Subgroups — #26203976

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253967	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253963	C₂	1	2	yes	maximal, complemented
3	#20253814	C₂	1	2	yes	maximal, complemented
4	#20243005	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243002	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242992	C₂	1	2	yes	maximal, complemented
7	#20242981	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242979	C₂	1	2	yes	maximal, complemented
9	#20242977	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094044	—	2	4	no	—
13	#15689673	C₂²	1	2	yes	—
14	#15689528	C₂²	1	1	yes	characteristic
15	#15689526	C₂²	1	2	yes	—
16	#15689505	—	2	4	no	—
17	#15689502	C₂²	1	2	yes	—
18	#15689500	C₂²	1	1	yes	characteristic
19	#15689499	C₂²	1	2	yes	—
20	#15689499	C₂²	1	2	yes	—
21	#15689497	C₂²	1	2	yes	—
22	#15689495	C₂²	1	2	yes	—
23	#15689471	C₂²	1	2	yes	—
24	#15689470	C₂²	1	2	yes	—
25	#15689469	C₂²	1	2	yes	—
26	#15689359	—	2	2	no	—
27	#15689357	—	2	4	no	—
28	C₄×D₃₇₆	—	2	2	no	—
29	C₄×#10625334	—	2	4	no	—
30	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₄₇×#532	C₂²	1	2	yes	—
37	C₄₇×#500	C₂²	1	1	yes	characteristic
38	C₄₇×#493	C₂²	1	1	yes	characteristic
39	C₄₇×#486	C₂²	1	2	yes	—
40	C₄₇×#466	—	2	2	no	—
41	C₄×D₈×C₄₇	—	2	2	no	—
42	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625350	—	2	2	no	—
44	#13094044	—	4	16	no	—
45	C₂×#10625337	—	2	4	no	—
46	C₂×#10625335	—	2	4	no	—
47	C₂×#10625336	—	2	4	no	—
48	C₂×D₃₇₆	—	2	2	no	—
49	C₂×#10625334	—	2	4	no	—
50	C₂×#10625334	—	2	4	no	—
51	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
52	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
53	#13094024	C₂³	1	2	yes	—
54	#13094021	C₂³	1	2	yes	—
55	#13094012	—	2	2	no	—
56	#13093998	—	2	2	no	—
57	#13093998	—	2	2	no	—
58	C₄×D₁₈₈	—	2	2	no	—
59	C₄×D₁₈₈	—	2	2	no	—
60	#13093958	—	2	2	no	—
61	#13093957	—	2	4	no	—
62	#13093946	—	2	4	no	—
63	#13093945	—	2	4	no	—
64	#13093944	—	2	4	no	—
65	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
66	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
67	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×#41×C₄₇	—	2	2	no	—
72	C₂×D₈×C₄₇	—	2	2	no	—
73	C₂²×C₈×C₄₇	D₄	1	2	yes	—
74	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
75	#127×C₄₇	C₂³	1	2	yes	—
76	#124×C₄₇	—	2	2	no	—
77	#121×C₄₇	—	2	2	no	—
78	#121×C₄₇	—	2	2	no	—
79	C₄×D₄×C₄₇	—	2	2	no	—
80	C₄×D₄×C₄₇	—	2	2	no	—
81	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	#107×C₄₇	—	2	2	no	—
83	#102×C₄₇	—	2	4	no	—
84	C₄×C₈×C₄₇	D₄	1	2	yes	—
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	#41×C₄₇	—	4	8	no	—
88	C₂×Q₈×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	2	2	no	—
92	C₂×D₄×C₄₇	—	4	2	no	—
93	C₂²×C₄×C₄₇	—	4	2	no	—
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₂²⋊D₄₇	—	4	2	no	—
98	#10625350	—	4	8	no	—
99	C₂×D₁₈₈	—	2	2	no	—
100	C₂×D₁₈₈	—	2	2	no	—
101	C₂×D₁₈₈	—	2	2	no	—
102	C₂×C₄×D₄₇	—	4	2	no	—
103	C₂×Dic₉₄	—	2	2	no	—
104	D₈×C₄₇	—	4	4	no	—
105	C₂×C₈×C₄₇	—	2	4	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₂²⋊C₄×C₄₇	—	4	2	no	—
111	C₂²⋊C₄×C₄₇	—	4	2	no	—
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
116	#10625337	—	4	8	no	—
117	#10625336	—	4	8	no	—
118	#10625335	—	4	8	no	—
119	#10625334	—	4	8	no	—
120	#10625334	—	4	8	no	—
121	(C₂×D₄₇)⋊C₄	—	4	2	no	—
122	(C₂×D₄₇)⋊C₄	—	4	2	no	—
123	C₄⋊Dic₄₇	—	2	2	no	—
124	C₄⋊Dic₄₇	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	4	no	—
126	C₂×C₄₇⋊C₈	D₈	1	8	yes	complemented
127	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
128	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
129	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
130	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
131	D₃₇₆	—	4	4	no	—
132	C₂³×C₄₇	—	4	2	no	—
133	C₂³×C₄₇	#127	1	1	yes	characteristic
134	C₂²×D₄₇	—	4	2	no	—
135	Q₈×C₄₇	—	4	4	no	—
136	D₄×C₄₇	—	4	4	no	—
137	D₄×C₄₇	—	4	4	no	—
138	D₄×C₄₇	—	4	4	no	—
139	D₄×C₄₇	—	8	4	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	8	4	no	—
144	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₄₇	—	8	4	no	—
152	C₂×Dic₄₇	—	4	2	no	—
153	D₁₈₈	—	4	4	no	—
154	D₁₈₈	—	4	4	no	—
155	D₁₈₈	—	4	4	no	—
156	C₄×D₄₇	—	8	4	no	—
157	Dic₉₄	—	4	4	no	—
158	C₈×C₄₇	—	2	2	no	—
159	C₈×C₄₇	#135	1	4	yes	complemented
160	C₄₇⋊C₈	—	2	2	no	—
161	C₄₇⋊C₈	C₂×D₈	1	4	yes	—
162	C₄₇⋊C₈	C₂×D₈	1	4	yes	—
163	C₄₇⋊C₈	#135	1	4	yes	complemented
164	#19603	—	47	1	no	maximal, Sylow
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	—	2	1	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	—	8	2	no	—
169	C₂²×C₄₇	—	8	2	no	—
170	C₂²×C₄₇	#493	1	2	yes	—
171	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
172	C₂×D₄₇	—	8	2	no	—
173	C₂×D₄₇	—	8	2	no	—
174	C₂×D₄₇	—	8	2	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	8	2	no	—
178	C₄×C₄₇	#493	1	2	yes	—
179	C₄×C₄₇	C₂²×D₈	1	2	yes	—
180	C₄×C₄₇	C₂×#135	1	2	yes	—
181	C₄×C₄₇	C₂×#135	1	2	yes	—
182	Dic₄₇	—	8	2	no	—
183	#3169	—	47	1	no	—
184	#3169	—	47	2	no	—
185	#3169	D₄₇	1	1	yes	characteristic, complemented
186	#3147	—	47	2	no	—
187	#1713	—	47	1	no	—
188	#1712	—	47	1	no	—
189	#1709	—	47	2	no	—
190	#1706	—	47	1	no	—
191	#1706	—	47	1	no	—
192	#1706	—	47	2	no	—
193	C₂×C₈²	—	47	1	no	—
194	C₂×C₄₇	—	2	2	no	—
195	C₂×C₄₇	—	16	2	no	—
196	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
197	C₂×C₄₇	#3147	1	1	yes	characteristic
198	C₂×C₄₇	#3147	1	1	yes	characteristic
199	D₄₇	—	16	2	no	—
200	C₂×#135	—	94	4	no	—
201	#532	—	47	2	no	—
202	#532	C₂×D₄₇	1	2	yes	—
203	#500	—	47	2	no	—
204	#500	—	47	1	no	—
205	#500	C₂×D₄₇	1	1	yes	characteristic
206	#495	—	94	4	no	—
207	#494	—	47	2	no	—
208	#493	—	47	2	no	—
209	#493	—	47	2	no	—
210	#493	—	47	1	no	—
211	#493	C₂×D₄₇	1	1	yes	characteristic
212	#492	—	47	2	no	—
213	#492	—	47	2	no	—
214	#486	—	47	2	no	—
215	#486	—	47	2	no	—
216	#486	—	47	2	no	—
217	#486	C₂×D₄₇	1	2	yes	—
218	#466	—	2	2	no	—
219	#466	—	94	4	no	—
220	#466	—	94	2	no	—
221	C₄×D₈	—	2	2	no	—
222	C₄×D₈	—	94	4	no	—
223	C₄×D₈	—	94	2	no	—
224	C₂×C₄×C₈	—	47	1	no	—
225	C₂×C₄×C₈	—	47	1	no	—
226	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
227	C₈²	—	47	1	no	—
228	C₈²	—	47	1	no	—
229	C₈²	—	47	1	no	—
230	C₈²	—	47	1	no	—
231	C₄₇	#19603	1	1	yes	characteristic, complemented, Sylow
232	C₂×#41	—	2	2	no	—
233	C₂×#41	—	94	2	no	—
234	#135	—	188	16	no	—
235	C₂×Q₁₆	—	94	4	no	—
236	C₂×SD₁₆	—	94	4	no	—
237	C₂×SD₁₆	—	94	4	no	—
238	C₂×D₈	—	2	2	no	—
239	C₂×D₈	—	94	4	no	—
240	C₂×D₈	—	94	4	no	—
241	C₂×D₈	—	94	2	no	—
242	C₂²×C₈	—	47	2	no	—
243	C₂²×C₈	—	47	2	no	—
244	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
245	C₄⋊Q₈	—	47	2	no	—
246	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
247	#127	—	47	2	no	—
248	#127	C₂²×D₄₇	1	2	yes	—
249	#124	—	2	2	no	—
250	#124	—	94	2	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	#121	—	94	2	no	—
254	#121	—	94	2	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	2	2	no	—
257	C₄×D₄	—	94	2	no	—
258	C₄×D₄	—	94	2	no	—
259	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
260	#107	—	2	2	no	—
261	#107	—	94	4	no	—
262	#107	—	94	2	no	—
263	#102	—	2	4	no	—
264	#102	—	94	4	no	—
265	#102	—	94	4	no	—
266	#102	—	94	4	no	—
267	C₄×C₈	—	47	2	no	—
268	C₄×C₈	—	47	2	no	—
269	C₄×C₈	—	47	1	no	—
270	C₄×C₈	—	47	1	no	—
271	C₄×C₈	—	47	1	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
274	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
276	#41	—	4	8	no	—
277	#41	—	188	8	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	94	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	94	2	no	—
285	C₂×D₄	—	94	2	no	—
286	C₂×D₄	—	94	2	no	—
287	C₂×D₄	—	188	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	—	188	2	no	—
290	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
293	Q₁₆	—	188	8	no	—
294	SD₁₆	—	188	8	no	—
295	SD₁₆	—	188	8	no	—
296	D₈	—	4	4	no	—
297	D₈	—	188	8	no	—
298	D₈	—	188	8	no	—
299	D₈	—	188	4	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	47	8	no	—
302	C₂×C₈	—	47	2	no	—
303	C₂×C₈	—	47	2	no	—
304	C₂×C₈	—	47	2	no	—
305	C₂×C₈	—	47	2	no	—
306	C₂×C₈	—	94	4	no	—
307	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
308	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	94	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	—	188	2	no	—
316	C₂²⋊C₄	—	188	2	no	—
317	C₄²	D₄×D₄₇	1	1	yes	characteristic
318	C₄²	D₄×D₄₇	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
320	C₄²	C₂³×D₄₇	1	1	yes	characteristic
321	C₂³	—	4	2	no	—
322	C₂³	—	188	2	no	—
323	C₂³	#13094020	1	1	yes	characteristic
324	Q₈	—	4	4	no	—
325	Q₈	—	188	4	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	4	4	no	—
329	D₄	—	8	4	no	—
330	D₄	—	376	4	no	—
331	D₄	—	188	4	no	—
332	D₄	—	188	4	no	—
333	D₄	—	188	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	4	no	—
338	C₂×C₄	—	376	4	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	#13094020	1	1	yes	characteristic
341	C₂×C₄	#13094020	1	1	yes	characteristic
342	C₂×C₄	#13094020	1	1	yes	characteristic
343	C₂×C₄	D₈×D₄₇	1	4	yes	—
344	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
345	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	47	4	no	—
349	C₈	—	47	4	no	—
350	C₈	—	47	4	no	—
351	C₈	—	94	2	no	—
352	C₈	#13094044	1	4	yes	complemented
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	376	2	no	—
359	C₂²	—	376	2	no	—
360	C₂²	—	376	2	no	—
361	C₂²	#15689498	1	2	yes	—
362	C₂²	C₂×#13094020	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	376	2	no	—
367	C₄	#15689498	1	2	yes	—
368	C₄	C₂×D₈×D₄₇	1	2	yes	—
369	C₄	#135×D₄₇	1	2	yes	—
370	C₄	C₂×#13094044	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	752	2	no	—
374	C₂	C₂×#15689498	1	1	yes	characteristic
375	C₂	#20253812	1	1	yes	characteristic
376	C₂	#20253814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3169	C₂
socle	C₂²×C₄₇	C₂×#127
hypercenter	#3169	D₄₇
hypocenter	C₄₇	#19603
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20253967	C₂	semidirect
2. #20253963	C₂	semidirect
3. #20253814	C₂	semidirect
4. #20243005	C₂	semidirect
5. #20243002	C₂	semidirect
6. #20242992	C₂	semidirect
7. #20242981	C₂	semidirect
8. #20242979	C₂	semidirect
9. #20242977	C₂	semidirect
10. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
11. C₄₇×#3169	C₂	semidirect
126. C₂×C₄₇⋊C₈	D₈	semidirect
159. C₈×C₄₇	#135	semidirect
163. C₄₇⋊C₈	#135	semidirect
185. #3169	D₄₇	semidirect
231. C₄₇	#19603	semidirect
352. C₈	#13094044	semidirect

Maximal subgroups

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

subgroup	index
#20253967	2
#20253963	2
#20253814	2
#20243005	2
#20243002	2
#20242992	2
#20242981	2
#20242979	2
#20242977	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#3169	2
#19603	47