Subgroups — #26111923

Order 12032.

644 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251740	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688868	C₂	1	2	yes	maximal, complemented
4	#20241272	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×#13094003	—	2	1	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	#15689029	—	2	1	no	—
11	#15689021	—	2	1	no	—
12	#15689021	C₄	1	2	yes	—
13	#15688993	C₂²	1	1	yes	characteristic
14	#15688983	—	2	1	no	—
15	#15688983	C₄	1	2	yes	—
16	#15688878	—	2	1	no	—
17	#15688868	—	2	4	no	—
18	C₂³×C₈×C₄₇	—	2	1	no	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
23	C₂²×Dic₉₄	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	#13094024	—	2	2	no	—
27	#13094003	—	4	2	no	—
28	#13094003	—	4	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	2	no	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
39	C₄₇⋊#95	—	2	2	no	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	—	2	4	no	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	—	4	2	no	—
47	C₂²×C₈×C₄₇	—	4	2	no	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	—	2	4	no	—
55	C₄⋊C₈×C₄₇	—	2	2	no	—
56	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
57	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	4	1	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	D₈	1	2	yes	—
71	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂×Dic₉₄	—	4	2	no	—
80	C₂×Dic₉₄	—	4	1	no	—
81	C₂×Dic₉₄	—	4	1	no	—
82	C₂×Dic₉₄	—	4	2	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	4	8	no	—
85	C₂×C₈×C₄₇	—	4	2	no	—
86	C₂×C₈×C₄₇	—	4	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	1	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	1	no	—
91	C₄²×C₄₇	C₂×D₄	1	2	yes	—
92	C₄²×C₄₇	#41	1	2	yes	—
93	C₂²⋊Dic₄₇	—	4	4	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	4	1	no	—
96	C₄⋊Dic₄₇	—	4	1	no	—
97	C₄⋊Dic₄₇	—	4	1	no	—
98	C₄⋊Dic₄₇	—	4	1	no	—
99	C₄⋊Dic₄₇	—	4	1	no	—
100	C₄⋊Dic₄₇	—	4	1	no	—
101	C₄⋊Dic₄₇	—	4	2	no	—
102	Dic₄₇⋊C₄	—	4	4	no	—
103	C₂×C₄₇⋊C₈	—	4	2	no	—
104	C₂×C₄₇⋊C₈	—	4	2	no	—
105	C₂×C₄₇⋊C₈	—	4	2	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	—	2	1	no	—
108	C₂³×C₄₇	—	2	1	no	—
109	C₂³×C₄₇	—	2	1	no	—
110	C₂³×C₄₇	#101	1	2	yes	—
111	C₂³×C₄₇	#102	1	2	yes	—
112	C₂³×C₄₇	#103	1	2	yes	—
113	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	#102	1	2	yes	—
131	C₂×C₄×C₄₇	#103	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#121	1	2	yes	—
135	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#135	1	2	yes	—
141	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	Dic₉₄	—	8	2	no	—
150	Dic₉₄	—	8	2	no	—
151	C₈×C₄₇	—	4	4	no	—
152	C₄₇⋊C₈	—	8	2	no	—
153	#9129	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	—	2	2	no	—
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	2	2	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	#332	1	4	yes	complemented
161	C₂²×C₄₇	#386	1	1	yes	characteristic
162	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
163	C₂²×C₄₇	#413	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
166	C₂²×C₄₇	#416	1	1	yes	characteristic
167	C₂²×C₄₇	#417	1	1	yes	characteristic
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	4	1	no	—
172	C₄×C₄₇	#450	1	1	yes	characteristic
173	C₄×C₄₇	#451	1	1	yes	characteristic
174	C₄×C₄₇	#465	1	2	yes	—
175	C₄×C₄₇	#466	1	2	yes	—
176	C₄×C₄₇	#483	1	1	yes	characteristic
177	C₄×C₄₇	#484	1	1	yes	characteristic
178	Dic₄₇	—	8	2	no	—
179	Dic₄₇	—	8	2	no	—
180	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
181	#2857	—	47	1	no	—
182	#1797	—	47	1	no	—
183	C₂×#332	—	47	2	no	—
184	#1296	—	47	2	no	—
185	C₂×C₄₇	—	2	4	no	—
186	C₂×C₄₇	C₂×#332	1	1	yes	characteristic
187	C₂×C₄₇	#1542	1	1	yes	characteristic
188	C₂×C₄₇	#1547	1	1	yes	characteristic
189	C₂×C₄₇	#1552	1	1	yes	characteristic
190	C₂×C₄₇	#1872	1	1	yes	characteristic
191	C₂×C₄₇	#1893	1	1	yes	characteristic
192	C₂×C₄₇	#1901	1	1	yes	characteristic
193	C₂³×C₈	—	2	1	no	—
194	C₂×C₄⋊Q₈	—	47	1	no	—
195	C₂×C₂²⋊Q₈	—	94	1	no	—
196	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
197	C₂×C₄⋊C₈	—	2	1	no	—
198	C₂×C₄⋊C₈	—	2	4	no	—
199	C₂×C₄⋊C₈	—	47	2	no	—
200	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
201	C₂×C₂²⋊C₈	—	94	1	no	—
202	#391	—	94	1	no	—
203	#389	—	47	2	no	—
204	#389	—	94	1	no	—
205	#384	—	47	1	no	—
206	#380	—	47	2	no	—
207	#380	—	94	1	no	—
208	#336	—	94	1	no	—
209	#332	—	94	4	no	—
210	C₄₇	#9129	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₈	—	94	1	no	—
212	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
213	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
214	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	4	2	no	—
218	C₂²×C₈	—	4	2	no	—
219	C₂²×C₈	—	94	2	no	—
220	C₄⋊Q₈	—	94	2	no	—
221	C₄⋊Q₈	—	94	2	no	—
222	C₂²⋊Q₈	—	188	2	no	—
223	C₂²⋊Q₈	—	188	2	no	—
224	C₂×C₄⋊C₄	—	47	2	no	—
225	C₂×C₄⋊C₄	—	47	2	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	C₂×C₄⋊C₄	—	94	1	no	—
228	C₂×C₄⋊C₄	—	94	1	no	—
229	C₂×C₄⋊C₄	—	94	1	no	—
230	C₂×C₂²⋊C₄	—	94	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
235	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	C₄⋊C₈	—	2	4	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	94	2	no	—
240	C₄⋊C₈	—	94	2	no	—
241	C₄⋊C₈	C₂²⋊D₄₇	1	4	yes	—
242	C₂²⋊C₈	—	188	2	no	—
243	#95	—	94	2	no	—
244	#95	—	94	2	no	—
245	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
246	C₂×Q₈	—	188	2	no	—
247	C₂×Q₈	—	188	1	no	—
248	C₂×Q₈	—	188	1	no	—
249	C₂×Q₈	—	188	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	94	2	no	—
260	C₂²×C₄	—	94	2	no	—
261	C₂²×C₄	#10625325	1	2	yes	—
262	C₂²×C₄	D₃₇₆	1	2	yes	—
263	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
266	C₂²×C₄	#10625335	1	2	yes	—
267	C₂²×C₄	#10625337	1	2	yes	—
268	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	4	8	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	188	2	no	—
276	C₂×C₈	—	188	2	no	—
277	C₂×C₈	—	188	2	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄⋊C₄	—	188	4	no	—
281	C₄⋊C₄	—	188	1	no	—
282	C₄⋊C₄	—	188	1	no	—
283	C₄⋊C₄	—	188	1	no	—
284	C₄⋊C₄	—	188	1	no	—
285	C₄⋊C₄	—	188	1	no	—
286	C₄⋊C₄	—	188	1	no	—
287	C₂²⋊C₄	—	188	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	#10625350	1	2	yes	—
293	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	#13093942	1	2	yes	—
299	C₂³	#13093946	1	2	yes	—
300	C₂³	#13093947	1	2	yes	—
301	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
302	Q₈	—	376	2	no	—
303	Q₈	—	376	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	188	2	no	—
321	C₂×C₄	—	188	2	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	—	188	2	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	#13093946	1	2	yes	—
327	C₂×C₄	#13093947	1	2	yes	—
328	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
329	C₂×C₄	#13093993	1	1	yes	characteristic
330	C₂×C₄	#13093998	1	2	yes	—
331	C₂×C₄	#13094007	1	1	yes	characteristic
332	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
333	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
334	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
335	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
336	C₂×C₄	#13094046	1	2	yes	—
337	C₂×C₄	#13094050	1	1	yes	characteristic
338	C₂×C₄	#13094055	1	1	yes	characteristic
339	C₈	—	4	4	no	—
340	C₈	—	376	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	#15688868	1	4	yes	complemented
348	C₂²	#15689002	1	1	yes	characteristic
349	C₂²	C₂×#13093942	1	1	yes	characteristic
350	C₂²	#15689105	1	1	yes	characteristic
351	C₂²	C₂×#13093946	1	1	yes	characteristic
352	C₂²	C₂×#13093947	1	1	yes	characteristic
353	C₂²	#15689120	1	1	yes	characteristic
354	C₂²	#15689122	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	376	2	no	—
360	C₄	—	376	2	no	—
361	C₄	#15689274	1	1	yes	characteristic
362	C₄	#15689279	1	1	yes	characteristic
363	C₄	#15689352	1	2	yes	—
364	C₄	#15689359	1	2	yes	—
365	C₄	#15689453	1	1	yes	characteristic
366	C₄	#15689458	1	1	yes	characteristic
367	C₂	—	2	4	no	—
368	C₂	C₂×#15688868	1	1	yes	characteristic
369	C₂	#20242046	1	1	yes	characteristic
370	C₂	#20242063	1	1	yes	characteristic
371	C₂	#20242086	1	1	yes	characteristic
372	C₂	#20243695	1	1	yes	characteristic
373	C₂	#20243816	1	1	yes	characteristic
374	C₂	#20243856	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#9129
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#15688868	C₂	semidirect
4. #20241272	C₂	semidirect
160. C₂²×C₄₇	#332	semidirect
210. C₄₇	#9129	semidirect
347. C₂²	#15688868	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
#20243368	2
C₂×#15688868	2
#20241272	2
C₂²×C₄⋊C₈×C₄₇	2
#9129	47