Subgroups — #26122095

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688992	C₂	1	2	yes	maximal
3	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688885	C₂	1	2	yes	maximal
5	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×Dic₄₇⋊C₄	—	2	4	no	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	—	2	4	no	—
11	C₂×C₈×Dic₄₇	C₄	1	4	yes	complemented
12	#15688992	—	2	4	no	—
13	#15688992	C₄	1	8	yes	complemented
14	#15688886	—	2	2	no	—
15	#15688885	—	2	4	no	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#340	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	2	no	—
20	C₂³×Dic₄₇	D₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×Dic₄₇⋊C₄	—	2	16	no	—
33	C₂×Dic₄₇⋊C₄	—	4	8	no	—
34	C₂×Dic₄₇⋊C₄	—	4	8	no	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
38	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
39	C₈×Dic₄₇	—	2	8	no	—
40	C₂³×C₄×C₄₇	—	2	2	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	C₂²⋊C₄	1	8	yes	—
75	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
76	C₂²×Dic₄₇	D₈	1	4	yes	—
77	C₂²×Dic₄₇	SD₁₆	1	8	yes	—
78	C₂²×Dic₄₇	Q₁₆	1	4	yes	—
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	4	no	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	complemented
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₄⋊C₄×C₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	Dic₄₇⋊C₄	—	4	32	no	—
93	C₄×Dic₄₇	C₄²	1	4	yes	—
94	C₄×Dic₄₇	C₂²⋊C₄	1	4	yes	—
95	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	complemented
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₇	#124	1	2	yes	—
104	C₂³×C₄₇	#127	1	1	yes	characteristic
105	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×D₈	1	2	yes	—
107	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
108	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	8	no	—
111	C₂×C₄×C₄₇	—	8	4	no	—
112	C₂×C₄×C₄₇	—	8	2	no	—
113	C₂×C₄×C₄₇	—	8	2	no	—
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	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
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×Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	#121	1	2	yes	—
126	C₂×C₄×C₄₇	#121	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	#125	1	2	yes	—
130	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	8	no	—
134	C₂×Dic₄₇	—	8	4	no	—
135	C₂×Dic₄₇	—	8	2	no	—
136	C₂×Dic₄₇	—	8	2	no	—
137	C₂×Dic₄₇	#95	1	8	yes	—
138	C₂×Dic₄₇	#102	1	16	yes	—
139	C₂×Dic₄₇	#103	1	16	yes	—
140	C₂×Dic₄₇	#106	1	8	yes	—
141	C₂×Dic₄₇	#107	1	8	yes	—
142	C₈×C₄₇	—	2	4	no	—
143	C₄₇⋊C₈	—	2	4	no	—
144	#10841	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
146	C₂²×C₄₇	#382	1	1	yes	characteristic
147	C₂²×C₄₇	#384	1	1	yes	characteristic
148	C₂²×C₄₇	#390	1	2	yes	—
149	C₂²×C₄₇	#391	1	2	yes	—
150	C₂²×C₄₇	C₂×#102	1	4	yes	—
151	C₂²×C₄₇	C₂×#103	1	4	yes	—
152	C₂²×C₄₇	C₂×#106	1	2	yes	—
153	C₂²×C₄₇	C₂×#107	1	2	yes	—
154	C₂²×C₄₇	#486	1	4	yes	—
155	C₂²×C₄₇	#487	1	4	yes	—
156	C₂²×C₄₇	#492	1	2	yes	—
157	C₂²×C₄₇	#493	1	1	yes	characteristic
158	C₂²×C₄₇	#494	1	1	yes	characteristic
159	C₂²×C₄₇	#498	1	2	yes	—
160	C₂²×C₄₇	#500	1	2	yes	—
161	C₄×C₄₇	—	8	4	no	—
162	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₄₇	#380	1	1	yes	characteristic
165	C₄×C₄₇	#380	1	1	yes	characteristic
166	C₄×C₄₇	#389	1	2	yes	—
167	C₄×C₄₇	#389	1	2	yes	—
168	Dic₄₇	—	8	4	no	—
169	Dic₄₇	#340	1	16	yes	complemented
170	C₂²×C₄×C₈	—	47	1	no	—
171	C₂×#384	—	47	2	no	—
172	C₂×#340	—	47	1	no	—
173	C₂×#340	—	47	2	no	—
174	C₂×#340	D₄₇	1	1	yes	characteristic
175	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#340	1	2	yes	—
177	C₂×C₄₇	#1834	1	2	yes	—
178	C₂×C₄₇	#1844	1	1	yes	characteristic
179	C₂×C₄₇	#1845	1	1	yes	characteristic
180	C₂×C₄₇	#1956	1	2	yes	—
181	C₂×C₄₇	#1957	1	2	yes	—
182	C₂×C₄₇	#1990	1	4	yes	—
183	C₂³×C₈	—	47	1	no	—
184	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	—	47	2	no	—
186	C₂²×C₄⋊C₄	—	94	4	no	—
187	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
188	C₂²×C₄²	—	47	1	no	—
189	C₂×C₄×C₈	—	47	4	no	—
190	C₂×C₄×C₈	—	94	4	no	—
191	#384	—	47	8	no	—
192	#384	—	94	4	no	—
193	#340	—	2	2	no	—
194	#340	—	94	4	no	—
195	#340	—	94	2	no	—
196	C₄₇	#10841	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	2	2	no	—
198	C₂³×C₄	—	47	2	no	—
199	C₂³×C₄	—	94	2	no	—
200	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	47	2	no	—
204	C₂²×C₈	—	47	2	no	—
205	C₂²×C₈	—	47	2	no	—
206	C₂²×C₈	—	94	2	no	—
207	C₂²×C₈	—	94	2	no	—
208	C₂²×C₈	Dic₉₄	1	2	yes	—
209	C₂²×C₈	C₄×D₄₇	1	2	yes	—
210	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	47	4	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄⋊C₄	—	94	2	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₄⋊C₄	—	94	2	no	—
221	C₂×C₄⋊C₄	—	94	16	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	—	188	8	no	—
224	C₂×C₄⋊C₄	—	188	8	no	—
225	C₂×C₄⋊C₄	C₄×D₄₇	1	4	yes	—
226	C₂×C₄²	—	47	4	no	—
227	C₂×C₄²	—	47	2	no	—
228	C₂×C₄²	—	47	4	no	—
229	C₂×C₄²	—	47	2	no	—
230	C₄×C₈	—	94	8	no	—
231	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	47	8	no	—
238	C₂²×C₄	—	47	4	no	—
239	C₂²×C₄	—	47	4	no	—
240	C₂²×C₄	—	47	8	no	—
241	C₂²×C₄	—	47	4	no	—
242	C₂²×C₄	—	94	4	no	—
243	C₂²×C₄	—	188	2	no	—
244	C₂²×C₄	—	188	2	no	—
245	C₂²×C₄	—	188	4	no	—
246	C₂²×C₄	—	188	4	no	—
247	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
250	C₂²×C₄	#10625350	1	2	yes	—
251	C₂²×C₄	D₄×D₄₇	1	2	yes	—
252	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	#10625352	1	2	yes	—
254	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	47	4	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	—	94	4	no	—
266	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	complemented
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	94	4	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	188	32	no	—
274	C₄⋊C₄	—	188	2	no	—
275	C₄⋊C₄	—	188	2	no	—
276	C₄²	—	47	4	no	—
277	C₄²	—	47	4	no	—
278	C₄²	—	47	8	no	—
279	C₂³	C₂²⋊C₄×D₄₇	1	2	yes	—
280	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
281	C₂³	#13094010	1	2	yes	—
282	C₂³	#13094020	1	1	yes	characteristic
283	C₂³	#13094023	1	1	yes	characteristic
284	C₂³	D₈×D₄₇	1	2	yes	—
285	C₂³	SD₁₆×D₄₇	1	4	yes	—
286	C₂³	Q₁₆×D₄₇	1	2	yes	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	—	47	8	no	—
293	C₂×C₄	—	47	16	no	—
294	C₂×C₄	—	47	16	no	—
295	C₂×C₄	—	47	8	no	—
296	C₂×C₄	—	47	8	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	8	no	—
302	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
306	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
307	C₂×C₄	#13093980	1	1	yes	characteristic
308	C₂×C₄	#13093981	1	1	yes	characteristic
309	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
310	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
311	C₂×C₄	#13093989	1	2	yes	—
312	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
313	C₂×C₄	#13093996	1	2	yes	—
314	C₂×C₄	#13093997	1	2	yes	—
315	C₂×C₄	#13094001	1	2	yes	—
316	C₂×C₄	#13094002	1	2	yes	—
317	C₂×C₄	#13094014	1	2	yes	—
318	C₂×C₄	#13094022	1	1	yes	characteristic
319	C₂×C₄	#13094023	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	94	4	no	—
322	C₂²	#95×D₄₇	1	1	yes	characteristic
323	C₂²	#15688987	1	1	yes	characteristic
324	C₂²	#15688992	1	1	yes	characteristic
325	C₂²	#15689023	1	2	yes	—
326	C₂²	#15689027	1	2	yes	—
327	C₂²	#102×D₄₇	1	4	yes	—
328	C₂²	#103×D₄₇	1	4	yes	—
329	C₂²	#106×D₄₇	1	2	yes	—
330	C₂²	#107×D₄₇	1	2	yes	—
331	C₂²	#15689468	1	4	yes	—
332	C₂²	#15689472	1	4	yes	—
333	C₂²	#15689494	1	2	yes	—
334	C₂²	#15689498	1	1	yes	characteristic
335	C₂²	#15689501	1	1	yes	characteristic
336	C₂²	#15689520	1	2	yes	—
337	C₂²	#15689527	1	2	yes	—
338	C₄	—	8	4	no	—
339	C₄	—	47	16	no	—
340	C₄	—	376	4	no	—
341	C₄	#15688974	1	1	yes	characteristic
342	C₄	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
343	C₄	#15688981	1	1	yes	characteristic
344	C₄	#15688982	1	1	yes	characteristic
345	C₄	#15689018	1	2	yes	—
346	C₄	#15689020	1	2	yes	—
347	C₂	Dic₄₇⋊#95	1	1	yes	characteristic
348	C₂	#340×D₄₇	1	2	yes	—
349	C₂	#20243514	1	2	yes	—
350	C₂	#20243543	1	1	yes	characteristic
351	C₂	#20243546	1	1	yes	characteristic
352	C₂	#20244188	1	2	yes	—
353	C₂	#20244192	1	2	yes	—
354	C₂	#20244347	1	4	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	#10841
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
11. C₂×C₈×Dic₄₇	C₄	semidirect
13. #15688992	C₄	semidirect
83. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
100. C₂×C₄₇⋊C₈	C₄⋊C₄	semidirect
169. Dic₄₇	#340	semidirect
196. C₄₇	#10841	semidirect
266. C₂×C₈	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
C₂×#15688992	2
C₂×#15688886	2
C₂×#15688885	2
C₂×C₄₇×#340	2
#10841	47