Subgroups — #26104243

Order 12032.

596 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094003	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	#20241213	C₂	1	2	yes	maximal, complemented
4	#20241212	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	—	2	1	no	—
7	C₂×#13094003	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₉₄	C₄	1	1	yes	characteristic
9	#15689029	C₂²	1	1	yes	characteristic
10	#15689021	C₂²	1	1	yes	characteristic
11	#15689007	—	2	1	no	—
12	#15688996	—	2	1	no	—
13	#15688983	C₄	1	1	yes	characteristic
14	C₄×Dic₄₇⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
17	#15688878	—	2	1	no	—
18	#15688878	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₄×C₈×C₄₇	—	2	1	no	—
22	C₄₇×#336	C₂²	1	2	yes	—
23	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×C₄₇⋊C₈	—	4	2	no	—
28	#13094003	—	2	2	no	—
29	#13094003	C₈	1	4	yes	complemented
30	C₄×Dic₉₄	—	4	2	no	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₄₇⋊C₄	—	2	1	no	—
35	C₂×Dic₄₇⋊C₄	—	2	1	no	—
36	C₂×Dic₄₇⋊C₄	—	2	1	no	—
37	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
38	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄×Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	C₄₇⋊#95	—	2	1	no	—
42	C₄₇⋊#95	—	2	1	no	—
43	C₄₇⋊#95	—	2	1	no	—
44	C₄₇⋊#95	—	2	1	no	—
45	C₄₇⋊#95	—	2	1	no	—
46	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	4	2	no	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²⋊C₈×C₄₇	—	2	4	no	—
56	C₄×C₈×C₄₇	—	4	2	no	—
57	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	4	1	no	—
63	C₂²×C₄×C₄₇	—	4	1	no	—
64	C₂²×C₄×C₄₇	—	4	1	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
70	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	1	no	—
76	C₂²×Dic₄₇	—	2	1	no	—
77	C₂²×Dic₄₇	—	2	1	no	—
78	C₂²×Dic₄₇	—	2	1	no	—
79	C₂×Dic₉₄	—	2	1	no	—
80	C₂×Dic₉₄	—	4	1	no	—
81	C₂×Dic₉₄	—	4	1	no	—
82	C₂×Dic₉₄	C₂×C₈	1	2	yes	complemented
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	—	2	4	no	—
91	C₂²⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	2	1	no	—
93	C₄⋊Dic₄₇	—	4	1	no	—
94	C₄⋊Dic₄₇	—	4	1	no	—
95	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	complemented
96	Dic₄₇⋊C₄	—	2	4	no	—
97	Dic₄₇⋊C₄	—	4	2	no	—
98	Dic₄₇⋊C₄	—	4	2	no	—
99	C₄×Dic₄₇	—	4	2	no	—
100	C₄×Dic₄₇	—	4	2	no	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	4	2	no	—
103	C₂×C₄₇⋊C₈	—	4	8	no	—
104	C₂×C₄₇⋊C₈	—	4	2	no	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	—	2	1	no	—
108	C₂³×C₄₇	—	2	1	no	—
109	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
110	C₂³×C₄₇	#103	1	2	yes	—
111	C₂³×C₄₇	#104	1	2	yes	—
112	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	1	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₄₇	#99	1	2	yes	—
131	C₂×C₄×C₄₇	#103	1	2	yes	—
132	C₂×C₄×C₄₇	#104	1	2	yes	—
133	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#135	1	2	yes	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	4	1	no	—
141	C₂×Dic₄₇	—	4	1	no	—
142	C₂×Dic₄₇	—	4	1	no	—
143	C₂×Dic₄₇	—	4	1	no	—
144	C₂×Dic₄₇	—	4	1	no	—
145	C₂×Dic₄₇	—	4	1	no	—
146	C₂×Dic₄₇	—	4	1	no	—
147	C₂×Dic₄₇	—	4	1	no	—
148	C₂×Dic₄₇	—	4	1	no	—
149	C₂×Dic₄₇	—	4	1	no	—
150	Dic₉₄	—	4	2	no	—
151	Dic₉₄	—	4	2	no	—
152	C₈×C₄₇	—	4	4	no	—
153	C₄₇⋊C₈	—	4	4	no	—
154	#7566	—	47	1	no	maximal, Sylow
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	—	2	2	no	—
161	C₂²×C₄₇	Q₈⋊C₈	1	4	yes	—
162	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
164	C₂²×C₄₇	#413	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
166	C₂²×C₄₇	#416	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
168	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	2	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	2	1	no	—
174	C₄×C₄₇	—	2	2	no	—
175	C₄×C₄₇	—	4	1	no	—
176	C₄×C₄₇	#328	1	4	yes	—
177	Dic₄₇	—	4	2	no	—
178	Dic₄₇	—	4	2	no	—
179	Dic₄₇	—	8	1	no	—
180	Dic₄₇	—	8	1	no	—
181	C₄×C₂²⋊Q₈	—	47	1	no	—
182	C₄×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
183	C₂×#336	—	47	1	no	—
184	#1272	—	47	2	no	—
185	#1271	—	47	2	no	—
186	C₂×C₄₇	—	2	4	no	—
187	C₂×C₄₇	#1459	1	1	yes	characteristic
188	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
189	C₂×C₄₇	#1476	1	1	yes	characteristic
190	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
191	C₂×C₄₇	#1501	1	1	yes	characteristic
192	C₂×C₄₇	#1524	1	1	yes	characteristic
193	C₂×C₄₇	#1542	1	1	yes	characteristic
194	C₂³×C₈	—	94	1	no	—
195	C₂×C₂²⋊Q₈	—	47	1	no	—
196	C₂×C₄×Q₈	—	47	1	no	—
197	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
198	C₂×C₂²⋊C₈	—	2	1	no	—
199	C₂×C₄×C₈	—	2	1	no	—
200	#391	—	47	1	no	—
201	#389	—	47	1	no	—
202	#387	—	94	1	no	—
203	#385	—	94	1	no	—
204	#380	—	47	1	no	—
205	C₄×C₄⋊C₄	—	47	1	no	—
206	C₄×C₄⋊C₄	—	94	1	no	—
207	C₄×C₂²⋊C₄	—	94	1	no	—
208	#336	—	47	2	no	—
209	#336	—	94	1	no	—
210	#336	C₂×D₄₇	1	2	yes	—
211	C₄₇	#7566	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	47	1	no	—
213	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
214	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
215	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	94	4	no	—
218	C₂²×C₈	—	94	2	no	—
219	C₂²×C₈	—	188	2	no	—
220	C₂²×C₈	—	188	2	no	—
221	C₂²⋊Q₈	—	47	4	no	—
222	C₂²⋊Q₈	—	94	2	no	—
223	C₄×Q₈	—	188	2	no	—
224	C₂×C₄⋊C₄	—	47	1	no	—
225	C₂×C₄⋊C₄	—	47	1	no	—
226	C₂×C₄⋊C₄	—	47	1	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	1	no	—
231	C₂×C₂²⋊C₄	—	94	1	no	—
232	C₂×C₄²	—	4	2	no	—
233	C₂×C₄²	—	94	1	no	—
234	C₂×C₄²	—	94	1	no	—
235	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	C₂²⋊C₈	—	2	4	no	—
240	C₄×C₈	—	4	2	no	—
241	#95	—	94	1	no	—
242	#95	—	94	1	no	—
243	#95	—	94	1	no	—
244	#95	—	94	1	no	—
245	#95	—	94	1	no	—
246	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
247	C₂×Q₈	—	47	2	no	—
248	C₂×Q₈	—	94	1	no	—
249	C₂×Q₈	—	188	1	no	—
250	C₂×Q₈	—	188	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	94	1	no	—
259	C₂²×C₄	—	94	1	no	—
260	C₂²×C₄	—	94	1	no	—
261	C₂²×C₄	—	94	1	no	—
262	C₂²×C₄	C₈×D₄₇	1	2	yes	—
263	C₂²×C₄	#10625324	1	2	yes	—
264	C₂²×C₄	#10625325	1	2	yes	—
265	C₂²×C₄	Dic₁₈₈	1	2	yes	—
266	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
267	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
268	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	94	4	no	—
276	C₂×C₈	—	188	2	no	—
277	C₂×C₈	—	188	8	no	—
278	C₂×C₈	—	188	2	no	—
279	C₄⋊C₄	—	47	2	no	—
280	C₄⋊C₄	—	94	4	no	—
281	C₄⋊C₄	—	94	1	no	—
282	C₄⋊C₄	—	188	1	no	—
283	C₄⋊C₄	—	188	1	no	—
284	C₄⋊C₄	—	188	2	no	—
285	C₄⋊C₄	—	188	2	no	—
286	C₂²⋊C₄	—	94	4	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	C₄²	—	188	2	no	—
293	C₄²	—	188	2	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
299	C₂³	#13093949	1	2	yes	—
300	C₂³	#13093952	1	2	yes	—
301	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
302	Q₈	—	188	2	no	—
303	Q₈	—	188	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	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	188	1	no	—
323	C₂×C₄	—	188	1	no	—
324	C₂×C₄	—	188	1	no	—
325	C₂×C₄	—	188	1	no	—
326	C₂×C₄	—	188	1	no	—
327	C₂×C₄	—	188	1	no	—
328	C₂×C₄	—	188	1	no	—
329	C₂×C₄	—	188	1	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	—	188	1	no	—
332	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
333	C₂×C₄	#13093938	1	2	yes	—
334	C₂×C₄	#13093949	1	2	yes	—
335	C₂×C₄	#13093950	1	2	yes	—
336	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
337	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
339	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
340	C₂×C₄	#13094044	1	2	yes	—
341	C₈	—	4	4	no	—
342	C₈	—	188	4	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	#15688855	1	4	yes	—
350	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
351	C₂²	C₂×#13093938	1	1	yes	characteristic
352	C₂²	#15689105	1	1	yes	characteristic
353	C₂²	C₂×#13093949	1	1	yes	characteristic
354	C₂²	#15689117	1	1	yes	characteristic
355	C₂²	C₂×#13093950	1	1	yes	characteristic
356	C₂²	C₂×#13093952	1	1	yes	characteristic
357	C₄	—	2	2	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	1	no	—
364	C₄	—	376	1	no	—
365	C₄	—	376	1	no	—
366	C₄	—	188	2	no	—
367	C₄	—	188	2	no	—
368	C₄	#15688861	1	4	yes	—
369	C₂	—	2	4	no	—
370	C₂	#20241704	1	1	yes	characteristic
371	C₂	C₂×#15688855	1	1	yes	characteristic
372	C₂	#20241768	1	1	yes	characteristic
373	C₂	C₂×#15688861	1	1	yes	characteristic
374	C₂	#20241877	1	1	yes	characteristic
375	C₂	#20241977	1	1	yes	characteristic
376	C₂	#20242043	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#7566
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20241213	C₂	semidirect
4. #20241212	C₂	semidirect
29. #13094003	C₈	semidirect
82. C₂×Dic₉₄	C₂×C₈	semidirect
95. C₄⋊Dic₄₇	C₂×C₈	semidirect
211. C₄₇	#7566	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094003	2
C₂×#15688878	2
#20241213	2
#20241212	2
C₄×C₂²⋊C₈×C₄₇	2
#7566	47