229	#41	—	188	1	no	—
230	#41	—	188	1	no	—
231	C₂×Q₈	—	94	1	no	—
232	C₂×D₄	—	94	1	no	—
233	C₂×D₄	—	188	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	94	1	no	—
236	C₂²×C₄	—	188	1	no	—
237	C₂²×C₄	—	188	1	no	—
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	Q₁₆	—	47	2	no	—
240	Q₁₆	—	94	1	no	—
241	SD₁₆	—	94	2	no	—
242	SD₁₆	—	94	2	no	—
243	D₈	—	47	2	no	—
244	D₈	—	94	1	no	—
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	188	4	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	47	1	no	—
250	C₂×C₈	—	47	1	no	—
251	C₂×C₈	—	47	2	no	—
252	C₂×C₈	—	47	2	no	—
253	C₂×C₈	—	188	4	no	—
254	C₂×C₈	#10625350	1	1	yes	characteristic
255	C₂×C₈	#10625350	1	1	yes	characteristic
256	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
257	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
258	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	94	2	no	—
263	C₄⋊C₄	—	188	4	no	—
264	C₄⋊C₄	—	188	4	no	—
265	C₄⋊C₄	—	188	1	no	—
266	C₄²	—	47	2	no	—
267	C₄²	—	94	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	94	2	no	—
270	C₂³	—	188	1	no	—
271	C₂³	#13094051	1	1	yes	characteristic
272	Q₈	—	94	2	no	—
273	Q₈	—	188	1	no	—
274	D₄	—	94	2	no	—
275	D₄	—	188	2	no	—
276	D₄	—	188	2	no	—
277	D₄	—	188	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	376	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	1	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	#13094002	1	1	yes	characteristic
288	C₂×C₄	#13094002	1	1	yes	characteristic
289	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
291	C₂×C₄	#13094051	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	47	2	no	—
295	C₈	—	47	2	no	—
296	C₈	#13094002	1	1	yes	characteristic
297	C₈	#13094002	1	1	yes	characteristic
298	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
299	C₈	C₂×#10625350	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	376	2	no	—
302	C₂²	—	188	2	no	—
303	C₂²	#15689412	1	1	yes	characteristic
304	C₂²	#15689412	1	1	yes	characteristic
305	C₂²	C₂×#13094051	1	1	yes	characteristic
306	C₄	—	8	2	no	—
307	C₄	—	376	2	no	—
308	C₄	—	188	2	no	—
309	C₄	—	188	2	no	—
310	C₄	#15689412	1	1	yes	characteristic
311	C₄	#15689412	1	1	yes	characteristic
312	C₄	C₂×#13094002	1	1	yes	characteristic
313	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	376	2	no	—
317	C₂	#20245563	1	2	yes	direct product factor
318	C₂	C₂×#15689412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689412
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#365	C₂
socle	C₂²×C₄₇	#476
prime radical	C₂×#13094046	C₂²
hypercenter	C₂×#365	D₄₇
hypocenter	C₄₇	C₂×#2231
LCS	G ▷ C₂×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
1. C₂×#15689523	C₂	semidirect
3. #20245563	C₂	direct (N × H)
4. C₂×#15688947	C₂	semidirect
5. C₂×#15688943	C₂	semidirect
8. C₂×C₄₇×#365	C₂	semidirect
14. #15689523	C₂²	semidirect
21. #15688947	C₂²	semidirect
22. #15688943	C₂²	semidirect
28. C₄₇×#365	C₂²	semidirect
159. C₂×#365	D₄₇	semidirect
165. C₂×C₄₇	#2231	semidirect
184. #365	C₂×D₄₇	semidirect
188. C₄₇	C₂×#2231	semidirect
317. C₂	#20245563	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×#15689523	2
C₂×#15689239	2
#20245563	2
C₂×#15688947	2
C₂×#15688943	2
C₂×#15688934	2
C₂×#15688932	2
C₂×C₄₇×#365	2
C₂×#2231	47

Subgroups — C2×#20245563 #26371013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245563 #26371013