Subgroups — #26369293

Order 12032.

636 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256490	C₂	1	1	yes	maximal, characteristic
2	#20251996	C₂	1	1	yes	maximal, characteristic
3	#20251980	C₂	1	1	yes	maximal, characteristic
4	#20245160	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245159	C₂	1	4	yes	maximal, complemented
6	C₂×#15688947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688946	C₂	1	1	yes	maximal, characteristic, complemented
8	#365×D₄₇	C₂	1	4	yes	maximal, complemented
9	C₄₇×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689889	—	2	1	no	—
11	C₂×#13094030	C₂²	1	1	yes	characteristic
12	#15689603	—	2	1	no	—
13	#15689541	C₄	1	4	yes	complemented
14	#15689539	C₄	1	4	yes	complemented
15	C₂×#13093964	C₂²	1	1	yes	characteristic
16	M₃₂×D₄₇	C₂²	1	4	yes	—
17	#15689171	—	2	1	no	—
18	#15689170	—	2	2	no	—
19	C₂×#13093960	C₂²	1	1	yes	characteristic
20	#108×D₄₇	C₂²	1	4	yes	—
21	#15689166	—	2	1	no	—
22	#15689164	—	2	2	no	—
23	#15689158	C₂²	1	2	yes	—
24	C₂×#13093956	C₂²	1	1	yes	characteristic
25	#106×D₄₇	C₂²	1	2	yes	—
26	#15688947	C₂²	1	4	yes	—
27	#15688946	C₂²	1	4	yes	—
28	#15688945	C₂²	1	2	yes	—
29	#15688945	C₂²	1	2	yes	—
30	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#427	C₂²	1	1	yes	characteristic
33	C₄₇×#365	C₂²	1	2	yes	—
34	C₄₇×#365	C₂²	1	2	yes	—
35	C₂×#10625350	D₄	1	1	yes	characteristic
36	#13094032	—	4	2	no	—
37	#13094030	—	4	2	no	—
38	#13094030	C₂×C₄	1	2	yes	—
39	#13094030	C₂×C₄	1	2	yes	—
40	#13094030	Q₈	1	2	yes	—
41	#13094030	Q₈	1	2	yes	—
42	C₂×#10625329	—	2	1	no	—
43	C₂×#10625329	D₄	1	1	yes	characteristic
44	C₂×#10625324	D₄	1	2	yes	—
45	M₁₆×D₄₇	—	2	4	no	—
46	C₂×C₈×D₄₇	C₂³	1	2	yes	—
47	C₄×Dic₉₄	—	4	1	no	—
48	C₄×D₁₈₈	—	4	1	no	—
49	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
50	#13093982	—	2	2	no	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₄⋊C₄×D₄₇	—	2	2	no	—
53	#13093964	C₂×C₄	1	2	yes	complemented
54	#13093964	C₂³	1	2	yes	—
55	#13093963	C₂×C₄	1	8	yes	complemented
56	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
57	C₁₆×D₄₇	C₂×C₄	1	8	yes	complemented
58	#13093960	—	2	1	no	—
59	#13093960	C₂³	1	2	yes	—
60	#13093959	—	2	4	no	—
61	#13093957	—	2	2	no	—
62	#13093956	—	2	1	no	—
63	#13093956	C₂³	1	2	yes	—
64	#13093955	—	2	2	no	—
65	C₂×M₁₆×C₄₇	—	2	1	no	—
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	#117×C₄₇	—	2	1	no	—
68	M₃₂×C₄₇	C₂×C₄	1	2	yes	complemented
69	M₃₂×C₄₇	C₂³	1	2	yes	—
70	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	complemented
71	#108×C₄₇	—	2	1	no	—
72	#108×C₄₇	C₂³	1	2	yes	—
73	#107×C₄₇	—	2	1	no	—
74	#106×C₄₇	C₂³	1	1	yes	characteristic
75	#106×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₄₇	SD₁₆	1	2	yes	—
78	C₂²×Dic₄₇	—	4	1	no	—
79	#10625350	C₄⋊C₄	1	2	yes	—
80	#10625350	C₄⋊C₄	1	2	yes	—
81	#10625350	SD₁₆	1	2	yes	—
82	#10625350	SD₁₆	1	2	yes	—
83	C₂×D₁₈₈	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×D₄₇	—	4	2	no	—
85	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
86	C₂×Dic₉₄	SD₁₆	1	1	yes	characteristic
87	M₁₆×C₄₇	—	2	2	no	—
88	M₁₆×C₄₇	—	4	1	no	—
89	C₂×C₈×C₄₇	—	4	1	no	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₄₇	—	2	1	no	—
97	C₄⋊C₄×C₄₇	—	2	1	no	—
98	C₂²⋊C₄×C₄₇	—	4	1	no	—
99	C₄²×C₄₇	—	4	1	no	—
100	(C₂×D₄₇)⋊C₄	—	4	2	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	4	1	no	—
103	Dic₄₇⋊C₄	—	4	2	no	—
104	C₄×Dic₄₇	—	4	2	no	—
105	#10625329	—	2	2	no	—
106	#10625329	—	4	1	no	—
107	#10625329	C₄⋊C₄	1	2	yes	—
108	#10625329	C₄⋊C₄	1	2	yes	—
109	C₂×C₄₇⋊C₈	—	4	1	no	—
110	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
111	#10625324	—	4	4	no	—
112	#10625324	C₄⋊C₄	1	4	yes	—
113	#10625324	C₄⋊C₄	1	4	yes	—
114	C₈×D₄₇	—	4	4	no	—
115	C₈×D₄₇	C₂²×C₄	1	4	yes	—
116	C₈×D₄₇	C₂×Q₈	1	4	yes	—
117	C₁₆×C₄₇	C₂²×C₄	1	4	yes	—
118	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	—
119	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂²×D₄₇	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₄₇	#106	1	4	yes	—
130	C₂²⋊D₄₇	#106	1	4	yes	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	16	1	no	—
133	C₂×Dic₄₇	C₂×SD₁₆	1	2	yes	—
134	D₁₈₈	#106	1	2	yes	—
135	D₁₈₈	#106	1	2	yes	—
136	C₄×D₄₇	—	16	2	no	—
137	C₄×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
138	C₄×D₄₇	C₂×SD₁₆	1	4	yes	—
139	Dic₉₄	#106	1	2	yes	—
140	Dic₉₄	#106	1	2	yes	—
141	C₈×C₄₇	—	4	2	no	—
142	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
145	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
146	C₄₇⋊C₈	—	4	2	no	—
147	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
148	#33404	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	4	1	no	—
150	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
152	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
153	C₂×D₄₇	—	4	2	no	—
154	C₂×D₄₇	C₂×#106	1	4	yes	—
155	C₄×C₄₇	—	8	1	no	—
156	C₄×C₄₇	—	8	1	no	—
157	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
158	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
159	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	C₂×#106	1	4	yes	—
163	#3407	—	47	1	no	—
164	#2916	—	47	1	no	—
165	#2912	—	47	1	no	—
166	#2155	—	47	1	no	—
167	#2155	—	47	4	no	—
168	#2155	D₄₇	1	1	yes	characteristic, complemented
169	C₂×#365	—	47	1	no	—
170	C₂×#365	—	47	1	no	—
171	C₂×#365	—	47	4	no	—
172	C₂×C₄₇	—	2	1	no	—
173	C₂×C₄₇	—	2	1	no	—
174	C₂×C₄₇	—	2	1	no	—
175	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
176	D₄₇	—	2	4	no	—
177	#568	—	94	1	no	—
178	C₂×#131	—	47	1	no	—
179	#520	—	94	1	no	—
180	#504	—	47	4	no	—
181	#504	—	47	4	no	—
182	C₂×M₃₂	—	47	1	no	—
183	C₂×M₃₂	—	47	4	no	—
184	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
185	#430	—	94	2	no	—
186	#430	—	94	1	no	—
187	C₂×#108	—	47	1	no	—
188	C₂×#108	—	47	4	no	—
189	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
190	#428	—	94	2	no	—
191	#428	—	94	1	no	—
192	#427	—	47	2	no	—
193	#427	C₂×D₄₇	1	1	yes	characteristic
194	C₂×#106	—	47	1	no	—
195	C₂×#106	—	47	2	no	—
196	#365	—	47	4	no	—
197	#365	—	47	4	no	—
198	#365	—	47	2	no	—
199	#365	—	47	2	no	—
200	#365	C₂×D₄₇	1	2	yes	—
201	#365	C₂×D₄₇	1	2	yes	—
202	C₄₇	#33404	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	—	47	1	no	—
204	#131	—	47	2	no	—
205	#131	—	47	2	no	—
206	#131	—	47	2	no	—
207	#131	—	47	2	no	—
208	#131	—	188	2	no	—
209	#131	—	188	2	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	47	1	no	—
212	C₂×M₁₆	—	47	2	no	—
213	C₂×M₁₆	—	94	4	no	—
214	C₂×M₁₆	—	94	1	no	—
215	C₂²×C₈	—	47	2	no	—
216	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
217	C₄×Q₈	—	188	1	no	—
218	C₄×D₄	—	188	1	no	—
219	C₄×D₄	—	188	2	no	—
220	#117	—	2	1	no	—
221	#117	—	94	2	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	—	94	1	no	—
224	M₃₂	—	47	2	no	—
225	M₃₂	—	47	2	no	—
226	M₃₂	—	47	8	no	—
227	M₃₂	C₄×D₄₇	1	2	yes	complemented
228	M₃₂	C₂²×D₄₇	1	2	yes	—
229	C₂×C₁₆	—	47	2	no	—
230	C₂×C₁₆	—	47	8	no	—
231	C₂×C₁₆	C₄×D₄₇	1	2	yes	complemented
232	#108	—	2	1	no	—
233	#108	—	47	2	no	—
234	#108	—	94	4	no	—
235	#108	—	94	1	no	—
236	#108	C₂²×D₄₇	1	2	yes	—
237	#107	—	2	1	no	—
238	#107	—	94	2	no	—
239	#106	—	47	2	no	—
240	#106	—	94	2	no	—
241	#106	—	94	1	no	—
242	#106	C₂²×D₄₇	1	1	yes	characteristic
243	#106	C₂²×D₄₇	1	1	yes	characteristic
244	#41	—	47	2	no	—
245	#41	—	47	2	no	—
246	#41	—	47	2	no	—
247	#41	—	47	2	no	—
248	C₂×Q₈	—	47	1	no	—
249	C₂×D₄	—	47	2	no	—
250	C₂×D₄	—	47	1	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	188	2	no	—
253	C₂²×C₄	—	188	1	no	—
254	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	47	2	no	—
258	M₁₆	—	47	2	no	—
259	M₁₆	—	47	4	no	—
260	M₁₆	—	47	4	no	—
261	M₁₆	—	94	2	no	—
262	M₁₆	—	188	4	no	—
263	M₁₆	—	188	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	47	2	no	—
266	C₂×C₈	—	47	4	no	—
267	C₂×C₈	—	47	4	no	—
268	C₂×C₈	—	188	4	no	—
269	C₂×C₈	—	188	1	no	—
270	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	94	2	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄⋊C₄	—	188	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	188	2	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	188	2	no	—
285	C₁₆	—	47	4	no	—
286	C₁₆	C₂×C₄×D₄₇	1	4	yes	—
287	C₂³	—	47	2	no	—
288	C₂³	SD₁₆×D₄₇	1	1	yes	characteristic
289	Q₈	—	47	2	no	—
290	Q₈	—	47	2	no	—
291	D₄	—	47	4	no	—
292	D₄	—	47	4	no	—
293	D₄	—	47	2	no	—
294	D₄	—	47	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	47	2	no	—
298	C₂×C₄	—	47	4	no	—
299	C₂×C₄	—	47	4	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	752	2	no	—
302	C₂×C₄	—	752	1	no	—
303	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	47	4	no	—
311	C₈	—	188	2	no	—
312	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
313	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
314	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
315	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
316	C₂²	—	4	1	no	—
317	C₂²	—	47	4	no	—
318	C₂²	—	188	2	no	—
319	C₂²	#106×D₄₇	1	1	yes	characteristic
320	C₂²	#106×D₄₇	1	1	yes	characteristic
321	C₂²	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	47	4	no	—
325	C₄	—	376	2	no	—
326	C₄	#106×D₄₇	1	1	yes	characteristic
327	C₄	#106×D₄₇	1	1	yes	characteristic
328	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	94	4	no	—
334	C₂	C₂×#106×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#106×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2155	C₂
socle	C₂×C₄₇	C₂²×#106
prime radical	C₂×#10625350	D₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2155	D₄₇
hypocenter	C₄₇	#33404
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
4. #20245160	C₂	semidirect
5. #20245159	C₂	semidirect
6. C₂×#15688947	C₂	semidirect
7. C₂×#15688946	C₂	semidirect
8. #365×D₄₇	C₂	semidirect
9. C₄₇×#2155	C₂	semidirect
13. #15689541	C₄	semidirect
14. #15689539	C₄	semidirect
53. #13093964	C₂×C₄	semidirect
55. #13093963	C₂×C₄	semidirect
56. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
57. C₁₆×D₄₇	C₂×C₄	semidirect
68. M₃₂×C₄₇	C₂×C₄	semidirect
70. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
168. #2155	D₄₇	semidirect
202. C₄₇	#33404	semidirect
227. 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
#20256490	2
#20251996	2
#20251980	2
#20245160	2
#20245159	2
C₂×#15688947	2
C₂×#15688946	2
#365×D₄₇	2
C₄₇×#2155	2
#33404	47