Subgroups — #26186831

Order 12032.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253948	C₂	1	1	yes	maximal, characteristic
2	#20252323	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252099	C₂	1	1	yes	maximal, characteristic
4	#20242293	C₂	1	4	yes	maximal, complemented
5	#20242203	C₂	1	4	yes	maximal, complemented
6	#20242146	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241840	C₂	1	1	yes	maximal, characteristic
8	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094032	—	2	1	no	—
11	#15689673	C₂²	1	1	yes	characteristic
12	#15689519	C₄	1	1	yes	characteristic
13	#15689497	C₄	1	1	yes	characteristic
14	#15689473	C₄	1	1	yes	characteristic
15	#15689471	C₄	1	1	yes	characteristic
16	#15689354	C₄	1	4	yes	—
17	#15689254	—	2	1	no	—
18	#15689254	C₂²	1	2	yes	—
19	Q₈×C₄₇⋊C₈	—	2	1	no	—
20	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	—
21	C₄×#10625336	C₂²	1	4	yes	—
22	#15689200	—	2	1	no	—
23	#15689200	C₂²	1	2	yes	—
24	#15689188	—	2	1	no	—
25	#15689188	C₂²	1	2	yes	—
26	#15689150	—	2	1	no	—
27	#15689147	C₂²	1	1	yes	characteristic
28	#15689091	—	2	1	no	—
29	#15689079	—	2	1	no	—
30	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
32	#15688875	C₂²	1	4	yes	—
33	#15688852	C₂²	1	4	yes	—
34	#15688845	C₂²	1	2	yes	—
35	#15688845	C₂²	1	2	yes	—
36	C₄₇×#533	C₂²	1	1	yes	characteristic
37	C₄₇×#423	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×Q₈⋊C₈	C₂²	1	4	yes	—
40	C₂×#10625350	—	2	1	no	—
41	C₂×#10625336	C₂×C₄	1	4	yes	—
42	#13094032	—	4	4	no	—
43	C₂×#10625329	—	2	1	no	—
44	C₂×#10625324	—	2	2	no	—
45	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
48	C₂×C₈×D₄₇	—	2	2	no	—
49	#13094024	C₂×C₄	1	1	yes	characteristic
50	#13094021	C₂×C₄	1	1	yes	characteristic
51	#13094012	—	2	1	no	—
52	#13093998	—	2	1	no	—
53	#13093998	C₂×C₄	1	2	yes	—
54	C₄×D₁₈₈	—	2	1	no	—
55	C₄×D₁₈₈	C₂³	1	2	yes	—
56	#13093955	C₂×C₄	1	4	yes	—
57	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
58	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
59	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
60	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
61	#13093948	C₂×C₄	1	4	yes	—
62	#13093944	C₂×C₄	1	4	yes	—
63	(C₂×D₄₇)⋊C₈	—	2	2	no	—
64	(C₂×D₄₇)⋊C₈	—	2	2	no	—
65	#13093935	—	2	2	no	—
66	C₄×C₄₇⋊C₈	—	2	2	no	—
67	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
70	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
71	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
72	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₄₇	—	2	1	no	—
75	C₂²×C₈×C₄₇	—	2	1	no	—
76	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
78	#125×C₄₇	—	2	1	no	—
79	C₂²⋊Q₈×C₄₇	—	2	1	no	—
80	C₂²⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
81	C₄×Q₈×C₄₇	—	2	1	no	—
82	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
83	#117×C₄₇	—	2	1	no	—
84	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₄₇	—	2	1	no	—
86	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
87	C₄×C₈×C₄₇	—	2	1	no	—
88	C₄×C₈×C₄₇	C₂³	1	2	yes	—
89	C₂×Q₈×C₄₇	—	2	1	no	—
90	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
94	C₂×C₂²⋊D₄₇	—	2	2	no	—
95	#10625350	—	4	2	no	—
96	#10625350	—	4	2	no	—
97	C₂×D₁₈₈	—	2	1	no	—
98	C₂×D₁₈₈	—	2	1	no	—
99	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
100	C₂×C₄×D₄₇	—	2	2	no	—
101	C₂×Dic₉₄	—	2	1	no	—
102	M₁₆×C₄₇	—	4	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	4	2	no	—
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₄⋊C₄×C₄₇	—	2	1	no	—
108	C₄⋊C₄×C₄₇	—	2	1	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
112	C₂²⋊C₄×C₄₇	—	2	2	no	—
113	C₄²×C₄₇	—	2	2	no	—
114	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	#41	1	1	yes	characteristic
117	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
118	#10625336	—	2	8	no	—
119	(C₂×D₄₇)⋊C₄	—	2	2	no	—
120	(C₂×D₄₇)⋊C₄	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	1	no	—
122	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
123	#10625329	—	4	2	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	4	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
129	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
134	#10625324	—	4	4	no	—
135	C₈×D₄₇	—	4	4	no	—
136	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₄₇	—	2	2	no	—
138	Q₈×C₄₇	—	2	4	no	—
139	Q₈×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₄₇	#137	1	2	yes	—
149	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₄₇	—	4	4	no	—
153	C₂×Dic₄₇	—	2	2	no	—
154	D₁₈₈	—	2	4	no	—
155	D₁₈₈	—	4	2	no	—
156	D₁₈₈	—	4	2	no	—
157	C₄×D₄₇	—	4	2	no	—
158	C₄×D₄₇	—	4	2	no	—
159	Dic₉₄	—	4	2	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₈×C₄₇	—	4	2	no	—
162	C₄₇⋊C₈	—	2	2	no	—
163	C₄₇⋊C₈	—	2	1	no	—
164	C₄₇⋊C₈	—	2	1	no	—
165	C₄₇⋊C₈	—	2	1	no	—
166	C₄₇⋊C₈	—	2	1	no	—
167	C₄₇⋊C₈	—	4	2	no	—
168	C₄₇⋊C₈	#131	1	4	yes	complemented
169	#17960	—	47	1	no	maximal, Sylow
170	C₂²×C₄₇	—	2	1	no	—
171	C₂²×C₄₇	—	2	1	no	—
172	C₂²×C₄₇	#440	1	2	yes	—
173	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₄₇	—	4	2	no	—
175	C₂×D₄₇	—	4	2	no	—
176	C₂×D₄₇	—	4	2	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	4	2	no	—
180	C₄×C₄₇	—	4	2	no	—
181	C₄×C₄₇	#440	1	2	yes	—
182	C₄×C₄₇	#444	1	2	yes	—
183	C₄×C₄₇	C₂×#131	1	2	yes	—
184	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
186	Dic₄₇	—	4	2	no	—
187	#3167	—	47	1	no	—
188	#2960	—	47	1	no	—
189	#2931	—	47	1	no	—
190	#1592	—	47	4	no	—
191	SD₁₆⋊C₈	—	47	4	no	—
192	#1567	—	47	1	no	—
193	#1493	D₄₇	1	1	yes	characteristic, complemented
194	#1492	—	47	1	no	—
195	C₂×#322	—	47	1	no	—
196	C₂×C₄₇	—	2	2	no	—
197	C₂×C₄₇	#2931	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#440	1	1	yes	characteristic
199	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
200	D₄₇	—	8	2	no	—
201	C₂×#131	—	94	1	no	—
202	#533	C₂×D₄₇	1	1	yes	characteristic
203	#532	—	47	1	no	—
204	#498	—	47	1	no	—
205	#492	—	47	1	no	—
206	#487	—	47	1	no	—
207	#486	—	47	1	no	—
208	#465	—	47	4	no	—
209	#446	—	47	2	no	—
210	#446	—	94	1	no	—
211	C₈×Q₈	—	47	2	no	—
212	C₈×Q₈	—	94	1	no	—
213	C₄×SD₁₆	—	47	4	no	—
214	#436	—	47	2	no	—
215	#436	—	94	1	no	—
216	C₈×D₄	—	47	2	no	—
217	C₈×D₄	—	94	1	no	—
218	#424	—	94	1	no	—
219	#423	—	47	1	no	—
220	#423	C₂×D₄₇	1	1	yes	characteristic
221	#408	—	94	1	no	—
222	#405	—	94	1	no	—
223	C₂×C₄×C₈	—	47	1	no	—
224	C₂×C₄×C₈	—	47	1	no	—
225	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
226	#335	—	47	4	no	—
227	Q₈⋊C₈	C₂×D₄₇	1	4	yes	—
228	D₄⋊C₈	—	47	4	no	—
229	#322	—	47	2	no	—
230	#322	—	47	2	no	—
231	C₄₇	#17960	1	1	yes	characteristic, complemented, Sylow
232	C₂×#41	—	94	1	no	—
233	C₂×SD₁₆	—	47	4	no	—
234	#131	—	188	4	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	94	2	no	—
237	C₂×M₁₆	—	94	1	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	47	2	no	—
240	C₂²×C₈	—	47	1	no	—
241	C₂²×C₈	—	47	1	no	—
242	C₂²×C₈	—	94	2	no	—
243	C₄⋊Q₈	—	47	1	no	—
244	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
245	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
246	#127	—	47	1	no	—
247	#125	—	2	1	no	—
248	#124	—	94	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	C₂²⋊Q₈	C₄×D₄₇	1	2	yes	—
251	#121	—	47	2	no	—
252	#121	—	94	1	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
255	C₄×D₄	—	47	2	no	—
256	C₄×D₄	—	94	1	no	—
257	#117	—	2	1	no	—
258	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
259	#106	—	47	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	94	2	no	—
263	C₄⋊C₈	—	94	2	no	—
264	C₄⋊C₈	—	94	1	no	—
265	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
266	#103	—	47	4	no	—
267	#102	—	47	4	no	—
268	C₂²⋊C₈	—	94	2	no	—
269	C₂²⋊C₈	—	94	2	no	—
270	#97	—	94	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	—	47	1	no	—
274	C₄×C₈	—	47	1	no	—
275	C₄×C₈	—	47	1	no	—
276	C₄×C₈	—	47	2	no	—
277	C₄×C₈	—	47	1	no	—
278	C₄×C₈	—	47	1	no	—
279	C₄×C₈	—	94	2	no	—
280	C₄×C₈	C₂²×D₄₇	1	2	yes	—
281	#41	—	188	2	no	—
282	#41	—	188	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	94	1	no	—
285	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
286	C₂×D₄	—	47	2	no	—
287	C₂×D₄	—	94	2	no	—
288	C₂×D₄	—	94	1	no	—
289	C₂×D₄	—	94	1	no	—
290	C₂²×C₄	—	94	2	no	—
291	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
293	C₂²×C₄	#10625352	1	1	yes	characteristic
294	SD₁₆	—	94	8	no	—
295	M₁₆	—	4	2	no	—
296	M₁₆	—	188	4	no	—
297	M₁₆	—	188	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	47	2	no	—
302	C₂×C₈	—	47	2	no	—
303	C₂×C₈	—	47	1	no	—
304	C₂×C₈	—	47	1	no	—
305	C₂×C₈	—	47	1	no	—
306	C₂×C₈	—	47	1	no	—
307	C₂×C₈	—	94	4	no	—
308	C₂×C₈	—	94	2	no	—
309	C₂×C₈	—	94	2	no	—
310	C₂×C₈	—	94	2	no	—
311	C₂×C₈	—	188	4	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	47	2	no	—
318	C₄⋊C₄	—	94	1	no	—
319	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
325	C₄²	D₄×D₄₇	1	1	yes	characteristic
326	C₄²	#10625352	1	1	yes	characteristic
327	C₄²	C₂³×D₄₇	1	1	yes	characteristic
328	C₂³	—	94	2	no	—
329	C₂³	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	188	2	no	—
333	D₄	—	94	4	no	—
334	D₄	—	188	4	no	—
335	D₄	—	188	2	no	—
336	D₄	—	188	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	94	2	no	—
342	C₂×C₄	—	188	2	no	—
343	C₂×C₄	—	188	2	no	—
344	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
345	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
346	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
347	C₂×C₄	#13094051	1	2	yes	—
348	C₂×C₄	#13094056	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
350	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
351	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	47	4	no	—
355	C₈	—	94	2	no	—
356	C₈	—	94	1	no	—
357	C₈	—	94	1	no	—
358	C₈	—	94	1	no	—
359	C₈	—	94	1	no	—
360	C₈	—	188	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	188	2	no	—
364	C₂²	—	188	2	no	—
365	C₂²	—	188	2	no	—
366	C₂²	#15689226	1	2	yes	—
367	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	188	2	no	—
373	C₄	#15689226	1	2	yes	—
374	C₄	#15689242	1	2	yes	—
375	C₄	#131×D₄₇	1	2	yes	—
376	C₄	C₂×#13094051	1	1	yes	characteristic
377	C₄	C₂×#13094056	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	376	2	no	—
380	C₂	#20252096	1	1	yes	characteristic
381	C₂	C₂×#15689226	1	1	yes	characteristic
382	C₂	C₂×#15689242	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1493	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1493	D₄₇
hypocenter	C₄₇	#17960
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1493

Complements

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

N	H ≅ G/N	type
2. #20252323	C₂	semidirect
4. #20242293	C₂	semidirect
5. #20242203	C₂	semidirect
6. #20242146	C₂	semidirect
8. C₂×#15688845	C₂	semidirect
9. C₄₇×#1493	C₂	semidirect
168. C₄₇⋊C₈	#131	semidirect
193. #1493	D₄₇	semidirect
231. C₄₇	#17960	semidirect

Maximal subgroups

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

subgroup	index
#20253948	2
#20252323	2
#20252099	2
#20242293	2
#20242203	2
#20242146	2
#20241840	2
C₂×#15688845	2
C₄₇×#1493	2
#17960	47