Subgroups — C₂×#20255187 #27092503

Order 12032.

2766 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689924	C₂	1	4	yes	maximal, complemented
2	#20255187	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689390	C₂	1	4	yes	maximal, complemented
4	C₂×#15689290	C₂	1	4	yes	maximal, complemented
5	C₂×#15689241	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094069	C₂²	1	4	yes	complemented
8	#15689924	—	2	16	no	—
9	#15689924	C₂²	1	32	yes	complemented
10	C₂×#13094050	C₂²	1	8	yes	complemented
11	C₂×#13094048	—	2	8	no	—
12	C₂×#13094046	C₂²	1	2	yes	—
13	C₂×#13094043	—	2	4	no	—
14	C₂×D₈×D₄₇	—	2	4	no	—
15	C₂×#13094030	C₂²	1	4	yes	complemented
16	#15689390	C₂²	1	32	yes	complemented
17	#15689290	C₂²	1	32	yes	complemented
18	#15689241	C₂²	1	16	yes	complemented
19	C₂×#13093960	C₂²	1	2	yes	complemented
20	C₂×#13093952	C₂²	1	4	yes	complemented
21	C₂×#13093941	C₂²	1	4	yes	complemented
22	C₂²×D₈×C₄₇	C₂²	1	4	yes	complemented
23	C₂×#127×C₄₇	C₂²	1	2	yes	complemented
24	C₄₇×#493	C₂²	1	8	yes	complemented
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
26	#13094069	—	2	16	no	—
27	#13094069	—	4	4	no	—
28	#13094069	C₂³	1	16	yes	complemented
29	C₂×#10625350	C₂³	1	4	yes	—
30	C₂×#10625352	—	2	8	no	—
31	C₂²×C₂²⋊D₄₇	—	4	4	no	—
32	C₂×D₄×D₄₇	—	2	8	no	—
33	#13094050	—	2	16	no	—
34	#13094050	C₂³	1	32	yes	complemented
35	#13094048	—	2	64	no	—
36	#13094046	D₄	1	8	yes	complemented
37	#13094046	C₂³	1	8	yes	—
38	#13094043	—	2	32	no	—
39	C₂×Dic₁₈₈	D₄	1	2	yes	complemented
40	C₂×#10625325	D₄	1	4	yes	complemented
41	C₂×#10625335	—	2	8	no	—
42	C₂×D₃₇₆	D₄	1	2	yes	complemented
43	C₂×#10625334	—	2	8	no	—
44	D₈×D₄₇	—	2	32	no	—
45	#13094030	—	2	8	no	—
46	#13094030	C₂³	1	16	yes	complemented
47	C₂×#10625329	C₂³	1	4	yes	—
48	C₂×#10625324	—	2	4	no	—
49	C₂×C₈×D₄₇	—	2	4	no	—
50	#13093960	C₂³	1	8	yes	complemented
51	#13093952	C₂³	1	16	yes	complemented
52	#13093941	C₂³	1	16	yes	complemented
53	C₂²×D₄×C₄₇	—	4	2	no	—
54	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
55	C₂×D₈×C₄₇	—	2	4	no	—
56	C₂×D₈×C₄₇	—	2	4	no	—
57	C₂×D₈×C₄₇	—	2	8	no	—
58	C₂×D₈×C₄₇	C₂³	1	16	yes	complemented
59	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
60	#127×C₄₇	—	4	2	no	—
61	#127×C₄₇	C₂³	1	8	yes	complemented
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂³	1	4	yes	complemented
64	C₂⁴×C₄₇	—	2	4	no	—
65	C₂³×D₄₇	—	4	4	no	—
66	C₂×D₄×C₄₇	—	2	8	no	—
67	C₂×D₄×C₄₇	—	2	8	no	—
68	C₂×D₄×C₄₇	—	2	4	no	—
69	C₂×D₄×C₄₇	—	4	8	no	—
70	C₂×D₄×C₄₇	—	8	8	no	—
71	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₇	—	2	4	no	—
75	C₂×C₂²⋊D₄₇	—	4	8	no	—
76	C₂×C₂²⋊D₄₇	—	4	16	no	—
77	C₂×C₂²⋊D₄₇	—	4	16	no	—
78	C₂²×Dic₄₇	—	4	4	no	—
79	#10625352	—	2	32	no	—
80	#10625352	—	4	16	no	—
81	D₄×D₄₇	—	2	32	no	—
82	D₄×D₄₇	—	4	16	no	—
83	#10625350	—	2	4	no	—
84	#10625350	—	2	4	no	—
85	#10625350	C₂×D₄	1	8	yes	—
86	#10625350	C₂⁴	1	8	yes	—
87	C₂×D₁₈₈	C₂×D₄	1	4	yes	—
88	C₂×C₄×D₄₇	—	2	4	no	—
89	C₂×Dic₉₄	C₂×D₄	1	4	yes	—
90	D₈×C₄₇	—	2	16	no	—
91	D₈×C₄₇	—	2	16	no	—
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
96	C₄²×C₄₇	—	2	1	no	—
97	C₄²×C₄₇	C₂⁴	1	2	yes	—
98	#10625335	—	2	32	no	—
99	#10625334	—	2	32	no	—
100	#10625329	—	2	4	no	—
101	#10625329	C₂⁴	1	8	yes	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	Dic₁₈₈	C₂×D₄	1	8	yes	complemented
104	D₃₇₆	C₂×D₄	1	8	yes	complemented
105	#10625325	C₂×D₄	1	16	yes	complemented
106	#10625324	—	2	16	no	—
107	C₈×D₄₇	—	2	16	no	—
108	C₂³×C₄₇	—	2	8	no	—
109	C₂³×C₄₇	—	4	4	no	—
110	C₂³×C₄₇	—	4	4	no	—
111	C₂³×C₄₇	—	8	8	no	—
112	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×D₄₇	—	2	4	no	—
114	C₂²×D₄₇	—	4	16	no	—
115	C₂²×D₄₇	—	16	8	no	—
116	D₄×C₄₇	—	2	16	no	—
117	D₄×C₄₇	—	4	8	no	—
118	D₄×C₄₇	—	8	16	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
125	C₂²⋊D₄₇	—	2	8	no	—
126	C₂²⋊D₄₇	—	2	8	no	—
127	C₂²⋊D₄₇	—	4	32	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	4	16	no	—
130	D₁₈₈	—	2	4	no	—
131	D₁₈₈	C₂²×D₄	1	8	yes	—
132	C₄×D₄₇	—	2	8	no	—
133	C₄×D₄₇	—	2	8	no	—
134	Dic₉₄	—	2	4	no	—
135	Dic₉₄	C₂²×D₄	1	8	yes	—
136	C₈×C₄₇	C₂²×D₄	1	4	yes	—
137	C₈×C₄₇	C₂²×D₄	1	4	yes	—
138	C₄₇⋊C₈	—	2	8	no	—
139	C₂×#3292	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	4	8	no	—
142	C₂²×C₄₇	—	8	8	no	—
143	C₂²×C₄₇	—	8	8	no	—
144	C₂²×C₄₇	—	16	4	no	—
145	C₂²×C₄₇	D₄²	1	1	yes	characteristic
146	C₂²×C₄₇	D₄²	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×D₄₇	—	2	8	no	—
149	C₂×D₄₇	—	4	8	no	—
150	C₂×D₄₇	—	16	16	no	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	D₄²	1	2	yes	—
153	C₄×C₄₇	C₂³×D₄	1	2	yes	—
154	Dic₄₇	—	2	8	no	—
155	C₂×#576	—	47	4	no	—
156	#3292	—	47	16	no	—
157	C₂×#493	D₄₇	1	1	yes	characteristic, complemented
158	C₂×#472	—	47	4	no	—
159	C₂×#453	—	47	4	no	—
160	C₂×#443	—	47	2	no	—
161	C₂×C₄₇	—	2	1	no	—
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	—	8	8	no	—
164	C₂×C₄₇	#3292	1	2	yes	complemented
165	C₂×C₄₇	C₂×D₄²	1	1	yes	characteristic
166	D₄₇	—	4	8	no	—
167	C₂×H(5,𝔽₂)	—	47	4	no	—
168	#576	—	47	32	no	—
169	#576	—	94	16	no	—
170	C₂×AGL(1,ℤ/8)	—	47	8	no	—
171	C₂×AGL(1,ℤ/8)	—	94	8	no	—
172	C₂×#135	—	47	2	no	—
173	C₂×#135	—	94	4	no	—
174	C₂²×D₈	—	94	4	no	—
175	C₂²×D₈	C₂×D₄₇	1	4	yes	complemented
176	C₂×#131	—	47	4	no	—
177	C₂×#127	C₂×D₄₇	1	2	yes	complemented
178	#493	C₂×D₄₇	1	8	yes	complemented
179	#472	—	47	32	no	—
180	#453	—	47	32	no	—
181	#443	—	47	16	no	—
182	C₂×#108	—	47	2	no	—
183	C₂×#104	—	47	4	no	—
184	C₂×#100	—	47	4	no	—
185	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic, complemented
186	C₄₇	C₂×#3292	1	1	yes	characteristic, complemented, Sylow
187	H(5,𝔽₂)	—	47	16	no	—
188	H(5,𝔽₂)	—	94	16	no	—
189	H(5,𝔽₂)	—	188	4	no	—
190	C₂×#41	—	47	4	no	—
191	C₂×#41	—	94	8	no	—
192	C₂²×D₄	—	4	2	no	—
193	C₂²×D₄	—	94	8	no	—
194	C₂²×D₄	—	188	4	no	—
195	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
196	AGL(1,ℤ/8)	—	47	32	no	—
197	AGL(1,ℤ/8)	—	94	64	no	—
198	AGL(1,ℤ/8)	—	94	16	no	—
199	#135	—	47	8	no	—
200	#135	—	47	8	no	—
201	#135	—	94	32	no	—
202	C₂×Q₁₆	—	47	2	no	—
203	C₂×SD₁₆	—	47	4	no	—
204	C₂×SD₁₆	—	94	8	no	—
205	C₂×D₈	—	2	4	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	2	8	no	—
208	C₂×D₈	—	47	2	no	—
209	C₂×D₈	—	94	32	no	—
210	C₂×D₈	—	94	8	no	—
211	C₂×D₈	C₂²×D₄₇	1	16	yes	complemented
212	#131	—	47	16	no	—
213	#131	—	94	8	no	—
214	C₂×M₁₆	—	47	4	no	—
215	C₂×M₁₆	—	94	4	no	—
216	C₂²×C₈	—	94	4	no	—
217	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
218	#127	—	4	2	no	—
219	#127	C₂²×D₄₇	1	8	yes	complemented
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	#108	—	47	8	no	—
222	#104	—	47	16	no	—
223	#100	—	47	16	no	—
224	C₄×C₈	C₂²×D₄₇	1	4	yes	complemented
225	C₂⁴	—	2	4	no	—
226	C₂⁴	—	188	4	no	—
227	#41	—	47	8	no	—
228	#41	—	47	8	no	—
229	#41	—	94	4	no	—
230	#41	—	94	4	no	—
231	#41	—	94	32	no	—
232	#41	—	188	16	no	—
233	C₂×Q₈	—	47	4	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	4	8	no	—
238	C₂×D₄	—	8	8	no	—
239	C₂×D₄	—	47	4	no	—
240	C₂×D₄	—	94	32	no	—
241	C₂×D₄	—	94	4	no	—
242	C₂×D₄	—	188	16	no	—
243	C₂×D₄	—	188	8	no	—
244	C₂×D₄	—	188	16	no	—
245	C₂×D₄	—	188	16	no	—
246	C₂×D₄	C₂³×D₄₇	1	8	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	94	4	no	—
249	C₂²×C₄	—	188	4	no	—
250	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
251	Q₁₆	—	47	8	no	—
252	SD₁₆	—	47	16	no	—
253	SD₁₆	—	94	32	no	—
254	D₈	—	2	16	no	—
255	D₈	—	2	16	no	—
256	D₈	—	47	8	no	—
257	D₈	—	94	32	no	—
258	M₁₆	—	47	8	no	—
259	M₁₆	—	94	4	no	—
260	M₁₆	—	94	16	no	—
261	C₂×C₈	—	94	4	no	—
262	C₂×C₈	—	94	16	no	—
263	C₂×C₈	D₄×D₄₇	1	2	yes	—
264	C₂×C₈	D₄×D₄₇	1	2	yes	—
265	C₂×C₈	D₄×D₄₇	1	4	yes	—
266	C₂×C₈	C₂³×D₄₇	1	4	yes	—
267	C₄²	—	2	1	no	—
268	C₄²	C₂³×D₄₇	1	2	yes	—
269	C₂³	—	2	8	no	—
270	C₂³	—	4	4	no	—
271	C₂³	—	4	4	no	—
272	C₂³	—	8	8	no	—
273	C₂³	—	94	4	no	—
274	C₂³	—	188	16	no	—
275	C₂³	—	752	8	no	—
276	C₂³	#13094069	1	1	yes	characteristic
277	Q₈	—	47	8	no	—
278	Q₈	—	94	4	no	—
279	D₄	—	2	16	no	—
280	D₄	—	4	8	no	—
281	D₄	—	8	16	no	—
282	D₄	—	47	8	no	—
283	D₄	—	94	8	no	—
284	D₄	—	94	8	no	—
285	D₄	—	94	4	no	—
286	D₄	—	188	32	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	94	8	no	—
291	C₂×C₄	—	94	8	no	—
292	C₂×C₄	—	188	16	no	—
293	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
294	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
295	C₂×C₄	#13094069	1	1	yes	characteristic
296	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
297	C₈	—	94	8	no	—
298	C₈	C₂×D₄×D₄₇	1	4	yes	—
299	C₈	C₂×D₄×D₄₇	1	4	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	8	8	no	—
303	C₂²	—	8	8	no	—
304	C₂²	—	16	4	no	—
305	C₂²	—	94	8	no	—
306	C₂²	—	188	8	no	—
307	C₂²	—	752	16	no	—
308	C₂²	#15689758	1	1	yes	characteristic
309	C₂²	#15689758	1	1	yes	characteristic
310	C₂²	C₂×#13094069	1	1	yes	characteristic
311	C₄	—	2	4	no	—
312	C₄	—	94	8	no	—
313	C₄	#15689758	1	2	yes	—
314	C₄	C₂²×D₄×D₄₇	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	8	no	—
318	C₂	—	188	8	no	—
319	C₂	#20255187	1	2	yes	direct product factor
320	C₂	C₂×#15689758	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689758
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#493	C₂
socle	C₂²×C₄₇	D₄²
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	C₂×#3292
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#15689924	C₂	semidirect
2. #20255187	C₂	direct (N × H)
3. C₂×#15689390	C₂	semidirect
4. C₂×#15689290	C₂	semidirect
5. C₂×#15689241	C₂	semidirect
6. C₂×C₄₇×#493	C₂	semidirect
7. C₂×#13094069	C₂²	semidirect
9. #15689924	C₂²	semidirect
10. C₂×#13094050	C₂²	semidirect
15. C₂×#13094030	C₂²	semidirect
16. #15689390	C₂²	semidirect
17. #15689290	C₂²	semidirect
18. #15689241	C₂²	semidirect
19. C₂×#13093960	C₂²	semidirect
20. C₂×#13093952	C₂²	semidirect
21. C₂×#13093941	C₂²	semidirect
22. C₂²×D₈×C₄₇	C₂²	semidirect
23. C₂×#127×C₄₇	C₂²	semidirect
24. C₄₇×#493	C₂²	semidirect
25. C₂×C₄×C₈×C₄₇	C₂²	semidirect
28. #13094069	C₂³	semidirect
34. #13094050	C₂³	semidirect
36. #13094046	D₄	semidirect
39. C₂×Dic₁₈₈	D₄	semidirect
40. C₂×#10625325	D₄	semidirect
42. C₂×D₃₇₆	D₄	semidirect
46. #13094030	C₂³	semidirect
50. #13093960	C₂³	semidirect
51. #13093952	C₂³	semidirect
52. #13093941	C₂³	semidirect
58. C₂×D₈×C₄₇	C₂³	semidirect
61. #127×C₄₇	C₂³	semidirect
63. C₄×C₈×C₄₇	C₂³	semidirect
103. Dic₁₈₈	C₂×D₄	semidirect
104. D₃₇₆	C₂×D₄	semidirect
105. #10625325	C₂×D₄	semidirect
157. C₂×#493	D₄₇	semidirect
164. C₂×C₄₇	#3292	semidirect
175. C₂²×D₈	C₂×D₄₇	semidirect
177. C₂×#127	C₂×D₄₇	semidirect
178. #493	C₂×D₄₇	semidirect
185. C₂×C₄×C₈	C₂×D₄₇	semidirect
186. C₄₇	C₂×#3292	semidirect
211. C₂×D₈	C₂²×D₄₇	semidirect
219. #127	C₂²×D₄₇	semidirect
224. C₄×C₈	C₂²×D₄₇	semidirect
319. C₂	#20255187	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×#15689924	2
#20255187	2
C₂×#15689390	2
C₂×#15689290	2
C₂×#15689241	2
C₂×C₄₇×#493	2
C₂×#3292	47

Subgroups — C2×#20255187 #27092503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20255187 #27092503