Subgroups — #26119077

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689497	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689471	C₂	1	1	yes	maximal, characteristic
3	#20244350	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244350	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244194	C₂	1	1	yes	maximal, characteristic
6	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625336	—	2	2	no	—
9	C₂×#13094021	C₂²	1	1	yes	characteristic
10	#15689497	—	2	2	no	—
11	#15689497	—	2	2	no	—
12	#15689471	—	2	2	no	—
13	#15689471	—	2	2	no	—
14	C₂×#13093946	—	2	2	no	—
15	C₂×#13093944	—	2	2	no	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	#15688993	C₂²	1	1	yes	characteristic
19	#15688885	—	2	2	no	—
20	#15688885	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#487	—	2	2	no	—
24	C₄₇×#487	—	2	2	no	—
25	C₂×#103×C₄₇	—	2	2	no	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×D₁₈₈	—	2	1	no	—
28	C₂²×D₁₈₈	—	2	1	no	—
29	C₂²×D₁₈₈	—	2	1	no	—
30	C₂×#10625336	—	2	8	no	—
31	C₂×#10625336	—	4	4	no	—
32	C₂×#10625336	—	4	4	no	—
33	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
34	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
35	#13094021	—	2	1	no	—
36	#13094021	—	2	1	no	—
37	#13094021	—	2	1	no	—
38	#13094021	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄⋊Dic₄₇	—	2	1	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	#13093946	—	4	4	no	—
43	#13093944	—	2	8	no	—
44	C₄×C₄₇⋊C₈	—	2	1	no	—
45	C₄×C₄₇⋊C₈	—	2	1	no	—
46	C₄×C₄₇⋊C₈	—	2	1	no	—
47	C₄×C₄₇⋊C₈	—	2	1	no	—
48	C₂²×Q₈×C₄₇	—	2	1	no	—
49	C₂²×Q₈×C₄₇	—	2	1	no	—
50	C₂²×C₈×C₄₇	D₄	1	2	yes	—
51	C₄⋊Q₈×C₄₇	—	2	1	no	—
52	C₄⋊Q₈×C₄₇	—	2	1	no	—
53	C₄⋊Q₈×C₄₇	—	2	1	no	—
54	C₄⋊Q₈×C₄₇	—	2	1	no	—
55	C₄⋊Q₈×C₄₇	—	2	1	no	—
56	C₄⋊Q₈×C₄₇	—	2	1	no	—
57	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
58	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#103×C₄₇	—	2	8	no	—
65	C₄×C₈×C₄₇	—	2	1	no	—
66	C₄×C₈×C₄₇	—	2	1	no	—
67	C₂³×D₄₇	—	4	1	no	—
68	C₂×Q₈×C₄₇	—	2	2	no	—
69	C₂×Q₈×C₄₇	—	2	2	no	—
70	C₂×Q₈×C₄₇	—	4	1	no	—
71	C₂×Q₈×C₄₇	—	4	1	no	—
72	C₂×Q₈×C₄₇	—	4	2	no	—
73	C₂×Q₈×C₄₇	—	4	2	no	—
74	C₂×Q₈×C₄₇	—	4	2	no	—
75	C₂×Q₈×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	—	4	1	no	—
77	C₂²×C₄×C₄₇	—	4	1	no	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	4	1	no	—
82	C₂×D₁₈₈	—	2	2	no	—
83	C₂×D₁₈₈	—	2	2	no	—
84	C₂×D₁₈₈	—	2	2	no	—
85	C₂×D₁₈₈	—	4	2	no	—
86	C₂×D₁₈₈	—	4	2	no	—
87	C₂×D₁₈₈	—	4	2	no	—
88	C₂×D₁₈₈	—	4	2	no	—
89	C₂×D₁₈₈	—	4	2	no	—
90	C₂×D₁₈₈	—	4	2	no	—
91	C₂×D₁₈₈	—	4	1	no	—
92	C₂×D₁₈₈	—	4	1	no	—
93	C₂×D₁₈₈	—	4	1	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	—	4	1	no	—
102	C₄⋊C₄×C₄₇	—	4	1	no	—
103	C₄⋊C₄×C₄₇	—	4	1	no	—
104	C₄⋊C₄×C₄₇	—	4	1	no	—
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₇	#41	1	1	yes	characteristic
109	#10625336	—	4	16	no	—
110	C₄⋊Dic₄₇	—	4	2	no	—
111	C₄⋊Dic₄₇	—	4	2	no	—
112	C₄⋊Dic₄₇	—	4	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
119	C₂³×C₄₇	#127	1	1	yes	characteristic
120	C₂²×D₄₇	—	4	2	no	—
121	C₂²×D₄₇	—	8	1	no	—
122	C₂²×D₄₇	—	8	1	no	—
123	C₂²×D₄₇	—	8	1	no	—
124	C₂²×D₄₇	—	8	1	no	—
125	C₂²×D₄₇	—	8	1	no	—
126	C₂²×D₄₇	—	8	1	no	—
127	Q₈×C₄₇	—	4	4	no	—
128	Q₈×C₄₇	—	4	4	no	—
129	Q₈×C₄₇	—	8	2	no	—
130	Q₈×C₄₇	—	8	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	8	1	no	—
134	C₂×C₄×C₄₇	—	8	1	no	—
135	C₂×C₄×C₄₇	—	8	1	no	—
136	C₂×C₄×C₄₇	—	8	1	no	—
137	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₄₇	#135	1	2	yes	—
145	C₂×C₄×C₄₇	#135	1	2	yes	—
146	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₄₇	#137	1	2	yes	—
149	C₂×Dic₄₇	—	8	1	no	—
150	C₂×Dic₄₇	—	8	1	no	—
151	C₂×Dic₄₇	—	8	1	no	—
152	D₁₈₈	—	4	4	no	—
153	D₁₈₈	—	4	4	no	—
154	D₁₈₈	—	4	4	no	—
155	D₁₈₈	—	8	2	no	—
156	D₁₈₈	—	8	2	no	—
157	D₁₈₈	—	8	2	no	—
158	C₈×C₄₇	—	4	2	no	—
159	C₄₇⋊C₈	—	2	4	no	—
160	C₄₇⋊C₈	—	4	2	no	—
161	#10310	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	#392	1	1	yes	characteristic
163	C₂²×C₄₇	#492	1	1	yes	characteristic
164	C₂²×C₄₇	#495	1	1	yes	characteristic
165	C₂²×C₄₇	#495	1	1	yes	characteristic
166	C₂²×C₄₇	#496	1	1	yes	characteristic
167	C₂²×C₄₇	#496	1	1	yes	characteristic
168	C₂²×C₄₇	#497	1	1	yes	characteristic
169	C₂×D₄₇	—	8	2	no	—
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	8	2	no	—
172	C₂×D₄₇	—	16	1	no	—
173	C₂×D₄₇	—	16	1	no	—
174	C₂×D₄₇	—	16	1	no	—
175	C₂×D₄₇	—	16	1	no	—
176	C₄×C₄₇	—	8	2	no	—
177	C₄×C₄₇	—	8	2	no	—
178	C₄×C₄₇	#449	1	2	yes	—
179	C₄×C₄₇	#450	1	2	yes	—
180	C₄×C₄₇	#452	1	2	yes	—
181	C₄×C₄₇	#460	1	2	yes	—
182	C₄×C₄₇	#463	1	2	yes	—
183	C₄×C₄₇	#464	1	2	yes	—
184	Dic₄₇	—	16	1	no	—
185	C₂×#492	—	47	1	no	—
186	C₂×#487	D₄₇	1	1	yes	characteristic, complemented
187	C₂×#486	—	47	1	no	—
188	#1990	—	47	1	no	—
189	#1990	—	47	1	no	—
190	#1957	—	47	1	no	—
191	#1273	—	47	1	no	—
192	C₂×C₄₇	#1708	1	1	yes	characteristic
193	C₂×C₄₇	#1714	1	1	yes	characteristic
194	C₂×C₄₇	#1717	1	1	yes	characteristic
195	C₂×C₄₇	#1719	1	1	yes	characteristic
196	C₂×C₄₇	#2012	1	1	yes	characteristic
197	C₂×C₄₇	#2014	1	1	yes	characteristic
198	C₂×C₄₇	#2015	1	1	yes	characteristic
199	D₄₇	—	16	2	no	—
200	C₂²×SD₁₆	—	94	2	no	—
201	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
202	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
203	C₂×#127	—	47	1	no	—
204	#492	—	94	2	no	—
205	#492	—	94	2	no	—
206	#487	—	2	2	no	—
207	#487	—	2	2	no	—
208	#486	—	94	2	no	—
209	#486	—	94	2	no	—
210	C₂×#103	—	2	2	no	—
211	C₂×#102	—	94	2	no	—
212	C₂×#102	—	94	2	no	—
213	C₂×C₄×C₈	—	47	1	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
216	#384	—	47	1	no	—
217	#340	—	94	2	no	—
218	#340	—	94	2	no	—
219	C₄₇	#10310	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×D₄	—	94	1	no	—
223	C₂²×D₄	—	94	1	no	—
224	C₂²×D₄	—	94	1	no	—
225	C₂×SD₁₆	—	94	8	no	—
226	C₂×SD₁₆	—	188	4	no	—
227	C₂×SD₁₆	—	188	4	no	—
228	C₂²×C₈	—	47	2	no	—
229	C₂²×C₈	—	47	2	no	—
230	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
238	C₄⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
239	#127	—	47	2	no	—
240	#127	—	94	1	no	—
241	#127	—	94	1	no	—
242	#127	—	94	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	94	1	no	—
248	C₂×C₄⋊C₄	—	94	1	no	—
249	C₂×C₄⋊C₄	—	94	1	no	—
250	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
251	#103	—	2	8	no	—
252	#102	—	94	8	no	—
253	#102	—	188	4	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	94	1	no	—
257	C₄×C₈	—	94	1	no	—
258	C₄×C₈	—	94	1	no	—
259	C₄×C₈	—	94	1	no	—
260	C₂⁴	—	188	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×D₄	—	94	2	no	—
270	C₂×D₄	—	94	2	no	—
271	C₂×D₄	—	94	2	no	—
272	C₂×D₄	—	188	1	no	—
273	C₂×D₄	—	188	1	no	—
274	C₂×D₄	—	188	1	no	—
275	C₂×D₄	—	188	2	no	—
276	C₂×D₄	—	188	2	no	—
277	C₂×D₄	—	188	2	no	—
278	C₂×D₄	—	188	2	no	—
279	C₂×D₄	—	188	2	no	—
280	C₂×D₄	—	188	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	188	1	no	—
284	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
287	SD₁₆	—	188	16	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	47	4	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	94	2	no	—
294	C₂×C₈	—	94	2	no	—
295	C₂×C₈	—	94	2	no	—
296	C₂×C₈	—	94	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	188	2	no	—
306	C₄⋊C₄	—	188	2	no	—
307	C₄⋊C₄	—	188	2	no	—
308	C₄²	D₄×D₄₇	1	1	yes	characteristic
309	C₄²	#10625352	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₂³	—	376	1	no	—
313	C₂³	—	376	1	no	—
314	C₂³	—	376	1	no	—
315	C₂³	—	376	1	no	—
316	C₂³	—	376	1	no	—
317	C₂³	—	376	1	no	—
318	C₂³	—	188	2	no	—
319	C₂³	#13094020	1	1	yes	characteristic
320	Q₈	—	4	4	no	—
321	Q₈	—	4	4	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	8	2	no	—
324	D₄	—	376	2	no	—
325	D₄	—	376	2	no	—
326	D₄	—	376	2	no	—
327	D₄	—	188	4	no	—
328	D₄	—	188	4	no	—
329	D₄	—	188	4	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	1	no	—
333	C₂×C₄	—	8	1	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	376	1	no	—
337	C₂×C₄	—	376	1	no	—
338	C₂×C₄	—	376	1	no	—
339	C₂×C₄	#13093992	1	1	yes	characteristic
340	C₂×C₄	#13093992	1	1	yes	characteristic
341	C₂×C₄	#13093993	1	1	yes	characteristic
342	C₂×C₄	#13093995	1	1	yes	characteristic
343	C₂×C₄	#13093996	1	1	yes	characteristic
344	C₂×C₄	#13093996	1	1	yes	characteristic
345	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
346	C₂×C₄	#13094044	1	2	yes	—
347	C₂×C₄	#13094047	1	2	yes	—
348	C₂×C₄	#13094049	1	2	yes	—
349	C₂×C₄	#13094051	1	2	yes	—
350	C₂×C₄	#13094056	1	2	yes	—
351	C₈	—	4	2	no	—
352	C₈	—	94	4	no	—
353	C₈	—	188	2	no	—
354	C₂²	—	376	2	no	—
355	C₂²	—	376	2	no	—
356	C₂²	—	376	2	no	—
357	C₂²	—	752	1	no	—
358	C₂²	—	752	1	no	—
359	C₂²	—	752	1	no	—
360	C₂²	—	752	1	no	—
361	C₂²	#15689031	1	1	yes	characteristic
362	C₂²	#15689494	1	1	yes	characteristic
363	C₂²	#15689505	1	1	yes	characteristic
364	C₂²	#15689508	1	1	yes	characteristic
365	C₂²	#15689509	1	1	yes	characteristic
366	C₂²	#15689513	1	1	yes	characteristic
367	C₂²	#15689516	1	1	yes	characteristic
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	752	1	no	—
371	C₄	#15689268	1	2	yes	—
372	C₄	#15689272	1	2	yes	—
373	C₄	#15689284	1	2	yes	—
374	C₄	#15689321	1	2	yes	—
375	C₄	#15689335	1	2	yes	—
376	C₄	#15689347	1	2	yes	—
377	C₂	—	752	2	no	—
378	C₂	#20242991	1	1	yes	characteristic
379	C₂	#20243012	1	1	yes	characteristic
380	C₂	#20243030	1	1	yes	characteristic
381	C₂	#20243042	1	1	yes	characteristic
382	C₂	#20244465	1	1	yes	characteristic
383	C₂	#20244474	1	1	yes	characteristic
384	C₂	#20244479	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#487	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	C₂×#487	D₄₇
hypocenter	C₄₇	#10310
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
3. #20244350	C₂	semidirect
4. #20244350	C₂	semidirect
6. C₄₇⋊#1273	C₂	semidirect
7. C₂×C₄₇×#487	C₂	semidirect
186. C₂×#487	D₄₇	semidirect
219. C₄₇	#10310	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689497	2
C₂×#15689471	2
#20244350	2
#20244194	2
C₄₇⋊#1273	2
C₂×C₄₇×#487	2
#10310	47