Subgroups — C₂×#20245578 #26371047

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689356	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689160	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245578	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688951	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×(C₂×D₄₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093995	—	2	1	no	—
11	C₂×#13093982	—	2	1	no	—
12	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#15689356	—	2	2	no	—
14	#15689356	C₂²	1	4	yes	—
15	#15689160	—	2	2	no	—
16	#15689160	C₂²	1	4	yes	complemented
17	C₂×#13093958	C₂²	1	1	yes	characteristic
18	C₂×#13093958	C₂²	1	1	yes	characteristic
19	C₂×#13093955	—	2	1	no	—
20	C₂×#13093945	—	2	1	no	—
21	#15688951	C₂²	1	4	yes	complemented
22	#15688948	C₂²	1	4	yes	complemented
23	#15688924	C₂²	1	4	yes	complemented
24	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#357	C₂²	1	4	yes	complemented
29	C₂²×C₂²⋊D₄₇	—	4	1	no	—
30	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₈×D₄₇	—	2	2	no	—
33	C₂×C₈×D₄₇	—	2	2	no	—
34	C₂×C₈×D₄₇	D₄	1	2	yes	—
35	C₂×C₈×D₄₇	C₂³	1	2	yes	—
36	#13093995	—	2	4	no	—
37	#13093995	—	4	2	no	—
38	#13093982	—	2	4	no	—
39	#13093982	—	4	2	no	—
40	C₂×C₄⋊Dic₄₇	—	2	1	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
43	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₄₇	—	4	1	no	—
45	C₂×C₄₇⋊C₁₆	—	2	2	no	—
46	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
47	#13093958	—	2	1	no	—
48	#13093958	—	2	1	no	—
49	#13093958	C₂³	1	2	yes	—
50	#13093958	C₂³	1	2	yes	—
51	#13093955	—	2	4	no	—
52	#13093945	—	2	4	no	—
53	C₂²×D₄×C₄₇	—	2	1	no	—
54	C₂×D₈×C₄₇	—	2	1	no	—
55	C₂×D₈×C₄₇	—	2	2	no	—
56	C₂×D₈×C₄₇	—	2	2	no	—
57	C₂×D₈×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₂×C₁₆×C₄₇	—	2	2	no	—
61	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
62	#107×C₄₇	—	2	1	no	—
63	#107×C₄₇	C₂³	1	2	yes	—
64	C₂⁴×C₄₇	—	4	1	no	—
65	C₂³×D₄₇	D₈	1	1	yes	characteristic
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	—	4	2	no	—
68	C₂×D₄×C₄₇	—	4	1	no	—
69	C₂×D₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	—	4	1	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₄₇	—	4	4	no	—
73	C₂×C₂²⋊D₄₇	—	8	4	no	—
74	C₂²×Dic₄₇	—	4	1	no	—
75	C₂²×Dic₄₇	—	4	1	no	—
76	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×C₄×D₄₇	—	2	2	no	—
79	C₂×C₄×D₄₇	D₈	1	2	yes	—
80	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
81	D₈×C₄₇	—	2	2	no	—
82	D₈×C₄₇	—	2	2	no	—
83	D₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	#41	1	2	yes	—
86	C₂×C₈×C₄₇	#41	1	2	yes	—
87	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	4	1	no	—
90	C₂²⋊Dic₄₇	—	4	4	no	—
91	(C₂×D₄₇)⋊C₄	—	4	4	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	4	1	no	—
95	C₄⋊Dic₄₇	—	4	1	no	—
96	C₄×Dic₄₇	—	4	4	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
99	C₈×D₄₇	—	2	4	no	—
100	C₈×D₄₇	—	2	2	no	—
101	C₈×D₄₇	—	2	2	no	—
102	C₁₆×C₄₇	—	2	2	no	—
103	C₄₇⋊C₁₆	—	2	2	no	—
104	C₂³×C₄₇	—	4	2	no	—
105	C₂³×C₄₇	—	8	2	no	—
106	C₂³×C₄₇	—	8	2	no	—
107	C₂³×C₄₇	—	8	2	no	—
108	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
109	C₂²×D₄₇	—	2	2	no	—
110	C₂²×D₄₇	—	2	2	no	—
111	C₂²×D₄₇	—	2	2	no	—
112	C₂²×D₄₇	C₂×D₈	1	2	yes	—
113	D₄×C₄₇	—	4	2	no	—
114	D₄×C₄₇	—	4	2	no	—
115	D₄×C₄₇	—	8	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	8	2	no	—
118	C₂×C₄×C₄₇	#123	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#137	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂²⋊D₄₇	—	8	8	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	8	2	no	—
127	C₂×Dic₄₇	—	8	2	no	—
128	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
129	C₄×D₄₇	—	2	2	no	—
130	C₄×D₄₇	—	2	2	no	—
131	C₄×D₄₇	—	2	4	no	—
132	C₈×C₄₇	#123	1	2	yes	—
133	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₄₇⋊C₈	—	2	2	no	—
136	C₂×#2234	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	8	2	no	—
138	C₂²×C₄₇	—	8	2	no	—
139	C₂²×C₄₇	—	8	2	no	—
140	C₂²×C₄₇	—	16	4	no	—
141	C₂²×C₄₇	#480	1	2	yes	—
142	C₂²×C₄₇	#508	1	2	yes	—
143	C₂²×C₄₇	#509	1	2	yes	—
144	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₄₇	—	2	2	no	—
146	C₂×D₄₇	—	2	2	no	—
147	C₂×D₄₇	—	2	2	no	—
148	C₂×D₄₇	—	4	4	no	—
149	C₄×C₄₇	—	8	2	no	—
150	C₄×C₄₇	#480	1	2	yes	—
151	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
152	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
153	Dic₄₇	—	2	2	no	—
154	Dic₄₇	—	8	2	no	—
155	Dic₄₇	—	8	2	no	—
156	C₂×#466	—	47	1	no	—
157	C₂×#427	—	47	1	no	—
158	#2234	—	47	8	no	—
159	C₂×#367	—	47	1	no	—
160	C₂×#366	—	47	1	no	—
161	C₂×#357	—	47	1	no	—
162	C₂×#357	D₄₇	1	1	yes	characteristic, complemented
163	C₂×C₂²⋊C₁₆	—	47	1	no	—
164	C₂×C₄₇	—	16	2	no	—
165	C₂×C₄₇	#2234	1	4	yes	complemented
166	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
169	D₄₇	—	4	2	no	—
170	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂³×C₈	—	47	1	no	—
172	C₂×#121	—	94	1	no	—
173	C₂×#117	—	94	1	no	—
174	C₂²×C₁₆	—	47	1	no	—
175	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
176	#466	—	47	4	no	—
177	#466	—	94	2	no	—
178	#427	—	47	4	no	—
179	#427	—	94	2	no	—
180	C₂×#107	—	47	1	no	—
181	C₂×#107	—	47	1	no	—
182	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#106	—	94	1	no	—
184	C₂×#102	—	94	1	no	—
185	#367	—	47	4	no	—
186	#366	—	47	4	no	—
187	#357	—	47	4	no	—
188	#357	C₂×D₄₇	1	4	yes	complemented
189	C₂²⋊C₁₆	—	47	4	no	—
190	C₄₇	C₂×#2234	1	1	yes	characteristic, complemented, Sylow
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	—	188	1	no	—
193	C₂³×C₄	—	47	1	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₂²×D₄₇	1	2	yes	—
198	C₂²×C₈	—	47	1	no	—
199	C₂²×C₈	—	47	2	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	C₂²×C₈	—	94	2	no	—
203	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
204	#121	—	94	4	no	—
205	#121	—	188	2	no	—
206	#117	—	94	4	no	—
207	#117	—	188	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	94	1	no	—
210	C₂×C₄⋊C₄	—	94	1	no	—
211	C₂×C₂²⋊C₄	—	188	1	no	—
212	C₂×C₂²⋊C₄	—	188	1	no	—
213	C₂×C₄²	—	188	1	no	—
214	C₂×C₁₆	—	2	2	no	—
215	C₂×C₁₆	—	47	2	no	—
216	C₂×C₁₆	—	94	2	no	—
217	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
218	#107	—	2	1	no	—
219	#107	—	47	2	no	—
220	#107	—	47	2	no	—
221	#107	—	94	1	no	—
222	#107	—	94	1	no	—
223	#107	C₂²×D₄₇	1	2	yes	—
224	#106	—	94	4	no	—
225	#102	—	94	4	no	—
226	C₂⁴	—	4	1	no	—
227	C₂⁴	—	47	1	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	376	4	no	—
233	C₂×D₄	—	188	4	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	47	1	no	—
236	C₂²×C₄	—	47	2	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	94	2	no	—
239	C₂²×C₄	—	94	2	no	—
240	C₂²×C₄	—	188	1	no	—
241	C₂²×C₄	—	188	1	no	—
242	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	—	47	2	no	—
247	C₂×C₈	—	94	2	no	—
248	C₂×C₈	—	94	4	no	—
249	C₂×C₈	—	94	2	no	—
250	C₂×C₈	—	94	2	no	—
251	C₂×C₈	#10625350	1	2	yes	—
252	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
253	C₂×C₈	#10625352	1	2	yes	—
254	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	94	2	no	—
258	C₄⋊C₄	—	94	2	no	—
259	C₄⋊C₄	—	188	1	no	—
260	C₄⋊C₄	—	188	1	no	—
261	C₂²⋊C₄	—	188	4	no	—
262	C₂²⋊C₄	—	188	4	no	—
263	C₄²	—	188	4	no	—
264	C₁₆	—	2	2	no	—
265	C₁₆	—	94	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	8	2	no	—
268	C₂³	—	8	2	no	—
269	C₂³	—	8	2	no	—
270	C₂³	—	47	2	no	—
271	C₂³	—	94	2	no	—
272	C₂³	—	94	2	no	—
273	C₂³	—	94	2	no	—
274	C₂³	D₈×D₄₇	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	2	no	—
278	D₄	—	376	8	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	47	2	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	94	2	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	376	2	no	—
287	C₂×C₄	—	376	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	#13094006	1	2	yes	—
291	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
292	C₂×C₄	#13094053	1	2	yes	—
293	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
294	C₈	—	94	2	no	—
295	C₈	#13094006	1	2	yes	—
296	C₈	C₂×#10625352	1	1	yes	characteristic
297	C₈	C₂×#10625350	1	1	yes	characteristic
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	4	no	—
302	C₂²	—	94	2	no	—
303	C₂²	—	94	2	no	—
304	C₂²	—	94	2	no	—
305	C₂²	—	188	4	no	—
306	C₂²	#15689435	1	2	yes	—
307	C₂²	#15689554	1	2	yes	—
308	C₂²	#15689557	1	2	yes	—
309	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
310	C₄	—	8	2	no	—
311	C₄	—	94	2	no	—
312	C₄	—	376	2	no	—
313	C₄	—	376	2	no	—
314	C₄	#15689435	1	2	yes	—
315	C₄	C₂×#13094006	1	1	yes	characteristic
316	C₄	C₂×#13094053	1	1	yes	characteristic
317	C₂	—	16	2	no	—
318	C₂	—	188	2	no	—
319	C₂	#20245578	1	4	yes	direct product factor
320	C₂	C₂×#15689435	1	1	yes	characteristic
321	C₂	C₂×#15689554	1	1	yes	characteristic
322	C₂	C₂×#15689557	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#357	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂×#15689356	C₂
hypercenter	C₂×#357	D₄₇
hypocenter	C₄₇	C₂×#2234
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
2. C₂×#15689160	C₂	semidirect
3. #20245578	C₂	direct (N × H)
4. C₂×#15688951	C₂	semidirect
5. C₂×#15688948	C₂	semidirect
6. C₂×#15688924	C₂	semidirect
7. C₂×(C₂×D₄₇)⋊C₁₆	C₂	semidirect
8. C₂×C₄₇×#357	C₂	semidirect
16. #15689160	C₂²	semidirect
21. #15688951	C₂²	semidirect
22. #15688948	C₂²	semidirect
23. #15688924	C₂²	semidirect
24. (C₂×D₄₇)⋊C₁₆	C₂²	semidirect
28. C₄₇×#357	C₂²	semidirect
162. C₂×#357	D₄₇	semidirect
165. C₂×C₄₇	#2234	semidirect
188. #357	C₂×D₄₇	semidirect
190. C₄₇	C₂×#2234	semidirect
319. C₂	#20245578	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×#15689356	2
C₂×#15689160	2
#20245578	2
C₂×#15688951	2
C₂×#15688948	2
C₂×#15688924	2
C₂×(C₂×D₄₇)⋊C₁₆	2
C₂×C₄₇×#357	2
C₂×#2234	47

Subgroups — C2×#20245578 #26371047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245578 #26371047