Subgroups — #26111956

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	#20251740	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688871	C₂	1	2	yes	maximal, complemented
4	#20241272	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×#13094003	C₄	1	2	yes	—
8	#15689180	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
11	C₂×#13093935	—	2	1	no	—
12	#15689029	C₄	1	2	yes	—
13	#15689021	—	2	1	no	—
14	#15689021	—	2	1	no	—
15	#15688993	C₂²	1	1	yes	characteristic
16	#15688983	—	2	1	no	—
17	#15688983	—	2	1	no	—
18	#15688871	—	2	4	no	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
22	C₄₇×#336	—	2	1	no	—
23	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	#13094024	—	2	2	no	—
27	#13094003	—	4	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	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×Dic₄₇⋊C₄	—	2	2	no	—
34	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
38	#13093935	—	2	4	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	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₇	—	2	2	no	—
52	C₄⋊C₈×C₄₇	—	2	2	no	—
53	C₂²⋊C₈×C₄₇	—	4	2	no	—
54	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	4	1	no	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂×Dic₉₄	—	2	4	no	—
75	C₂×Dic₉₄	—	4	2	no	—
76	C₂×Dic₉₄	—	4	2	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	C₂×D₄	1	2	yes	—
85	C₄²×C₄₇	#41	1	2	yes	—
86	C₂²⋊Dic₄₇	—	4	4	no	—
87	C₄⋊Dic₄₇	—	4	1	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	1	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	4	1	no	—
92	C₄⋊Dic₄₇	—	4	1	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	Dic₄₇⋊C₄	—	4	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	—	2	1	no	—
104	C₂³×C₄₇	#100	1	2	yes	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
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	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#121	1	2	yes	—
125	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#135	1	2	yes	—
127	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#137	1	2	yes	—
129	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#137	1	2	yes	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	Dic₉₄	—	8	4	no	—
138	C₈×C₄₇	—	8	2	no	—
139	C₄₇⋊C₈	—	4	4	no	—
140	#9136	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	4	1	no	—
145	C₂²×C₄₇	—	4	1	no	—
146	C₂²×C₄₇	—	4	1	no	—
147	C₂²×C₄₇	—	4	1	no	—
148	C₂²×C₄₇	#386	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
150	C₂²×C₄₇	#413	1	1	yes	characteristic
151	C₂²×C₄₇	#416	1	1	yes	characteristic
152	C₂²×C₄₇	#417	1	1	yes	characteristic
153	C₂²×C₄₇	#419	1	1	yes	characteristic
154	C₂²×C₄₇	#419	1	1	yes	characteristic
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	4	1	no	—
157	C₄×C₄₇	—	4	1	no	—
158	C₄×C₄₇	—	4	1	no	—
159	C₄×C₄₇	#452	1	2	yes	—
160	C₄×C₄₇	#468	1	2	yes	—
161	C₄×C₄₇	#470	1	2	yes	—
162	C₄×C₄₇	#485	1	2	yes	—
163	Dic₄₇	—	8	2	no	—
164	Dic₄₇	—	8	2	no	—
165	C₂×#432	—	47	1	no	—
166	#2857	—	47	1	no	—
167	#1797	D₄₇	1	1	yes	characteristic
168	C₂×#333	—	47	2	no	—
169	#1296	—	47	2	no	—
170	C₂×C₄₇	—	4	2	no	—
171	C₂×C₄₇	#1551	1	1	yes	characteristic
172	C₂×C₄₇	#1554	1	1	yes	characteristic
173	C₂×C₄₇	#1558	1	1	yes	characteristic
174	C₂×C₄₇	#1559	1	1	yes	characteristic
175	C₂×C₄₇	#1873	1	1	yes	characteristic
176	C₂×C₄₇	#1894	1	1	yes	characteristic
177	C₂×C₄₇	#1900	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	—	47	1	no	—
179	C₂×C₂²⋊Q₈	—	47	2	no	—
180	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
181	#432	—	94	4	no	—
182	C₂×C₄⋊C₈	—	47	2	no	—
183	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
184	C₂×C₂²⋊C₈	—	2	1	no	—
185	C₂×C₂²⋊C₈	—	94	1	no	—
186	C₂×#97	—	94	1	no	—
187	#391	—	47	2	no	—
188	#389	—	94	1	no	—
189	#389	—	94	1	no	—
190	#384	—	47	1	no	—
191	#380	—	94	1	no	—
192	#380	—	94	1	no	—
193	#336	—	2	1	no	—
194	#333	—	94	4	no	—
195	C₄₇	#9136	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	47	2	no	—
197	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
198	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	C₄⋊Q₈	—	94	2	no	—
203	C₄⋊Q₈	—	94	2	no	—
204	C₂²⋊Q₈	—	188	4	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	1	no	—
208	C₂×C₄⋊C₄	—	94	1	no	—
209	C₂×C₄⋊C₄	—	94	1	no	—
210	C₂×C₄⋊C₄	—	94	1	no	—
211	C₂×C₂²⋊C₄	—	94	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	47	4	no	—
221	C₄⋊C₈	—	94	2	no	—
222	C₂²⋊C₈	—	4	2	no	—
223	C₂²⋊C₈	—	94	4	no	—
224	#97	—	94	4	no	—
225	#95	—	94	2	no	—
226	#95	—	94	2	no	—
227	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
228	C₂×Q₈	—	94	4	no	—
229	C₂×Q₈	—	188	2	no	—
230	C₂×Q₈	—	188	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	94	2	no	—
243	C₂²×C₄	—	94	2	no	—
244	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	188	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₄⋊C₄	—	188	2	no	—
257	C₄⋊C₄	—	188	4	no	—
258	C₄⋊C₄	—	188	1	no	—
259	C₄⋊C₄	—	188	2	no	—
260	C₄⋊C₄	—	188	1	no	—
261	C₄⋊C₄	—	188	2	no	—
262	C₄⋊C₄	—	188	1	no	—
263	C₄⋊C₄	—	188	1	no	—
264	C₂²⋊C₄	—	188	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	D₄×D₄₇	1	2	yes	—
270	C₄²	#10625352	1	2	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	#13093940	1	2	yes	—
278	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
279	Q₈	—	376	4	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	188	2	no	—
297	C₂×C₄	—	188	2	no	—
298	C₂×C₄	—	188	2	no	—
299	C₂×C₄	—	188	2	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
303	C₂×C₄	#13093994	1	1	yes	characteristic
304	C₂×C₄	#13093996	1	2	yes	—
305	C₂×C₄	#13094008	1	1	yes	characteristic
306	C₂×C₄	#13094046	1	2	yes	—
307	C₂×C₄	#13094050	1	1	yes	characteristic
308	C₂×C₄	#13094054	1	2	yes	—
309	C₂×C₄	#13094055	1	1	yes	characteristic
310	C₂×C₄	#13094057	1	2	yes	—
311	C₈	—	8	2	no	—
312	C₈	—	188	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	#15689000	1	1	yes	characteristic
321	C₂²	C₂×#13093940	1	1	yes	characteristic
322	C₂²	#15689105	1	1	yes	characteristic
323	C₂²	#15689120	1	1	yes	characteristic
324	C₂²	#15689122	1	1	yes	characteristic
325	C₂²	#15689131	1	1	yes	characteristic
326	C₂²	#15689133	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	376	2	no	—
332	C₄	—	376	2	no	—
333	C₄	#15689287	1	2	yes	—
334	C₄	#15689367	1	2	yes	—
335	C₄	#15689379	1	2	yes	—
336	C₄	#15689465	1	2	yes	—
337	C₂	—	4	2	no	—
338	C₂	#20242081	1	1	yes	characteristic
339	C₂	#20242096	1	1	yes	characteristic
340	C₂	#20242113	1	1	yes	characteristic
341	C₂	#20242116	1	1	yes	characteristic
342	C₂	#20243698	1	1	yes	characteristic
343	C₂	#20243819	1	1	yes	characteristic
344	C₂	#20243849	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	C₄₇×#1797	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#9136
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#15688871	C₂	semidirect
4. #20241272	C₂	semidirect
195. C₄₇	#9136	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
#20251740	2
C₂×#15688871	2
#20241272	2
C₄₇×#1797	2
#9136	47