Subgroups — #26216842

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689077	C₂	1	1	yes	maximal, characteristic
2	#20245996	C₂	1	2	yes	maximal
3	#20243552	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243523	C₂	1	2	yes	maximal, complemented
5	#20243418	C₂	1	2	yes	maximal
6	#20243410	C₂	1	2	yes	maximal
7	#20243288	C₂	1	2	yes	maximal, complemented
8	#106×Dic₄₇	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
12	C₂×#13093981	C₂²	1	1	yes	characteristic
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093956	C₂²	1	2	yes	—
15	#15689077	C₄	1	8	yes	complemented
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	C₂×#13093934	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15689013	—	2	4	no	—
20	#15689004	—	2	4	no	—
21	#15688989	C₂²	1	2	yes	—
22	#15688980	C₂²	1	4	yes	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
24	#15688885	C₂²	1	4	yes	—
25	#15688885	C₂²	1	4	yes	—
26	C₂×#106×C₄₇	C₂²	1	2	yes	—
27	C₂×#106×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#384	C₂²	1	2	yes	—
30	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
31	C₂×#10625324	—	2	2	no	—
32	C₂×#10625324	—	2	2	no	—
33	C₂×#10625324	C₂×C₄	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₄₇	—	2	2	no	—
37	C₂×C₈×D₄₇	—	2	2	no	—
38	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
39	#13093981	C₂×C₄	1	4	yes	—
40	#13093981	Q₈	1	4	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
42	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
43	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
44	C₂×(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	—	2	4	no	—
46	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
47	#13093958	—	2	4	no	—
48	#13093956	—	2	4	no	—
49	#13093935	C₂×C₄	1	4	yes	—
50	#13093934	C₂×C₄	1	4	yes	—
51	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
52	C₄₇⋊#95	—	2	4	no	—
53	C₄₇⋊#95	—	2	4	no	—
54	C₄₇⋊#95	—	2	4	no	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#106×C₄₇	—	2	4	no	—
61	#106×C₄₇	—	2	4	no	—
62	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
63	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₄₇	—	2	1	no	—
71	C₂×C₄×D₄₇	—	2	1	no	—
72	C₂×C₄×D₄₇	—	2	1	no	—
73	C₂×C₄×D₄₇	—	2	1	no	—
74	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	#41	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	2	yes	—
80	C₂×C₈×C₄₇	#41	1	2	yes	—
81	C₂×C₈×C₄₇	#41	1	2	yes	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	8	4	no	—
85	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
87	(C₂×D₄₇)⋊C₄	—	8	4	no	—
88	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	8	yes	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
92	C₄×Dic₄₇	—	8	4	no	—
93	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
94	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
95	C₂×C₄₇⋊C₈	—	2	1	no	—
96	C₂×C₄₇⋊C₈	—	2	1	no	—
97	C₂×C₄₇⋊C₈	—	2	1	no	—
98	C₂×C₄₇⋊C₈	—	2	1	no	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
101	#10625324	—	2	8	no	—
102	C₈×D₄₇	—	2	8	no	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂²×D₄₇	—	2	1	no	—
105	C₂²×D₄₇	—	2	1	no	—
106	C₂²×D₄₇	—	2	4	no	—
107	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	1	no	—
120	C₂×Dic₄₇	—	2	1	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₄×D₄₇	—	2	2	no	—
128	C₄×D₄₇	—	2	2	no	—
129	C₄×D₄₇	—	2	2	no	—
130	C₄×D₄₇	—	2	2	no	—
131	C₈×C₄₇	#117	1	4	yes	—
132	C₈×C₄₇	#117	1	4	yes	—
133	C₄₇⋊C₈	—	2	2	no	—
134	C₄₇⋊C₈	—	2	2	no	—
135	#20820	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	#532	1	1	yes	characteristic
138	C₂²×C₄₇	#532	1	1	yes	characteristic
139	C₂²×C₄₇	#577	1	4	yes	—
140	C₂×D₄₇	—	2	4	no	—
141	C₂×D₄₇	—	2	2	no	—
142	C₂×D₄₇	—	4	2	no	—
143	C₂×D₄₇	—	4	2	no	—
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	—	8	4	no	—
146	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
147	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
148	C₄×C₄₇	#533	1	1	yes	characteristic
149	C₄×C₄₇	#533	1	1	yes	characteristic
150	Dic₄₇	—	2	4	no	—
151	Dic₄₇	—	2	2	no	—
152	Dic₄₇	—	8	4	no	—
153	C₂×#405	—	47	1	no	—
154	#2314	—	47	2	no	—
155	#1846	—	47	1	no	—
156	#1844	D₄₇	1	1	yes	characteristic, complemented
157	#1836	—	47	2	no	—
158	#1809	—	47	2	no	—
159	#1807	—	47	2	no	—
160	#1776	—	47	2	no	—
161	C₄×#106	—	47	2	no	—
162	C₂×C₄₇	#2285	1	1	yes	characteristic
163	C₂×C₄₇	#2912	1	2	yes	—
164	C₂×C₄₇	#3173	1	2	yes	—
165	C₂×C₄₇	#3174	1	2	yes	—
166	D₄₇	—	4	2	no	—
167	C₂²×M₁₆	—	47	1	no	—
168	C₂³×C₈	—	47	1	no	—
169	C₂×#117	—	47	1	no	—
170	C₂×#107	—	47	2	no	—
171	C₂×#106	—	47	2	no	—
172	C₂×#106	C₂×D₄₇	1	2	yes	—
173	C₂×#106	C₂×D₄₇	1	2	yes	—
174	#405	—	47	8	no	—
175	C₂×#97	—	47	1	no	—
176	C₂×#97	—	47	1	no	—
177	C₂×C₄×C₈	—	47	1	no	—
178	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
179	#388	—	94	4	no	—
180	#387	—	94	4	no	—
181	#384	C₂×D₄₇	1	2	yes	—
182	#382	—	47	2	no	—
183	#380	—	47	4	no	—
184	C₄×C₄⋊C₄	—	47	4	no	—
185	#340	—	47	4	no	—
186	#340	—	47	4	no	—
187	C₄₇	#20820	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	47	1	no	—
189	C₂×M₁₆	—	47	4	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂²×C₈	—	47	1	no	—
193	C₂²×C₈	—	47	1	no	—
194	C₂²×C₈	—	47	4	no	—
195	C₂²×C₈	—	94	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	#117	—	47	4	no	—
200	#117	—	47	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	47	2	no	—
203	C₂×C₄⋊C₄	—	47	4	no	—
204	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
205	C₂×C₂²⋊C₄	—	47	2	no	—
206	C₂×C₂²⋊C₄	—	94	4	no	—
207	C₂×C₄²	—	47	1	no	—
208	C₂×C₄²	—	94	4	no	—
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	#107	—	94	4	no	—
211	#106	—	2	4	no	—
212	#106	—	2	4	no	—
213	#106	—	94	4	no	—
214	#97	—	47	4	no	—
215	#97	—	47	4	no	—
216	C₄×C₈	—	47	4	no	—
217	C₄×C₈	C₄×D₄₇	1	4	yes	—
218	#95	—	94	4	no	—
219	#95	—	94	4	no	—
220	#95	—	94	4	no	—
221	C₂⁴	—	47	1	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	47	1	no	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	—	47	2	no	—
226	C₂²×C₄	—	47	2	no	—
227	C₂²×C₄	—	94	1	no	—
228	C₂²×C₄	—	94	1	no	—
229	C₂²×C₄	—	94	1	no	—
230	C₂²×C₄	—	94	1	no	—
231	C₂²×C₄	—	94	4	no	—
232	C₂²×C₄	D₄×D₄₇	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	M₁₆	—	94	8	no	—
235	C₂×C₈	—	47	2	no	—
236	C₂×C₈	—	47	2	no	—
237	C₂×C₈	—	94	1	no	—
238	C₂×C₈	—	94	1	no	—
239	C₂×C₈	—	94	1	no	—
240	C₂×C₈	—	94	1	no	—
241	C₂×C₈	—	94	8	no	—
242	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
244	C₂×C₈	#10625352	1	2	yes	—
245	C₂×C₈	#10625352	1	2	yes	—
246	C₂×C₈	#10625354	1	2	yes	—
247	C₂×C₈	#10625354	1	2	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	8	4	no	—
251	C₄⋊C₄	—	47	8	no	—
252	C₄⋊C₄	—	94	4	no	—
253	C₄⋊C₄	—	94	4	no	—
254	C₂²⋊C₄	—	47	8	no	—
255	C₂²⋊C₄	—	376	4	no	—
256	C₄²	—	47	2	no	—
257	C₄²	—	47	2	no	—
258	C₄²	—	376	4	no	—
259	C₄²	C₂×C₄×D₄₇	1	2	yes	—
260	C₄²	Q₈×D₄₇	1	2	yes	—
261	C₂³	—	47	2	no	—
262	C₂³	—	94	1	no	—
263	C₂³	—	94	4	no	—
264	C₂³	—	94	1	no	—
265	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	47	4	no	—
271	C₂×C₄	—	47	2	no	—
272	C₂×C₄	—	94	1	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	1	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	94	2	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
283	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
285	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
286	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
287	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
289	C₈	—	94	2	no	—
290	C₈	—	94	2	no	—
291	C₈	#13093982	1	4	yes	—
292	C₈	#13093982	1	4	yes	—
293	C₂²	—	94	2	no	—
294	C₂²	—	94	4	no	—
295	C₂²	—	188	2	no	—
296	C₂²	—	188	2	no	—
297	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
298	C₂²	#15689671	1	1	yes	characteristic
299	C₂²	#15689674	1	1	yes	characteristic
300	C₂²	#15689929	1	4	yes	—
301	C₄	—	2	4	no	—
302	C₄	—	8	4	no	—
303	C₄	—	94	2	no	—
304	C₄	—	94	4	no	—
305	C₄	—	376	4	no	—
306	C₄	C₂×#13093982	1	1	yes	characteristic
307	C₄	C₂×#13093982	1	1	yes	characteristic
308	C₄	#15689678	1	1	yes	characteristic
309	C₄	#15689678	1	1	yes	characteristic
310	C₂	—	188	2	no	—
311	C₂	#20245793	1	1	yes	characteristic
312	C₂	#20251979	1	2	yes	—
313	C₂	#20254007	1	2	yes	—
314	C₂	#20254013	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#20820
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
3. #20243552	C₂	semidirect
4. #20243523	C₂	semidirect
7. #20243288	C₂	semidirect
8. #106×Dic₄₇	C₂	semidirect
9. C₄₇×#1844	C₂	semidirect
15. #15689077	C₄	semidirect
156. #1844	D₄₇	semidirect
187. C₄₇	#20820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689077	2
#20245996	2
#20243552	2
#20243523	2
#20243418	2
#20243410	2
#20243288	2
#106×Dic₄₇	2
C₄₇×#1844	2
#20820	47