Subgroups — #26215317

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689247	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093989	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₈×Dic₄₇⋊C₄	C₂	1	2	yes	maximal
6	#20243249	C₂	1	2	yes	maximal
7	C₄⋊C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₇⋊C₈	C₂	1	2	yes	maximal
9	#20243133	C₂	1	1	yes	maximal, characteristic
10	#20243132	C₂	1	2	yes	maximal
11	C₈×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₂×#13093989	C₄	1	2	yes	—
14	C₈×Dic₉₄	C₄	1	8	yes	complemented
15	#15689247	C₄	1	8	yes	complemented
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
24	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
25	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
26	#15688974	C₄	1	4	yes	—
27	C₄²×Dic₄₇	C₂²	1	2	yes	—
28	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
29	#15688878	C₂²	1	2	yes	—
30	#15688877	C₂²	1	1	yes	characteristic
31	#15688877	C₂²	1	2	yes	—
32	#15688877	C₂²	1	1	yes	characteristic
33	#15688877	C₂²	1	1	yes	characteristic
34	#15688877	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#336	C₂²	1	2	yes	—
40	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
45	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
46	#13093989	C₈	1	16	yes	complemented
47	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
50	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
57	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
58	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
59	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
60	C₄×C₄₇⋊C₈	—	4	2	no	—
61	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
62	C₈×Dic₄₇	—	4	2	no	—
63	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
64	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
65	C₄₇⋊#95	C₂×C₄	1	4	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
73	C₄×C₈×C₄₇	—	4	2	no	—
74	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
75	#95×C₄₇	C₂×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₉₄	C₄²	1	4	yes	—
85	C₂×Dic₉₄	C₂×C₈	1	8	yes	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	1	no	—
88	C₂×C₈×C₄₇	—	2	1	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
95	C₄⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
100	C₄⋊Dic₄₇	C₄²	1	4	yes	—
101	Dic₄₇⋊C₄	C₄²	1	8	yes	—
102	Dic₄₇⋊C₄	C₂×C₈	1	16	yes	—
103	C₄×Dic₄₇	—	2	1	no	—
104	C₄×Dic₄₇	—	2	1	no	—
105	C₄×Dic₄₇	—	2	1	no	—
106	C₄×Dic₄₇	—	2	1	no	—
107	C₄×Dic₄₇	—	4	2	no	—
108	C₄×Dic₄₇	—	4	4	no	—
109	C₄×Dic₄₇	C₂×C₈	1	8	yes	—
110	C₄×Dic₄₇	C₂×C₈	1	16	yes	—
111	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
112	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	1	no	—
115	C₂×C₄₇⋊C₈	—	2	1	no	—
116	C₂×C₄₇⋊C₈	—	2	1	no	—
117	C₂×C₄₇⋊C₈	—	2	1	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
122	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	—
129	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
142	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
143	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
144	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
145	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
146	Dic₉₄	C₄×C₈	1	16	yes	complemented
147	C₈×C₄₇	—	2	2	no	—
148	C₈×C₄₇	—	2	4	no	—
149	C₈×C₄₇	—	2	2	no	—
150	C₄₇⋊C₈	—	2	2	no	—
151	C₄₇⋊C₈	—	2	2	no	—
152	C₄₇⋊C₈	—	2	4	no	—
153	#20569	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
155	C₂²×C₄₇	#519	1	1	yes	characteristic
156	C₂²×C₄₇	#520	1	1	yes	characteristic
157	C₂²×C₄₇	C₂³×C₈	1	2	yes	—
158	C₂²×C₄₇	#568	1	2	yes	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	8	no	—
161	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
162	C₄×C₄₇	#519	1	1	yes	characteristic
163	C₄×C₄₇	#519	1	1	yes	characteristic
164	C₄×C₄₇	#520	1	1	yes	characteristic
165	C₄×C₄₇	#520	1	1	yes	characteristic
166	Dic₄₇	—	2	4	no	—
167	Dic₄₇	—	2	4	no	—
168	Dic₄₇	—	2	4	no	—
169	Dic₄₇	C₂×C₄×C₈	1	8	yes	—
170	C₂×C₈×Q₈	—	47	1	no	—
171	C₂×C₈×Q₈	—	47	1	no	—
172	C₄²×Q₈	—	47	1	no	—
173	C₈×C₄⋊C₄	—	47	2	no	—
174	C₈×C₄⋊C₄	—	47	1	no	—
175	C₈×C₄⋊C₄	—	47	2	no	—
176	C₈×C₄⋊C₄	D₄₇	1	1	yes	characteristic
177	C₄×C₄⋊C₈	—	47	1	no	—
178	C₄×C₄⋊C₈	—	47	2	no	—
179	#1744	—	47	1	no	—
180	#1744	—	47	2	no	—
181	C₂×C₄₇	#2276	1	1	yes	characteristic
182	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
183	C₂×C₄₇	#2875	1	1	yes	characteristic
184	C₂×C₄₇	#2982	1	2	yes	—
185	C₂×C₄₇	#2988	1	2	yes	—
186	C₂×C₄×Q₈	—	47	2	no	—
187	C₂×C₄×Q₈	—	47	1	no	—
188	C₈×Q₈	—	47	8	no	—
189	C₈×Q₈	—	47	8	no	—
190	C₂×C₄⋊C₈	—	47	2	no	—
191	C₂×C₄⋊C₈	—	47	2	no	—
192	C₂×C₄⋊C₈	—	47	1	no	—
193	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
194	C₂×C₄×C₈	—	47	2	no	—
195	C₂×C₄×C₈	—	47	2	no	—
196	C₂×C₄×C₈	—	47	1	no	—
197	C₂×C₄×C₈	—	47	2	no	—
198	C₂×C₄×C₈	—	47	2	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
200	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
201	C₄×C₄⋊C₄	—	47	2	no	—
202	C₄×C₄⋊C₄	—	47	2	no	—
203	C₄×C₄⋊C₄	—	47	4	no	—
204	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
205	C₄³	—	47	2	no	—
206	C₄³	—	47	1	no	—
207	#336	—	47	1	no	—
208	#336	—	47	1	no	—
209	#336	—	47	2	no	—
210	#336	—	47	1	no	—
211	#336	—	47	1	no	—
212	#336	—	47	2	no	—
213	#336	C₂×D₄₇	1	2	yes	—
214	C₄₇	#20569	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₈	—	47	1	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₂²×C₈	—	47	1	no	—
218	C₂²×C₈	—	47	2	no	—
219	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
222	C₄×Q₈	—	47	16	no	—
223	C₄×Q₈	—	47	4	no	—
224	C₄×Q₈	—	47	4	no	—
225	C₂×C₄⋊C₄	—	47	2	no	—
226	C₂×C₄⋊C₄	—	47	4	no	—
227	C₂×C₄⋊C₄	—	47	1	no	—
228	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
229	C₂×C₄²	—	47	4	no	—
230	C₂×C₄²	—	47	2	no	—
231	C₂×C₄²	—	47	2	no	—
232	C₂×C₄²	—	47	1	no	—
233	C₂×C₄²	—	47	1	no	—
234	C₂×C₄²	—	47	2	no	—
235	C₂×C₄²	—	47	4	no	—
236	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₄⋊C₈	—	47	8	no	—
239	C₄⋊C₈	—	47	8	no	—
240	C₄⋊C₈	—	47	4	no	—
241	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
242	C₄×C₈	—	4	2	no	—
243	C₄×C₈	—	47	8	no	—
244	C₄×C₈	—	47	8	no	—
245	C₄×C₈	—	47	4	no	—
246	C₄×C₈	—	188	2	no	—
247	C₄×C₈	—	188	2	no	—
248	C₄×C₈	C₄×D₄₇	1	4	yes	—
249	#95	—	47	4	no	—
250	#95	C₄×D₄₇	1	2	yes	—
251	C₂×Q₈	—	47	4	no	—
252	C₂×Q₈	—	47	8	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	47	2	no	—
256	C₂²×C₄	—	47	2	no	—
257	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
259	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	47	2	no	—
268	C₂×C₈	—	47	2	no	—
269	C₂×C₈	—	47	4	no	—
270	C₂×C₈	—	94	1	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	94	1	no	—
273	C₂×C₈	—	94	1	no	—
274	C₂×C₈	—	94	1	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
277	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
278	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
279	C₄⋊C₄	—	47	8	no	—
280	C₄⋊C₄	—	47	4	no	—
281	C₄⋊C₄	—	47	16	no	—
282	C₄⋊C₄	C₈×D₄₇	1	8	yes	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	47	16	no	—
286	C₄²	—	47	8	no	—
287	C₄²	—	47	4	no	—
288	C₄²	—	47	4	no	—
289	C₄²	—	94	1	no	—
290	C₄²	—	94	1	no	—
291	C₄²	—	94	1	no	—
292	C₄²	—	94	1	no	—
293	C₄²	—	188	4	no	—
294	C₄²	—	188	2	no	—
295	C₄²	C₂×C₄×D₄₇	1	2	yes	—
296	C₄²	C₂×C₄×D₄₇	1	2	yes	—
297	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	Q₈	—	47	16	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	47	4	no	—
303	C₂×C₄	—	47	8	no	—
304	C₂×C₄	—	47	4	no	—
305	C₂×C₄	—	47	4	no	—
306	C₂×C₄	—	47	4	no	—
307	C₂×C₄	—	94	2	no	—
308	C₂×C₄	—	94	2	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	94	4	no	—
312	C₂×C₄	C₄²×D₄₇	1	2	yes	—
313	C₂×C₄	C₄²×D₄₇	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₄₇	1	8	yes	—
315	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
316	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
318	C₂×C₄	#13094069	1	1	yes	characteristic
319	C₂×C₄	#13094071	1	1	yes	characteristic
320	C₂×C₄	#13094071	1	1	yes	characteristic
321	C₂×C₄	#13094072	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	4	no	—
324	C₈	—	2	2	no	—
325	C₈	—	94	2	no	—
326	C₈	—	94	2	no	—
327	C₈	—	94	4	no	—
328	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
329	C₂²	#15689599	1	1	yes	characteristic
330	C₂²	#15689602	1	1	yes	characteristic
331	C₂²	C₂²×C₈×D₄₇	1	2	yes	—
332	C₂²	#15689888	1	2	yes	—
333	C₄	—	2	8	no	—
334	C₄	—	2	2	no	—
335	C₄	—	47	8	no	—
336	C₄	—	94	4	no	—
337	C₄	—	94	4	no	—
338	C₄	—	94	4	no	—
339	C₄	C₄×C₈×D₄₇	1	4	yes	—
340	C₄	#15689599	1	1	yes	characteristic
341	C₄	#15689600	1	1	yes	characteristic
342	C₄	#15689602	1	1	yes	characteristic
343	C₄	#15689604	1	1	yes	characteristic
344	C₂	#20245752	1	1	yes	characteristic
345	C₂	C₂×C₄×C₈×D₄₇	1	1	yes	characteristic
346	C₂	#20251815	1	1	yes	characteristic
347	C₂	#20252467	1	2	yes	—
348	C₂	#20252495	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Fitting	C₈×C₄⋊C₄×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂³×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#20569
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₈×Dic₉₄	C₄	semidirect
15. #15689247	C₄	semidirect
46. #13093989	C₈	semidirect
146. Dic₉₄	C₄×C₈	semidirect
214. C₄₇	#20569	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉₄	2
C₂×#15689247	2
C₄×#13093989	2
C₄⋊C₄×C₄₇⋊C₈	2
C₈×Dic₄₇⋊C₄	2
#20243249	2
C₄⋊C₈×Dic₄₇	2
C₄×Dic₄₇⋊C₈	2
#20243133	2
#20243132	2
C₈×C₄⋊C₄×C₄₇	2
#20569	47