Subgroups — #26122040

Order 12032.

686 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×#15689078	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688891	C₂	1	2	yes	maximal
3	#20243108	C₂	1	2	yes	maximal
4	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	#15689078	—	2	2	no	—
10	#15689078	C₄	1	4	yes	complemented
11	C₂×#13093934	—	2	1	no	—
12	C₂×C₈×Dic₄₇	—	2	1	no	—
13	#15688891	C₄	1	8	yes	complemented
14	#15688886	—	2	2	no	—
15	#15688886	—	2	2	no	—
16	#15688886	—	2	1	no	—
17	#15688886	—	2	1	no	—
18	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#341	—	2	2	no	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂×#10625329	—	2	1	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	C₂×C₄	1	2	yes	—
25	C₂×#10625329	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	2	1	no	—
27	#13093983	—	2	1	no	—
28	#13093983	—	2	1	no	—
29	#13093983	C₂×C₄	1	2	yes	—
30	#13093983	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	#13093934	—	2	4	no	—
42	C₈×Dic₄₇	—	2	4	no	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	—	2	2	no	—
49	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	D₄	1	2	yes	—
57	C₂⁴×C₄₇	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×Dic₄₇	—	2	1	no	—
69	C₂²×Dic₄₇	—	2	1	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	8	2	no	—
74	C₂²×Dic₄₇	—	8	2	no	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	4	2	no	—
78	M₁₆×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	4	2	no	—
86	C₂×C₈×C₄₇	—	4	2	no	—
87	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
89	C₂²⋊Dic₄₇	—	2	2	no	—
90	C₂²⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	1	no	—
92	C₄⋊Dic₄₇	—	2	1	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	4	2	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
98	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	#10625329	—	2	2	no	—
102	#10625329	—	2	4	no	—
103	#10625329	C₄²	1	4	yes	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	4	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	1	no	—
108	C₂³×C₄₇	—	2	1	no	—
109	C₂³×C₄₇	#117	1	1	yes	characteristic
110	C₂³×C₄₇	#117	1	2	yes	—
111	C₂³×C₄₇	#121	1	1	yes	characteristic
112	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂³×C₄₇	#126	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#117	1	2	yes	—
120	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#123	1	2	yes	—
128	C₂×C₄×C₄₇	#123	1	2	yes	—
129	C₂×C₄×C₄₇	#124	1	2	yes	—
130	C₂×C₄×C₄₇	#125	1	2	yes	—
131	C₂×C₄×C₄₇	#126	1	2	yes	—
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	1	no	—
136	C₂×Dic₄₇	—	4	1	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	8	4	no	—
139	C₂×Dic₄₇	—	8	2	no	—
140	C₂×Dic₄₇	—	8	2	no	—
141	C₂×Dic₄₇	—	8	4	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	—	2	2	no	—
144	C₈×C₄₇	—	4	4	no	—
145	C₄₇⋊C₈	—	2	4	no	—
146	#10827	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	2	2	no	—
155	C₂²×C₄₇	#376	1	1	yes	characteristic
156	C₂²×C₄₇	#380	1	1	yes	characteristic
157	C₂²×C₄₇	#383	1	1	yes	characteristic
158	C₂²×C₄₇	#387	1	2	yes	—
159	C₂²×C₄₇	#388	1	2	yes	—
160	C₂²×C₄₇	#471	1	2	yes	—
161	C₂²×C₄₇	#472	1	1	yes	characteristic
162	C₂²×C₄₇	#473	1	1	yes	characteristic
163	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
164	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₄₇	#381	1	1	yes	characteristic
166	C₄×C₄₇	#382	1	1	yes	characteristic
167	C₄×C₄₇	#387	1	2	yes	—
168	C₄×C₄₇	#388	1	2	yes	—
169	Dic₄₇	—	4	2	no	—
170	Dic₄₇	—	4	2	no	—
171	Dic₄₇	—	8	4	no	—
172	C₂×#405	—	47	1	no	—
173	C₂×#343	—	47	2	no	—
174	C₂×#341	D₄₇	1	1	yes	characteristic
175	#1736	—	47	2	no	—
176	C₂×#340	—	47	1	no	—
177	C₂×C₄₇	—	2	1	no	—
178	C₂×C₄₇	—	2	1	no	—
179	C₂×C₄₇	—	2	2	no	—
180	C₂×C₄₇	#1434	1	1	yes	characteristic
181	C₂×C₄₇	#1815	1	1	yes	characteristic
182	C₂×C₄₇	#1816	1	1	yes	characteristic
183	C₂×C₄₇	#1930	1	2	yes	—
184	C₂×C₄₇	#1931	1	2	yes	—
185	C₂²×M₁₆	—	47	1	no	—
186	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
187	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
188	C₂×#117	—	47	1	no	—
189	C₂²×C₄⋊C₄	—	47	2	no	—
190	#405	—	47	4	no	—
191	#405	—	94	2	no	—
192	C₂×#97	—	94	1	no	—
193	C₂×C₄×C₈	—	94	1	no	—
194	#343	—	47	8	no	—
195	#341	—	2	2	no	—
196	#340	—	94	2	no	—
197	#340	—	94	2	no	—
198	#340	—	94	1	no	—
199	#340	—	94	1	no	—
200	C₄₇	#10827	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₄	—	94	2	no	—
202	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	47	4	no	—
209	C₂×M₁₆	—	47	2	no	—
210	C₂×M₁₆	—	94	2	no	—
211	C₂×M₁₆	—	94	1	no	—
212	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	94	1	no	—
217	C₂²×C₈	C₄×D₄₇	1	2	yes	—
218	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
219	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
221	#117	—	47	2	no	—
222	#117	—	47	2	no	—
223	#117	—	94	1	no	—
224	#117	—	94	1	no	—
225	C₂×C₄⋊C₄	—	47	1	no	—
226	C₂×C₄⋊C₄	—	47	1	no	—
227	C₂×C₄⋊C₄	—	47	4	no	—
228	C₂×C₄⋊C₄	—	94	2	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	94	2	no	—
231	C₂×C₄⋊C₄	—	94	2	no	—
232	C₂×C₄⋊C₄	—	94	2	no	—
233	C₂×C₂²⋊C₄	—	94	1	no	—
234	C₂×C₄²	—	94	1	no	—
235	#97	—	94	4	no	—
236	C₄×C₈	—	94	4	no	—
237	C₂⁴	#10625354	1	1	yes	characteristic
238	C₂²×C₄	—	94	1	no	—
239	C₂²×C₄	—	94	1	no	—
240	C₂²×C₄	—	94	4	no	—
241	C₂²×C₄	—	376	2	no	—
242	C₂²×C₄	—	376	2	no	—
243	C₂²×C₄	—	188	2	no	—
244	C₂²×C₄	—	188	2	no	—
245	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
248	C₂²×C₄	#10625350	1	2	yes	—
249	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
250	C₂²×C₄	#10625352	1	1	yes	characteristic
251	C₂²×C₄	#10625352	1	2	yes	—
252	C₂²×C₄	#10625352	1	2	yes	—
253	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	4	2	no	—
258	M₁₆	—	4	2	no	—
259	M₁₆	—	47	4	no	—
260	M₁₆	—	94	2	no	—
261	M₁₆	—	94	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	—	94	2	no	—
271	C₂×C₈	—	94	4	no	—
272	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
273	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
274	C₄⋊C₄	—	47	2	no	—
275	C₄⋊C₄	—	47	2	no	—
276	C₄⋊C₄	—	94	1	no	—
277	C₄⋊C₄	—	94	1	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	94	4	no	—
280	C₄⋊C₄	—	188	2	no	—
281	C₄⋊C₄	—	188	2	no	—
282	C₂²⋊C₄	—	94	2	no	—
283	C₂²⋊C₄	—	94	2	no	—
284	C₄²	—	94	2	no	—
285	C₄²	—	94	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#13093982	1	1	yes	characteristic
290	C₂³	#13093982	1	2	yes	—
291	C₂³	#13093999	1	1	yes	characteristic
292	C₂³	#13094000	1	1	yes	characteristic
293	C₂³	#13094019	1	2	yes	—
294	C₂×C₄	—	94	2	no	—
295	C₂×C₄	—	94	2	no	—
296	C₂×C₄	—	376	4	no	—
297	C₂×C₄	—	376	2	no	—
298	C₂×C₄	—	376	2	no	—
299	C₂×C₄	—	376	4	no	—
300	C₂×C₄	—	188	4	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	—	188	1	no	—
303	C₂×C₄	—	188	1	no	—
304	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	#13093980	1	1	yes	characteristic
308	C₂×C₄	#13093980	1	1	yes	characteristic
309	C₂×C₄	#13093980	1	2	yes	—
310	C₂×C₄	#13093981	1	1	yes	characteristic
311	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
312	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
313	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
314	C₂×C₄	#13093989	1	2	yes	—
315	C₂×C₄	#13093995	1	1	yes	characteristic
316	C₂×C₄	#13094004	1	1	yes	characteristic
317	C₂×C₄	#13094005	1	2	yes	—
318	C₂×C₄	#13094008	1	2	yes	—
319	C₂×C₄	#13094010	1	2	yes	—
320	C₂×C₄	#13094014	1	2	yes	—
321	C₂×C₄	#13094017	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	4	4	no	—
325	C₈	—	94	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	#15688968	1	1	yes	characteristic
335	C₂²	#15688980	1	1	yes	characteristic
336	C₂²	#15688990	1	1	yes	characteristic
337	C₂²	#15689004	1	2	yes	—
338	C₂²	#15689015	1	2	yes	—
339	C₂²	#15689387	1	2	yes	—
340	C₂²	#15689394	1	1	yes	characteristic
341	C₂²	#15689399	1	1	yes	characteristic
342	C₄	—	376	4	no	—
343	C₄	—	188	2	no	—
344	C₄	—	188	2	no	—
345	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
346	C₄	#15688974	1	1	yes	characteristic
347	C₄	#15688984	1	1	yes	characteristic
348	C₄	#15688987	1	1	yes	characteristic
349	C₄	#15689008	1	2	yes	—
350	C₄	#15689011	1	2	yes	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	#20241612	1	1	yes	characteristic
355	C₂	#20243444	1	1	yes	characteristic
356	C₂	#20243449	1	1	yes	characteristic
357	C₂	#20244044	1	2	yes	—
358	C₂	#20244051	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093999
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#10827
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
10. #15689078	C₄	semidirect
13. #15688891	C₄	semidirect
200. C₄₇	#10827	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689078	2
C₂×#15688891	2
#20243108	2
C₂×#15688886	2
C₂×C₄₇×#341	2
#10827	47