Subgroups — #26121696

Order 12032.

1170 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689107	C₂	1	2	yes	maximal
2	C₂×#15688888	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₇⋊#338	C₂	1	2	yes	maximal, complemented
4	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	#15689107	—	2	4	no	—
7	#15689107	C₄	1	8	yes	—
8	C₂×#13093943	—	2	2	no	—
9	C₂×#13093941	—	2	2	no	—
10	#15688888	—	2	4	no	—
11	C₄₇⋊#338	—	2	4	no	—
12	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#131×C₄₇	—	2	4	no	—
14	C₂×#131×C₄₇	—	2	4	no	—
15	C₂×#131×C₄₇	—	2	4	no	—
16	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
17	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₄₇	C₄	1	2	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	C₂×C₄	1	8	yes	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	#13093943	—	2	8	no	—
28	#13093941	—	2	8	no	—
29	C₂×#41×C₄₇	—	2	2	no	—
30	C₂×#41×C₄₇	—	2	2	no	—
31	C₂×#41×C₄₇	—	2	2	no	—
32	C₂×#41×C₄₇	—	2	2	no	—
33	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×#41×C₄₇	D₄	1	4	yes	—
35	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×D₄×C₄₇	Q₈	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	#131×C₄₇	—	2	32	no	—
41	C₂×M₁₆×C₄₇	—	2	4	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	4	no	—
45	C₂×M₁₆×C₄₇	—	2	4	no	—
46	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
47	C₂²×C₈×C₄₇	—	2	4	no	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	—	2	4	no	—
51	C₂²×C₈×C₄₇	—	2	4	no	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	D₄	1	4	yes	—
56	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
57	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	#41×C₄₇	—	2	8	no	—
59	#41×C₄₇	—	2	8	no	—
60	#41×C₄₇	—	2	8	no	—
61	#41×C₄₇	—	2	8	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	2	no	—
64	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	—	2	4	no	—
67	C₂×D₄×C₄₇	—	2	2	no	—
68	C₂×D₄×C₄₇	—	2	2	no	—
69	C₂×D₄×C₄₇	C₂²⋊C₄	1	4	yes	—
70	C₂×D₄×C₄₇	C₄⋊C₄	1	8	yes	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	M₁₆×C₄₇	—	2	16	no	—
87	M₁₆×C₄₇	—	2	8	no	—
88	C₂×C₈×C₄₇	—	2	16	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	1	no	—
95	C₂×C₈×C₄₇	—	2	8	no	—
96	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
98	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
99	#10625329	—	2	8	no	—
100	#10625329	—	2	8	no	—
101	#10625329	—	4	4	no	—
102	#10625329	—	4	4	no	—
103	C₂×C₄₇⋊C₈	—	2	8	no	—
104	C₂×C₄₇⋊C₈	—	4	4	no	—
105	C₂×C₄₇⋊C₈	—	4	4	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	4	2	no	—
109	C₂³×C₄₇	—	4	4	no	—
110	C₂³×C₄₇	#95	1	4	yes	—
111	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	Q₈×C₄₇	—	2	8	no	—
116	D₄×C₄₇	—	2	16	no	—
117	D₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	#95	1	4	yes	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
133	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
134	C₂×C₄×C₄₇	#120	1	4	yes	—
135	C₂×C₄×C₄₇	#121	1	2	yes	—
136	C₂×C₄×C₄₇	#121	1	2	yes	—
137	C₂×C₄×C₄₇	#123	1	4	yes	—
138	C₈×C₄₇	—	2	2	no	—
139	C₈×C₄₇	—	2	4	no	—
140	C₈×C₄₇	—	2	2	no	—
141	C₄₇⋊C₈	—	4	8	no	—
142	#10760	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	—	2	4	no	—
146	C₂²×C₄₇	—	4	2	no	—
147	C₂²×C₄₇	—	4	2	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	—	4	4	no	—
150	C₂²×C₄₇	—	4	4	no	—
151	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
152	C₂²×C₄₇	#380	1	1	yes	characteristic
153	C₂²×C₄₇	#380	1	1	yes	characteristic
154	C₂²×C₄₇	#385	1	4	yes	—
155	C₄×C₄₇	—	2	4	no	—
156	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₄₇	#379	1	1	yes	characteristic
159	C₄×C₄₇	#381	1	1	yes	characteristic
160	C₄×C₄₇	#386	1	4	yes	—
161	C₂²×#131	D₄₇	1	1	yes	characteristic
162	C₂×#413	—	47	2	no	—
163	C₂×#341	—	47	2	no	—
164	C₂×#338	—	47	2	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	4	no	—
168	C₂×C₄₇	—	4	4	no	—
169	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
170	C₂×C₄₇	#1813	1	2	yes	—
171	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
172	C₂×#131	—	2	4	no	—
173	C₂×#131	—	2	4	no	—
174	C₂×#131	—	2	4	no	—
175	C₂²×M₁₆	—	47	4	no	—
176	C₂²×M₁₆	Dic₄₇	1	2	yes	—
177	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
178	C₂³×C₈	Dic₄₇	1	2	yes	—
179	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
180	#413	—	47	8	no	—
181	#413	—	94	4	no	—
182	C₂×#101	—	94	2	no	—
183	C₂×#100	—	94	2	no	—
184	#341	—	94	4	no	—
185	#338	—	94	4	no	—
186	C₄₇	#10760	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	2	2	no	—
188	C₂×#41	—	2	2	no	—
189	C₂×#41	—	2	2	no	—
190	C₂×#41	—	2	2	no	—
191	C₂×#41	C₄×D₄₇	1	4	yes	—
192	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
193	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
194	C₂²×D₄	Dic₉₄	1	2	yes	—
195	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	#131	—	2	32	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	4	no	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	47	8	no	—
205	C₂×M₁₆	—	94	4	no	—
206	C₂×M₁₆	—	94	4	no	—
207	C₂×M₁₆	—	94	4	no	—
208	C₂×M₁₆	—	94	4	no	—
209	C₂×M₁₆	—	94	4	no	—
210	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
211	C₂²×C₈	—	2	4	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	—	94	4	no	—
217	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
219	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
221	#101	—	94	8	no	—
222	#100	—	94	8	no	—
223	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
224	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
225	#41	—	2	8	no	—
226	#41	—	2	8	no	—
227	#41	—	2	8	no	—
228	#41	—	2	8	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	Dic₄₇⋊C₄	1	8	yes	—
237	C₂×D₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
243	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
244	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
245	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
246	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
247	C₂²×C₄	#10625350	1	2	yes	—
248	C₂²×C₄	#10625350	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
252	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
253	M₁₆	—	2	16	no	—
254	M₁₆	—	2	8	no	—
255	M₁₆	—	94	8	no	—
256	M₁₆	—	94	8	no	—
257	M₁₆	—	188	4	no	—
258	M₁₆	—	188	4	no	—
259	C₂×C₈	—	2	16	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	8	no	—
267	C₂×C₈	—	94	8	no	—
268	C₂×C₈	—	188	4	no	—
269	C₂×C₈	—	188	4	no	—
270	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
271	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
272	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	4	no	—
277	C₂³	C₄₇⋊#95	1	4	yes	—
278	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
279	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
280	C₂³	#13094003	1	2	yes	—
281	C₂³	#13094003	1	2	yes	—
282	Q₈	—	2	8	no	—
283	D₄	—	2	16	no	—
284	D₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	C₄₇⋊#95	1	4	yes	—
292	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
294	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
297	C₂×C₄	#13093983	1	1	yes	characteristic
298	C₂×C₄	#13093983	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
300	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
301	C₂×C₄	#13093993	1	4	yes	—
302	C₂×C₄	#13093998	1	2	yes	—
303	C₂×C₄	#13093998	1	2	yes	—
304	C₂×C₄	#13094007	1	4	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	—	2	2	no	—
308	C₈	—	188	8	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
318	C₂²	#15688983	1	1	yes	characteristic
319	C₂²	#15688983	1	1	yes	characteristic
320	C₂²	#15688996	1	4	yes	—
321	C₄	—	2	4	no	—
322	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
323	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
324	C₄	#15688979	1	1	yes	characteristic
325	C₄	#15688986	1	1	yes	characteristic
326	C₄	#15689002	1	4	yes	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	4	no	—
330	C₂	—	4	4	no	—
331	C₂	#20241611	1	1	yes	characteristic
332	C₂	#20243438	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#10760
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#15688888	C₂	semidirect
3. C₂×C₄₇⋊#338	C₂	semidirect
186. C₄₇	#10760	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689107	2
C₂×#15688888	2
C₂×C₄₇⋊#338	2
C₂²×#131×C₄₇	2
#10760	47