Subgroups — #26204210

Order 12032.

858 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254505	C₂	1	2	yes	maximal, complemented
2	#20253965	C₂	1	1	yes	maximal, characteristic
3	#500×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253842	C₂	1	2	yes	maximal, complemented
5	#20243061	C₂	1	2	yes	maximal, complemented
6	#20243057	C₂	1	2	yes	maximal, complemented
7	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243005	C₂	1	1	yes	maximal, characteristic, complemented
9	#20242977	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625324	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094053	—	2	4	no	—
13	#15689671	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₄₇	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689528	C₂²	1	2	yes	—
18	#15689527	—	2	2	no	—
19	#15689526	C₂²	1	2	yes	—
20	#15689526	C₂²	1	2	yes	—
21	#15689514	—	2	4	no	—
22	#15689503	C₂²	1	2	yes	—
23	#15689502	C₂²	1	2	yes	—
24	#15689495	C₂²	1	2	yes	—
25	#15689469	C₂²	1	2	yes	—
26	#15689469	C₂²	1	2	yes	—
27	#15689435	—	2	4	no	—
28	#15689401	—	2	4	no	—
29	#15689356	—	2	2	no	—
30	D₈×Dic₄₇	—	2	2	no	—
31	#15689174	C₂²	1	1	yes	characteristic
32	#107×D₄₇	—	2	2	no	—
33	C₄×#10625324	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
35	#15688845	C₂²	1	1	yes	characteristic
36	#15688844	C₂²	1	1	yes	characteristic
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄₇×#493	C₂²	1	1	yes	characteristic
40	C₄₇×#492	C₂²	1	2	yes	—
41	C₄₇×#486	C₂²	1	2	yes	—
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625352	—	2	2	no	—
44	C₂×Q₈×D₄₇	—	2	2	no	—
45	#13094053	—	4	16	no	—
46	C₂×Dic₁₈₈	—	2	4	no	—
47	C₂×#10625337	—	2	4	no	—
48	C₂×#10625335	—	2	4	no	—
49	C₂×#10625324	D₄	1	2	yes	—
50	C₂×C₈×D₄₇	D₄	1	2	yes	—
51	#13094024	C₂³	1	2	yes	—
52	#13094024	C₂³	1	2	yes	—
53	#13094023	—	2	2	no	—
54	#13094023	—	2	2	no	—
55	#13094022	—	2	2	no	—
56	#13094011	—	2	2	no	—
57	#13093995	—	2	2	no	—
58	#13093995	—	2	2	no	—
59	D₄×Dic₄₇	—	2	2	no	—
60	D₄×Dic₄₇	—	2	2	no	—
61	C₄⋊C₄×D₄₇	—	2	2	no	—
62	C₄⋊C₄×D₄₇	—	2	2	no	—
63	C₄²×D₄₇	C₂³	1	1	yes	characteristic
64	#13093958	—	2	2	no	—
65	#13093958	—	2	2	no	—
66	#13093958	—	2	4	no	—
67	#13093957	—	2	4	no	—
68	#13093957	—	2	4	no	—
69	Dic₄₇⋊C₈	D₄	1	2	yes	—
70	#13093945	—	2	4	no	—
71	#13093945	—	2	4	no	—
72	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
73	#13093934	D₄	1	2	yes	—
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₇	D₄	1	2	yes	—
78	C₂×SD₁₆×C₄₇	—	2	4	no	—
79	C₂×D₈×C₄₇	—	2	2	no	—
80	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
81	#127×C₄₇	C₂³	1	2	yes	—
82	#107×C₄₇	—	2	2	no	—
83	#102×C₄₇	—	2	4	no	—
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×C₂²⋊D₄₇	—	4	2	no	—
92	C₂²×Dic₄₇	—	4	2	no	—
93	Q₈×D₄₇	—	4	8	no	—
94	#10625352	—	4	8	no	—
95	C₂×C₄×D₄₇	—	4	2	no	—
96	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₉₄	—	2	2	no	—
100	C₂×Dic₉₄	—	2	2	no	—
101	C₂×Dic₉₄	—	4	2	no	—
102	SD₁₆×C₄₇	—	4	8	no	—
103	D₈×C₄₇	—	4	4	no	—
104	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₄₇	—	4	2	no	—
111	C₂²⋊Dic₄₇	—	4	2	no	—
112	#10625337	—	4	8	no	—
113	#10625335	—	4	8	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	Dic₄₇⋊C₄	—	4	2	no	—
119	Dic₄₇⋊C₄	—	4	2	no	—
120	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
124	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
125	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
126	Dic₁₈₈	—	4	8	no	—
127	#10625324	D₈	1	8	yes	complemented
128	C₈×D₄₇	—	2	4	no	—
129	C₂³×C₄₇	—	4	2	no	—
130	C₂²×D₄₇	#127	1	1	yes	characteristic
131	Q₈×C₄₇	—	4	4	no	—
132	D₄×C₄₇	—	4	4	no	—
133	D₄×C₄₇	—	4	4	no	—
134	D₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₇	—	8	4	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	8	4	no	—
143	C₂×Dic₄₇	#127	1	1	yes	characteristic
144	C₂×Dic₄₇	#127	1	1	yes	characteristic
145	C₂×Dic₄₇	#127	1	1	yes	characteristic
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	8	4	no	—
149	C₄×D₄₇	C₂×D₈	1	4	yes	—
150	Dic₉₄	—	4	4	no	—
151	Dic₉₄	—	4	4	no	—
152	Dic₉₄	—	8	4	no	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	C₂×D₈	1	4	yes	—
155	C₈×C₄₇	#137	1	4	yes	complemented
156	C₄₇⋊C₈	—	2	2	no	—
157	C₄₇⋊C₈	—	2	2	no	—
158	C₄₇⋊C₈	C₂×D₈	1	4	yes	—
159	#19630	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	8	2	no	—
161	C₂²×C₄₇	—	8	2	no	—
162	C₂²×C₄₇	—	8	2	no	—
163	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
164	C₂×D₄₇	—	2	1	no	—
165	C₂×D₄₇	—	2	1	no	—
166	C₂×D₄₇	#493	1	2	yes	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	C₂²×D₈	1	2	yes	—
169	C₄×C₄₇	C₂×#137	1	2	yes	—
170	C₄×C₄₇	#578	1	2	yes	—
171	Dic₄₇	—	2	1	no	—
172	Dic₄₇	—	2	1	no	—
173	Dic₄₇	—	8	2	no	—
174	Dic₄₇	—	8	2	no	—
175	Dic₄₇	#493	1	2	yes	—
176	#3226	—	47	2	no	—
177	#3169	—	47	1	no	—
178	C₂×#500	—	47	1	no	—
179	#3152	—	47	2	no	—
180	#1722	—	47	2	no	—
181	#1722	—	47	2	no	—
182	#1713	—	47	1	no	—
183	#1713	—	47	1	no	—
184	#1706	—	47	1	no	—
185	#1706	D₄₇	1	1	yes	characteristic, complemented
186	C₈×M₁₆	—	47	1	no	—
187	C₂×C₄₇	—	16	2	no	—
188	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
189	C₂×C₄₇	#3152	1	1	yes	characteristic
190	C₂×C₄₇	#3156	1	1	yes	characteristic
191	D₄₇	—	2	2	no	—
192	C₂×#137	—	94	4	no	—
193	#532	—	47	2	no	—
194	C₂×C₄⋊Q₈	—	47	2	no	—
195	#500	—	47	2	no	—
196	#500	—	47	2	no	—
197	#500	—	47	1	no	—
198	#500	—	47	1	no	—
199	#500	—	47	2	no	—
200	#500	—	94	2	no	—
201	#500	C₂×D₄₇	1	1	yes	characteristic
202	#497	—	94	4	no	—
203	#494	—	47	2	no	—
204	#494	—	47	2	no	—
205	#493	C₂×D₄₇	1	1	yes	characteristic
206	#492	—	47	2	no	—
207	#492	C₂×D₄₇	1	2	yes	—
208	#486	—	47	2	no	—
209	#486	—	47	2	no	—
210	#486	C₂×D₄₇	1	2	yes	—
211	#480	—	94	4	no	—
212	#474	—	94	4	no	—
213	#466	—	94	2	no	—
214	C₄×D₈	—	94	2	no	—
215	#431	—	47	1	no	—
216	C₂×#107	—	94	2	no	—
217	C₄×M₁₆	—	47	1	no	—
218	C₂×C₄×C₈	—	47	1	no	—
219	#322	—	47	1	no	—
220	#322	—	47	1	no	—
221	C₈²	—	47	1	no	—
222	C₈²	C₂×D₄₇	1	1	yes	characteristic
223	C₄₇	#19630	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	94	2	no	—
225	C₂²×Q₈	—	94	2	no	—
226	#137	—	188	16	no	—
227	C₂×Q₁₆	—	94	4	no	—
228	C₂×Q₁₆	—	94	4	no	—
229	C₂×SD₁₆	—	2	4	no	—
230	C₂×SD₁₆	—	94	4	no	—
231	C₂×D₈	—	2	2	no	—
232	C₂×M₁₆	—	47	2	no	—
233	C₂²×C₈	—	47	2	no	—
234	C₄⋊Q₈	—	47	2	no	—
235	C₄⋊Q₈	—	47	2	no	—
236	C₄⋊Q₈	—	94	2	no	—
237	C₄⋊Q₈	—	94	2	no	—
238	C₄⋊Q₈	—	94	2	no	—
239	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
240	#127	C₂²×D₄₇	1	2	yes	—
241	#124	—	94	2	no	—
242	#121	—	94	2	no	—
243	#121	—	94	2	no	—
244	C₄×D₄	—	94	2	no	—
245	C₄×D₄	—	94	2	no	—
246	C₂×C₄⋊C₄	—	94	2	no	—
247	C₂×C₄⋊C₄	—	94	2	no	—
248	C₂×C₄²	—	47	1	no	—
249	#107	—	2	2	no	—
250	#107	—	94	4	no	—
251	#107	—	94	4	no	—
252	#107	—	94	2	no	—
253	#107	—	94	2	no	—
254	#107	—	94	4	no	—
255	C₄⋊C₈	—	47	2	no	—
256	#102	—	2	4	no	—
257	#102	—	94	4	no	—
258	#102	—	94	4	no	—
259	C₂²⋊C₈	—	47	2	no	—
260	#97	—	47	2	no	—
261	C₄×C₈	—	47	2	no	—
262	C₄×C₈	—	47	1	no	—
263	C₄×C₈	—	47	1	no	—
264	C₄×C₈	—	47	1	no	—
265	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
268	#41	—	188	8	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	94	2	no	—
271	C₂×Q₈	—	94	2	no	—
272	C₂×Q₈	—	188	2	no	—
273	C₂×Q₈	—	188	8	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	188	2	no	—
278	C₂²×C₄	—	47	1	no	—
279	C₂²×C₄	—	47	1	no	—
280	C₂²×C₄	—	47	1	no	—
281	C₂²×C₄	—	188	2	no	—
282	C₂²×C₄	—	188	2	no	—
283	Q₁₆	—	188	8	no	—
284	Q₁₆	—	188	8	no	—
285	SD₁₆	—	4	8	no	—
286	SD₁₆	—	188	8	no	—
287	D₈	—	4	4	no	—
288	M₁₆	—	47	8	no	—
289	C₂×C₈	—	47	2	no	—
290	C₂×C₈	—	47	2	no	—
291	C₂×C₈	—	47	2	no	—
292	C₂×C₈	—	94	4	no	—
293	C₂×C₈	D₄×D₄₇	1	2	yes	—
294	C₂×C₈	D₄×D₄₇	1	2	yes	—
295	C₂×C₈	D₄×D₄₇	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	94	2	no	—
302	C₄⋊C₄	—	188	2	no	—
303	C₄⋊C₄	—	188	2	no	—
304	C₂²⋊C₄	—	188	2	no	—
305	C₂²⋊C₄	—	188	2	no	—
306	C₄²	—	47	1	no	—
307	C₄²	—	47	1	no	—
308	C₄²	—	47	1	no	—
309	C₄²	C₂³×D₄₇	1	1	yes	characteristic
310	C₂³	—	4	2	no	—
311	C₂³	—	47	1	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	376	4	no	—
314	Q₈	—	188	4	no	—
315	Q₈	—	188	4	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	4	4	no	—
319	D₄	—	376	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	47	1	no	—
322	C₂×C₄	—	47	4	no	—
323	C₂×C₄	—	47	1	no	—
324	C₂×C₄	—	47	1	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	376	4	no	—
328	C₂×C₄	—	376	4	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	47	4	no	—
336	C₈	—	94	2	no	—
337	C₈	—	94	2	no	—
338	C₈	D₈×D₄₇	1	4	yes	—
339	C₈	#13094053	1	4	yes	complemented
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	47	2	no	—
344	C₂²	—	94	1	no	—
345	C₂²	—	94	1	no	—
346	C₂²	#127×D₄₇	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	47	2	no	—
349	C₄	—	94	1	no	—
350	C₄	—	94	1	no	—
351	C₄	—	376	2	no	—
352	C₄	—	376	2	no	—
353	C₄	C₂×D₈×D₄₇	1	2	yes	—
354	C₄	C₂×#13094053	1	2	yes	—
355	C₄	#15689937	1	2	yes	—
356	C₂	—	16	2	no	—
357	C₂	—	94	2	no	—
358	C₂	#493×D₄₇	1	1	yes	characteristic
359	C₂	#20253842	1	1	yes	characteristic
360	C₂	#20253878	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1706	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#20253965	C₂
hypercenter	#1706	D₄₇
hypocenter	C₄₇	#19630
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #20254505	C₂	semidirect
3. #500×D₄₇	C₂	semidirect
4. #20253842	C₂	semidirect
5. #20243061	C₂	semidirect
6. #20243057	C₂	semidirect
7. #20243006	C₂	semidirect
8. #20243005	C₂	semidirect
9. #20242977	C₂	semidirect
10. C₈×#10625324	C₂	semidirect
11. C₄₇×#1706	C₂	semidirect
127. #10625324	D₈	semidirect
155. C₈×C₄₇	#137	semidirect
185. #1706	D₄₇	semidirect
223. C₄₇	#19630	semidirect
339. C₈	#13094053	semidirect

Maximal subgroups

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

subgroup	index
#20254505	2
#20253965	2
#500×D₄₇	2
#20253842	2
#20243061	2
#20243057	2
#20243006	2
#20243005	2
#20242977	2
C₈×#10625324	2
C₄₇×#1706	2
#19630	47