Subgroups — #26216818

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245783	C₂	1	2	yes	maximal
3	#20243549	C₂	1	2	yes	maximal, complemented
4	#20243517	C₂	1	2	yes	maximal, complemented
5	#20243415	C₂	1	2	yes	maximal
6	#20243413	C₂	1	1	yes	maximal, characteristic
7	#20243280	C₂	1	2	yes	maximal, complemented
8	#106×Dic₄₇	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1807	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×#13093957	C₂²	1	4	yes	—
13	C₂×#13093956	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	8	yes	complemented
15	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688990	—	2	2	no	—
20	#15688988	C₂²	1	4	yes	—
21	#15688984	—	2	2	no	—
22	#15688983	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
24	C₂²⋊C₄×Dic₄₇	—	2	2	no	—
25	#15688968	—	2	2	no	—
26	#15688886	C₂²	1	2	yes	—
27	#15688885	C₂²	1	4	yes	—
28	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#106×C₄₇	C₂²	1	2	yes	—
30	C₄₇×#380	C₂²	1	2	yes	—
31	C₄₇×#340	C₂²	1	2	yes	—
32	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
33	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	—	2	4	no	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
39	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	4	no	—
41	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
47	#13093957	—	2	8	no	—
48	#13093956	—	2	4	no	—
49	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
50	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
51	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
52	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₄₇⋊#95	—	2	4	no	—
55	C₄₇⋊#95	—	2	4	no	—
56	C₄₇⋊#95	—	2	4	no	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	—	2	4	no	—
59	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
64	#106×C₄₇	—	2	4	no	—
65	#95×C₄₇	—	2	2	no	—
66	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
78	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
79	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	8	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	#41	1	4	yes	—
88	C₄⋊C₄×C₄₇	—	2	4	no	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₂²⋊C₄×C₄₇	—	4	4	no	—
91	C₂²⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄×Dic₄₇	—	4	8	no	—
95	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
96	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
97	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
98	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
99	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
100	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	8	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₄₇⋊C₈	#41	1	4	yes	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₄₇	#135	1	4	yes	—
110	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	2	8	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×Dic₄₇	#135	1	4	yes	—
136	C₈×C₄₇	—	2	2	no	—
137	C₈×C₄₇	#117	1	4	yes	—
138	C₄₇⋊C₈	—	2	2	no	—
139	C₄₇⋊C₈	#117	1	4	yes	—
140	#20818	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	#427	1	4	yes	—
145	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	#532	1	2	yes	—
147	C₂²×C₄₇	C₂×#135	1	2	yes	—
148	C₂²×C₄₇	C₂×#135	1	2	yes	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	8	4	no	—
151	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
152	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₄₇	#533	1	2	yes	—
154	Dic₄₇	—	2	2	no	—
155	Dic₄₇	—	2	2	no	—
156	Dic₄₇	—	2	2	no	—
157	Dic₄₇	—	8	4	no	—
158	Dic₄₇	#427	1	4	yes	—
159	C₂²×C₄×C₈	—	47	1	no	—
160	#2283	—	47	2	no	—
161	#1846	—	47	2	no	—
162	#1835	—	47	2	no	—
163	#1808	—	47	2	no	—
164	#1807	—	47	1	no	—
165	#1807	D₄₇	1	1	yes	characteristic
166	C₄×#107	—	47	2	no	—
167	C₄×#106	—	47	2	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	#2285	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
172	C₂×C₄₇	#3168	1	4	yes	—
173	C₂³×C₈	—	47	1	no	—
174	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
175	C₂²×C₄²	—	47	1	no	—
176	C₂×#107	—	47	4	no	—
177	C₂×#106	—	47	2	no	—
178	C₂×#106	C₂×D₄₇	1	2	yes	—
179	C₂×C₄×C₈	—	47	8	no	—
180	C₂×C₄×C₈	—	47	1	no	—
181	C₂×C₄×C₈	—	47	1	no	—
182	C₂×C₄×C₈	—	47	1	no	—
183	C₂×C₄×C₈	—	47	1	no	—
184	#383	—	94	2	no	—
185	#382	—	47	4	no	—
186	#381	—	94	2	no	—
187	#380	—	47	2	no	—
188	#380	C₂×D₄₇	1	2	yes	—
189	C₄×C₄⋊C₄	—	47	4	no	—
190	C₄×C₂²⋊C₄	—	94	2	no	—
191	#376	—	94	2	no	—
192	#340	—	47	2	no	—
193	#340	—	47	4	no	—
194	#340	C₂×D₄₇	1	2	yes	—
195	C₄₇	#20818	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	47	1	no	—
197	C₂³×C₄	—	47	1	no	—
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	47	1	no	—
201	C₂²×C₈	—	47	4	no	—
202	C₂²×C₈	—	47	1	no	—
203	C₂²×C₈	—	94	4	no	—
204	C₂²×C₈	C₄×D₄₇	1	4	yes	—
205	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	47	4	no	—
208	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	94	2	no	—
211	C₂×C₄²	—	47	4	no	—
212	C₂×C₄²	—	47	1	no	—
213	C₂×C₄²	—	47	1	no	—
214	C₂×C₄²	—	47	4	no	—
215	C₂×C₄²	—	47	1	no	—
216	C₂×C₄²	—	47	1	no	—
217	C₂×C₄²	—	94	4	no	—
218	#107	—	94	8	no	—
219	#106	—	2	4	no	—
220	#106	—	94	4	no	—
221	C₄×C₈	—	47	4	no	—
222	C₄×C₈	—	47	4	no	—
223	C₄×C₈	—	47	4	no	—
224	C₄×C₈	—	47	4	no	—
225	#95	—	2	2	no	—
226	#95	—	94	4	no	—
227	#95	—	94	4	no	—
228	#95	—	94	4	no	—
229	#95	—	94	2	no	—
230	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	47	1	no	—
235	C₂²×C₄	—	47	4	no	—
236	C₂²×C₄	—	47	4	no	—
237	C₂²×C₄	—	47	1	no	—
238	C₂²×C₄	—	47	1	no	—
239	C₂²×C₄	—	47	1	no	—
240	C₂²×C₄	—	94	4	no	—
241	C₂²×C₄	—	94	4	no	—
242	C₂²×C₄	—	94	4	no	—
243	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
244	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
245	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	8	no	—
249	C₂×C₈	—	47	2	no	—
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	47	2	no	—
252	C₂×C₈	—	94	8	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
256	C₂×C₈	#10625352	1	4	yes	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	94	4	no	—
260	C₄⋊C₄	—	94	4	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	188	4	no	—
263	C₄²	—	47	4	no	—
264	C₄²	—	47	4	no	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	2	no	—
267	C₄²	—	47	2	no	—
268	C₄²	—	47	2	no	—
269	C₄²	—	188	8	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
273	C₂³	#13094045	1	4	yes	—
274	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	47	2	no	—
283	C₂×C₄	—	47	4	no	—
284	C₂×C₄	—	47	2	no	—
285	C₂×C₄	—	47	2	no	—
286	C₂×C₄	—	47	2	no	—
287	C₂×C₄	—	94	8	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	94	2	no	—
291	C₂×C₄	—	94	8	no	—
292	C₂×C₄	—	188	4	no	—
293	C₂×C₄	—	188	4	no	—
294	C₂×C₄	—	188	4	no	—
295	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
298	C₂×C₄	#41×D₄₇	1	2	yes	—
299	C₂×C₄	C₂×#10625352	1	2	yes	—
300	C₈	—	2	2	no	—
301	C₈	—	47	4	no	—
302	C₈	—	94	2	no	—
303	C₈	#13093980	1	4	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	8	no	—
307	C₂²	#15689158	1	4	yes	—
308	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
309	C₂²	#15689668	1	2	yes	—
310	C₂²	#135×D₄₇	1	2	yes	—
311	C₂²	C₂×#13094045	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	8	4	no	—
314	C₄	—	47	4	no	—
315	C₄	—	94	2	no	—
316	C₄	—	94	2	no	—
317	C₄	—	94	2	no	—
318	C₄	—	376	4	no	—
319	C₄	#117×D₄₇	1	1	yes	characteristic
320	C₄	C₂×#13093980	1	1	yes	characteristic
321	C₄	#15689675	1	2	yes	—
322	C₂	—	2	4	no	—
323	C₂	#20245790	1	1	yes	characteristic
324	C₂	#427×D₄₇	1	1	yes	characteristic
325	C₂	C₂×#15689158	1	1	yes	characteristic
326	C₂	#20253951	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1807	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1807	D₄₇
hypocenter	C₄₇	#20818
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1807

Complements

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

N	H ≅ G/N	type
3. #20243549	C₂	semidirect
4. #20243517	C₂	semidirect
7. #20243280	C₂	semidirect
8. #106×Dic₄₇	C₂	semidirect
14. C₂×C₈×Dic₄₇	C₄	semidirect
195. C₄₇	#20818	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
#20245783	2
#20243549	2
#20243517	2
#20243415	2
#20243413	2
#20243280	2
#106×Dic₄₇	2
C₄₇×#1807	2
#20818	47