Subgroups — #26125180

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688886	C₂	1	2	yes	maximal
3	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
4	#20241618	C₂	1	2	yes	maximal
5	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×#13093958	—	2	2	no	—
9	C₂×#13093958	—	2	2	no	—
10	C₂×#13093958	C₄	1	4	yes	complemented
11	C₂×C₄₇⋊#95	—	2	4	no	—
12	#15688886	—	2	4	no	—
13	#15688886	—	2	2	no	—
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#336	—	2	2	no	—
18	C₂³×Dic₄₇	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	Q₈	1	4	yes	—
32	#13093958	—	2	8	no	—
33	C₄₇⋊#95	—	4	8	no	—
34	C₂³×C₄×C₄₇	D₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	—	2	1	no	—
39	C₂²×C₈×C₄₇	—	2	1	no	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₄₇	D₄	1	2	yes	—
44	C₂²×C₈×C₄₇	D₄	1	2	yes	—
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	—	2	1	no	—
47	C₂×C₄²×C₄₇	—	2	2	no	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	1	no	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	D₈	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	4	2	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	—	4	1	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
86	C₄²×C₄₇	—	4	1	no	—
87	C₄²×C₄₇	—	4	1	no	—
88	C₄²×C₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	—	4	2	no	—
95	C₄⋊Dic₄₇	—	4	2	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₄₇	#121	1	1	yes	characteristic
99	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₄₇	#123	1	2	yes	—
101	C₂³×C₄₇	#127	1	1	yes	characteristic
102	C₂³×C₄₇	C₂×D₈	1	2	yes	—
103	C₂³×C₄₇	#135	1	2	yes	—
104	C₂³×C₄₇	#137	1	4	yes	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#117	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	2	yes	—
121	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#125	1	2	yes	—
124	C₂×C₄×C₄₇	#125	1	2	yes	—
125	C₂×C₄×C₄₇	#126	1	2	yes	—
126	C₂×C₄×C₄₇	#126	1	2	yes	—
127	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	8	no	—
131	C₂×Dic₄₇	—	4	8	no	—
132	C₂×Dic₄₇	—	8	4	no	—
133	C₂×Dic₄₇	—	8	4	no	—
134	C₂×Dic₄₇	—	8	2	no	—
135	C₂×Dic₄₇	—	8	2	no	—
136	C₂×Dic₄₇	—	8	2	no	—
137	C₂×Dic₄₇	—	8	2	no	—
138	C₈×C₄₇	—	2	4	no	—
139	C₈×C₄₇	—	4	2	no	—
140	#11379	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	#387	1	2	yes	—
142	C₂²×C₄₇	#390	1	1	yes	characteristic
143	C₂²×C₄₇	#397	1	1	yes	characteristic
144	C₂²×C₄₇	#398	1	1	yes	characteristic
145	C₂²×C₄₇	#399	1	2	yes	—
146	C₂²×C₄₇	C₄×D₈	1	2	yes	—
147	C₂²×C₄₇	#441	1	2	yes	—
148	C₂²×C₄₇	#466	1	2	yes	—
149	C₂²×C₄₇	#470	1	2	yes	—
150	C₂²×C₄₇	#474	1	4	yes	—
151	C₂²×C₄₇	#479	1	4	yes	—
152	C₂²×C₄₇	#480	1	4	yes	—
153	C₂²×C₄₇	#485	1	4	yes	—
154	C₂²×C₄₇	#493	1	1	yes	characteristic
155	C₂²×C₄₇	#495	1	1	yes	characteristic
156	C₂²×C₄₇	#497	1	2	yes	—
157	C₄×C₄₇	—	4	4	no	—
158	C₄×C₄₇	#381	1	1	yes	characteristic
159	C₄×C₄₇	#387	1	2	yes	—
160	C₄×C₄₇	#391	1	1	yes	characteristic
161	C₄×C₄₇	#397	1	1	yes	characteristic
162	C₄×C₄₇	#398	1	1	yes	characteristic
163	C₄×C₄₇	#400	1	2	yes	—
164	Dic₄₇	—	8	4	no	—
165	Dic₄₇	—	8	4	no	—
166	C₂²×#107	—	47	1	no	—
167	C₂×#340	—	47	2	no	—
168	C₂×#340	—	47	1	no	—
169	C₂×#336	D₄₇	1	1	yes	characteristic
170	C₄⋊#95	—	47	2	no	—
171	C₂×C₄₇	#1445	1	1	yes	characteristic
172	C₂×C₄₇	#1918	1	4	yes	—
173	C₂×C₄₇	#1970	1	1	yes	characteristic
174	C₂×C₄₇	#1971	1	1	yes	characteristic
175	C₂×C₄₇	#2075	1	2	yes	—
176	C₂×C₄₇	#2086	1	2	yes	—
177	C₂×C₄₇	#2093	1	2	yes	—
178	C₂×C₄₇	#2097	1	2	yes	—
179	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
180	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	—	47	2	no	—
182	C₂²×C₄⋊C₄	—	47	2	no	—
183	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
184	C₂×#107	—	47	4	no	—
185	C₂×#107	—	94	2	no	—
186	C₂×#107	—	94	2	no	—
187	C₂×#95	—	94	4	no	—
188	#340	—	94	4	no	—
189	#340	—	94	2	no	—
190	#336	—	2	2	no	—
191	C₄₇	#11379	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	94	2	no	—
193	C₂³×C₄	—	94	2	no	—
194	C₂³×C₄	D₁₈₈	1	2	yes	—
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	C₄×D₄₇	1	2	yes	—
203	C₂²×C₈	D₁₈₈	1	2	yes	—
204	C₂²×C₈	D₁₈₈	1	2	yes	—
205	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	47	4	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	—	94	2	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	—	94	2	no	—
215	C₂×C₄⋊C₄	—	94	2	no	—
216	C₂×C₄⋊C₄	—	94	2	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	1	no	—
222	#107	—	94	8	no	—
223	#95	—	188	8	no	—
224	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	94	4	no	—
230	C₂²×C₄	—	94	4	no	—
231	C₂²×C₄	—	188	2	no	—
232	C₂²×C₄	—	188	2	no	—
233	C₂²×C₄	—	188	2	no	—
234	C₂²×C₄	—	188	2	no	—
235	C₂²×C₄	—	188	4	no	—
236	C₂²×C₄	—	188	4	no	—
237	C₂²×C₄	—	188	4	no	—
238	C₂²×C₄	—	188	4	no	—
239	C₂²×C₄	D₃₇₆	1	4	yes	—
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂²×C₄	#10625350	1	1	yes	characteristic
244	C₂²×C₄	#10625350	1	1	yes	characteristic
245	C₂²×C₄	#10625352	1	2	yes	—
246	C₂²×C₄	#10625352	1	2	yes	—
247	C₂²×C₄	#10625352	1	2	yes	—
248	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
260	C₄⋊C₄	—	94	4	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄⋊C₄	—	94	4	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₄⋊C₄	—	188	2	no	—
265	C₄⋊C₄	—	188	2	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₄⋊C₄	—	188	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	2	no	—
271	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
272	C₂³	#13093998	1	1	yes	characteristic
273	C₂³	#13094004	1	2	yes	—
274	C₂³	#13094008	1	2	yes	—
275	C₂³	#13094021	1	1	yes	characteristic
276	C₂³	C₂×D₃₇₆	1	2	yes	—
277	C₂³	#13094046	1	2	yes	—
278	C₂³	#13094057	1	4	yes	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	376	2	no	—
285	C₂×C₄	—	376	2	no	—
286	C₂×C₄	—	376	2	no	—
287	C₂×C₄	—	376	2	no	—
288	C₂×C₄	—	376	4	no	—
289	C₂×C₄	—	376	4	no	—
290	C₂×C₄	—	188	4	no	—
291	C₂×C₄	—	188	4	no	—
292	C₂×C₄	—	188	8	no	—
293	C₂×C₄	—	188	8	no	—
294	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
295	C₂×C₄	#13093980	1	2	yes	—
296	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
297	C₂×C₄	#13093989	1	2	yes	—
298	C₂×C₄	#13093998	1	1	yes	characteristic
299	C₂×C₄	#13094004	1	2	yes	—
300	C₂×C₄	#13094004	1	2	yes	—
301	C₂×C₄	#13094005	1	1	yes	characteristic
302	C₂×C₄	#13094005	1	1	yes	characteristic
303	C₂×C₄	#13094008	1	2	yes	—
304	C₂×C₄	#13094008	1	2	yes	—
305	C₂×C₄	#13094011	1	1	yes	characteristic
306	C₂×C₄	#13094012	1	1	yes	characteristic
307	C₂×C₄	#13094014	1	2	yes	—
308	C₂×C₄	#13094014	1	2	yes	—
309	C₂×C₄	#13094017	1	2	yes	—
310	C₂×C₄	#13094017	1	2	yes	—
311	C₂×C₄	#13094022	1	1	yes	characteristic
312	C₈	—	2	4	no	—
313	C₈	—	4	2	no	—
314	C₂²	#15689008	1	2	yes	—
315	C₂²	#15689025	1	1	yes	characteristic
316	C₂²	#15689048	1	1	yes	characteristic
317	C₂²	#15689053	1	1	yes	characteristic
318	C₂²	#15689060	1	2	yes	—
319	C₂²	C₄×D₃₇₆	1	2	yes	—
320	C₂²	#15689231	1	2	yes	—
321	C₂²	#15689359	1	2	yes	—
322	C₂²	#15689381	1	2	yes	—
323	C₂²	#15689404	1	4	yes	—
324	C₂²	#15689431	1	4	yes	—
325	C₂²	#15689437	1	4	yes	—
326	C₂²	#15689465	1	4	yes	—
327	C₂²	#15689500	1	1	yes	characteristic
328	C₂²	#15689507	1	1	yes	characteristic
329	C₂²	#15689517	1	2	yes	—
330	C₄	—	4	4	no	—
331	C₄	—	376	4	no	—
332	C₄	—	376	4	no	—
333	C₄	#15688984	1	1	yes	characteristic
334	C₄	#15689008	1	2	yes	—
335	C₄	#15689028	1	1	yes	characteristic
336	C₄	#15689048	1	1	yes	characteristic
337	C₄	#15689053	1	1	yes	characteristic
338	C₄	#15689063	1	2	yes	—
339	C₂	#20241655	1	1	yes	characteristic
340	C₂	#20243960	1	4	yes	—
341	C₂	#20244254	1	1	yes	characteristic
342	C₂	#20244259	1	1	yes	characteristic
343	C₂	#20244811	1	2	yes	—
344	C₂	#20244876	1	2	yes	—
345	C₂	#20244915	1	2	yes	—
346	C₂	#20244935	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#11379
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
10. C₂×#13093958	C₄	semidirect
191. C₄₇	#11379	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093958	2
C₂×#15688886	2
#20241618	2
C₂×C₄₇×#336	2
#11379	47