Subgroups — #26109674

Order 12032.

588 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	1	yes	maximal, characteristic
2	#20241615	C₂	1	1	yes	maximal, characteristic
3	#20241268	C₂	1	4	yes	maximal
4	C₄₇×#1817	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	—	2	1	no	—
6	#15688993	C₄	1	1	yes	characteristic, complemented
7	#15688993	C₄	1	1	yes	characteristic, complemented
8	#15688989	—	2	1	no	—
9	#15688983	—	2	1	no	—
10	#15688983	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₄₇	—	2	1	no	—
12	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	complemented
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	C₂×C₄₇⋊#95	—	2	1	no	—
15	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#336	—	2	1	no	—
19	C₄₇×#336	—	2	1	no	—
20	C₂³×Dic₄₇	—	2	2	no	—
21	#13093983	—	4	4	no	—
22	C₂×C₄⋊Dic₄₇	—	2	1	no	—
23	C₂×C₄⋊Dic₄₇	—	2	1	no	—
24	C₂×C₄⋊Dic₄₇	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₄₇⋊#95	—	2	4	no	—
31	C₄₇⋊#95	—	2	2	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₄₇⋊#95	—	4	2	no	—
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	—	2	4	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂²⋊C₈×C₄₇	—	2	4	no	—
45	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	—	4	1	no	—
52	C₂²×C₄×C₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
58	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	4	yes	—
62	C₂²×Dic₄₇	—	2	4	no	—
63	C₂²×Dic₄₇	—	2	2	no	—
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂²×Dic₄₇	—	4	2	no	—
66	C₂²×Dic₄₇	—	4	2	no	—
67	C₂²×Dic₄₇	—	4	2	no	—
68	C₂²×Dic₄₇	—	4	2	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	4	2	no	—
72	C₂×C₈×C₄₇	—	4	2	no	—
73	C₂×C₈×C₄₇	—	4	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₂²⋊Dic₄₇	—	4	4	no	—
79	C₄⋊Dic₄₇	—	4	2	no	—
80	C₄⋊Dic₄₇	—	4	2	no	—
81	C₄⋊Dic₄₇	—	4	2	no	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	C₄⋊Dic₄₇	—	4	2	no	—
84	C₄⋊Dic₄₇	—	4	2	no	—
85	C₄×Dic₄₇	—	4	4	no	—
86	C₂³×C₄₇	—	2	1	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	#123	1	2	yes	—
92	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	#103	1	2	yes	—
113	C₂×C₄×C₄₇	#103	1	2	yes	—
114	C₂×C₄×C₄₇	#104	1	2	yes	—
115	C₂×C₄×C₄₇	#117	1	2	yes	—
116	C₂×C₄×C₄₇	#123	1	2	yes	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	8	2	no	—
127	C₂×Dic₄₇	—	8	2	no	—
128	C₂×Dic₄₇	—	8	2	no	—
129	C₂×Dic₄₇	—	8	2	no	—
130	C₈×C₄₇	—	8	2	no	—
131	#8717	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	#381	1	1	yes	characteristic
141	C₂²×C₄₇	#410	1	1	yes	characteristic
142	C₂²×C₄₇	#410	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
144	C₂²×C₄₇	#418	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
146	C₂²×C₄₇	#421	1	1	yes	characteristic
147	C₂²×C₄₇	#481	1	2	yes	—
148	C₂²×C₄₇	#484	1	2	yes	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	4	1	no	—
155	Dic₄₇	—	8	4	no	—
156	Dic₄₇	—	8	2	no	—
157	Dic₄₇	—	8	2	no	—
158	#2285	—	47	1	no	—
159	#1817	D₄₇	1	1	yes	characteristic
160	#1434	—	47	1	no	—
161	#1294	—	47	4	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	#1503	1	1	yes	characteristic
165	C₂×C₄₇	#1505	1	1	yes	characteristic
166	C₂×C₄₇	#1514	1	1	yes	characteristic
167	C₂×C₄₇	#1518	1	1	yes	characteristic
168	C₂×C₄₇	#1820	1	1	yes	characteristic
169	C₂×C₄₇	#1824	1	1	yes	characteristic
170	C₂×C₄₇	#1826	1	1	yes	characteristic
171	C₂×#117	—	94	1	no	—
172	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
174	#384	—	47	1	no	—
175	#384	—	47	1	no	—
176	#382	—	94	1	no	—
177	#380	—	47	2	no	—
178	#380	—	94	1	no	—
179	C₄×C₄⋊C₄	—	47	2	no	—
180	C₄×C₄⋊C₄	—	94	1	no	—
181	C₂×#95	—	47	2	no	—
182	C₂×#95	—	94	1	no	—
183	C₂×#95	—	94	1	no	—
184	#336	—	2	1	no	—
185	#336	—	2	1	no	—
186	C₄₇	#8717	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	—	94	2	no	—
188	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
189	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
190	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂²×C₈	—	2	2	no	—
192	#117	—	188	4	no	—
193	C₂×C₄⋊C₄	—	47	2	no	—
194	C₂×C₄⋊C₄	—	47	2	no	—
195	C₂×C₄⋊C₄	—	94	1	no	—
196	C₂×C₄⋊C₄	—	94	1	no	—
197	C₂×C₄⋊C₄	—	94	1	no	—
198	C₂×C₄⋊C₄	—	94	1	no	—
199	C₂×C₂²⋊C₄	—	94	2	no	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂²⋊C₈	—	2	4	no	—
207	#95	—	94	4	no	—
208	#95	—	94	2	no	—
209	#95	—	94	2	no	—
210	#95	—	94	4	no	—
211	#95	—	188	2	no	—
212	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	—	94	2	no	—
222	C₂²×C₄	—	94	2	no	—
223	C₂²×C₄	—	188	2	no	—
224	C₂²×C₄	—	188	2	no	—
225	C₂²×C₄	—	188	2	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	—	188	2	no	—
228	C₂²×C₄	—	188	2	no	—
229	C₂²×C₄	#10625325	1	2	yes	—
230	C₂²×C₄	Dic₁₈₈	1	2	yes	—
231	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	#10625352	1	4	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₄⋊C₄	—	188	2	no	—
242	C₄⋊C₄	—	188	2	no	—
243	C₄⋊C₄	—	188	2	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄⋊C₄	—	188	2	no	—
247	C₂²⋊C₄	—	188	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	188	4	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
258	C₂³	#13094008	1	2	yes	—
259	C₂³	C₂×#10625325	1	1	yes	characteristic
260	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
261	C₂³	#13094052	1	2	yes	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	376	2	no	—
280	C₂×C₄	—	376	2	no	—
281	C₂×C₄	—	376	2	no	—
282	C₂×C₄	—	376	2	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	#13093949	1	2	yes	—
293	C₂×C₄	#13093949	1	2	yes	—
294	C₂×C₄	#13093950	1	2	yes	—
295	C₂×C₄	#13093980	1	2	yes	—
296	C₂×C₄	#13094005	1	2	yes	—
297	C₈	—	8	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	#15688984	1	1	yes	characteristic
307	C₂²	#15689096	1	1	yes	characteristic
308	C₂²	#15689096	1	1	yes	characteristic
309	C₂²	C₂×#13093949	1	1	yes	characteristic
310	C₂²	#15689129	1	1	yes	characteristic
311	C₂²	C₂×#13093950	1	1	yes	characteristic
312	C₂²	#15689139	1	1	yes	characteristic
313	C₂²	#15689442	1	2	yes	—
314	C₂²	#15689459	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	376	4	no	—
322	C₄	—	376	2	no	—
323	C₄	—	376	2	no	—
324	C₂	—	2	2	no	—
325	C₂	—	2	2	no	—
326	C₂	#20241888	1	1	yes	characteristic
327	C₂	#20241897	1	1	yes	characteristic
328	C₂	#20241936	1	1	yes	characteristic
329	C₂	#20241953	1	1	yes	characteristic
330	C₂	#20243463	1	1	yes	characteristic
331	C₂	#20243479	1	1	yes	characteristic
332	C₂	#20243485	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1817	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1817	D₄₇
hypocenter	C₄₇	#8717
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

Complements

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

N	H ≅ G/N	type
6. #15688993	C₄	semidirect
7. #15688993	C₄	semidirect
12. C₄×C₄⋊Dic₄₇	C₄	semidirect
186. C₄₇	#8717	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
#20241615	2
#20241268	2
C₄₇×#1817	2
#8717	47