Subgroups — #26226690

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245997	C₂	1	2	yes	maximal
2	#20244158	C₂	1	2	yes	maximal, complemented
3	#20244158	C₂	1	2	yes	maximal, complemented
4	#20244152	C₂	1	2	yes	maximal, complemented
5	#20244147	C₂	1	2	yes	maximal, complemented
6	#20243421	C₂	1	2	yes	maximal
7	#20243417	C₂	1	1	yes	maximal, characteristic
8	#20243413	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093958	C₂²	1	2	yes	—
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093956	C₂²	1	2	yes	—
13	C₂×#13093956	C₂²	1	2	yes	—
14	#15689016	—	2	2	no	—
15	#15689016	—	2	2	no	—
16	#15689009	—	2	2	no	—
17	#15689009	—	2	2	no	—
18	#15688993	C₂²	1	2	yes	—
19	#15688989	C₂²	1	2	yes	—
20	#15688983	C₂²	1	2	yes	—
21	#15688983	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
24	#15688886	C₂²	1	1	yes	characteristic
25	#15688886	C₂²	1	2	yes	—
26	#15688886	C₂²	1	1	yes	characteristic
27	#15688886	C₂²	1	1	yes	characteristic
28	#15688886	C₂²	1	1	yes	characteristic
29	#15688886	C₂²	1	2	yes	—
30	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#424	C₄	1	8	yes	complemented
34	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
43	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	C₂×C₄×Dic₄₇	—	2	2	no	—
46	#13093958	—	4	2	no	—
47	#13093958	—	4	2	no	—
48	#13093956	—	4	2	no	—
49	#13093956	—	4	2	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₄₇⋊#95	—	2	2	no	—
55	C₄₇⋊#95	—	2	2	no	—
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₄₇	—	4	2	no	—
58	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₄₇	—	4	2	no	—
60	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	#117×C₄₇	C₂×C₄	1	4	yes	—
65	#117×C₄₇	Q₈	1	4	yes	—
66	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
70	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
71	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
72	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	2	2	no	—
82	C₂²×Dic₄₇	—	2	2	no	—
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	2	no	—
85	M₁₆×C₄₇	—	4	4	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	4	4	no	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
95	C₄⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
96	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
98	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
99	C₂²⋊Dic₄₇	—	8	2	no	—
100	C₂²⋊Dic₄₇	—	8	2	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	8	2	no	—
110	C₄⋊Dic₄₇	—	8	2	no	—
111	C₄×Dic₄₇	—	8	2	no	—
112	C₄×Dic₄₇	—	8	2	no	—
113	C₂³×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	—	2	1	no	—
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	—	2	1	no	—
117	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
133	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	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	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₈×C₄₇	—	2	2	no	—
147	C₈×C₄₇	—	2	2	no	—
148	C₈×C₄₇	—	2	4	no	—
149	#21966	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	4	2	no	—
154	C₂²×C₄₇	—	4	2	no	—
155	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄₇	#537	1	2	yes	—
157	C₂²×C₄₇	#576	1	1	yes	characteristic
158	C₂²×C₄₇	#577	1	2	yes	—
159	C₂²×C₄₇	#578	1	1	yes	characteristic
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	#518	1	1	yes	characteristic
164	C₄×C₄₇	#519	1	1	yes	characteristic
165	C₄×C₄₇	#544	1	2	yes	—
166	Dic₄₇	—	8	2	no	—
167	Dic₄₇	—	8	2	no	—
168	Dic₄₇	—	8	2	no	—
169	Dic₄₇	—	8	2	no	—
170	C₂×#424	D₄₇	1	1	yes	characteristic
171	#2314	—	47	2	no	—
172	#1949	—	47	2	no	—
173	#1949	—	47	2	no	—
174	#1948	—	47	2	no	—
175	#1947	—	47	2	no	—
176	#1809	—	47	2	no	—
177	#1808	—	47	1	no	—
178	#1807	—	47	1	no	—
179	C₂×C₄₇	—	4	2	no	—
180	C₂×C₄₇	#2316	1	1	yes	characteristic
181	C₂×C₄₇	#2912	1	1	yes	characteristic
182	C₂×C₄₇	#2913	1	1	yes	characteristic
183	C₂×C₄₇	#3257	1	2	yes	—
184	C₂×C₄₇	#3258	1	2	yes	—
185	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
186	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
187	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
188	C₂×#107	—	47	2	no	—
189	C₂×#107	—	47	2	no	—
190	C₂×#106	—	47	2	no	—
191	C₂×#106	—	47	2	no	—
192	#424	Dic₄₇	1	8	yes	complemented
193	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
194	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
195	#388	—	94	2	no	—
196	#388	—	94	2	no	—
197	#387	—	94	2	no	—
198	#387	—	94	2	no	—
199	#384	—	47	2	no	—
200	#382	—	47	2	no	—
201	#380	—	47	2	no	—
202	#380	—	47	2	no	—
203	C₄×C₄⋊C₄	—	47	2	no	—
204	C₄×C₄⋊C₄	—	47	2	no	—
205	#340	—	47	1	no	—
206	#340	—	47	1	no	—
207	#340	—	47	1	no	—
208	#340	—	47	1	no	—
209	#340	—	47	2	no	—
210	#340	—	47	2	no	—
211	C₄₇	#21966	1	1	yes	characteristic, complemented, Sylow
212	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×M₁₆	—	4	2	no	—
214	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
215	C₂²×C₈	—	4	2	no	—
216	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
217	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
220	#117	Dic₉₄	1	4	yes	—
221	#117	C₂×Dic₄₇	1	4	yes	—
222	C₂×C₄⋊C₄	—	47	2	no	—
223	C₂×C₄⋊C₄	—	47	2	no	—
224	C₂×C₄⋊C₄	—	47	2	no	—
225	C₂×C₄⋊C₄	—	47	2	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
229	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
230	C₂×C₂²⋊C₄	—	94	2	no	—
231	C₂×C₂²⋊C₄	—	94	2	no	—
232	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
233	C₂×C₄²	—	94	2	no	—
234	C₂×C₄²	—	94	2	no	—
235	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
236	#107	—	188	2	no	—
237	#107	—	188	2	no	—
238	#106	—	188	2	no	—
239	#106	—	188	2	no	—
240	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
241	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
242	#95	—	94	2	no	—
243	#95	—	94	2	no	—
244	#95	—	94	2	no	—
245	#95	—	94	2	no	—
246	#95	—	94	2	no	—
247	#95	—	94	2	no	—
248	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	—	94	2	no	—
253	C₂²×C₄	—	94	2	no	—
254	C₂²×C₄	—	94	2	no	—
255	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
256	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
258	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
260	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
261	M₁₆	—	4	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
268	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
269	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
270	C₄⋊C₄	—	94	2	no	—
271	C₄⋊C₄	—	94	2	no	—
272	C₄⋊C₄	—	94	2	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₄⋊C₄	—	94	2	no	—
275	C₄⋊C₄	—	94	2	no	—
276	C₄⋊C₄	—	94	2	no	—
277	C₄⋊C₄	—	94	2	no	—
278	C₄⋊C₄	—	376	2	no	—
279	C₄⋊C₄	—	376	2	no	—
280	C₄⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
281	C₄⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
282	C₂²⋊C₄	—	376	2	no	—
283	C₂²⋊C₄	—	376	2	no	—
284	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
285	C₄²	—	376	2	no	—
286	C₄²	—	376	2	no	—
287	C₄²	C₂×Dic₉₄	1	4	yes	—
288	C₄²	C₂²×Dic₄₇	1	4	yes	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
294	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
316	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
317	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
318	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
320	C₂×C₄	#13094069	1	2	yes	—
321	C₂×C₄	#13094072	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	4	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
331	C₂²	#15689704	1	2	yes	—
332	C₂²	#15689928	1	1	yes	characteristic
333	C₂²	#15689935	1	2	yes	—
334	C₂²	#15689940	1	1	yes	characteristic
335	C₄	—	2	4	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	2	no	—
338	C₄	—	376	2	no	—
339	C₄	—	376	2	no	—
340	C₄	—	376	2	no	—
341	C₄	—	376	2	no	—
342	C₄	#15689598	1	1	yes	characteristic
343	C₄	#15689601	1	1	yes	characteristic
344	C₄	#15689754	1	2	yes	—
345	C₂	—	4	2	no	—
346	C₂	#20246006	1	1	yes	characteristic
347	C₂	#20251983	1	1	yes	characteristic
348	C₂	#20251988	1	1	yes	characteristic
349	C₂	#20254801	1	2	yes	—
350	C₂	#20254816	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#424	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#424	D₄₇
hypocenter	C₄₇	#21966
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
2. #20244158	C₂	semidirect
3. #20244158	C₂	semidirect
4. #20244152	C₂	semidirect
5. #20244147	C₂	semidirect
33. C₄₇×#424	C₄	semidirect
192. #424	Dic₄₇	semidirect
211. C₄₇	#21966	semidirect

Maximal subgroups

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

subgroup	index
#20245997	2
#20244158	2
#20244152	2
#20244147	2
#20243421	2
#20243417	2
#20243413	2
C₂×C₄₇×#424	2
#21966	47