Subgroups — #26122039

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688891	C₂	1	2	yes	maximal
2	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
3	#20243109	C₂	1	2	yes	maximal
4	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	#15688891	C₄	1	8	yes	complemented
9	#15688888	—	2	2	no	—
10	#15688886	—	2	1	no	—
11	#15688886	—	2	1	no	—
12	#15688885	—	2	2	no	—
13	#15688885	—	2	2	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#405	—	2	2	no	—
18	C₄₇×#405	C₄	1	4	yes	complemented
19	C₂×#97×C₄₇	—	2	1	no	—
20	C₂×C₄×C₈×C₄₇	—	2	1	no	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	C₂×C₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	1	no	—
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₇	—	2	1	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	#117×C₄₇	—	2	1	no	—
48	#117×C₄₇	—	2	1	no	—
49	#117×C₄₇	C₂×C₄	1	2	yes	—
50	#117×C₄₇	Q₈	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₄²×C₄₇	—	2	1	no	—
55	#97×C₄₇	—	2	4	no	—
56	C₄×C₈×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	8	2	no	—
74	C₂²×Dic₄₇	—	8	2	no	—
75	M₁₆×C₄₇	—	2	2	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	C₄²	1	4	yes	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₄₇	—	2	1	no	—
88	C₄⋊C₄×C₄₇	—	2	1	no	—
89	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
91	C₂²⋊C₄×C₄₇	—	2	2	no	—
92	C₂²⋊C₄×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	—	4	2	no	—
98	C₄⋊Dic₄₇	—	4	2	no	—
99	#10625329	—	2	4	no	—
100	#10625329	—	2	4	no	—
101	#10625329	—	4	2	no	—
102	#10625329	—	4	2	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	—	4	2	no	—
105	C₂×C₄₇⋊C₈	—	4	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	1	no	—
108	C₂³×C₄₇	—	2	1	no	—
109	C₂³×C₄₇	#117	1	2	yes	—
110	C₂³×C₄₇	#117	1	1	yes	characteristic
111	C₂³×C₄₇	#121	1	1	yes	characteristic
112	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂³×C₄₇	#126	1	2	yes	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	1	no	—
118	C₂×C₄×C₄₇	—	4	1	no	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	2	yes	—
123	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#123	1	2	yes	—
133	C₂×C₄×C₄₇	#123	1	2	yes	—
134	C₂×C₄×C₄₇	#124	1	2	yes	—
135	C₂×C₄×C₄₇	#125	1	2	yes	—
136	C₂×C₄×C₄₇	#126	1	2	yes	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	8	4	no	—
139	C₂×Dic₄₇	—	8	2	no	—
140	C₂×Dic₄₇	—	8	2	no	—
141	C₂×Dic₄₇	—	8	4	no	—
142	C₈×C₄₇	—	2	4	no	—
143	C₈×C₄₇	—	2	2	no	—
144	C₈×C₄₇	—	2	2	no	—
145	C₄₇⋊C₈	—	4	4	no	—
146	#10827	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	2	2	no	—
155	C₂²×C₄₇	#376	1	1	yes	characteristic
156	C₂²×C₄₇	#380	1	1	yes	characteristic
157	C₂²×C₄₇	#383	1	1	yes	characteristic
158	C₂²×C₄₇	#387	1	2	yes	—
159	C₂²×C₄₇	#388	1	2	yes	—
160	C₂²×C₄₇	#471	1	2	yes	—
161	C₂²×C₄₇	#472	1	1	yes	characteristic
162	C₂²×C₄₇	#473	1	1	yes	characteristic
163	C₄×C₄₇	—	4	2	no	—
164	C₄×C₄₇	—	4	2	no	—
165	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₄×C₄₇	#381	1	1	yes	characteristic
168	C₄×C₄₇	#382	1	1	yes	characteristic
169	C₄×C₄₇	#387	1	2	yes	—
170	C₄×C₄₇	#388	1	2	yes	—
171	Dic₄₇	—	8	4	no	—
172	C₂×#405	D₄₇	1	1	yes	characteristic
173	C₂×#343	—	47	2	no	—
174	C₂×#341	—	47	1	no	—
175	#1736	—	47	2	no	—
176	C₂×#340	—	47	1	no	—
177	C₂×C₄₇	—	2	1	no	—
178	C₂×C₄₇	—	2	1	no	—
179	C₂×C₄₇	—	2	2	no	—
180	C₂×C₄₇	#1434	1	1	yes	characteristic
181	C₂×C₄₇	#1815	1	1	yes	characteristic
182	C₂×C₄₇	#1816	1	1	yes	characteristic
183	C₂×C₄₇	#1930	1	2	yes	—
184	C₂×C₄₇	#1931	1	2	yes	—
185	C₂²×M₁₆	—	47	2	no	—
186	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
187	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
188	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
189	C₂²×C₄⋊C₄	—	47	2	no	—
190	#405	—	2	2	no	—
191	#405	Dic₄₇	1	4	yes	complemented
192	C₂×#97	—	2	1	no	—
193	C₂×C₄×C₈	—	2	1	no	—
194	#343	—	47	8	no	—
195	#341	—	94	2	no	—
196	#340	—	94	1	no	—
197	#340	—	94	1	no	—
198	#340	—	94	2	no	—
199	#340	—	94	2	no	—
200	C₄₇	#10827	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₄	—	94	2	no	—
202	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	47	4	no	—
206	C₂×M₁₆	—	94	2	no	—
207	C₂×M₁₆	—	94	2	no	—
208	C₂×M₁₆	—	94	2	no	—
209	C₂×M₁₆	—	94	2	no	—
210	C₂×M₁₆	—	94	2	no	—
211	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
212	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	94	2	no	—
217	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
218	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
221	#117	—	2	1	no	—
222	#117	—	2	1	no	—
223	#117	Dic₉₄	1	2	yes	—
224	#117	C₂×Dic₄₇	1	2	yes	—
225	C₂×C₄⋊C₄	—	47	4	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	C₂×C₄⋊C₄	—	94	2	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	94	2	no	—
231	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
232	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
233	C₂×C₂²⋊C₄	—	2	1	no	—
234	C₂×C₄²	—	2	1	no	—
235	#97	—	2	4	no	—
236	C₄×C₈	—	2	4	no	—
237	C₂⁴	#10625350	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	94	4	no	—
241	C₂²×C₄	—	376	2	no	—
242	C₂²×C₄	—	376	2	no	—
243	C₂²×C₄	—	188	2	no	—
244	C₂²×C₄	—	188	2	no	—
245	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
248	C₂²×C₄	#10625350	1	1	yes	characteristic
249	C₂²×C₄	#10625350	1	2	yes	—
250	C₂²×C₄	#10625350	1	2	yes	—
251	C₂²×C₄	#10625350	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	94	4	no	—
258	M₁₆	—	94	4	no	—
259	M₁₆	—	188	2	no	—
260	M₁₆	—	188	2	no	—
261	M₁₆	C₄×Dic₄₇	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	94	4	no	—
270	C₂×C₈	—	188	2	no	—
271	C₂×C₈	—	188	2	no	—
272	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
273	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	94	4	no	—
277	C₄⋊C₄	—	94	4	no	—
278	C₄⋊C₄	—	188	2	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
281	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#13093981	1	2	yes	—
290	C₂³	#13093983	1	1	yes	characteristic
291	C₂³	#13093998	1	1	yes	characteristic
292	C₂³	#13094003	1	1	yes	characteristic
293	C₂³	#13094018	1	2	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	376	4	no	—
300	C₂×C₄	—	376	2	no	—
301	C₂×C₄	—	376	2	no	—
302	C₂×C₄	—	376	4	no	—
303	C₂×C₄	—	188	4	no	—
304	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
307	C₂×C₄	#13093981	1	2	yes	—
308	C₂×C₄	#13093983	1	1	yes	characteristic
309	C₂×C₄	#13093983	1	1	yes	characteristic
310	C₂×C₄	#13093983	1	1	yes	characteristic
311	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
312	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
313	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
314	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
315	C₂×C₄	#13093998	1	1	yes	characteristic
316	C₂×C₄	#13094003	1	1	yes	characteristic
317	C₂×C₄	#13094007	1	2	yes	—
318	C₂×C₄	#13094007	1	2	yes	—
319	C₂×C₄	#13094012	1	2	yes	—
320	C₂×C₄	#13094016	1	2	yes	—
321	C₂×C₄	#13094018	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	188	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	#15688969	1	1	yes	characteristic
335	C₂²	#15688983	1	1	yes	characteristic
336	C₂²	#15688991	1	1	yes	characteristic
337	C₂²	#15689007	1	2	yes	—
338	C₂²	#15689014	1	2	yes	—
339	C₂²	#15689386	1	2	yes	—
340	C₂²	#15689393	1	1	yes	characteristic
341	C₂²	#15689398	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	376	4	no	—
345	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
346	C₄	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
347	C₄	#15688986	1	1	yes	characteristic
348	C₄	#15688989	1	1	yes	characteristic
349	C₄	#15689007	1	2	yes	—
350	C₄	#15689014	1	2	yes	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	#20241615	1	1	yes	characteristic
355	C₂	#20243447	1	1	yes	characteristic
356	C₂	#20243452	1	1	yes	characteristic
357	C₂	#20244043	1	2	yes	—
358	C₂	#20244050	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093998
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#10827
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. #15688891	C₄	semidirect
18. C₄₇×#405	C₄	semidirect
191. #405	Dic₄₇	semidirect
200. C₄₇	#10827	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688891	2
C₂×#15688888	2
#20243109	2
C₂×#15688886	2
C₂×C₄₇×#405	2
#10827	47