Subgroups — #26131325

Order 12032.

490 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	#20251810	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688908	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688906	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094032	C₄	1	1	yes	characteristic
9	C₂×#13094030	C₂²	1	1	yes	characteristic
10	#15689591	C₄	1	1	yes	characteristic
11	C₂×#13093964	C₂²	1	1	yes	characteristic
12	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#15689080	—	2	1	no	—
14	#15689079	C₄	1	1	yes	characteristic
15	#15689077	—	2	1	no	—
16	#15689075	—	2	1	no	—
17	M₁₆×Dic₄₇	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₄₇	—	2	1	no	—
20	#15688908	C₄	1	4	yes	complemented
21	#15688906	C₄	1	4	yes	complemented
22	#15688898	—	2	1	no	—
23	C₁₆×Dic₄₇	—	2	1	no	—
24	C₁₆×Dic₄₇	—	2	1	no	—
25	C₁₆×Dic₄₇	—	2	1	no	—
26	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#405	C₂²	1	1	yes	characteristic
29	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
30	#13094032	—	2	2	no	—
31	#13094032	C₈	1	4	yes	—
32	#13094030	—	2	1	no	—
33	#13094030	—	2	1	no	—
34	#13094030	C₂×C₄	1	2	yes	—
35	#13094030	C₂×C₄	1	2	yes	—
36	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
37	C₂×#10625324	—	2	1	no	—
38	C₂×#10625324	—	2	1	no	—
39	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₄₇	—	2	1	no	—
41	C₂×C₈×D₄₇	—	2	1	no	—
42	#13093989	C₈	1	2	yes	—
43	#13093986	C₈	1	2	yes	—
44	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
45	#13093981	—	2	1	no	—
46	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₄₇	—	2	1	no	—
48	#13093964	—	2	1	no	—
49	#13093964	C₂×C₄	1	2	yes	complemented
50	C₂×C₄₇⋊C₁₆	—	2	1	no	—
51	C₂×C₄₇⋊C₁₆	—	2	1	no	—
52	C₂×C₄₇⋊C₁₆	—	2	1	no	—
53	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
54	#13093935	—	2	1	no	—
55	#13093934	—	2	1	no	—
56	C₄×C₄₇⋊C₈	—	2	1	no	—
57	C₈×Dic₄₇	—	2	1	no	—
58	C₈×Dic₄₇	D₄	1	1	yes	characteristic
59	C₈×Dic₄₇	D₄	1	1	yes	characteristic
60	C₈×Dic₄₇	D₄	1	1	yes	characteristic
61	C₈×Dic₄₇	D₄	1	1	yes	characteristic
62	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₄₇	—	2	1	no	—
66	M₃₂×C₄₇	C₂×C₄	1	2	yes	complemented
67	C₂×C₁₆×C₄₇	—	2	1	no	—
68	C₂×C₁₆×C₄₇	—	2	1	no	—
69	C₂×C₁₆×C₄₇	—	2	1	no	—
70	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	complemented
71	#97×C₄₇	—	2	1	no	—
72	C₄×C₈×C₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₇	—	2	1	no	—
75	C₂²×Dic₄₇	M₁₆	1	2	yes	—
76	#10625350	—	2	1	no	—
77	#10625350	—	2	1	no	—
78	#10625350	C₄²	1	2	yes	—
79	#10625350	C₂×C₈	1	2	yes	—
80	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₄₇	—	2	1	no	—
82	C₂×C₄×D₄₇	—	2	2	no	—
83	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
84	M₁₆×C₄₇	—	2	1	no	—
85	M₁₆×C₄₇	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₄₇	—	2	1	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
92	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
94	C₂²⋊C₄×C₄₇	—	2	1	no	—
95	C₄²×C₄₇	—	2	1	no	—
96	(C₂×D₄₇)⋊C₄	—	2	2	no	—
97	Dic₄₇⋊C₄	—	2	2	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
102	#10625329	—	2	1	no	—
103	#10625329	C₄²	1	2	yes	—
104	C₂×C₄₇⋊C₈	—	2	1	no	—
105	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
108	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
109	#10625324	—	2	2	no	—
110	#10625324	—	2	2	no	—
111	#10625324	—	2	4	no	—
112	C₈×D₄₇	—	2	2	no	—
113	C₈×D₄₇	—	2	2	no	—
114	C₈×D₄₇	—	2	4	no	—
115	C₁₆×C₄₇	—	2	2	no	—
116	C₁₆×C₄₇	—	2	2	no	—
117	C₄₇⋊C₁₆	—	2	2	no	—
118	C₄₇⋊C₁₆	—	2	2	no	—
119	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂²×D₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂²⋊D₄₇	—	2	2	no	—
129	C₂²⋊D₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	2	1	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
133	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
134	D₁₈₈	—	2	1	no	—
135	D₁₈₈	C₄×C₈	1	2	yes	—
136	C₄×D₄₇	—	2	2	no	—
137	C₄×D₄₇	—	2	2	no	—
138	C₄×D₄₇	—	4	4	no	—
139	Dic₉₄	—	2	1	no	—
140	Dic₉₄	C₄×C₈	1	2	yes	—
141	C₈×C₄₇	—	2	2	no	—
142	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
143	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
144	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
145	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
146	C₄₇⋊C₈	—	2	2	no	—
147	C₄₇⋊C₈	C₂²⋊C₈	1	2	yes	—
148	C₄₇⋊C₈	C₂²⋊C₈	1	2	yes	—
149	C₄×#350	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₂²×C₄₇	#431	1	1	yes	characteristic
154	C₂×D₄₇	—	2	2	no	—
155	C₂×D₄₇	—	4	2	no	—
156	C₄×C₄₇	—	4	2	no	—
157	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
159	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₄₇	#431	1	1	yes	characteristic
161	Dic₄₇	—	2	2	no	—
162	Dic₄₇	—	2	2	no	—
163	Dic₄₇	#350	1	4	yes	complemented
164	C₄×#131	—	47	1	no	—
165	C₂×#350	—	47	1	no	—
166	C₂×#350	—	47	1	no	—
167	C₄×M₃₂	—	47	1	no	—
168	C₂×C₄×C₁₆	—	47	1	no	—
169	#1379	—	47	1	no	—
170	#1379	D₄₇	1	1	yes	characteristic, complemented
171	C₂×C₄₇	—	2	1	no	—
172	C₂×C₄₇	—	2	1	no	—
173	C₂×C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#350	1	2	yes	—
175	D₄₇	—	4	2	no	—
176	C₂×#131	—	47	1	no	—
177	C₂×#131	—	47	1	no	—
178	C₄×#41	—	47	1	no	—
179	C₂×M₃₂	—	47	1	no	—
180	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₁₆	—	47	1	no	—
182	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
183	#405	—	47	1	no	—
184	#405	—	94	1	no	—
185	#405	—	94	1	no	—
186	#405	C₂×D₄₇	1	1	yes	characteristic
187	C₄×M₁₆	—	47	1	no	—
188	C₄×M₁₆	—	94	1	no	—
189	C₂×C₄×C₈	—	47	1	no	—
190	C₂×C₄×C₈	—	94	1	no	—
191	#350	—	47	4	no	—
192	#350	—	47	4	no	—
193	#346	—	94	1	no	—
194	C₄×C₁₆	—	94	1	no	—
195	C₄×C₁₆	—	94	1	no	—
196	C₄×C₁₆	—	94	1	no	—
197	C₄₇	C₄×#350	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	47	1	no	—
199	#131	—	47	4	no	—
200	#131	—	47	2	no	—
201	#131	—	47	2	no	—
202	#131	—	94	2	no	—
203	#131	—	94	1	no	—
204	#131	—	94	1	no	—
205	C₂×M₁₆	—	47	1	no	—
206	C₂×M₁₆	—	94	1	no	—
207	C₂×M₁₆	—	94	1	no	—
208	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	94	1	no	—
211	C₂²×C₈	—	94	1	no	—
212	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	C₄×Q₈	—	47	2	no	—
214	C₄×D₄	—	47	2	no	—
215	C₄×D₄	—	94	2	no	—
216	#117	—	94	1	no	—
217	#117	C₄×D₄₇	1	1	yes	characteristic
218	C₂×C₄²	—	47	1	no	—
219	C₂×C₄²	—	94	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	47	2	no	—
222	M₃₂	—	94	1	no	—
223	M₃₂	C₄×D₄₇	1	2	yes	complemented
224	C₂×C₁₆	—	2	1	no	—
225	C₂×C₁₆	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	47	2	no	—
228	C₂×C₁₆	—	94	1	no	—
229	C₂×C₁₆	—	94	1	no	—
230	C₂×C₁₆	—	94	1	no	—
231	C₂×C₁₆	C₄×D₄₇	1	2	yes	complemented
232	#97	—	2	1	no	—
233	#97	—	94	1	no	—
234	#97	—	94	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	47	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	C₄×C₈	—	47	1	no	—
240	C₄×C₈	—	94	1	no	—
241	C₄×C₈	—	94	1	no	—
242	#41	—	47	2	no	—
243	#41	—	47	2	no	—
244	#41	—	94	1	no	—
245	#41	—	94	1	no	—
246	C₂×Q₈	—	47	1	no	—
247	C₂×D₄	—	47	1	no	—
248	C₂×D₄	—	94	1	no	—
249	C₂²×C₄	—	47	2	no	—
250	C₂²×C₄	—	94	2	no	—
251	C₂²×C₄	—	94	1	no	—
252	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	47	2	no	—
255	M₁₆	—	94	4	no	—
256	M₁₆	—	94	1	no	—
257	M₁₆	—	94	2	no	—
258	M₁₆	—	94	2	no	—
259	M₁₆	C₈×D₄₇	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	47	1	no	—
262	C₂×C₈	—	47	1	no	—
263	C₂×C₈	—	47	2	no	—
264	C₂×C₈	—	47	2	no	—
265	C₂×C₈	—	94	4	no	—
266	C₂×C₈	—	94	1	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
272	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₈	#10625352	1	1	yes	characteristic
274	C₂×C₈	#10625354	1	1	yes	characteristic
275	C₄⋊C₄	—	94	2	no	—
276	C₄⋊C₄	C₈×D₄₇	1	2	yes	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	94	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	47	1	no	—
281	C₄²	—	47	1	no	—
282	C₄²	—	47	2	no	—
283	C₄²	—	94	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	94	2	no	—
287	C₁₆	—	94	2	no	—
288	C₂³	—	94	1	no	—
289	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
290	Q₈	—	47	2	no	—
291	Q₈	—	94	1	no	—
292	D₄	—	47	2	no	—
293	D₄	—	94	2	no	—
294	D₄	—	94	2	no	—
295	D₄	—	94	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	47	2	no	—
298	C₂×C₄	—	47	2	no	—
299	C₂×C₄	—	94	4	no	—
300	C₂×C₄	—	94	1	no	—
301	C₂×C₄	—	94	2	no	—
302	C₂×C₄	—	94	2	no	—
303	C₂×C₄	—	188	4	no	—
304	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	#13093982	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	47	2	no	—
312	C₈	—	47	2	no	—
313	C₈	—	94	2	no	—
314	C₈	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
315	C₈	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
316	C₈	#13093986	1	1	yes	characteristic
317	C₈	#13093986	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	94	2	no	—
320	C₂²	—	188	2	no	—
321	C₂²	#15689075	1	1	yes	characteristic
322	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
323	C₂²	#15689175	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	47	4	no	—
326	C₄	—	94	2	no	—
327	C₄	—	94	2	no	—
328	C₄	#15688970	1	1	yes	characteristic
329	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
330	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
331	C₄	#15689175	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	188	2	no	—
335	C₂	#20243145	1	1	yes	characteristic
336	C₂	#350×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1379	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1379	D₄₇
hypocenter	C₄₇	C₄×#350
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₄₇	C₂	semidirect
5. C₂×C₁₆×Dic₄₇	C₂	semidirect
6. #20241477	C₂	semidirect
7. C₄₇×#1379	C₂	semidirect
20. #15688908	C₄	semidirect
21. #15688906	C₄	semidirect
49. #13093964	C₂×C₄	semidirect
53. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
66. M₃₂×C₄₇	C₂×C₄	semidirect
70. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
163. Dic₄₇	#350	semidirect
170. #1379	D₄₇	semidirect
197. C₄₇	C₄×#350	semidirect
223. M₃₂	C₄×D₄₇	semidirect
231. C₂×C₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20251810	2
C₂×#15688908	2
C₂×#15688906	2
M₃₂×Dic₄₇	2
C₂×C₁₆×Dic₄₇	2
#20241477	2
C₄₇×#1379	2
C₄×#350	47