Subgroups — C₂×#20245576 #26371045

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×#15689161	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245576	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688950	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688924	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#466	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	—	2	1	no	—
10	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#15689161	—	2	2	no	—
13	#15689161	C₂²	1	4	yes	complemented
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093956	—	2	1	no	—
17	#15688950	C₂²	1	4	yes	complemented
18	#15688948	C₂²	1	4	yes	complemented
19	#15688924	C₂²	1	4	yes	complemented
20	#15688924	C₂²	1	4	yes	complemented
21	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
22	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#121×C₄₇	—	2	1	no	—
25	C₄₇×#466	—	2	2	no	—
26	C₄₇×#466	C₂²	1	4	yes	—
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₇	—	2	1	no	—
29	#13093983	—	2	4	no	—
30	#13093983	—	4	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
34	C₂×C₄×Dic₄₇	—	4	1	no	—
35	C₂×C₄₇⋊C₁₆	—	2	2	no	—
36	C₂×C₄₇⋊C₁₆	—	2	2	no	—
37	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
38	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
39	#13093958	—	2	1	no	—
40	#13093958	—	2	1	no	—
41	#13093958	C₂³	1	2	yes	—
42	#13093958	C₂³	1	2	yes	—
43	#13093956	—	2	4	no	—
44	C₂²×D₄×C₄₇	—	2	1	no	—
45	C₂²×D₄×C₄₇	—	4	1	no	—
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂×D₈×C₄₇	—	2	1	no	—
48	C₂×D₈×C₄₇	—	2	2	no	—
49	C₂×D₈×C₄₇	—	2	2	no	—
50	C₂×D₈×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	D₄	1	2	yes	—
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	#121×C₄₇	—	2	4	no	—
58	#121×C₄₇	—	4	2	no	—
59	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
61	#107×C₄₇	—	2	1	no	—
62	#107×C₄₇	C₂³	1	2	yes	—
63	#102×C₄₇	—	2	4	no	—
64	C₂⁴×C₄₇	—	4	1	no	—
65	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂×D₄×C₄₇	—	4	1	no	—
70	C₂×D₄×C₄₇	—	4	2	no	—
71	C₂×D₄×C₄₇	—	8	4	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	D₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	4	1	no	—
80	C₂²×Dic₄₇	—	4	1	no	—
81	D₈×C₄₇	—	2	2	no	—
82	D₈×C₄₇	—	2	2	no	—
83	D₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	#41	1	2	yes	—
91	C₂×C₈×C₄₇	#41	1	2	yes	—
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	—	4	1	no	—
95	C₂²⋊C₄×C₄₇	—	4	4	no	—
96	C₂²⋊Dic₄₇	—	4	4	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	4	1	no	—
100	C₄⋊Dic₄₇	—	4	1	no	—
101	C₄×Dic₄₇	—	4	4	no	—
102	C₄₇⋊C₁₆	—	2	2	no	—
103	C₄₇⋊C₁₆	—	2	2	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	—	8	2	no	—
109	C₂³×C₄₇	—	8	2	no	—
110	C₂³×C₄₇	—	8	2	no	—
111	C₂³×C₄₇	C₂×D₈	1	2	yes	—
112	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
113	D₄×C₄₇	—	4	2	no	—
114	D₄×C₄₇	—	4	2	no	—
115	D₄×C₄₇	—	8	8	no	—
116	D₄×C₄₇	—	8	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	8	2	no	—
123	C₂×C₄×C₄₇	#123	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#137	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	8	2	no	—
131	C₂×Dic₄₇	—	8	2	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	#123	1	2	yes	—
134	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₂×#2234	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	4	4	no	—
141	C₂²×C₄₇	—	8	2	no	—
142	C₂²×C₄₇	—	8	2	no	—
143	C₂²×C₄₇	—	8	2	no	—
144	C₂²×C₄₇	—	16	4	no	—
145	C₂²×C₄₇	#480	1	2	yes	—
146	C₂²×C₄₇	#508	1	2	yes	—
147	C₂²×C₄₇	#509	1	2	yes	—
148	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	8	2	no	—
151	C₄×C₄₇	#480	1	2	yes	—
152	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
154	Dic₄₇	—	8	2	no	—
155	Dic₄₇	—	8	2	no	—
156	C₂×#466	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
163	C₂×C₂²⋊C₁₆	—	47	1	no	—
164	C₂×C₄₇	—	4	2	no	—
165	C₂×C₄₇	—	16	2	no	—
166	C₂×C₄₇	#2234	1	4	yes	complemented
167	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
170	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
172	C₂×#121	—	2	1	no	—
173	C₂×#117	—	94	1	no	—
174	C₂²×C₁₆	—	47	1	no	—
175	C₂²×C₁₆	—	47	1	no	—
176	#466	—	2	2	no	—
177	#466	C₂×D₄₇	1	4	yes	—
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	—	2	1	no	—
185	#367	—	47	4	no	—
186	#366	—	47	4	no	—
187	#357	—	47	4	no	—
188	#357	—	47	4	no	—
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	—	4	1	no	—
193	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	C₂²×D₄₇	1	2	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
202	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
203	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
204	#121	—	2	4	no	—
205	#121	—	4	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	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	188	1	no	—
213	C₂×C₄²	—	188	1	no	—
214	C₂×C₁₆	—	47	2	no	—
215	C₂×C₁₆	—	47	2	no	—
216	C₂×C₁₆	—	94	2	no	—
217	C₂×C₁₆	—	94	2	no	—
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	—	2	4	no	—
226	C₂⁴	—	4	1	no	—
227	C₂⁴	#10625334	1	1	yes	characteristic
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	188	1	no	—
238	C₂²×C₄	—	188	1	no	—
239	C₂²×C₄	#10625334	1	2	yes	—
240	C₂²×C₄	#10625334	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	#10625352	1	2	yes	—
251	C₂×C₈	#10625352	1	2	yes	—
252	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
253	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
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	—	4	4	no	—
262	C₂²⋊C₄	—	188	4	no	—
263	C₄²	—	188	4	no	—
264	C₁₆	—	94	2	no	—
265	C₁₆	—	94	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	C₂×#10625334	1	2	yes	—
274	C₂³	C₂×#10625334	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	8	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	376	2	no	—
286	C₂×C₄	—	376	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	#13094005	1	2	yes	—
290	C₂×C₄	C₂×#10625334	1	2	yes	—
291	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
292	C₂×C₄	#13094054	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	#13094005	1	2	yes	—
296	C₈	C₂×#10625352	1	1	yes	characteristic
297	C₈	C₂×#10625352	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	16	4	no	—
306	C₂²	#15689434	1	2	yes	—
307	C₂²	#15689553	1	2	yes	—
308	C₂²	#15689559	1	2	yes	—
309	C₂²	C₂²×#10625334	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	376	2	no	—
313	C₄	—	376	2	no	—
314	C₄	#15689434	1	2	yes	—
315	C₄	C₂×#13094005	1	1	yes	characteristic
316	C₄	C₂×#13094054	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	16	2	no	—
319	C₂	#20245576	1	4	yes	direct product factor
320	C₂	C₂×#15689434	1	1	yes	characteristic
321	C₂	C₂×#15689553	1	1	yes	characteristic
322	C₂	C₂×#15689559	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₂×C₄₇×#466	C₂
socle	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#466	D₄₇
hypocenter	C₄₇	C₂×#2234
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×#15689161	C₂	semidirect
2. #20245576	C₂	direct (N × H)
3. C₂×#15688950	C₂	semidirect
4. C₂×#15688948	C₂	semidirect
5. C₂×#15688924	C₂	semidirect
6. C₂×#15688924	C₂	semidirect
7. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
13. #15689161	C₂²	semidirect
17. #15688950	C₂²	semidirect
18. #15688948	C₂²	semidirect
19. #15688924	C₂²	semidirect
20. #15688924	C₂²	semidirect
21. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
166. C₂×C₄₇	#2234	semidirect
190. C₄₇	C₂×#2234	semidirect
319. C₂	#20245576	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×#15689161	2
#20245576	2
C₂×#15688950	2
C₂×#15688948	2
C₂×#15688924	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂×C₄₇×#466	2
C₂×#2234	47

Subgroups — C2×#20245576 #26371045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245576 #26371045