Subgroups — #26220957

Order 12032.

1062 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	1	yes	maximal, characteristic
2	#20251807	C₂	1	1	yes	maximal, characteristic
3	#20244382	C₂	1	4	yes	maximal, complemented
4	#20244378	C₂	1	4	yes	maximal, complemented
5	#20243127	C₂	1	2	yes	maximal
6	C₂×#15688892	C₂	1	2	yes	maximal
7	C₂×C₄₇×#544	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625329	C₂²	1	2	yes	—
10	#15689150	—	2	4	no	—
11	#15689147	C₂²	1	4	yes	—
12	#15689079	—	2	4	no	—
13	C₄×#10625329	C₂²	1	4	yes	—
14	#15688892	C₂²	1	4	yes	—
15	#15688892	C₂²	1	8	yes	—
16	#15688892	C₂²	1	4	yes	—
17	C₄₇×#544	—	2	4	no	—
18	C₄₇×#544	—	2	4	no	—
19	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
20	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
21	C₂×#125×C₄₇	C₄	1	2	yes	—
22	C₂×#125×C₄₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₄₇	C₄	1	4	yes	—
24	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	8	no	—
31	C₂×#10625329	Q₈	1	8	yes	complemented
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂×#10625329	C₂³	1	4	yes	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
37	#13093935	—	2	4	no	—
38	C₄×C₄₇⋊C₈	—	2	4	no	—
39	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₄₇	—	2	8	no	—
43	C₄⋊Q₈×C₄₇	—	2	4	no	—
44	C₄⋊Q₈×C₄₇	—	2	4	no	—
45	#125×C₄₇	—	2	8	no	—
46	#125×C₄₇	—	2	4	no	—
47	#125×C₄₇	—	2	4	no	—
48	C₂²⋊Q₈×C₄₇	—	2	16	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	D₄	1	4	yes	—
52	#117×C₄₇	D₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
68	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	8	no	—
70	C₂×Q₈×C₄₇	—	8	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	8	2	no	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₄₇	#41	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	2	8	no	—
86	C₄⋊C₄×C₄₇	—	2	8	no	—
87	C₄⋊C₄×C₄₇	—	2	8	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	2	1	no	—
91	C₄⋊C₄×C₄₇	—	2	1	no	—
92	C₄⋊C₄×C₄₇	—	2	1	no	—
93	C₄⋊C₄×C₄₇	—	2	1	no	—
94	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
95	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
96	C₂²⋊C₄×C₄₇	—	2	4	no	—
97	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
98	C₄²×C₄₇	—	2	4	no	—
99	C₄²×C₄₇	C₂×D₄	1	4	yes	—
100	C₄²×C₄₇	C₂×D₄	1	4	yes	—
101	#10625329	—	2	8	no	—
102	#10625329	—	2	4	no	—
103	#10625329	—	2	8	no	—
104	#10625329	—	4	4	no	—
105	#10625329	C₂×Q₈	1	8	yes	—
106	C₂×C₄₇⋊C₈	—	2	8	no	—
107	C₂×C₄₇⋊C₈	—	2	4	no	—
108	C₂×C₄₇⋊C₈	—	2	4	no	—
109	C₂×C₄₇⋊C₈	—	4	4	no	—
110	C₂³×C₄₇	—	4	1	no	—
111	C₂³×C₄₇	—	4	1	no	—
112	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
114	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₄₇	—	8	8	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	8	4	no	—
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
129	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
130	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
131	C₂×C₄×C₄₇	#124	1	4	yes	—
132	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
137	C₄₇⋊C₈	—	2	8	no	—
138	C₄₇⋊C₈	—	4	4	no	—
139	#21338	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	4	4	no	—
145	C₂²×C₄₇	#391	1	4	yes	—
146	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	4	4	no	—
151	C₄×C₄₇	—	4	2	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	#391	1	4	yes	—
154	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
155	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
156	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×#544	D₄₇	1	1	yes	characteristic
158	#2905	—	47	1	no	—
159	#2873	—	47	1	no	—
160	#1998	—	47	4	no	—
161	#1997	—	47	4	no	—
162	#1742	—	47	2	no	—
163	C₂×#343	—	47	2	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	2	4	no	—
166	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
167	C₂×C₄₇	#2888	1	1	yes	characteristic
168	C₂×C₄₇	#2889	1	1	yes	characteristic
169	C₂²×M₁₆	—	47	2	no	—
170	C₂²×M₁₆	—	47	2	no	—
171	#544	—	2	4	no	—
172	#544	—	2	4	no	—
173	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
174	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
175	C₂×#125	Dic₄₇	1	2	yes	—
176	C₂×#125	C₂×D₄₇	1	2	yes	—
177	C₂×C₂²⋊Q₈	Dic₄₇	1	4	yes	—
178	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
179	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
181	#424	—	94	4	no	—
182	#423	—	47	4	no	—
183	#405	—	94	4	no	—
184	C₄×M₁₆	—	47	4	no	—
185	#343	—	47	8	no	—
186	#343	—	47	4	no	—
187	#343	—	47	4	no	—
188	C₄₇	#21338	1	1	yes	characteristic, complemented, Sylow
189	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
190	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
191	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	C₂×M₁₆	—	47	8	no	—
193	C₂×M₁₆	—	47	4	no	—
194	C₂×M₁₆	—	47	4	no	—
195	C₂×M₁₆	—	94	4	no	—
196	C₂×M₁₆	—	94	2	no	—
197	C₂×M₁₆	—	94	2	no	—
198	C₂×M₁₆	—	94	8	no	—
199	C₂²×C₈	—	94	2	no	—
200	C₂²×C₈	—	94	2	no	—
201	C₄⋊Q₈	—	2	8	no	—
202	C₄⋊Q₈	—	2	4	no	—
203	C₄⋊Q₈	—	2	4	no	—
204	#125	—	2	8	no	—
205	#125	—	2	4	no	—
206	#125	—	2	4	no	—
207	C₂²⋊Q₈	—	2	16	no	—
208	#117	—	2	2	no	—
209	#117	—	2	2	no	—
210	#117	C₂²⋊D₄₇	1	4	yes	—
211	#117	C₂²⋊D₄₇	1	4	yes	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
221	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
224	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
225	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
226	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
227	C₄⋊C₈	—	94	8	no	—
228	#97	—	94	4	no	—
229	C₄×C₈	—	94	4	no	—
230	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₂×Q₈	—	2	8	no	—
232	C₂×Q₈	—	8	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	8	2	no	—
235	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
236	C₂²×C₄	#10625354	1	4	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
238	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
239	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
240	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
244	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
245	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
246	M₁₆	—	47	8	no	—
247	M₁₆	—	94	8	no	—
248	M₁₆	—	94	4	no	—
249	M₁₆	—	94	8	no	—
250	M₁₆	—	188	4	no	—
251	C₂×C₈	—	94	8	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	188	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
266	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
269	C₄²	—	2	4	no	—
270	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
271	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
275	C₂³	Q₈×Dic₄₇	1	4	yes	—
276	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
277	Q₈	—	8	8	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
288	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
289	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
290	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
291	C₂×C₄	#13094000	1	4	yes	—
292	C₂×C₄	#13094000	1	4	yes	—
293	C₂×C₄	#13094013	1	4	yes	—
294	C₂×C₄	#13094022	1	4	yes	—
295	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
298	C₂×C₄	C₂×#10625354	1	2	yes	—
299	C₈	—	94	8	no	—
300	C₈	—	188	4	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	4	no	—
306	C₂²	#15689030	1	4	yes	—
307	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
308	C₂²	C₂×Q₈×Dic₄₇	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	#15689030	1	4	yes	—
315	C₄	C₂×#13094000	1	2	yes	—
316	C₄	C₂×#13094013	1	1	yes	characteristic
317	C₄	C₂×#13094022	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	—	2	2	no	—
320	C₂	C₂×#15689030	1	1	yes	characteristic
321	C₂	#20251867	1	1	yes	characteristic
322	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×#544	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#544	D₄₇
hypocenter	C₄₇	#21338
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#544

Complements

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

N	H ≅ G/N	type
3. #20244382	C₂	semidirect
4. #20244378	C₂	semidirect
31. C₂×#10625329	Q₈	semidirect
188. C₄₇	#21338	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
#20251807	2
#20244382	2
#20244378	2
#20243127	2
C₂×#15688892	2
C₂×C₄₇×#544	2
#21338	47