Subgroups — #26182211

Order 12032.

1098 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	#20252473	C₂	1	2	yes	maximal, complemented
2	#20241732	C₂	1	4	yes	maximal, complemented
3	#20241722	C₂	1	4	yes	maximal
4	#20241712	C₂	1	2	yes	maximal, complemented
5	#20241702	C₂	1	2	yes	maximal
6	C₂×C₄₇×#519	C₂	1	1	yes	maximal, characteristic
7	#15689254	—	2	8	no	—
8	#15689180	C₂²	1	8	yes	—
9	#15689176	C₂²	1	4	yes	—
10	#15688849	C₂²	1	4	yes	—
11	#15688849	C₂²	1	8	yes	—
12	#15688847	C₂²	1	4	yes	—
13	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
14	C₄₇×#519	—	2	8	no	—
15	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
16	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₇	C₄	1	2	yes	—
22	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
23	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₄₇	C₂²	1	2	yes	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
27	C₂²⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
28	#13093935	—	2	8	no	—
29	C₄×C₄₇⋊C₈	—	2	4	no	—
30	C₂²×Q₈×C₄₇	—	2	4	no	—
31	C₂²×Q₈×C₄₇	—	2	4	no	—
32	C₂²×Q₈×C₄₇	—	2	4	no	—
33	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₄₇	—	2	4	no	—
39	C₄×Q₈×C₄₇	—	2	8	no	—
40	C₄×Q₈×C₄₇	—	2	4	no	—
41	C₄×Q₈×C₄₇	—	2	4	no	—
42	C₄×Q₈×C₄₇	—	2	8	no	—
43	C₄×Q₈×C₄₇	D₄	1	8	yes	—
44	#117×C₄₇	—	2	8	no	—
45	#117×C₄₇	—	2	4	no	—
46	#117×C₄₇	—	2	4	no	—
47	#117×C₄₇	—	2	8	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
66	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×Q₈×C₄₇	—	2	32	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	2	2	no	—
70	C₂×Q₈×C₄₇	—	2	4	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	—	2	2	no	—
74	C₂×Q₈×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
76	C₂×Q₈×C₄₇	M₁₆	1	4	yes	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	4	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	4	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	—	2	4	no	—
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	4	no	—
99	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
100	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
101	C₂²⋊C₄×C₄₇	—	2	2	no	—
102	C₂²⋊C₄×C₄₇	—	2	4	no	—
103	C₂²⋊C₄×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	—	2	4	no	—
105	C₄²×C₄₇	—	2	4	no	—
106	C₄²×C₄₇	—	2	4	no	—
107	C₄²×C₄₇	—	2	4	no	—
108	C₄²×C₄₇	—	2	2	no	—
109	C₄²×C₄₇	C₂×D₄	1	4	yes	—
110	C₄²×C₄₇	C₂×D₄	1	8	yes	—
111	C₂×C₄₇⋊C₈	—	2	8	no	—
112	C₂³×C₄₇	—	2	1	no	—
113	C₂³×C₄₇	—	2	1	no	—
114	C₂³×C₄₇	—	4	1	no	—
115	C₂³×C₄₇	—	4	1	no	—
116	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
117	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	Q₈×C₄₇	—	2	16	no	—
119	Q₈×C₄₇	—	2	8	no	—
120	Q₈×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	—	2	16	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	8	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	4	1	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	1	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
140	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
141	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
143	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
144	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
147	C₄₇⋊C₈	—	4	8	no	—
148	#17371	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	—	4	2	no	—
154	C₂²×C₄₇	—	4	1	no	—
155	C₂²×C₄₇	—	4	1	no	—
156	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
158	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×C₄₇	—	2	4	no	—
160	C₄×C₄₇	—	2	4	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	2	4	no	—
165	C₄×C₄₇	—	4	2	no	—
166	C₄×C₄₇	—	4	1	no	—
167	C₄×C₄₇	—	4	1	no	—
168	C₄×C₄₇	#407	1	4	yes	—
169	C₂×#519	D₄₇	1	1	yes	characteristic
170	#2983	—	47	2	no	—
171	#1467	—	47	4	no	—
172	#1464	—	47	4	no	—
173	#1461	—	47	2	no	—
174	#1458	—	47	2	no	—
175	C₂×C₄₇	—	2	1	no	—
176	C₂×C₄₇	—	2	1	no	—
177	C₂×C₄₇	—	4	2	no	—
178	C₂×C₄₇	C₂×#407	1	1	yes	characteristic
179	C₂×C₄₇	#2884	1	1	yes	characteristic
180	C₂×C₄₇	#2889	1	1	yes	characteristic
181	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
182	#519	—	2	8	no	—
183	C₂×C₄×Q₈	Dic₄₇	1	2	yes	—
184	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
185	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
186	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
187	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
188	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
189	C₂×#117	Dic₄₇	1	2	yes	—
190	C₂×#117	Dic₄₇	1	1	yes	characteristic
191	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
192	C₂×#117	C₂×D₄₇	1	2	yes	—
193	#446	—	94	8	no	—
194	#432	—	47	8	no	—
195	#431	—	47	4	no	—
196	#324	—	47	4	no	—
197	#324	—	47	8	no	—
198	#323	—	47	4	no	—
199	C₄₇	#17371	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	2	4	no	—
201	C₂²×Q₈	—	2	4	no	—
202	C₂²×Q₈	—	2	4	no	—
203	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
204	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
206	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
207	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
208	C₄×Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	8	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	8	no	—
213	C₄×Q₈	C₂²⋊D₄₇	1	8	yes	—
214	#117	—	2	8	no	—
215	#117	—	2	4	no	—
216	#117	—	2	4	no	—
217	#117	—	2	8	no	—
218	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
219	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
221	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
224	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
226	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
227	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
228	C₂×C₂²⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
230	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
231	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
232	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
233	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
236	C₄⋊C₈	—	94	4	no	—
237	C₄⋊C₈	—	94	8	no	—
238	C₂²⋊C₈	—	47	8	no	—
239	#97	—	94	8	no	—
240	C₄×C₈	—	94	4	no	—
241	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₂×Q₈	—	2	32	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	#10625329	1	4	yes	—
251	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	#10625329	1	4	yes	—
258	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
266	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
267	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
268	C₂×C₈	—	94	8	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
276	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	4	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	4	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
286	C₄²	C₂×C₂²⋊D₄₇	1	8	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	C₂×#10625329	1	2	yes	—
292	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
293	Q₈	—	2	16	no	—
294	Q₈	—	2	8	no	—
295	Q₈	—	2	8	no	—
296	C₂×C₄	—	2	16	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
314	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
315	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
316	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
317	C₂×C₄	C₂×#10625329	1	4	yes	—
318	C₂×C₄	C₂×#10625329	1	2	yes	—
319	C₂×C₄	C₂×#10625329	1	2	yes	—
320	C₂×C₄	C₂×#10625329	1	2	yes	—
321	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
322	C₈	—	188	8	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
331	C₂²	C₂²×#10625329	1	1	yes	characteristic
332	C₂²	C₂²×#10625329	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	#15689087	1	4	yes	—
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	C₂×#15689087	1	1	yes	characteristic
347	C₂	#20251852	1	1	yes	characteristic
348	C₂	#20251871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#519	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#519	D₄₇
hypocenter	C₄₇	#17371
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
1. #20252473	C₂	semidirect
2. #20241732	C₂	semidirect
4. #20241712	C₂	semidirect
199. C₄₇	#17371	semidirect

Maximal subgroups

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

subgroup	index
#20252473	2
#20241732	2
#20241722	2
#20241712	2
#20241702	2
C₂×C₄₇×#519	2
#17371	47