Subgroups — #26104225

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	#20241214	C₂	1	2	yes	maximal
4	#20241211	C₂	1	2	yes	maximal
5	C₄₇×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	C₂²	1	2	yes	—
11	#15688878	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
13	C₄₇×#391	C₄	1	1	yes	characteristic
14	C₄₇×#391	C₂²	1	1	yes	characteristic
15	C₄₇×#389	C₄	1	1	yes	characteristic
16	C₄₇×#389	C₂²	1	1	yes	characteristic
17	C₄₇×#387	—	2	1	no	—
18	C₄₇×#387	—	2	1	no	—
19	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
21	C₄₇×#376	—	2	1	no	—
22	C₄³×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄×C₄₇⋊C₈	—	2	4	no	—
27	C₄×C₄₇⋊C₈	—	4	2	no	—
28	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₄₇	—	2	2	no	—
30	C₄×Q₈×C₄₇	C₈	1	4	yes	complemented
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	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	—	2	1	no	—
38	C₂×C₄²×C₄₇	—	2	1	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	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	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	#95×C₄₇	—	2	1	no	—
47	#95×C₄₇	—	2	1	no	—
48	#95×C₄₇	—	2	1	no	—
49	#95×C₄₇	—	2	1	no	—
50	#95×C₄₇	—	2	1	no	—
51	#95×C₄₇	—	2	1	no	—
52	#95×C₄₇	—	2	1	no	—
53	C₂×Q₈×C₄₇	—	2	1	no	—
54	C₂×Q₈×C₄₇	—	4	1	no	—
55	C₂×Q₈×C₄₇	—	4	1	no	—
56	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	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	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	2	1	no	—
70	C₄⋊C₄×C₄₇	—	4	1	no	—
71	C₄⋊C₄×C₄₇	—	4	1	no	—
72	C₄⋊C₄×C₄₇	—	4	2	no	—
73	C₄⋊C₄×C₄₇	—	4	2	no	—
74	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
75	C₄²×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	—	4	1	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₄²×C₄₇	C₂×C₈	1	2	yes	—
83	C₄²×C₄₇	M₁₆	1	2	yes	—
84	C₄²×C₄₇	SD₁₆	1	2	yes	—
85	C₄²×C₄₇	Q₁₆	1	2	yes	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	Q₈×C₄₇	—	4	2	no	—
94	Q₈×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	2	1	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	1	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	4	1	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	—	4	1	no	—
117	C₂×C₄×C₄₇	—	4	1	no	—
118	C₂×C₄×C₄₇	—	4	1	no	—
119	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₄₇	#103	1	2	yes	—
122	C₂×C₄×C₄₇	#103	1	2	yes	—
123	C₂×C₄×C₄₇	#104	1	2	yes	—
124	C₂×C₄×C₄₇	#104	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#135	1	2	yes	—
130	C₄₇⋊C₈	—	4	4	no	—
131	C₄₇⋊C₈	—	4	4	no	—
132	#7562	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₄₇	#410	1	1	yes	characteristic
135	C₂²×C₄₇	#413	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
137	C₂²×C₄₇	#416	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	4	2	no	—
149	C₄×C₄₇	—	4	2	no	—
150	C₄×C₄₇	—	8	1	no	—
151	C₄×C₄₇	—	8	1	no	—
152	C₄×C₄₇	Q₈⋊C₈	1	4	yes	—
153	#2295	D₄₇	1	1	yes	characteristic
154	C₄×C₄⋊C₈	—	47	1	no	—
155	#1744	—	47	1	no	—
156	#1272	—	47	2	no	—
157	#1271	—	47	2	no	—
158	C₂×C₄₇	#1458	1	1	yes	characteristic
159	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
160	C₂×C₄₇	#1476	1	1	yes	characteristic
161	C₂×C₄₇	#1500	1	1	yes	characteristic
162	C₂×C₄₇	#1503	1	1	yes	characteristic
163	C₂×C₄₇	#1524	1	1	yes	characteristic
164	C₂×C₄₇	#1542	1	1	yes	characteristic
165	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	94	1	no	—
167	C₂×C₄×C₈	—	94	1	no	—
168	C₂×C₄×C₈	—	94	1	no	—
169	#391	Dic₄₇	1	1	yes	characteristic
170	#391	C₂×D₄₇	1	1	yes	characteristic
171	#389	Dic₄₇	1	1	yes	characteristic
172	#389	C₂×D₄₇	1	1	yes	characteristic
173	#387	—	2	1	no	—
174	#387	—	2	1	no	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
177	#376	—	2	1	no	—
178	C₄³	C₂×D₄₇	1	1	yes	characteristic
179	#336	—	47	2	no	—
180	#336	—	47	2	no	—
181	#336	—	94	1	no	—
182	C₄₇	#7562	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
184	C₂²×C₈	—	94	2	no	—
185	C₂²×C₈	—	94	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	C₄₇⋊C₈	1	4	yes	complemented
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₄₇	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	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊C₈	—	94	4	no	—
204	C₄×C₈	—	94	4	no	—
205	C₄×C₈	—	188	2	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	#95	—	2	1	no	—
210	#95	—	2	1	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
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	C₂×C₈	—	94	4	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	188	2	no	—
231	C₂×C₈	—	188	2	no	—
232	C₂×C₈	—	188	2	no	—
233	C₂×C₈	—	188	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
249	C₄²	#10625329	1	2	yes	—
250	C₄²	#10625336	1	2	yes	—
251	C₄²	#10625337	1	2	yes	—
252	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	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	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
280	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
281	C₂×C₄	#13093948	1	2	yes	—
282	C₂×C₄	#13093948	1	2	yes	—
283	C₂×C₄	#13093951	1	2	yes	—
284	C₂×C₄	#13093951	1	2	yes	—
285	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
286	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
287	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
289	C₂×C₄	#13094044	1	2	yes	—
290	C₈	—	188	4	no	—
291	C₈	—	188	4	no	—
292	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
293	C₂²	#15689097	1	1	yes	characteristic
294	C₂²	#15689107	1	1	yes	characteristic
295	C₂²	C₂×#13093948	1	1	yes	characteristic
296	C₂²	#15689119	1	1	yes	characteristic
297	C₂²	C₂×#13093951	1	1	yes	characteristic
298	C₂²	C₂×#13093951	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	1	no	—
310	C₄	—	8	1	no	—
311	C₄	Q₈⋊(C₄₇⋊C₈)	1	4	yes	—
312	C₂	#20241702	1	1	yes	characteristic
313	C₂	C₂×Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
314	C₂	#20241770	1	1	yes	characteristic
315	C₂	#20241874	1	1	yes	characteristic
316	C₂	#20241887	1	1	yes	characteristic
317	C₂	#20241976	1	1	yes	characteristic
318	C₂	#20242045	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2295	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2295	D₄₇
hypocenter	C₄₇	#7562
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
30. C₄×Q₈×C₄₇	C₈	semidirect
182. C₄₇	#7562	semidirect
187. C₄×Q₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243134	2
#20241214	2
#20241211	2
C₄₇×#2295	2
#7562	47