Subgroups — #26104760

Order 12032.

562 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#20243082	C₂	1	1	yes	maximal, characteristic
3	#20241214	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2857	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#391	C₂²	1	2	yes	—
15	C₄₇×#389	—	2	1	no	—
16	C₄₇×#389	—	2	1	no	—
17	C₄₇×#384	C₄	1	1	yes	characteristic
18	C₄₇×#380	—	2	1	no	—
19	C₄₇×#380	—	2	1	no	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	4	2	no	—
22	C₂×#10625329	—	4	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₇	—	4	2	no	—
32	C₂²⋊Q₈×C₄₇	—	4	4	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	—	4	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#95×C₄₇	—	2	2	no	—
46	#95×C₄₇	—	2	2	no	—
47	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₄₇	—	2	4	no	—
49	C₂×Q₈×C₄₇	—	4	2	no	—
50	C₂×Q₈×C₄₇	—	4	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	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₄⋊C₄×C₄₇	—	4	2	no	—
71	C₄⋊C₄×C₄₇	—	4	2	no	—
72	C₄⋊C₄×C₄₇	—	4	4	no	—
73	C₄⋊C₄×C₄₇	—	4	2	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	4	4	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	4	1	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	#10625329	—	4	8	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	4	2	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	#103	1	2	yes	—
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₄₇	—	8	4	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	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	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	#99	1	2	yes	—
117	C₂×C₄×C₄₇	#100	1	2	yes	—
118	C₂×C₄×C₄₇	#103	1	2	yes	—
119	C₂×C₄×C₄₇	#104	1	2	yes	—
120	C₂×C₄×C₄₇	#104	1	2	yes	—
121	C₂×C₄×C₄₇	#131	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#137	1	2	yes	—
125	C₄₇⋊C₈	—	4	4	no	—
126	C₄₇⋊C₈	—	4	4	no	—
127	#7687	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	4	1	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	—	4	1	no	—
135	C₂²×C₄₇	#408	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
139	C₂²×C₄₇	#419	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	8	2	no	—
149	C₄×C₄₇	—	8	2	no	—
150	C₄×C₄₇	#328	1	4	yes	—
151	#2857	D₄₇	1	1	yes	characteristic
152	C₄×C₂²⋊C₈	—	47	1	no	—
153	#1728	—	47	1	no	—
154	#1272	—	47	4	no	—
155	C₂×C₄₇	—	4	2	no	—
156	C₂×C₄₇	#1471	1	1	yes	characteristic
157	C₂×C₄₇	#1485	1	1	yes	characteristic
158	C₂×C₄₇	#1493	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
160	C₂×C₄₇	#1508	1	1	yes	characteristic
161	C₂×C₄₇	#1525	1	1	yes	characteristic
162	C₂×C₄₇	#1548	1	1	yes	characteristic
163	C₂²×M₁₆	—	94	1	no	—
164	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
165	C₂×C₂²⋊Q₈	Dic₄₇	1	2	yes	—
166	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	—	94	1	no	—
168	C₂×C₄×C₈	—	94	1	no	—
169	#391	C₂×D₄₇	1	2	yes	—
170	#389	—	2	1	no	—
171	#389	—	2	1	no	—
172	#384	Dic₄₇	1	1	yes	characteristic
173	#380	—	2	1	no	—
174	#380	—	2	1	no	—
175	#336	—	47	2	no	—
176	#336	—	47	2	no	—
177	#336	—	94	1	no	—
178	C₄₇	#7687	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
180	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂×M₁₆	—	94	4	no	—
184	C₂×M₁₆	—	188	2	no	—
185	C₂×M₁₆	—	188	2	no	—
186	C₂²×C₈	—	94	2	no	—
187	C₂²×C₈	—	94	2	no	—
188	C₄⋊Q₈	—	4	2	no	—
189	C₂²⋊Q₈	—	4	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	—	4	2	no	—
198	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²⋊C₈	—	94	4	no	—
203	C₄×C₈	—	188	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	#10625336	1	2	yes	—
221	C₂²×C₄	#10625337	1	2	yes	—
222	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	M₁₆	—	188	8	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	94	4	no	—
231	C₂×C₈	—	188	2	no	—
232	C₂×C₈	—	188	2	no	—
233	C₂×C₈	—	188	2	no	—
234	C₂×C₈	—	188	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	#13093948	1	2	yes	—
253	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
254	Q₈	—	8	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#13093939	1	2	yes	—
276	C₂×C₄	#13093941	1	2	yes	—
277	C₂×C₄	#13093948	1	2	yes	—
278	C₂×C₄	#13093951	1	2	yes	—
279	C₂×C₄	#13093951	1	2	yes	—
280	C₂×C₄	#13094031	1	2	yes	—
281	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
283	C₂×C₄	#13094055	1	2	yes	—
284	C₈	—	188	4	no	—
285	C₈	—	188	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	#15689090	1	1	yes	characteristic
294	C₂²	C₂×#13093939	1	1	yes	characteristic
295	C₂²	C₂×#13093941	1	1	yes	characteristic
296	C₂²	C₂×#13093948	1	1	yes	characteristic
297	C₂²	#15689132	1	1	yes	characteristic
298	C₂²	C₂×#13093951	1	1	yes	characteristic
299	C₂²	C₂×#13093951	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	#15688860	1	4	yes	—
309	C₂	—	4	2	no	—
310	C₂	#20241748	1	1	yes	characteristic
311	C₂	#20241809	1	1	yes	characteristic
312	C₂	#20241843	1	1	yes	characteristic
313	C₂	C₂×#15688860	1	1	yes	characteristic
314	C₂	#20241908	1	1	yes	characteristic
315	C₂	#20241980	1	1	yes	characteristic
316	C₂	#20242069	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2857	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2857	D₄₇
hypocenter	C₄₇	#7687
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
3. #20241214	C₂	semidirect
178. C₄₇	#7687	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20243082	2
#20241214	2
C₄₇×#2857	2
#7687	47