Subgroups — #26109667

Order 12032.

588 conjugacy classes of subgroups.

The subgroup lattice has 338 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	#20243456	C₂	1	1	yes	maximal, characteristic
3	#20241267	C₂	1	4	yes	maximal
4	C₄₇×#1434	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	#15688993	C₄	1	1	yes	characteristic, complemented
8	#15688993	C₄	1	1	yes	characteristic, complemented
9	#15688989	—	2	1	no	—
10	#15688983	—	2	1	no	—
11	#15688983	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₄₇	—	2	1	no	—
13	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	complemented
14	#15688878	—	2	1	no	—
15	#15688878	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#95×C₄₇	—	2	1	no	—
18	C₂×#95×C₄₇	—	2	1	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	#13093983	—	4	2	no	—
22	#13093983	—	4	2	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	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₂³×C₄×C₄₇	—	2	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	C₂×C₄²×C₄₇	—	2	4	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	#95×C₄₇	—	2	4	no	—
44	#95×C₄₇	—	2	4	no	—
45	#95×C₄₇	—	4	2	no	—
46	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
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	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	—	4	1	no	—
54	C₂²×C₄×C₄₇	—	4	1	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	D₈	1	2	yes	—
66	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₂²⋊Dic₄₇	—	4	2	no	—
77	C₂²⋊Dic₄₇	—	4	2	no	—
78	C₄⋊Dic₄₇	—	4	2	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	4	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	#123	1	2	yes	—
94	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
95	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂³×C₄₇	#137	1	2	yes	—
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	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	8	2	no	—
118	C₂×C₄×C₄₇	—	8	2	no	—
119	C₂×C₄×C₄₇	—	8	2	no	—
120	C₂×C₄×C₄₇	—	8	2	no	—
121	C₂×C₄×C₄₇	#102	1	2	yes	—
122	C₂×C₄×C₄₇	#102	1	2	yes	—
123	C₂×C₄×C₄₇	#104	1	2	yes	—
124	C₂×C₄×C₄₇	#117	1	2	yes	—
125	C₂×C₄×C₄₇	#123	1	2	yes	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₄₇⋊C₈	—	8	2	no	—
133	#8716	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	#381	1	1	yes	characteristic
143	C₂²×C₄₇	#410	1	1	yes	characteristic
144	C₂²×C₄₇	#410	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
146	C₂²×C₄₇	#419	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
148	C₂²×C₄₇	#421	1	1	yes	characteristic
149	C₂²×C₄₇	#480	1	2	yes	—
150	C₂²×C₄₇	#483	1	2	yes	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	4	1	no	—
157	C₄×C₄₇	—	8	4	no	—
158	Dic₄₇	—	8	2	no	—
159	Dic₄₇	—	8	2	no	—
160	#2285	—	47	1	no	—
161	#1817	—	47	1	no	—
162	#1434	D₄₇	1	1	yes	characteristic
163	#1294	—	47	4	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	2	2	no	—
166	C₂×C₄₇	#1502	1	1	yes	characteristic
167	C₂×C₄₇	#1506	1	1	yes	characteristic
168	C₂×C₄₇	#1515	1	1	yes	characteristic
169	C₂×C₄₇	#1517	1	1	yes	characteristic
170	C₂×C₄₇	#1820	1	1	yes	characteristic
171	C₂×C₄₇	#1823	1	1	yes	characteristic
172	C₂×C₄₇	#1826	1	1	yes	characteristic
173	C₂×#117	—	94	1	no	—
174	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
175	C₂×C₂²⋊C₈	—	47	2	no	—
176	#384	—	47	1	no	—
177	#384	—	47	1	no	—
178	#382	—	94	1	no	—
179	#380	—	47	2	no	—
180	#380	—	94	1	no	—
181	C₄×C₄⋊C₄	—	47	2	no	—
182	C₄×C₄⋊C₄	—	94	1	no	—
183	C₂×#95	—	2	1	no	—
184	C₂×#95	—	2	1	no	—
185	C₂×#95	C₂×D₄₇	1	2	yes	—
186	#336	—	94	1	no	—
187	#336	—	94	1	no	—
188	C₄₇	#8716	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	—	2	2	no	—
190	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²×C₈	—	94	2	no	—
194	#117	—	188	2	no	—
195	#117	—	188	2	no	—
196	C₂×C₄⋊C₄	—	47	2	no	—
197	C₂×C₄⋊C₄	—	47	2	no	—
198	C₂×C₄⋊C₄	—	94	1	no	—
199	C₂×C₄⋊C₄	—	94	1	no	—
200	C₂×C₄⋊C₄	—	94	1	no	—
201	C₂×C₄⋊C₄	—	94	1	no	—
202	C₂×C₂²⋊C₄	—	94	2	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
207	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
208	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
209	C₂²⋊C₈	—	94	4	no	—
210	#95	—	2	4	no	—
211	#95	—	2	4	no	—
212	#95	—	4	2	no	—
213	#95	—	94	2	no	—
214	#95	—	94	2	no	—
215	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	94	2	no	—
231	C₂²×C₄	—	94	2	no	—
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×D₄₇)⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	#10625334	1	2	yes	—
237	C₂²×C₄	#10625336	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	#10625352	1	4	yes	—
241	C₂×C₈	—	188	2	no	—
242	C₂×C₈	—	188	2	no	—
243	C₂×C₈	—	188	2	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄⋊C₄	—	188	2	no	—
247	C₄⋊C₄	—	188	2	no	—
248	C₄⋊C₄	—	188	2	no	—
249	C₄⋊C₄	—	188	2	no	—
250	C₂²⋊C₄	—	188	2	no	—
251	C₂²⋊C₄	—	188	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	188	4	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
262	C₂³	#13094005	1	2	yes	—
263	C₂³	C₂×#10625334	1	1	yes	characteristic
264	C₂³	C₂×#10625336	1	1	yes	characteristic
265	C₂³	#13094054	1	2	yes	—
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	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	2	no	—
296	C₂×C₄	#13093944	1	2	yes	—
297	C₂×C₄	#13093944	1	2	yes	—
298	C₂×C₄	#13093952	1	2	yes	—
299	C₂×C₄	#13093980	1	2	yes	—
300	C₂×C₄	#13094008	1	2	yes	—
301	C₈	—	376	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#15688984	1	1	yes	characteristic
311	C₂²	#15689096	1	1	yes	characteristic
312	C₂²	#15689096	1	1	yes	characteristic
313	C₂²	C₂×#13093944	1	1	yes	characteristic
314	C₂²	#15689131	1	1	yes	characteristic
315	C₂²	C₂×#13093952	1	1	yes	characteristic
316	C₂²	#15689141	1	1	yes	characteristic
317	C₂²	#15689434	1	2	yes	—
318	C₂²	#15689451	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	8	4	no	—
326	C₄	—	376	2	no	—
327	C₄	—	376	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	2	no	—
330	C₂	#20241882	1	1	yes	characteristic
331	C₂	#20241899	1	1	yes	characteristic
332	C₂	#20241938	1	1	yes	characteristic
333	C₂	#20241947	1	1	yes	characteristic
334	C₂	#20243463	1	1	yes	characteristic
335	C₂	#20243472	1	1	yes	characteristic
336	C₂	#20243488	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×#1434	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1434	D₄₇
hypocenter	C₄₇	#8716
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
7. #15688993	C₄	semidirect
8. #15688993	C₄	semidirect
13. C₄×C₄⋊Dic₄₇	C₄	semidirect
188. C₄₇	#8716	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
#20243456	2
#20241267	2
C₄₇×#1434	2
#8716	47