Subgroups — #26106323

Order 12032.

742 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	#20245795	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
3	#20241268	C₂	1	4	yes	maximal
4	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	—	2	1	no	—
6	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
7	#15688993	C₄	1	2	yes	complemented
8	#15688989	—	2	2	no	—
9	#15688989	—	2	1	no	—
10	#15688989	C₄	1	4	yes	complemented
11	#15688983	—	2	1	no	—
12	#15688983	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₄₇	—	2	1	no	—
14	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	complemented
15	C₂×C₄₇⋊#95	—	2	1	no	—
16	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#431	—	2	2	no	—
19	C₄₇×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	#13093983	—	4	2	no	—
25	#13093983	—	4	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	4	2	no	—
30	C₂×C₄⋊Dic₄₇	—	4	2	no	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₄₇⋊#95	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	2	yes	—
50	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
51	C₄⋊C₈×C₄₇	—	2	4	no	—
52	C₂²⋊C₈×C₄₇	—	2	2	no	—
53	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	D₈	1	2	yes	—
67	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
69	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	—	2	4	no	—
78	C₂²×Dic₄₇	—	4	4	no	—
79	C₂²×Dic₄₇	—	4	4	no	—
80	C₂²×Dic₄₇	—	4	2	no	—
81	C₂²×Dic₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	4	4	no	—
84	C₄²×C₄₇	C₄²	1	2	yes	—
85	C₄²×C₄₇	C₄²	1	2	yes	—
86	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
92	C₂²⋊Dic₄₇	—	4	2	no	—
93	C₂²⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	—	4	8	no	—
96	C₄⋊Dic₄₇	—	4	4	no	—
97	C₄⋊Dic₄₇	—	4	4	no	—
98	C₄×Dic₄₇	—	4	4	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	—	4	1	no	—
117	C₂×C₄×C₄₇	—	4	1	no	—
118	C₂×C₄×C₄₇	#95	1	2	yes	—
119	C₂×C₄×C₄₇	#95	1	2	yes	—
120	C₂×C₄×C₄₇	#95	1	2	yes	—
121	C₂×C₄×C₄₇	#95	1	2	yes	—
122	C₂×C₄×C₄₇	#102	1	2	yes	—
123	C₂×C₄×C₄₇	#102	1	4	yes	—
124	C₂×C₄×C₄₇	#102	1	2	yes	—
125	C₂×C₄×C₄₇	#103	1	4	yes	—
126	C₂×C₄×C₄₇	#103	1	2	yes	—
127	C₂×C₄×C₄₇	#103	1	2	yes	—
128	C₂×C₄×C₄₇	#106	1	4	yes	—
129	C₂×C₄×C₄₇	#107	1	4	yes	—
130	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	—	4	4	no	—
140	C₂×Dic₄₇	—	8	4	no	—
141	C₂×Dic₄₇	—	8	4	no	—
142	C₈×C₄₇	—	4	4	no	—
143	#8026	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
153	C₂²×C₄₇	#410	1	2	yes	—
154	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
158	C₂²×C₄₇	#421	1	2	yes	—
159	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	#340	1	4	yes	—
168	C₄×C₄₇	#340	1	4	yes	—
169	Dic₄₇	—	8	4	no	—
170	Dic₄₇	—	8	4	no	—
171	C₂×#431	D₄₇	1	1	yes	characteristic
172	#2285	—	47	1	no	—
173	C₂×#382	—	47	1	no	—
174	#1294	—	47	4	no	—
175	C₂×C₄₇	—	2	1	no	—
176	C₂×C₄₇	—	2	1	no	—
177	C₂×C₄₇	—	2	1	no	—
178	C₂×C₄₇	—	2	1	no	—
179	C₂×C₄₇	#1517	1	2	yes	—
180	C₂×C₄₇	#1518	1	2	yes	—
181	C₂×C₄₇	#1733	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
183	C₂×C₄₇	#1740	1	1	yes	characteristic
184	C₂×#117	—	94	1	no	—
185	C₂²×C₄⋊C₄	—	94	1	no	—
186	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
187	#431	—	2	2	no	—
188	#431	Dic₄₇	1	4	yes	complemented
189	C₂×C₄⋊C₈	—	2	1	no	—
190	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
191	C₂×C₄×C₈	—	2	1	no	—
192	#384	—	47	2	no	—
193	#382	—	47	4	no	—
194	#382	—	94	2	no	—
195	#382	—	94	1	no	—
196	#380	—	47	2	no	—
197	#380	—	94	1	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	C₄×C₄⋊C₄	—	94	1	no	—
200	C₂×#95	—	47	2	no	—
201	C₂×#95	—	94	1	no	—
202	C₄₇	#8026	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	94	2	no	—
204	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
205	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	—	2	2	no	—
208	#117	—	188	2	no	—
209	#117	—	188	2	no	—
210	C₂×C₄⋊C₄	—	47	4	no	—
211	C₂×C₄⋊C₄	—	94	4	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	—	188	2	no	—
215	C₂×C₄⋊C₄	—	188	2	no	—
216	C₂×C₂²⋊C₄	—	94	2	no	—
217	C₂×C₄²	—	94	2	no	—
218	C₂×C₄²	Dic₉₄	1	2	yes	—
219	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₄×D₄₇	1	2	yes	—
221	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
223	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
226	C₄⋊C₈	—	2	4	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
229	C₄×C₈	—	2	4	no	—
230	#95	—	47	4	no	—
231	#95	—	94	4	no	—
232	#95	—	94	2	no	—
233	#95	—	94	2	no	—
234	#95	—	94	2	no	—
235	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	94	4	no	—
239	C₂²×C₄	—	94	4	no	—
240	C₂²×C₄	—	188	4	no	—
241	C₂²×C₄	—	188	4	no	—
242	C₂²×C₄	—	188	2	no	—
243	C₂²×C₄	—	188	2	no	—
244	C₂²×C₄	#10625325	1	2	yes	—
245	C₂²×C₄	#10625325	1	2	yes	—
246	C₂²×C₄	D₃₇₆	1	2	yes	—
247	C₂²×C₄	Dic₁₈₈	1	2	yes	—
248	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
249	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
250	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
252	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
253	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
254	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₄⋊C₄	—	188	4	no	—
265	C₄⋊C₄	—	188	4	no	—
266	C₄⋊C₄	—	188	8	no	—
267	C₄⋊C₄	—	188	4	no	—
268	C₂²⋊C₄	—	188	2	no	—
269	C₂²⋊C₄	—	188	2	no	—
270	C₄²	—	188	4	no	—
271	C₄²	C₄×Dic₄₇	1	2	yes	—
272	C₄²	C₄×Dic₄₇	1	2	yes	—
273	C₄²	Dic₄₇⋊C₄	1	2	yes	—
274	C₄²	Dic₄₇⋊C₄	1	2	yes	—
275	C₄²	Dic₄₇⋊C₄	1	2	yes	—
276	C₄²	Dic₄₇⋊C₄	1	2	yes	—
277	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
278	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
282	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
283	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
284	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
285	C₂³	C₂×#10625325	1	1	yes	characteristic
286	C₂³	C₂×#10625325	1	1	yes	characteristic
287	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	376	4	no	—
299	C₂×C₄	—	376	4	no	—
300	C₂×C₄	—	188	4	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	—	188	4	no	—
303	C₂×C₄	—	188	4	no	—
304	C₂×C₄	—	188	4	no	—
305	C₂×C₄	—	188	4	no	—
306	C₂×C₄	C₄₇⋊#95	1	2	yes	—
307	C₂×C₄	C₄₇⋊#95	1	2	yes	—
308	C₂×C₄	C₄₇⋊#95	1	2	yes	—
309	C₂×C₄	C₄₇⋊#95	1	2	yes	—
310	C₂×C₄	#13093946	1	2	yes	—
311	C₂×C₄	#13093946	1	4	yes	—
312	C₂×C₄	#13093946	1	2	yes	—
313	C₂×C₄	#13093949	1	4	yes	—
314	C₂×C₄	#13093949	1	2	yes	—
315	C₂×C₄	#13093949	1	2	yes	—
316	C₂×C₄	#13093956	1	4	yes	—
317	C₂×C₄	#13093958	1	4	yes	—
318	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
322	C₈	—	4	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
332	C₂²	#15689096	1	2	yes	—
333	C₂²	C₂×#13093946	1	1	yes	characteristic
334	C₂²	C₂×#13093946	1	1	yes	characteristic
335	C₂²	C₂×#13093949	1	1	yes	characteristic
336	C₂²	C₂×#13093949	1	1	yes	characteristic
337	C₂²	#15689139	1	2	yes	—
338	C₂²	C₂×#13093956	1	1	yes	characteristic
339	C₂²	C₂×#13093958	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	376	4	no	—
347	C₄	—	376	4	no	—
348	C₄	#15688886	1	4	yes	—
349	C₄	#15688886	1	4	yes	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	#20241949	1	2	yes	—
355	C₂	#20241953	1	2	yes	—
356	C₂	#20243099	1	1	yes	characteristic
357	C₂	C₂×#15688886	1	1	yes	characteristic
358	C₂	#20243120	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#8026
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
7. #15688993	C₄	semidirect
10. #15688989	C₄	semidirect
14. C₄×C₄⋊Dic₄₇	C₄	semidirect
19. C₄₇×#431	C₄	semidirect
188. #431	Dic₄₇	semidirect
202. C₄₇	#8026	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
C₂×#15688989	2
#20241268	2
C₂×C₄₇×#431	2
#8026	47