Subgroups — #26104412

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689021	C₂	1	1	yes	maximal, characteristic
2	#20243456	C₂	1	1	yes	maximal, characteristic
3	#20241210	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₄₇⋊C₄	—	2	1	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
9	#15689021	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	C₄	1	2	yes	—
12	#15689021	C₂²	1	2	yes	—
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	#15688878	—	2	1	no	—
15	#15688878	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₄₇	—	2	1	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	—	2	1	no	—
19	C₄₇×#336	C₂²	1	2	yes	—
20	C₂³×Dic₄₇	—	2	1	no	—
21	C₂³×Dic₄₇	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	1	no	—
24	C₂×C₄⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	4	2	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₇⋊C₄	—	2	2	no	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×Dic₄₇⋊C₄	—	4	4	no	—
32	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	4	no	—
39	C₂×M₁₆×C₄₇	—	4	2	no	—
40	C₂×M₁₆×C₄₇	—	4	2	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	—	4	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	4	2	no	—
68	C₂²×Dic₄₇	—	4	2	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	M₁₆×C₄₇	—	4	8	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₄²×C₄₇	—	4	1	no	—
78	C₄²×C₄₇	—	4	1	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	C₄⋊Dic₄₇	—	4	2	no	—
84	C₄⋊Dic₄₇	M₁₆	1	4	yes	complemented
85	Dic₄₇⋊C₄	—	4	4	no	—
86	Dic₄₇⋊C₄	—	4	4	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	#99	1	2	yes	—
95	C₂³×C₄₇	#103	1	2	yes	—
96	C₂³×C₄₇	#104	1	2	yes	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	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	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	#101	1	2	yes	—
116	C₂×C₄×C₄₇	#103	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	4	8	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	8	4	no	—
129	C₈×C₄₇	—	4	4	no	—
130	C₄₇⋊C₈	—	8	2	no	—
131	#7607	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	#328	1	4	yes	—
139	C₂²×C₄₇	#407	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
143	C₂²×C₄₇	#418	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
145	C₂²×C₄₇	#421	1	1	yes	characteristic
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	4	1	no	—
153	Dic₄₇	—	4	4	no	—
154	Dic₄₇	—	8	2	no	—
155	C₂×#389	—	47	1	no	—
156	#1817	—	47	1	no	—
157	#1728	D₄₇	1	1	yes	characteristic
158	#1271	—	47	4	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	#1463	1	1	yes	characteristic
161	C₂×C₄₇	#1487	1	1	yes	characteristic
162	C₂×C₄₇	#1490	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
164	C₂×C₄₇	#1510	1	1	yes	characteristic
165	C₂×C₄₇	#1528	1	1	yes	characteristic
166	C₂×C₄₇	#1549	1	1	yes	characteristic
167	C₂²×M₁₆	—	2	1	no	—
168	C₂²×C₄⋊C₄	—	47	1	no	—
169	C₂²×C₄⋊C₄	—	47	1	no	—
170	C₂²×C₄⋊C₄	—	94	1	no	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	C₂×C₂²⋊C₈	—	94	1	no	—
173	#389	—	47	2	no	—
174	#389	—	47	2	no	—
175	#389	—	94	1	no	—
176	#389	—	94	1	no	—
177	C₂×#95	—	94	1	no	—
178	#336	—	2	1	no	—
179	#336	—	47	2	no	—
180	#336	—	94	1	no	—
181	#336	C₂×D₄₇	1	2	yes	—
182	C₄₇	#7607	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	—	94	1	no	—
184	C₂³×C₄	—	94	1	no	—
185	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
186	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
187	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	4	2	no	—
190	C₂×M₁₆	—	4	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂×C₄⋊C₄	—	47	2	no	—
194	C₂×C₄⋊C₄	—	47	2	no	—
195	C₂×C₄⋊C₄	—	94	4	no	—
196	C₂×C₄⋊C₄	—	94	1	no	—
197	C₂×C₄⋊C₄	—	94	1	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄⋊C₄	—	188	2	no	—
201	C₂×C₄⋊C₄	—	188	4	no	—
202	C₂×C₄²	—	4	2	no	—
203	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²⋊C₈	—	188	2	no	—
208	#95	—	94	2	no	—
209	#95	—	94	2	no	—
210	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
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	—	94	4	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	—	94	2	no	—
222	C₂²×C₄	—	188	2	no	—
223	C₂²×C₄	—	188	2	no	—
224	C₂²×C₄	—	188	2	no	—
225	C₂²×C₄	—	188	2	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	#10625335	1	2	yes	—
231	C₂²×C₄	#10625337	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	M₁₆	—	4	8	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	188	2	no	—
240	C₂×C₈	—	188	2	no	—
241	C₂×C₈	—	188	2	no	—
242	C₄⋊C₄	—	47	4	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	94	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄⋊C₄	—	188	4	no	—
247	C₄⋊C₄	—	188	2	no	—
248	C₄⋊C₄	—	188	4	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	#13093938	1	2	yes	—
258	C₂³	#13093947	1	2	yes	—
259	C₂³	#13093952	1	2	yes	—
260	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
261	C₂×C₄	—	2	8	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	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	376	4	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	—	188	8	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	#13093942	1	2	yes	—
287	C₂×C₄	#13093947	1	2	yes	—
288	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
289	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
291	C₂×C₄	#13094052	1	2	yes	—
292	C₈	—	4	4	no	—
293	C₈	—	376	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	#15688859	1	4	yes	—
301	C₂²	#15689086	1	1	yes	characteristic
302	C₂²	C₂×#13093938	1	1	yes	characteristic
303	C₂²	C₂×#13093942	1	1	yes	characteristic
304	C₂²	C₂×#13093947	1	1	yes	characteristic
305	C₂²	#15689129	1	1	yes	characteristic
306	C₂²	C₂×#13093952	1	1	yes	characteristic
307	C₂²	#15689139	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	376	2	no	—
316	C₄	—	188	4	no	—
317	C₂	—	2	4	no	—
318	C₂	#20241717	1	1	yes	characteristic
319	C₂	#20241816	1	1	yes	characteristic
320	C₂	#20241832	1	1	yes	characteristic
321	C₂	C₂×#15688859	1	1	yes	characteristic
322	C₂	#20241918	1	1	yes	characteristic
323	C₂	#20241989	1	1	yes	characteristic
324	C₂	#20242074	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#7607
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
3. #20241210	C₂	semidirect
84. C₄⋊Dic₄₇	M₁₆	semidirect
182. C₄₇	#7607	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689021	2
#20243456	2
#20241210	2
C₄₇×#1728	2
#7607	47