Subgroups — #26369747

Order 12032.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689566	C₂	1	1	yes	maximal, characteristic
2	#20252306	C₂	1	1	yes	maximal, characteristic
3	#20252257	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245281	C₂	1	8	yes	maximal, complemented
5	#20245161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688940	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245125	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	#15689889	—	2	1	no	—
12	#15689600	—	2	1	no	—
13	#15689566	C₄	1	4	yes	complemented
14	#15689566	C₂²	1	4	yes	—
15	C₂×Dic₃₇₆	C₄	1	2	yes	complemented
16	C₂×#13093969	C₄	1	2	yes	complemented
17	#15689246	C₂²	1	4	yes	—
18	#15689241	—	2	2	no	—
19	#15689231	C₂²	1	4	yes	complemented
20	C₄×Dic₁₈₈	—	2	2	no	—
21	C₂×#13093960	C₂²	1	1	yes	characteristic
22	#15689161	C₂²	1	1	yes	characteristic
23	#15689141	—	2	1	no	—
24	#15689133	—	2	1	no	—
25	#15688946	C₂²	1	4	yes	complemented
26	#15688940	C₂²	1	4	yes	complemented
27	#15688934	C₂²	1	4	yes	—
28	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#405	C₂²	1	1	yes	characteristic
31	C₄₇×#347	C₂²	1	4	yes	complemented
32	C₂×#10625350	—	2	1	no	—
33	C₂²×Dic₉₄	—	2	1	no	—
34	#13094046	—	2	1	no	—
35	#13094046	—	2	1	no	—
36	#13094046	C₂×C₄	1	2	yes	—
37	#13094046	C₂³	1	2	yes	—
38	C₂×Dic₁₈₈	—	2	1	no	—
39	C₂×Dic₁₈₈	—	2	1	no	—
40	C₂×Dic₁₈₈	—	2	1	no	—
41	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
42	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
44	C₂×Dic₁₈₈	C₂³	1	2	yes	—
45	C₂×#10625325	—	2	1	no	—
46	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
47	#13094032	—	4	2	no	—
48	#13094030	—	2	4	no	—
49	C₂×#10625329	—	2	1	no	—
50	M₁₆×D₄₇	—	4	2	no	—
51	C₄×Dic₉₄	—	2	4	no	—
52	#13093989	—	4	2	no	—
53	#13093983	—	2	1	no	—
54	#13093980	—	4	2	no	—
55	Dic₃₇₆	C₂×C₄	1	8	yes	—
56	#13093969	C₂×C₄	1	8	yes	—
57	#13093960	—	2	1	no	—
58	#13093960	C₂³	1	2	yes	—
59	#13093958	—	2	1	no	—
60	#13093956	C₂³	1	2	yes	—
61	#13093952	—	2	4	no	—
62	#13093949	—	2	4	no	—
63	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	#117×C₄₇	D₄	1	1	yes	characteristic
66	M₃₂×C₄₇	—	2	1	no	—
67	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
68	M₃₂×C₄₇	C₂³	1	2	yes	—
69	M₃₂×C₄₇	C₂³	1	2	yes	—
70	C₂×C₁₆×C₄₇	—	2	1	no	—
71	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
72	#97×C₄₇	C₂³	1	2	yes	—
73	C₄×C₈×C₄₇	D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₄₇	—	4	1	no	—
76	C₂²×Dic₄₇	—	4	1	no	—
77	#10625350	—	2	2	no	—
78	#10625350	—	2	2	no	—
79	#10625350	—	4	1	no	—
80	#10625350	—	4	1	no	—
81	C₂×D₁₈₈	—	2	1	no	—
82	C₂×C₄×D₄₇	—	4	1	no	—
83	C₂×Dic₉₄	—	2	1	no	—
84	C₂×Dic₉₄	—	2	2	no	—
85	C₂×Dic₉₄	—	2	2	no	—
86	C₂×Dic₉₄	—	2	2	no	—
87	C₂×Dic₉₄	—	4	1	no	—
88	C₂×Dic₉₄	—	4	1	no	—
89	C₂×Dic₉₄	—	4	1	no	—
90	M₁₆×C₄₇	—	2	1	no	—
91	M₁₆×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
97	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₄₇	—	2	1	no	—
100	C₂²⋊C₄×C₄₇	—	2	1	no	—
101	C₄²×C₄₇	C₂×D₄	1	2	yes	—
102	C₂²⋊Dic₄₇	—	4	1	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	Dic₄₇⋊C₄	—	4	4	no	—
105	C₄×Dic₄₇	—	4	4	no	—
106	C₄×Dic₄₇	—	4	1	no	—
107	#10625329	—	2	2	no	—
108	#10625329	—	4	1	no	—
109	C₂×C₄₇⋊C₈	—	4	1	no	—
110	Dic₁₈₈	—	2	2	no	—
111	Dic₁₈₈	—	2	2	no	—
112	Dic₁₈₈	—	2	1	no	—
113	Dic₁₈₈	—	2	2	no	—
114	Dic₁₈₈	C₂²×C₄	1	4	yes	—
115	Dic₁₈₈	C₂²×C₄	1	2	yes	—
116	D₃₇₆	—	2	1	no	—
117	D₃₇₆	C₂²×C₄	1	2	yes	—
118	#10625325	—	2	2	no	—
119	#10625325	—	2	2	no	—
120	#10625324	—	4	4	no	—
121	C₈×D₄₇	—	4	4	no	—
122	C₁₆×C₄₇	—	2	2	no	—
123	C₁₆×C₄₇	C₂²×C₄	1	4	yes	—
124	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₄₇	—	4	2	no	—
134	C₂²⋊D₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	1	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	D₁₈₈	—	2	2	no	—
141	D₁₈₈	—	4	1	no	—
142	C₄×D₄₇	—	4	2	no	—
143	C₄×D₄₇	—	4	2	no	—
144	Dic₉₄	—	2	2	no	—
145	Dic₉₄	—	2	4	no	—
146	Dic₉₄	—	4	2	no	—
147	Dic₉₄	—	4	2	no	—
148	Dic₉₄	—	4	2	no	—
149	Dic₉₄	—	4	1	no	—
150	C₈×C₄₇	—	2	2	no	—
151	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
152	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
153	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₄₇⋊C₈	—	4	2	no	—
156	#33470	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	—	4	1	no	—
158	C₂²×C₄₇	#442	1	1	yes	characteristic
159	C₂²×C₄₇	#442	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×D₄₇	—	4	2	no	—
162	C₂×D₄₇	—	16	1	no	—
163	C₄×C₄₇	—	4	2	no	—
164	C₄×C₄₇	#442	1	1	yes	characteristic
165	C₄×C₄₇	#442	1	1	yes	characteristic
166	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	Dic₄₇	—	4	4	no	—
169	Dic₄₇	—	4	2	no	—
170	Dic₄₇	—	8	2	no	—
171	C₂×#510	—	47	1	no	—
172	#2957	—	47	1	no	—
173	#2950	—	47	1	no	—
174	#2180	—	47	8	no	—
175	#2155	—	47	1	no	—
176	C₂×#362	—	47	1	no	—
177	#2145	—	47	1	no	—
178	#2110	D₄₇	1	1	yes	characteristic, complemented
179	C₂×C₄₇	—	2	1	no	—
180	C₂×C₄₇	—	2	1	no	—
181	C₂×C₄₇	—	2	1	no	—
182	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
183	D₄₇	—	8	2	no	—
184	C₂×#135	—	47	1	no	—
185	C₂²×Q₁₆	—	47	1	no	—
186	#568	—	94	1	no	—
187	#519	—	94	1	no	—
188	#510	—	47	4	no	—
189	#510	—	47	4	no	—
190	C₂×Q₃₂	—	47	2	no	—
191	C₂×SD₃₂	—	47	2	no	—
192	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
193	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
194	#444	—	47	4	no	—
195	#443	—	94	2	no	—
196	#441	—	47	4	no	—
197	C₄×Q₁₆	—	94	2	no	—
198	C₂×#108	—	47	1	no	—
199	#427	—	47	1	no	—
200	#421	—	94	1	no	—
201	#419	—	94	1	no	—
202	#405	C₂×D₄₇	1	1	yes	characteristic
203	#365	—	47	4	no	—
204	#362	—	47	4	no	—
205	#360	—	47	4	no	—
206	#347	C₂×D₄₇	1	4	yes	complemented
207	C₄₇	#33470	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	94	1	no	—
209	C₂²×Q₈	—	94	1	no	—
210	#135	—	47	2	no	—
211	#135	—	47	2	no	—
212	#135	—	94	1	no	—
213	#135	—	94	1	no	—
214	C₂×Q₁₆	—	47	2	no	—
215	C₂×Q₁₆	—	47	2	no	—
216	C₂×Q₁₆	—	47	1	no	—
217	C₂×Q₁₆	—	47	2	no	—
218	C₂×Q₁₆	—	94	1	no	—
219	C₂×Q₁₆	—	94	1	no	—
220	C₂×Q₁₆	—	94	1	no	—
221	C₂×SD₁₆	—	94	1	no	—
222	C₂×D₈	—	47	1	no	—
223	#131	—	94	4	no	—
224	#131	—	188	2	no	—
225	C₂×M₁₆	—	94	1	no	—
226	C₂×M₁₆	—	188	2	no	—
227	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₄×Q₈	—	94	4	no	—
230	C₄×Q₈	—	188	2	no	—
231	#117	—	94	1	no	—
232	#117	—	188	2	no	—
233	#117	D₁₈₈	1	1	yes	characteristic
234	Q₃₂	—	47	8	no	—
235	SD₃₂	—	47	8	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	C₄×D₄₇	1	2	yes	—
238	M₃₂	C₂²×D₄₇	1	2	yes	—
239	M₃₂	C₂²×D₄₇	1	2	yes	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
242	#108	—	47	2	no	—
243	#108	—	94	1	no	—
244	#107	—	94	1	no	—
245	#106	—	47	2	no	—
246	#104	—	94	4	no	—
247	#103	—	94	4	no	—
248	#97	C₂²×D₄₇	1	2	yes	—
249	C₄×C₈	D₁₈₈	1	2	yes	—
250	#41	—	94	2	no	—
251	#41	—	94	2	no	—
252	#41	—	188	1	no	—
253	#41	—	188	1	no	—
254	C₂×Q₈	—	94	1	no	—
255	C₂×Q₈	—	94	2	no	—
256	C₂×Q₈	—	94	2	no	—
257	C₂×Q₈	—	94	2	no	—
258	C₂×Q₈	—	188	1	no	—
259	C₂×Q₈	—	188	1	no	—
260	C₂×Q₈	—	188	1	no	—
261	C₂×D₄	—	94	1	no	—
262	C₂×D₄	—	188	1	no	—
263	C₂²×C₄	—	188	1	no	—
264	C₂²×C₄	—	188	1	no	—
265	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
266	Q₁₆	—	47	4	no	—
267	Q₁₆	—	47	2	no	—
268	Q₁₆	—	94	2	no	—
269	Q₁₆	—	94	2	no	—
270	Q₁₆	—	94	1	no	—
271	Q₁₆	—	94	2	no	—
272	SD₁₆	—	94	2	no	—
273	SD₁₆	—	94	2	no	—
274	D₈	—	47	2	no	—
275	D₈	—	94	1	no	—
276	M₁₆	—	2	1	no	—
277	M₁₆	—	2	1	no	—
278	M₁₆	—	94	2	no	—
279	M₁₆	—	188	4	no	—
280	M₁₆	—	188	1	no	—
281	C₂×C₈	—	188	4	no	—
282	C₂×C₈	—	188	1	no	—
283	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
284	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
285	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
286	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
287	C₂×C₈	#10625350	1	1	yes	characteristic
288	C₂×C₈	#10625350	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	94	2	no	—
292	C₄⋊C₄	—	188	4	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	188	1	no	—
295	C₄²	—	188	4	no	—
296	C₄²	—	188	1	no	—
297	C₄²	C₂×D₁₈₈	1	2	yes	—
298	C₁₆	—	2	2	no	—
299	C₁₆	C₂×C₄×D₄₇	1	4	yes	—
300	C₂³	—	188	1	no	—
301	C₂³	#13094052	1	1	yes	characteristic
302	Q₈	—	94	2	no	—
303	Q₈	—	94	4	no	—
304	Q₈	—	188	2	no	—
305	Q₈	—	188	2	no	—
306	Q₈	—	188	1	no	—
307	Q₈	—	188	2	no	—
308	D₄	—	94	2	no	—
309	D₄	—	188	2	no	—
310	D₄	—	188	2	no	—
311	D₄	—	188	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	188	1	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
321	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
322	C₂×C₄	#13094052	1	1	yes	characteristic
323	C₂×C₄	#13094052	1	1	yes	characteristic
324	C₂×C₄	#13094052	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	188	2	no	—
328	C₈	C₄×D₁₈₈	1	1	yes	characteristic
329	C₈	C₄×D₁₈₈	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
331	C₈	C₂×#10625350	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	188	2	no	—
334	C₂²	—	752	1	no	—
335	C₂²	#15689236	1	1	yes	characteristic
336	C₂²	#15689236	1	1	yes	characteristic
337	C₂²	C₂×#13094052	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	376	2	no	—
340	C₄	—	188	2	no	—
341	C₄	—	188	4	no	—
342	C₄	#15689236	1	1	yes	characteristic
343	C₄	#15689236	1	1	yes	characteristic
344	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
345	C₄	C₂×#13094052	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	376	2	no	—
350	C₂	C₂×#15689236	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#15689236
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2110	C₂
socle	C₂×C₄₇	C₂×#442
prime radical	C₂×#13094046	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2110	D₄₇
hypocenter	C₄₇	#33470
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #20252257	C₂	semidirect
4. #20245281	C₂	semidirect
5. #20245161	C₂	semidirect
6. C₂×#15688940	C₂	semidirect
8. C₄₇×#2110	C₂	semidirect
13. #15689566	C₄	semidirect
15. C₂×Dic₃₇₆	C₄	semidirect
16. C₂×#13093969	C₄	semidirect
19. #15689231	C₂²	semidirect
25. #15688946	C₂²	semidirect
26. #15688940	C₂²	semidirect
31. C₄₇×#347	C₂²	semidirect
178. #2110	D₄₇	semidirect
206. #347	C₂×D₄₇	semidirect
207. C₄₇	#33470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689566	2
#20252306	2
#20252257	2
#20245281	2
#20245161	2
C₂×#15688940	2
#20245125	2
C₄₇×#2110	2
#33470	47