Subgroups — #26109642

Order 12032.

540 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241229	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	C₂×#117×C₄₇	—	2	1	no	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#384	C₄	1	1	yes	characteristic
14	C₄₇×#384	C₄	1	1	yes	characteristic
15	C₄₇×#382	—	2	1	no	—
16	C₄₇×#380	—	2	1	no	—
17	C₄₇×#380	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
25	C₄×C₄₇⋊C₈	—	2	4	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	#117×C₄₇	—	4	4	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	—	2	4	no	—
38	C₂×C₄²×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#95×C₄₇	—	2	2	no	—
44	#95×C₄₇	—	2	2	no	—
45	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	—	2	4	no	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₄⋊C₄×C₄₇	—	4	2	no	—
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₂²⋊C₄×C₄₇	—	4	4	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	4	4	no	—
75	C₄²×C₄₇	#41	1	4	yes	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	4	2	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	—	2	1	no	—
86	C₂³×C₄₇	—	2	1	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	#103	1	2	yes	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	#103	1	2	yes	—
110	C₂×C₄×C₄₇	#104	1	2	yes	—
111	C₂×C₄×C₄₇	#117	1	2	yes	—
112	C₂×C₄×C₄₇	#123	1	2	yes	—
113	C₂×C₄×C₄₇	#123	1	2	yes	—
114	C₂×C₄×C₄₇	#131	1	2	yes	—
115	C₂×C₄×C₄₇	#131	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₄₇⋊C₈	—	4	4	no	—
120	C₄₇⋊C₈	—	8	2	no	—
121	#8710	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	4	1	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	1	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	#381	1	1	yes	characteristic
130	C₂²×C₄₇	#408	1	1	yes	characteristic
131	C₂²×C₄₇	#408	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
133	C₂²×C₄₇	#418	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
135	C₂²×C₄₇	#421	1	1	yes	characteristic
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	4	1	no	—
140	C₄×C₄₇	—	8	2	no	—
141	C₄×C₄₇	—	8	2	no	—
142	C₄×C₄₇	#433	1	4	yes	—
143	C₄×C₄₇	#481	1	2	yes	—
144	C₄×C₄₇	#484	1	2	yes	—
145	#2285	D₄₇	1	1	yes	characteristic
146	#1797	—	47	1	no	—
147	C₄×C₂²⋊C₈	—	47	1	no	—
148	#1279	—	47	4	no	—
149	C₂×C₄₇	—	4	2	no	—
150	C₂×C₄₇	#1481	1	1	yes	characteristic
151	C₂×C₄₇	#1484	1	1	yes	characteristic
152	C₂×C₄₇	#1493	1	1	yes	characteristic
153	C₂×C₄₇	#1496	1	1	yes	characteristic
154	C₂×C₄₇	#1818	1	1	yes	characteristic
155	C₂×C₄₇	#1824	1	1	yes	characteristic
156	C₂×C₄₇	#1826	1	1	yes	characteristic
157	C₂×#117	—	2	1	no	—
158	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
159	C₂×C₄⋊C₈	—	47	2	no	—
160	C₂×C₂²⋊C₈	—	94	1	no	—
161	C₂×C₂²⋊C₈	—	94	1	no	—
162	C₂×C₄×C₈	—	47	2	no	—
163	#384	Dic₄₇	1	1	yes	characteristic
164	#384	Dic₄₇	1	1	yes	characteristic
165	#382	—	2	1	no	—
166	#380	—	2	1	no	—
167	#380	Dic₄₇	1	2	yes	—
168	C₄×C₄⋊C₄	—	2	1	no	—
169	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
170	#336	—	94	1	no	—
171	#336	—	94	1	no	—
172	C₄₇	#8710	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
174	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
176	C₂²×C₈	—	94	2	no	—
177	C₂²×C₈	—	94	2	no	—
178	#117	—	4	4	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	94	4	no	—
193	C₂²⋊C₈	—	94	4	no	—
194	C₂²⋊C₈	—	188	2	no	—
195	C₄×C₈	—	94	4	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	#10625336	1	2	yes	—
210	C₂²×C₄	#10625337	1	2	yes	—
211	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₈	—	94	4	no	—
218	C₂×C₈	—	188	2	no	—
219	C₂×C₈	—	188	2	no	—
220	C₂×C₈	—	188	2	no	—
221	C₂×C₈	—	188	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	4	4	no	—
234	C₄²	#10625350	1	4	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	#13093948	1	2	yes	—
242	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	#13093948	1	2	yes	—
263	C₂×C₄	#13093951	1	2	yes	—
264	C₂×C₄	#13093983	1	2	yes	—
265	C₂×C₄	#13094007	1	2	yes	—
266	C₂×C₄	#13094007	1	2	yes	—
267	C₂×C₄	#13094031	1	2	yes	—
268	C₂×C₄	#13094031	1	2	yes	—
269	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
270	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
271	C₂×C₄	#13094050	1	2	yes	—
272	C₈	—	376	2	no	—
273	C₈	—	188	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	#15688986	1	1	yes	characteristic
282	C₂²	#15689090	1	1	yes	characteristic
283	C₂²	#15689090	1	1	yes	characteristic
284	C₂²	C₂×#13093948	1	1	yes	characteristic
285	C₂²	#15689128	1	1	yes	characteristic
286	C₂²	C₂×#13093951	1	1	yes	characteristic
287	C₂²	#15689140	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	#15689184	1	4	yes	—
295	C₄	#15689441	1	2	yes	—
296	C₄	#15689458	1	2	yes	—
297	C₂	—	4	2	no	—
298	C₂	#20241791	1	1	yes	characteristic
299	C₂	#20241805	1	1	yes	characteristic
300	C₂	#20241843	1	1	yes	characteristic
301	C₂	#20241857	1	1	yes	characteristic
302	C₂	#20243459	1	1	yes	characteristic
303	C₂	#20243478	1	1	yes	characteristic
304	C₂	#20243487	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#8710
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20241229	C₂	semidirect
172. C₄₇	#8710	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20241229	2
C₄₇×#2285	2
#8710	47