Subgroups — #26115256

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 386 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	C₂×#15688861	C₂	1	4	yes	maximal
3	#20241618	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
8	#15689029	C₄	1	2	yes	complemented
9	#15689021	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15688993	C₄	1	1	yes	characteristic, complemented
12	#15688983	—	2	1	no	—
13	#15688983	—	2	1	no	—
14	C₂×C₄₇⋊#95	—	2	1	no	—
15	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
16	#15688861	—	2	8	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	—	2	1	no	—
20	C₄₇×#336	—	2	1	no	—
21	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
22	C₂³×Dic₄₇	—	2	2	no	—
23	#13094024	—	4	2	no	—
24	#13094003	—	2	4	no	—
25	#13094003	—	2	4	no	—
26	C₂×C₄⋊Dic₄₇	—	2	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	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	—	4	2	no	—
32	C₂×C₄⋊Dic₄₇	—	4	2	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	2	no	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₄₇⋊#95	—	2	4	no	—
39	C₄₇⋊#95	—	4	2	no	—
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	—	4	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂²⋊C₈×C₄₇	—	2	2	no	—
50	C₂²⋊C₈×C₄₇	D₄	1	4	yes	—
51	C₄×C₈×C₄₇	—	4	2	no	—
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂²×Dic₄₇	—	4	2	no	—
79	C₂²×Dic₄₇	—	4	2	no	—
80	C₂×Dic₉₄	—	2	4	no	—
81	C₂×Dic₉₄	—	4	2	no	—
82	C₂×Dic₉₄	—	4	2	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₂×C₈×C₄₇	—	4	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	4	1	no	—
89	C₄²×C₄₇	—	4	1	no	—
90	C₂²⋊Dic₄₇	—	4	2	no	—
91	C₂²⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	—	4	8	no	—
95	C₄⋊Dic₄₇	—	4	2	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄⋊Dic₄₇	—	4	2	no	—
98	Dic₄₇⋊C₄	—	4	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₇	#124	1	1	yes	characteristic
105	C₂³×C₄₇	#127	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
109	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
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	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	#99	1	2	yes	—
127	C₂×C₄×C₄₇	#103	1	2	yes	—
128	C₂×C₄×C₄₇	#103	1	2	yes	—
129	C₂×C₄×C₄₇	#103	1	2	yes	—
130	C₂×C₄×C₄₇	#103	1	2	yes	—
131	C₂×C₄×C₄₇	#104	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	#121	1	2	yes	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	8	2	no	—
144	C₂×Dic₄₇	—	8	2	no	—
145	C₂×Dic₄₇	—	8	2	no	—
146	C₂×Dic₄₇	—	8	2	no	—
147	Dic₉₄	—	8	4	no	—
148	C₈×C₄₇	—	4	4	no	—
149	#9679	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	—	2	1	no	—
155	C₂²×C₄₇	—	2	1	no	—
156	C₂²×C₄₇	—	2	1	no	—
157	C₂²×C₄₇	—	2	1	no	—
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	#390	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
164	C₂²×C₄₇	#410	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
166	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
168	C₂²×C₄₇	#421	1	1	yes	characteristic
169	C₂²×C₄₇	#487	1	1	yes	characteristic
170	C₂²×C₄₇	#487	1	1	yes	characteristic
171	C₂²×C₄₇	#492	1	1	yes	characteristic
172	C₂²×C₄₇	#494	1	1	yes	characteristic
173	C₄×C₄₇	—	2	2	no	—
174	C₄×C₄₇	—	2	2	no	—
175	C₄×C₄₇	—	2	2	no	—
176	C₄×C₄₇	—	2	2	no	—
177	C₄×C₄₇	—	2	2	no	—
178	C₄×C₄₇	#328	1	4	yes	complemented
179	Dic₄₇	—	8	4	no	—
180	Dic₄₇	—	8	2	no	—
181	Dic₄₇	—	8	2	no	—
182	#2857	—	47	1	no	—
183	C₄×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
184	C₂×#328	—	47	4	no	—
185	C₄⋊#95	—	47	1	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	2	1	no	—
189	C₂×C₄₇	—	2	1	no	—
190	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
191	C₂×C₄₇	#1503	1	1	yes	characteristic
192	C₂×C₄₇	#1518	1	1	yes	characteristic
193	C₂×C₄₇	#1520	1	1	yes	characteristic
194	C₂×C₄₇	#1956	1	1	yes	characteristic
195	C₂×C₄₇	#1964	1	1	yes	characteristic
196	C₂×C₄₇	#1980	1	1	yes	characteristic
197	C₂×C₄⋊Q₈	—	47	1	no	—
198	C₂×C₂²⋊Q₈	—	47	2	no	—
199	C₂²×C₄⋊C₄	—	94	1	no	—
200	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
201	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
202	C₂×C₄×C₈	—	2	1	no	—
203	#391	—	47	2	no	—
204	#389	—	94	1	no	—
205	#389	—	94	1	no	—
206	#384	—	47	1	no	—
207	#380	—	94	1	no	—
208	#380	—	94	1	no	—
209	C₂×#95	—	47	2	no	—
210	C₂×#95	—	94	1	no	—
211	#336	—	2	1	no	—
212	#328	—	94	8	no	—
213	C₄₇	#9679	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	47	2	no	—
215	C₂³×C₄	—	94	2	no	—
216	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
217	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
218	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
219	C₂²×C₈	—	2	2	no	—
220	C₄⋊Q₈	—	188	2	no	—
221	C₂²⋊Q₈	—	94	4	no	—
222	C₂²⋊Q₈	—	94	4	no	—
223	C₂×C₄⋊C₄	—	47	2	no	—
224	C₂×C₄⋊C₄	—	94	4	no	—
225	C₂×C₄⋊C₄	—	94	1	no	—
226	C₂×C₄⋊C₄	—	94	1	no	—
227	C₂×C₄⋊C₄	—	94	1	no	—
228	C₂×C₄⋊C₄	—	94	1	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	188	2	no	—
231	C₂×C₄⋊C₄	—	188	2	no	—
232	C₂×C₂²⋊C₄	—	94	2	no	—
233	C₂×C₄²	—	4	2	no	—
234	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂²⋊C₈	—	2	2	no	—
239	C₂²⋊C₈	D₁₈₈	1	4	yes	—
240	C₄×C₈	—	4	2	no	—
241	#95	—	94	2	no	—
242	#95	—	94	2	no	—
243	#95	—	94	4	no	—
244	#95	—	188	2	no	—
245	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
246	C₂×Q₈	—	94	4	no	—
247	C₂×Q₈	—	188	2	no	—
248	C₂×Q₈	—	188	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	94	4	no	—
256	C₂²×C₄	—	94	2	no	—
257	C₂²×C₄	—	94	2	no	—
258	C₂²×C₄	—	188	2	no	—
259	C₂²×C₄	—	188	2	no	—
260	C₂²×C₄	—	188	2	no	—
261	C₂²×C₄	—	188	2	no	—
262	C₂²×C₄	—	188	2	no	—
263	C₂²×C₄	—	188	2	no	—
264	C₂²×C₄	#10625325	1	2	yes	—
265	C₂²×C₄	#10625325	1	2	yes	—
266	C₂²×C₄	Dic₁₈₈	1	2	yes	—
267	C₂²×C₄	Dic₁₈₈	1	2	yes	—
268	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
270	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
271	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
274	C₂²×C₄	#10625350	1	2	yes	—
275	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	4	2	no	—
279	C₄⋊C₄	—	94	4	no	—
280	C₄⋊C₄	—	188	2	no	—
281	C₄⋊C₄	—	188	8	no	—
282	C₄⋊C₄	—	188	2	no	—
283	C₄⋊C₄	—	188	2	no	—
284	C₄⋊C₄	—	188	2	no	—
285	C₄⋊C₄	—	188	4	no	—
286	C₂²⋊C₄	—	188	2	no	—
287	C₂²⋊C₄	—	188	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	4	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
297	C₂³	#13094012	1	1	yes	characteristic
298	C₂³	#13094021	1	1	yes	characteristic
299	C₂³	C₂×#10625325	1	1	yes	characteristic
300	C₂³	C₂×#10625325	1	1	yes	characteristic
301	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
302	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
303	Q₈	—	376	4	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	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	376	2	no	—
321	C₂×C₄	—	376	2	no	—
322	C₂×C₄	—	376	2	no	—
323	C₂×C₄	—	376	2	no	—
324	C₂×C₄	—	188	4	no	—
325	C₂×C₄	—	188	4	no	—
326	C₂×C₄	—	188	4	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	#13093938	1	2	yes	—
334	C₂×C₄	#13093949	1	2	yes	—
335	C₂×C₄	#13093949	1	2	yes	—
336	C₂×C₄	#13093949	1	2	yes	—
337	C₂×C₄	#13093949	1	2	yes	—
338	C₂×C₄	#13093950	1	2	yes	—
339	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
340	C₂×C₄	#13093998	1	2	yes	—
341	C₈	—	4	4	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	#15689025	1	1	yes	characteristic
355	C₂²	C₂×#13093938	1	1	yes	characteristic
356	C₂²	#15689096	1	1	yes	characteristic
357	C₂²	C₂×#13093949	1	1	yes	characteristic
358	C₂²	C₂×#13093949	1	1	yes	characteristic
359	C₂²	C₂×#13093950	1	1	yes	characteristic
360	C₂²	#15689139	1	1	yes	characteristic
361	C₂²	#15689474	1	1	yes	characteristic
362	C₂²	#15689474	1	1	yes	characteristic
363	C₂²	#15689496	1	1	yes	characteristic
364	C₂²	#15689503	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	376	4	no	—
371	C₄	—	376	2	no	—
372	C₄	—	376	2	no	—
373	C₄	#15688861	1	4	yes	complemented
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	C₂×#15688861	1	1	yes	characteristic
379	C₂	#20241888	1	1	yes	characteristic
380	C₂	#20241953	1	1	yes	characteristic
381	C₂	#20241961	1	1	yes	characteristic
382	C₂	#20244190	1	1	yes	characteristic
383	C₂	#20244224	1	1	yes	characteristic
384	C₂	#20244307	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#9679
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
5. C₂×#13094024	C₄	semidirect
8. #15689029	C₄	semidirect
11. #15688993	C₄	semidirect
178. C₄×C₄₇	#328	semidirect
213. C₄₇	#9679	semidirect
373. C₄	#15688861	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
C₂×#15688861	2
#20241618	2
C₄×C₂²⋊C₈×C₄₇	2
#9679	47