Subgroups — #26214870

Order 12032.

998 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689244	C₂	1	2	yes	maximal
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
3	#20251817	C₂	1	2	yes	maximal, complemented
4	#20243275	C₂	1	2	yes	maximal, complemented
5	#20243224	C₂	1	4	yes	maximal, complemented
6	#20243201	C₂	1	2	yes	maximal, complemented
7	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₂×#13094032	C₂²	1	4	yes	—
11	C₄×#10625350	C₂²	1	2	yes	—
12	#15689244	C₄	1	16	yes	complemented
13	C₄×Dic₁₈₈	C₄	1	2	yes	complemented
14	C₄×#10625325	C₄	1	4	yes	complemented
15	C₄×D₃₇₆	C₄	1	2	yes	complemented
16	#15689161	C₂²	1	1	yes	characteristic
17	C₂×#13093950	C₂²	1	4	yes	—
18	#15689120	C₂²	1	2	yes	—
19	#15689081	C₂²	1	4	yes	—
20	#15689077	—	2	4	no	—
21	#15689075	—	2	2	no	—
22	C₂×#13093935	C₂²	1	1	yes	characteristic
23	C₂×#13093934	C₂²	1	2	yes	—
24	#97×D₄₇	—	2	2	no	—
25	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
26	#15688969	C₂²	1	1	yes	characteristic
27	#15688884	C₂²	1	4	yes	—
28	C₂×#108×C₄₇	C₂²	1	2	yes	—
29	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#341	C₂²	1	2	yes	—
32	C₂×#10625350	C₂³	1	2	yes	—
33	#13094046	C₂×C₄	1	8	yes	—
34	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#10625325	C₂×C₄	1	2	yes	—
36	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
37	#13094032	—	2	8	no	—
38	#13094032	C₂×C₄	1	16	yes	complemented
39	C₂×#10625324	—	2	4	no	—
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₂×C₈×D₄₇	—	2	4	no	—
42	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
43	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₄₇	—	2	4	no	—
45	#13093983	C₂³	1	2	yes	—
46	#13093981	—	2	2	no	—
47	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
48	C₄²×D₄₇	—	2	2	no	—
49	#13093958	C₂×C₄	1	2	yes	—
50	#13093956	C₂×C₄	1	2	yes	—
51	#13093950	C₂×C₄	1	16	yes	complemented
52	#13093949	C₂×C₄	1	4	yes	—
53	#13093946	C₂×C₄	1	4	yes	—
54	#13093935	D₄	1	2	yes	—
55	#13093935	D₄	1	2	yes	—
56	#13093934	—	2	4	no	—
57	#13093934	C₂×C₄	1	8	yes	complemented
58	C₈×Dic₄₇	—	2	4	no	—
59	C₈×Dic₄₇	—	2	4	no	—
60	C₄₇⋊#95	—	2	2	no	—
61	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	#108×C₄₇	C₂×C₄	1	8	yes	complemented
66	#97×C₄₇	—	2	2	no	—
67	C₄×C₈×C₄₇	—	2	2	no	—
68	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₂²⋊D₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	#10625350	—	2	4	no	—
75	#10625350	C₂²×C₄	1	8	yes	—
76	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×C₄×D₄₇	—	2	4	no	—
79	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
80	M₁₆×C₄₇	—	2	4	no	—
81	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²⋊Dic₄₇	—	2	2	no	—
92	(C₂×D₄₇)⋊C₄	—	2	4	no	—
93	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
94	Dic₄₇⋊C₄	—	2	4	no	—
95	C₄×Dic₄₇	—	2	4	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	#41	1	4	yes	—
102	Dic₁₈₈	C₄²	1	4	yes	—
103	D₃₇₆	C₄²	1	4	yes	—
104	#10625325	C₄²	1	8	yes	—
105	#10625324	—	2	16	no	—
106	C₈×D₄₇	—	2	16	no	—
107	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂²×D₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂²⋊D₄₇	—	2	8	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	D₁₈₈	—	2	2	no	—
122	D₁₈₈	C₂×C₄²	1	4	yes	—
123	C₄×D₄₇	—	2	8	no	—
124	C₄×D₄₇	—	8	4	no	—
125	Dic₉₄	—	2	2	no	—
126	Dic₉₄	C₂×C₄²	1	4	yes	—
127	C₈×C₄₇	—	2	8	no	—
128	C₈×C₄₇	—	2	2	no	—
129	C₈×C₄₇	C₂×C₄²	1	2	yes	—
130	C₈×C₄₇	C₂×C₄²	1	2	yes	—
131	C₄₇⋊C₈	C₄×D₄	1	8	yes	—
132	#20513	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₄×#41	1	2	yes	—
136	C₂×D₄₇	—	2	4	no	—
137	C₂×D₄₇	—	4	4	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
141	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
142	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
143	Dic₄₇	—	2	4	no	—
144	Dic₄₇	—	4	4	no	—
145	Dic₄₇	—	4	4	no	—
146	C₂×#444	—	47	2	no	—
147	C₄×#135	—	47	1	no	—
148	#2875	—	47	2	no	—
149	C₄×#108	D₄₇	1	1	yes	characteristic, complemented
150	#1773	—	47	2	no	—
151	#1764	—	47	4	no	—
152	#1759	—	47	2	no	—
153	#1726	—	47	1	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
156	C₂×C₄₇	#2957	1	2	yes	—
157	D₄₇	—	4	4	no	—
158	C₂×#135	—	47	1	no	—
159	C₂×#131	—	47	4	no	—
160	C₄×#41	—	47	2	no	—
161	#444	—	47	16	no	—
162	C₄×Q₁₆	—	47	2	no	—
163	C₄×SD₁₆	—	47	4	no	—
164	C₄×D₈	—	47	2	no	—
165	C₂×#108	C₂×D₄₇	1	2	yes	—
166	#427	—	47	1	no	—
167	C₂×#104	—	47	4	no	—
168	#416	—	47	2	no	—
169	#405	—	47	4	no	—
170	#405	—	94	4	no	—
171	C₄×M₁₆	—	94	2	no	—
172	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
173	C₂×#97	—	47	2	no	—
174	C₂×#97	—	47	1	no	—
175	C₂×#97	—	94	2	no	—
176	C₂×C₄×C₈	—	47	2	no	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	#376	—	47	1	no	—
179	#341	C₂×D₄₇	1	2	yes	—
180	#339	—	47	4	no	—
181	C₄₇	#20513	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	47	2	no	—
183	#135	—	47	8	no	—
184	C₂×Q₁₆	—	47	1	no	—
185	C₂×SD₁₆	—	47	2	no	—
186	C₂×D₈	—	47	1	no	—
187	#131	—	47	16	no	—
188	#131	—	94	8	no	—
189	C₂×M₁₆	—	94	4	no	—
190	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	94	4	no	—
193	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₄×Q₈	—	47	4	no	—
196	C₄×D₄	—	47	4	no	—
197	C₄×D₄	—	94	4	no	—
198	#117	—	47	2	no	—
199	#117	—	94	2	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	#108	C₄×D₄₇	1	8	yes	complemented
204	#107	—	47	2	no	—
205	#106	—	47	2	no	—
206	#104	—	47	16	no	—
207	#103	—	47	4	no	—
208	#102	—	47	4	no	—
209	#97	—	2	2	no	—
210	#97	—	47	8	no	—
211	#97	—	47	2	no	—
212	#97	—	47	2	no	—
213	#97	—	94	4	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	—	94	4	no	—
216	C₄×C₈	—	94	4	no	—
217	C₄×C₈	C₄×D₄₇	1	2	yes	—
218	C₄×C₈	C₄×D₄₇	1	2	yes	—
219	#95	—	94	2	no	—
220	#41	—	47	8	no	—
221	#41	—	94	4	no	—
222	C₂×Q₈	—	47	2	no	—
223	C₂×D₄	—	47	2	no	—
224	C₂×D₄	—	94	2	no	—
225	C₂²×C₄	—	94	2	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	4	no	—
228	C₂²×C₄	#10625354	1	2	yes	—
229	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
230	Q₁₆	—	47	4	no	—
231	SD₁₆	—	47	8	no	—
232	D₈	—	47	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	94	16	no	—
235	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	94	16	no	—
242	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
246	C₄⋊C₄	—	47	4	no	—
247	C₄⋊C₄	—	94	4	no	—
248	C₂²⋊C₄	—	94	2	no	—
249	C₂²⋊C₄	—	94	4	no	—
250	C₄²	—	47	4	no	—
251	C₄²	—	94	4	no	—
252	C₄²	—	94	2	no	—
253	C₄²	—	94	2	no	—
254	C₄²	C₂×C₄×D₄₇	1	2	yes	—
255	C₄²	D₄×D₄₇	1	1	yes	characteristic
256	C₄²	D₄×D₄₇	1	1	yes	characteristic
257	C₂³	—	94	2	no	—
258	C₂³	C₂×#10625354	1	1	yes	characteristic
259	Q₈	—	47	4	no	—
260	Q₈	—	94	2	no	—
261	D₄	—	47	4	no	—
262	D₄	—	94	8	no	—
263	D₄	—	94	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	94	2	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	94	4	no	—
268	C₂×C₄	—	94	4	no	—
269	C₂×C₄	—	376	4	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	#13093986	1	2	yes	—
272	C₂×C₄	#13093986	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
274	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	#41×D₄₇	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₈	—	2	2	no	—
279	C₈	—	47	8	no	—
280	C₈	C₄²×D₄₇	1	2	yes	—
281	C₈	C₄²×D₄₇	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	94	4	no	—
284	C₂²	—	188	4	no	—
285	C₂²	C₂×#13093986	1	1	yes	characteristic
286	C₂²	C₄×#10625354	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	94	4	no	—
290	C₄	—	188	4	no	—
291	C₄	—	188	4	no	—
292	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
293	C₄	C₄×D₄×D₄₇	1	2	yes	—
294	C₄	#15689591	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	188	4	no	—
297	C₂	C₄×#13093986	1	1	yes	characteristic
298	C₂	#20252304	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#20513
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. #20251817	C₂	semidirect
4. #20243275	C₂	semidirect
5. #20243224	C₂	semidirect
6. #20243201	C₂	semidirect
7. #20243074	C₂	semidirect
8. C₄×#108×C₄₇	C₂	semidirect
12. #15689244	C₄	semidirect
13. C₄×Dic₁₈₈	C₄	semidirect
14. C₄×#10625325	C₄	semidirect
15. C₄×D₃₇₆	C₄	semidirect
38. #13094032	C₂×C₄	semidirect
51. #13093950	C₂×C₄	semidirect
57. #13093934	C₂×C₄	semidirect
65. #108×C₄₇	C₂×C₄	semidirect
149. C₄×#108	D₄₇	semidirect
181. C₄₇	#20513	semidirect
203. #108	C₄×D₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689244	2
C₄×#13094046	2
#20251817	2
#20243275	2
#20243224	2
#20243201	2
#20243074	2
C₄×#108×C₄₇	2
#20513	47