Subgroups — #26371733

Order 12032.

794 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689565	C₂	1	2	yes	maximal, complemented
2	C₂×#15689556	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₃₂×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253876	C₂	1	2	yes	maximal, complemented
5	#20245657	C₂	1	2	yes	maximal, complemented
6	#20245656	C₂	1	2	yes	maximal, complemented
7	#20245650	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245649	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245639	C₂	1	1	yes	maximal, characteristic, complemented
10	#20244983	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094056	—	2	2	no	—
13	C₂×#13094047	C₂²	1	2	yes	—
14	C₂×Q₁₆×D₄₇	C₂²	1	2	yes	—
15	#15689698	—	2	2	no	—
16	#15689565	—	2	8	no	—
17	#15689556	—	2	4	no	—
18	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
19	C₂×#13093970	C₂²	1	2	yes	—
20	C₂×#13093970	C₂²	1	2	yes	—
21	Q₃₂×D₄₇	—	2	4	no	—
22	C₂×#13093969	C₂²	1	2	yes	—
23	C₂×#13093967	C₂²	1	2	yes	—
24	C₂×#13093967	C₂²	1	2	yes	—
25	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
26	C₂×#13093963	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
28	#15689516	—	2	2	no	—
29	#15689515	—	2	2	no	—
30	#15689508	C₂²	1	2	yes	—
31	#15689507	C₂²	1	2	yes	—
32	#15689501	C₂²	1	2	yes	—
33	#15689077	C₂²	1	1	yes	characteristic
34	#15688899	C₂²	1	1	yes	characteristic
35	#15688898	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
37	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
38	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#494	C₂²	1	2	yes	—
41	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×#10625354	—	2	2	no	—
43	C₂×Q₈×D₄₇	—	2	2	no	—
44	#13094056	—	4	8	no	—
45	#13094047	—	2	4	no	—
46	#13094047	—	2	4	no	—
47	C₂×Dic₁₈₈	C₂³	1	2	yes	—
48	C₂×#10625337	—	2	2	no	—
49	C₂×#10625337	—	2	2	no	—
50	Q₁₆×D₄₇	—	2	4	no	—
51	Q₁₆×D₄₇	—	2	4	no	—
52	C₂×#10625325	—	2	2	no	—
53	C₂×#10625336	—	2	2	no	—
54	C₂×#10625336	—	2	2	no	—
55	C₂×D₃₇₆	C₂³	1	2	yes	—
56	C₂×#10625324	D₄	1	1	yes	characteristic
57	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
58	#13094022	—	2	2	no	—
59	#13094013	—	2	2	no	—
60	#13094012	—	2	2	no	—
61	#13094009	—	4	4	no	—
62	#13094002	—	4	4	no	—
63	#13093981	D₄	1	1	yes	characteristic
64	Dic₃₇₆	—	2	2	no	—
65	#13093970	—	2	4	no	—
66	#13093970	—	2	4	no	—
67	#13093969	—	2	4	no	—
68	#13093967	—	2	4	no	—
69	#13093967	—	2	4	no	—
70	D₇₅₂	—	2	2	no	—
71	#13093963	D₄	1	4	yes	—
72	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₁₆×D₄₇	D₄	1	4	yes	—
75	#13093935	D₄	1	1	yes	characteristic
76	#13093934	D₄	1	1	yes	characteristic
77	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₄₇	—	2	2	no	—
79	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
80	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₄₇	—	2	2	no	—
82	Q₃₂×C₄₇	—	2	2	no	—
83	Q₃₂×C₄₇	—	2	2	no	—
84	Q₃₂×C₄₇	—	2	4	no	—
85	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₄₇	—	2	2	no	—
89	C₂×Q₈×C₄₇	—	2	2	no	—
90	#10625354	—	4	4	no	—
91	#10625354	—	4	4	no	—
92	Q₈×D₄₇	—	4	4	no	—
93	Q₈×D₄₇	—	4	4	no	—
94	C₂×D₁₈₈	—	2	2	no	—
95	C₂×D₁₈₈	—	4	2	no	—
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×C₄×D₄₇	—	4	2	no	—
98	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₉₄	—	2	2	no	—
100	C₂×Dic₉₄	—	4	2	no	—
101	Q₁₆×C₄₇	—	2	2	no	—
102	Q₁₆×C₄₇	—	2	2	no	—
103	Q₁₆×C₄₇	—	2	2	no	—
104	Q₁₆×C₄₇	—	2	2	no	—
105	Q₁₆×C₄₇	—	4	4	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₄₇	—	4	4	no	—
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	#10625337	—	4	4	no	—
111	#10625337	—	4	4	no	—
112	#10625336	—	4	4	no	—
113	#10625336	—	4	4	no	—
114	(C₂×D₄₇)⋊C₄	—	4	4	no	—
115	(C₂×D₄₇)⋊C₄	—	4	4	no	—
116	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
117	Dic₄₇⋊C₄	—	4	4	no	—
118	Dic₄₇⋊C₄	D₈	1	2	yes	—
119	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
122	Dic₁₈₈	—	2	2	no	—
123	Dic₁₈₈	—	2	2	no	—
124	D₃₇₆	—	2	2	no	—
125	D₃₇₆	—	2	2	no	—
126	#10625325	—	4	4	no	—
127	#10625324	D₈	1	4	yes	—
128	C₈×D₄₇	C₂×D₄	1	2	yes	—
129	C₈×D₄₇	C₂×D₄	1	2	yes	—
130	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
131	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
132	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
133	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
134	C₂²×D₄₇	—	4	2	no	—
135	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
136	Q₈×C₄₇	—	4	2	no	—
137	Q₈×C₄₇	—	4	2	no	—
138	Q₈×C₄₇	—	4	2	no	—
139	Q₈×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
146	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
147	D₁₈₈	—	4	2	no	—
148	D₁₈₈	—	4	2	no	—
149	D₁₈₈	—	8	4	no	—
150	C₄×D₄₇	—	8	4	no	—
151	C₄×D₄₇	—	8	4	no	—
152	C₄×D₄₇	#127	1	2	yes	—
153	C₄×D₄₇	C₂×D₈	1	2	yes	—
154	Dic₉₄	—	4	2	no	—
155	Dic₉₄	—	4	2	no	—
156	Dic₉₄	—	8	4	no	—
157	C₈×C₄₇	C₂×D₈	1	2	yes	—
158	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
159	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
160	C₄₇⋊C₈	#127	1	2	yes	—
161	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
162	#33721	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
164	C₂×D₄₇	—	2	2	no	—
165	C₂×D₄₇	—	8	2	no	—
166	C₂×D₄₇	—	8	4	no	—
167	C₂×D₄₇	#493	1	2	yes	—
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	8	2	no	—
170	C₄×C₄₇	—	8	2	no	—
171	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
172	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
173	Dic₄₇	—	2	2	no	—
174	Dic₄₇	—	8	2	no	—
175	Dic₄₇	#493	1	2	yes	—
176	C₂×#510	—	47	2	no	—
177	C₂×#508	—	47	1	no	—
178	C₂²×Q₃₂	—	47	1	no	—
179	#3156	—	47	2	no	—
180	#2251	—	47	2	no	—
181	#2251	—	47	2	no	—
182	#2249	—	47	1	no	—
183	#2249	—	47	1	no	—
184	#2247	—	47	1	no	—
185	#2247	D₄₇	1	1	yes	characteristic, complemented
186	#2108	—	47	1	no	—
187	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
188	C₂×C₄₇	#3417	1	2	yes	—
189	D₄₇	—	2	2	no	—
190	D₄₇	—	16	2	no	—
191	C₂×#137	—	94	2	no	—
192	C₂×#135	—	47	2	no	—
193	C₂²×Q₁₆	—	47	2	no	—
194	#536	—	94	2	no	—
195	#510	—	94	8	no	—
196	#508	—	94	4	no	—
197	C₂×Q₃₂	—	47	2	no	—
198	C₂×Q₃₂	—	47	2	no	—
199	C₂×Q₃₂	—	47	1	no	—
200	C₂×Q₃₂	—	94	4	no	—
201	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
202	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
203	C₂×Q₃₂	C₂×D₄₇	1	2	yes	—
204	C₂×SD₃₂	—	47	2	no	—
205	C₂×SD₃₂	—	47	2	no	—
206	C₂×SD₃₂	—	47	2	no	—
207	C₂×D₁₆	—	47	1	no	—
208	C₂×M₃₂	—	47	1	no	—
209	C₂²×C₁₆	—	47	1	no	—
210	#497	—	94	2	no	—
211	#497	—	94	2	no	—
212	#495	—	47	2	no	—
213	#495	—	47	2	no	—
214	#494	—	47	2	no	—
215	#494	C₂×D₄₇	1	2	yes	—
216	#405	—	47	1	no	—
217	#346	—	47	1	no	—
218	#346	—	47	1	no	—
219	C₄×C₁₆	—	47	1	no	—
220	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
221	C₄₇	#33721	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	94	2	no	—
223	C₂²×Q₈	—	94	2	no	—
224	#137	—	188	8	no	—
225	#135	—	94	4	no	—
226	#135	—	94	4	no	—
227	C₂×Q₁₆	—	2	2	no	—
228	C₂×Q₁₆	—	47	2	no	—
229	C₂×Q₁₆	—	94	4	no	—
230	C₂×Q₁₆	—	94	4	no	—
231	C₂×Q₁₆	—	94	2	no	—
232	C₂×Q₁₆	—	94	2	no	—
233	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
234	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
235	C₂×SD₁₆	—	94	2	no	—
236	C₂×SD₁₆	—	94	2	no	—
237	C₂×SD₁₆	—	94	2	no	—
238	C₂×D₈	—	47	2	no	—
239	C₂×M₁₆	—	47	1	no	—
240	C₂²×C₈	—	47	1	no	—
241	C₄⋊Q₈	—	2	2	no	—
242	C₄⋊Q₈	—	94	2	no	—
243	#124	—	94	2	no	—
244	#124	—	94	2	no	—
245	#123	—	188	4	no	—
246	C₂²⋊Q₈	—	188	4	no	—
247	#117	—	47	1	no	—
248	Q₃₂	—	2	2	no	—
249	Q₃₂	—	2	2	no	—
250	Q₃₂	—	2	4	no	—
251	Q₃₂	—	94	4	no	—
252	Q₃₂	—	94	4	no	—
253	Q₃₂	—	94	2	no	—
254	SD₃₂	—	94	4	no	—
255	SD₃₂	—	94	4	no	—
256	SD₃₂	—	94	4	no	—
257	D₁₆	—	94	2	no	—
258	M₃₂	—	47	4	no	—
259	C₂×C₁₆	—	47	1	no	—
260	C₂×C₁₆	—	47	1	no	—
261	C₂×C₁₆	—	47	4	no	—
262	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
263	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
264	#97	—	47	1	no	—
265	#97	—	47	1	no	—
266	C₄×C₈	—	47	1	no	—
267	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
268	#41	—	188	4	no	—
269	#41	—	188	4	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	94	2	no	—
273	C₂×Q₈	—	188	2	no	—
274	C₂×Q₈	—	188	4	no	—
275	C₂×Q₈	—	188	4	no	—
276	C₂×D₄	—	94	2	no	—
277	C₂×D₄	—	188	2	no	—
278	C₂²×C₄	—	47	1	no	—
279	C₂²×C₄	—	188	2	no	—
280	C₂²×C₄	—	188	2	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	2	2	no	—
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	2	2	no	—
285	Q₁₆	—	4	4	no	—
286	Q₁₆	—	94	2	no	—
287	Q₁₆	—	94	2	no	—
288	Q₁₆	—	188	4	no	—
289	Q₁₆	—	188	4	no	—
290	SD₁₆	—	188	4	no	—
291	SD₁₆	—	188	4	no	—
292	SD₁₆	—	188	4	no	—
293	D₈	—	94	2	no	—
294	D₈	—	94	2	no	—
295	M₁₆	—	47	4	no	—
296	C₂×C₈	—	47	1	no	—
297	C₂×C₈	—	47	1	no	—
298	C₂×C₈	—	47	2	no	—
299	C₂×C₈	—	47	2	no	—
300	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
301	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
302	C₄⋊C₄	—	4	4	no	—
303	C₄⋊C₄	—	47	2	no	—
304	C₄⋊C₄	—	188	4	no	—
305	C₂²⋊C₄	—	47	2	no	—
306	C₂²⋊C₄	—	188	4	no	—
307	C₂²⋊C₄	—	188	4	no	—
308	C₄²	—	47	1	no	—
309	C₄²	D₄×D₄₇	1	1	yes	characteristic
310	C₁₆	—	47	2	no	—
311	C₁₆	—	47	2	no	—
312	C₁₆	D₄×D₄₇	1	2	yes	—
313	C₁₆	D₄×D₄₇	1	2	yes	—
314	C₂³	—	47	1	no	—
315	C₂³	—	188	2	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	4	2	no	—
320	Q₈	—	376	4	no	—
321	Q₈	—	188	2	no	—
322	Q₈	—	188	2	no	—
323	D₄	—	376	4	no	—
324	D₄	—	188	2	no	—
325	D₄	—	188	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	47	1	no	—
329	C₂×C₄	—	47	2	no	—
330	C₂×C₄	—	47	2	no	—
331	C₂×C₄	—	47	2	no	—
332	C₂×C₄	—	376	4	no	—
333	C₂×C₄	—	376	4	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	D₈×D₄₇	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
337	C₈	—	47	2	no	—
338	C₈	—	47	2	no	—
339	C₈	D₈×D₄₇	1	2	yes	—
340	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
341	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
342	C₂²	—	47	2	no	—
343	C₂²	—	94	2	no	—
344	C₂²	—	376	4	no	—
345	C₂²	—	376	2	no	—
346	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	47	2	no	—
351	C₄	—	94	2	no	—
352	C₄	—	376	2	no	—
353	C₄	#127×D₄₇	1	1	yes	characteristic
354	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
355	C₂	—	94	2	no	—
356	C₂	—	752	2	no	—
357	C₂	#493×D₄₇	1	1	yes	characteristic
358	C₂	#20256537	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2247	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689556	C₂
hypercenter	#2247	D₄₇
hypocenter	C₄₇	#33721
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

Complements

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

N	H ≅ G/N	type
1. C₂×#15689565	C₂	semidirect
3. C₂×Q₃₂×D₄₇	C₂	semidirect
4. #20253876	C₂	semidirect
5. #20245657	C₂	semidirect
6. #20245656	C₂	semidirect
7. #20245650	C₂	semidirect
8. #20245649	C₂	semidirect
9. #20245639	C₂	semidirect
10. #20244983	C₂	semidirect
11. C₄₇×#2247	C₂	semidirect
185. #2247	D₄₇	semidirect
221. C₄₇	#33721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689565	2
C₂×#15689556	2
C₂×Q₃₂×D₄₇	2
#20253876	2
#20245657	2
#20245656	2
#20245650	2
#20245649	2
#20245639	2
#20244983	2
C₄₇×#2247	2
#33721	47