Subgroups — #26226826

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20244149	C₂	1	2	yes	maximal, complemented
4	#20244144	C₂	1	2	yes	maximal, complemented
5	#20243587	C₂	1	4	yes	maximal
6	#20243550	C₂	1	2	yes	maximal, complemented
7	#20243518	C₂	1	2	yes	maximal, complemented
8	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
9	#15689176	C₄	1	8	yes	complemented
10	C₂×#13093957	C₂²	1	4	yes	—
11	C₂×#13093955	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15688991	—	2	1	no	—
17	#15688989	C₂²	1	1	yes	characteristic
18	#15688989	C₂²	1	1	yes	characteristic
19	#15688986	—	2	1	no	—
20	#15688983	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
23	#15688969	—	2	1	no	—
24	#15688885	C₂²	1	4	yes	—
25	#15688885	C₂²	1	4	yes	—
26	C₂×#117×C₄₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#384	C₂²	1	2	yes	—
29	C₄₇×#382	C₂²	1	2	yes	—
30	C₄₇×#380	—	2	1	no	—
31	C₄₇×#380	—	2	1	no	—
32	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	#13093957	—	4	4	no	—
41	#13093955	—	4	4	no	—
42	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
44	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
45	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
46	C₄₇⋊#95	—	2	2	no	—
47	C₄₇⋊#95	—	2	1	no	—
48	C₄₇⋊#95	—	2	1	no	—
49	C₄₇⋊#95	—	2	1	no	—
50	C₄₇⋊#95	—	2	1	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	#117×C₄₇	—	4	2	no	—
56	#117×C₄₇	—	4	2	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄²×C₄₇	—	2	2	no	—
62	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	#95×C₄₇	—	2	2	no	—
69	#95×C₄₇	—	2	2	no	—
70	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	—	2	1	no	—
74	C₂²×C₄×C₄₇	—	2	1	no	—
75	C₂²×C₄×C₄₇	—	2	1	no	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	4	2	no	—
78	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₄₇	—	2	2	no	—
89	C₂²×Dic₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	4	4	no	—
93	C₄⋊C₄×C₄₇	—	4	4	no	—
94	C₂²⋊C₄×C₄₇	—	4	4	no	—
95	C₄²×C₄₇	—	4	4	no	—
96	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
97	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
101	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
102	C₂²⋊Dic₄₇	—	4	4	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	—	4	4	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	—	2	2	no	—
111	C₂×C₄₇⋊C₈	—	2	2	no	—
112	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
113	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
114	C₂³×C₄₇	—	2	1	no	—
115	C₂³×C₄₇	—	2	1	no	—
116	C₂³×C₄₇	—	2	2	no	—
117	C₂³×C₄₇	—	2	2	no	—
118	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	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	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	8	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	—	4	4	no	—
132	C₂×C₄×C₄₇	—	4	4	no	—
133	C₂×C₄×C₄₇	—	4	4	no	—
134	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₄₇	#135	1	4	yes	—
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₄₇⋊C₈	—	2	4	no	—
153	C₄₇⋊C₈	—	2	4	no	—
154	#21985	—	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	—	4	2	no	—
159	C₂²×C₄₇	—	4	2	no	—
160	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₄₇	#535	1	1	yes	characteristic
162	C₂²×C₄₇	#536	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#135	1	2	yes	—
164	C₂²×C₄₇	#575	1	2	yes	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	8	4	no	—
170	C₄×C₄₇	#427	1	4	yes	—
171	C₄×C₄₇	#520	1	1	yes	characteristic
172	C₄×C₄₇	#520	1	1	yes	characteristic
173	C₄×C₄₇	#544	1	1	yes	characteristic
174	C₄×C₄₇	#544	1	1	yes	characteristic
175	Dic₄₇	—	8	2	no	—
176	Dic₄₇	—	8	2	no	—
177	C₂×#431	—	47	1	no	—
178	#2285	D₄₇	1	1	yes	characteristic
179	#2283	—	47	1	no	—
180	#1948	—	47	2	no	—
181	#1947	—	47	2	no	—
182	#1854	—	47	4	no	—
183	#1846	—	47	2	no	—
184	#1835	—	47	2	no	—
185	C₂×C₄₇	—	4	2	no	—
186	C₂×C₄₇	#2317	1	1	yes	characteristic
187	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
188	C₂×C₄₇	#2911	1	1	yes	characteristic
189	C₂×C₄₇	#3212	1	2	yes	—
190	C₂×C₄₇	#3233	1	2	yes	—
191	C₂×#117	C₂×D₄₇	1	2	yes	—
192	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
193	#431	—	47	8	no	—
194	C₂×#107	—	47	4	no	—
195	C₂×#106	—	47	4	no	—
196	C₂×C₄⋊C₈	—	47	1	no	—
197	C₂×C₄⋊C₈	—	47	1	no	—
198	C₂×C₂²⋊C₈	—	47	2	no	—
199	C₂×C₄×C₈	—	47	2	no	—
200	#384	C₂×D₄₇	1	2	yes	—
201	#383	—	94	1	no	—
202	#382	—	47	1	no	—
203	#382	—	47	1	no	—
204	#382	C₂×D₄₇	1	2	yes	—
205	#381	—	94	1	no	—
206	#380	—	2	1	no	—
207	#380	—	2	1	no	—
208	#380	—	47	2	no	—
209	C₄×C₄⋊C₄	—	2	1	no	—
210	C₄×C₄⋊C₄	—	2	1	no	—
211	C₄×C₄⋊C₄	—	47	2	no	—
212	C₄×C₂²⋊C₄	—	94	1	no	—
213	#376	—	94	1	no	—
214	#340	—	47	4	no	—
215	#340	—	47	4	no	—
216	C₄₇	#21985	1	1	yes	characteristic, complemented, Sylow
217	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₈	—	47	2	no	—
221	C₂²×C₈	—	47	2	no	—
222	#117	—	4	2	no	—
223	#117	—	4	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	47	2	no	—
227	C₂×C₄⋊C₄	—	47	2	no	—
228	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	94	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	94	2	no	—
233	C₂×C₄²	Dic₉₄	1	4	yes	—
234	C₂×C₄²	C₄×D₄₇	1	4	yes	—
235	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	#107	—	188	4	no	—
240	#106	—	188	4	no	—
241	C₄⋊C₈	—	47	4	no	—
242	C₄⋊C₈	—	47	4	no	—
243	C₂²⋊C₈	—	47	8	no	—
244	C₄×C₈	—	47	8	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	94	2	no	—
248	#95	—	94	1	no	—
249	#95	—	94	1	no	—
250	#95	—	94	1	no	—
251	#95	—	94	1	no	—
252	#95	—	94	2	no	—
253	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	—	94	2	no	—
263	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
264	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
265	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
266	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
273	C₂×C₈	—	47	4	no	—
274	C₂×C₈	—	47	4	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	—	4	4	no	—
283	C₄⋊C₄	—	94	2	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₄⋊C₄	—	94	2	no	—
286	C₄⋊C₄	—	94	2	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	188	4	no	—
289	C₄²	—	4	4	no	—
290	C₄²	—	188	4	no	—
291	C₄²	Dic₄₇⋊C₄	1	4	yes	—
292	C₄²	Dic₄₇⋊C₄	1	4	yes	—
293	C₄²	C₂×Dic₉₄	1	2	yes	—
294	C₄²	C₂×Dic₉₄	1	2	yes	—
295	C₄²	C₂×C₄×D₄₇	1	2	yes	—
296	C₄²	C₂×C₄×D₄₇	1	2	yes	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
302	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	8	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	—	188	2	no	—
321	C₂×C₄	—	188	2	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
324	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
325	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
326	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
327	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
328	C₂×C₄	#13094044	1	4	yes	—
329	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
331	C₂×C₄	#13094069	1	1	yes	characteristic
332	C₂×C₄	#13094069	1	1	yes	characteristic
333	C₂×C₄	#13094072	1	1	yes	characteristic
334	C₂×C₄	#13094072	1	1	yes	characteristic
335	C₈	—	94	4	no	—
336	C₈	—	94	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
343	C₂²	#15689690	1	1	yes	characteristic
344	C₂²	#15689697	1	1	yes	characteristic
345	C₂²	C₂×#13094044	1	2	yes	—
346	C₂²	#15689921	1	2	yes	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	8	4	no	—
352	C₄	—	376	2	no	—
353	C₄	—	376	2	no	—
354	C₄	#15689159	1	4	yes	—
355	C₄	#15689604	1	1	yes	characteristic
356	C₄	#15689604	1	1	yes	characteristic
357	C₄	#15689754	1	1	yes	characteristic
358	C₄	#15689754	1	1	yes	characteristic
359	C₂	—	4	2	no	—
360	C₂	#20246012	1	1	yes	characteristic
361	C₂	C₂×#15689159	1	1	yes	characteristic
362	C₂	#20251977	1	1	yes	characteristic
363	C₂	#20254364	1	2	yes	—
364	C₂	#20254567	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#21985
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20244149	C₂	semidirect
4. #20244144	C₂	semidirect
6. #20243550	C₂	semidirect
7. #20243518	C₂	semidirect
9. #15689176	C₄	semidirect
216. C₄₇	#21985	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20245786	2
#20244149	2
#20244144	2
#20243587	2
#20243550	2
#20243518	2
C₄₇×#2285	2
#21985	47