Subgroups — #26110296

Order 12032.

420 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251950	C₂	1	1	yes	maximal, characteristic
2	#20243369	C₂	1	1	yes	maximal, characteristic
3	#20242162	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241700	C₂	1	1	yes	maximal, characteristic
5	#20241284	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₄₇	—	2	1	no	—
9	C₂×#13093981	C₂²	1	1	yes	characteristic
10	#15689176	C₂²	1	1	yes	characteristic
11	#15689148	—	2	2	no	—
12	#15689148	—	2	2	no	—
13	#15689147	C₂²	1	1	yes	characteristic
14	#15689146	—	2	1	no	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
17	#15688877	—	2	1	no	—
18	#15688875	—	2	2	no	—
19	#15688848	C₄	1	2	yes	—
20	#15688846	C₄	1	2	yes	—
21	C₄₇×#431	C₂²	1	1	yes	characteristic
22	C₄₇×#423	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#335	—	2	2	no	—
25	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
26	C₂×#10625329	—	2	1	no	—
27	C₂×#10625324	—	2	2	no	—
28	M₁₆×D₄₇	—	4	2	no	—
29	M₁₆×D₄₇	—	4	2	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	C₂×C₈×D₄₇	—	2	2	no	—
32	#13093981	—	2	1	no	—
33	#13093981	—	2	1	no	—
34	#13093981	—	2	1	no	—
35	#13093981	—	2	1	no	—
36	C₂×Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₄₇⋊C₄	Q₈	1	1	yes	characteristic
38	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×(C₂×D₄₇)⋊C₄	Q₈	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
46	Dic₄₇⋊C₈	—	2	4	no	—
47	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
48	(C₂×D₄₇)⋊C₈	—	4	2	no	—
49	C₄×C₄₇⋊C₈	—	2	1	no	—
50	C₂×M₁₆×C₄₇	—	2	1	no	—
51	C₂²×C₈×C₄₇	—	2	1	no	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	—	2	1	no	—
54	C₄⋊C₈×C₄₇	—	2	1	no	—
55	C₄⋊C₈×C₄₇	—	2	1	no	—
56	C₄⋊C₈×C₄₇	—	2	1	no	—
57	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
58	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₇	—	2	1	no	—
60	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₄₇	—	2	1	no	—
68	C₂×C₄×D₄₇	—	2	1	no	—
69	C₂×C₄×D₄₇	—	2	1	no	—
70	C₂×C₄×D₄₇	—	2	1	no	—
71	C₂×C₄×D₄₇	D₈	1	2	yes	—
72	C₂×C₄×D₄₇	Q₁₆	1	2	yes	—
73	M₁₆×C₄₇	—	4	1	no	—
74	M₁₆×C₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	4	1	no	—
78	C₂×C₈×C₄₇	—	4	1	no	—
79	C₂×C₈×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₇	#41	1	1	yes	characteristic
83	C₄²×C₄₇	#41	1	1	yes	characteristic
84	(C₂×D₄₇)⋊C₄	—	2	1	no	—
85	(C₂×D₄₇)⋊C₄	—	2	1	no	—
86	(C₂×D₄₇)⋊C₄	—	2	1	no	—
87	(C₂×D₄₇)⋊C₄	—	2	1	no	—
88	Dic₄₇⋊C₄	—	2	1	no	—
89	Dic₄₇⋊C₄	—	2	1	no	—
90	Dic₄₇⋊C₄	—	2	1	no	—
91	Dic₄₇⋊C₄	—	2	1	no	—
92	C₄×Dic₄₇	—	2	1	no	—
93	C₄×Dic₄₇	—	2	1	no	—
94	#10625329	—	4	1	no	—
95	#10625329	—	4	1	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	4	1	no	—
99	C₂×C₄₇⋊C₈	—	4	1	no	—
100	C₂×C₄₇⋊C₈	—	4	1	no	—
101	#10625324	—	4	4	no	—
102	C₈×D₄₇	—	4	4	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×D₄₇	—	2	1	no	—
105	C₂²×D₄₇	—	2	1	no	—
106	C₂²×D₄₇	—	4	1	no	—
107	C₂²×D₄₇	—	4	1	no	—
108	C₂²×D₄₇	#107	1	2	yes	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	1	no	—
122	C₂×Dic₄₇	—	2	1	no	—
123	C₂×Dic₄₇	—	2	1	no	—
124	C₂×Dic₄₇	—	2	1	no	—
125	C₂×Dic₄₇	—	4	1	no	—
126	C₂×Dic₄₇	—	4	1	no	—
127	C₂×Dic₄₇	#107	1	2	yes	—
128	C₄×D₄₇	—	2	2	no	—
129	C₄×D₄₇	—	2	2	no	—
130	C₄×D₄₇	—	2	2	no	—
131	C₄×D₄₇	—	2	2	no	—
132	C₈×C₄₇	—	4	2	no	—
133	C₈×C₄₇	—	4	2	no	—
134	C₈×C₄₇	—	8	1	no	—
135	C₄₇⋊C₈	—	4	2	no	—
136	C₄₇⋊C₈	—	4	2	no	—
137	C₄₇⋊C₈	—	8	1	no	—
138	#8861	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	#385	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
143	C₂²×C₄₇	#428	1	1	yes	characteristic
144	C₂×D₄₇	—	2	2	no	—
145	C₂×D₄₇	—	4	1	no	—
146	C₂×D₄₇	—	4	1	no	—
147	C₂×D₄₇	—	4	2	no	—
148	C₂×D₄₇	—	8	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	#447	1	1	yes	characteristic
154	C₄×C₄₇	#451	1	1	yes	characteristic
155	C₄×C₄₇	#480	1	1	yes	characteristic
156	C₄×C₄₇	#484	1	1	yes	characteristic
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	—	4	1	no	—
159	Dic₄₇	—	4	1	no	—
160	#2905	—	47	1	no	—
161	#1798	—	47	1	no	—
162	#1571	—	47	1	no	—
163	#1571	D₄₇	1	1	yes	characteristic, complemented
164	#1458	—	47	1	no	—
165	#1301	—	47	1	no	—
166	#1301	—	47	1	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	#1852	1	1	yes	characteristic
170	C₂×C₄₇	#1858	1	1	yes	characteristic
171	C₂×C₄₇	#1868	1	1	yes	characteristic
172	D₄₇	—	4	2	no	—
173	C₂²×M₁₆	—	94	1	no	—
174	C₂×#117	—	47	1	no	—
175	#431	—	47	1	no	—
176	#431	C₂×D₄₇	1	1	yes	characteristic
177	#424	—	94	2	no	—
178	#424	—	94	2	no	—
179	#423	—	47	1	no	—
180	#423	—	94	1	no	—
181	#423	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	47	1	no	—
183	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	—	94	1	no	—
185	#336	—	94	1	no	—
186	#335	—	2	2	no	—
187	#335	—	94	2	no	—
188	#324	—	47	2	no	—
189	#323	—	47	2	no	—
190	C₄₇	#8861	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	47	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	—	94	1	no	—
195	C₂×M₁₆	—	188	2	no	—
196	C₂×M₁₆	—	188	2	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂²×C₈	—	94	1	no	—
200	#117	—	94	1	no	—
201	#117	—	94	1	no	—
202	#117	—	94	1	no	—
203	#117	—	94	1	no	—
204	C₂×C₄⋊C₄	—	47	1	no	—
205	C₂×C₄⋊C₄	—	47	1	no	—
206	C₂×C₂²⋊C₄	—	47	1	no	—
207	C₂×C₂²⋊C₄	—	47	1	no	—
208	C₂×C₄²	—	47	1	no	—
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	47	2	no	—
215	C₄⋊C₈	—	94	4	no	—
216	C₄⋊C₈	—	94	1	no	—
217	C₄⋊C₈	—	94	1	no	—
218	C₄⋊C₈	—	94	1	no	—
219	C₄⋊C₈	—	94	1	no	—
220	C₄⋊C₈	C₂²⋊D₄₇	1	2	yes	—
221	C₂²⋊C₈	—	47	2	no	—
222	C₂²⋊C₈	—	188	2	no	—
223	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	94	1	no	—
226	C₂⁴	—	47	1	no	—
227	C₂²×C₄	—	47	1	no	—
228	C₂²×C₄	—	47	2	no	—
229	C₂²×C₄	—	47	2	no	—
230	C₂²×C₄	—	47	1	no	—
231	C₂²×C₄	—	47	1	no	—
232	C₂²×C₄	—	94	1	no	—
233	C₂²×C₄	—	94	1	no	—
234	C₂²×C₄	—	94	1	no	—
235	C₂²×C₄	—	94	1	no	—
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	188	4	no	—
242	M₁₆	—	188	1	no	—
243	M₁₆	—	188	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	94	2	no	—
250	C₂×C₈	—	94	2	no	—
251	C₂×C₈	—	188	4	no	—
252	C₂×C₈	—	188	1	no	—
253	C₂×C₈	—	188	1	no	—
254	C₂×C₈	—	188	1	no	—
255	C₄⋊C₄	—	94	1	no	—
256	C₄⋊C₄	—	94	1	no	—
257	C₄⋊C₄	—	94	1	no	—
258	C₄⋊C₄	—	94	1	no	—
259	C₂²⋊C₄	—	94	1	no	—
260	C₂²⋊C₄	—	94	1	no	—
261	C₂²⋊C₄	—	94	1	no	—
262	C₂²⋊C₄	—	94	1	no	—
263	C₄²	—	94	1	no	—
264	C₄²	—	94	1	no	—
265	C₄²	#10625350	1	1	yes	characteristic
266	C₄²	D₄×D₄₇	1	1	yes	characteristic
267	C₄²	#10625354	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₂³	—	47	2	no	—
270	C₂³	—	94	1	no	—
271	C₂³	—	94	1	no	—
272	C₂³	—	188	1	no	—
273	C₂³	—	188	1	no	—
274	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	47	2	no	—
278	C₂×C₄	—	94	1	no	—
279	C₂×C₄	—	94	1	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	94	2	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	94	2	no	—
284	C₂×C₄	—	94	1	no	—
285	C₂×C₄	—	94	1	no	—
286	C₂×C₄	—	188	1	no	—
287	C₂×C₄	—	188	1	no	—
288	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₂×C₄	#13093986	1	1	yes	characteristic
290	C₂×C₄	#13093992	1	1	yes	characteristic
291	C₂×C₄	#13094000	1	1	yes	characteristic
292	C₂×C₄	#13094002	1	1	yes	characteristic
293	C₂×C₄	#13094009	1	1	yes	characteristic
294	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
295	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
296	C₂×C₄	#13094048	1	1	yes	characteristic
297	C₂×C₄	#13094056	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	376	1	no	—
302	C₈	—	188	2	no	—
303	C₈	—	188	2	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	94	2	no	—
307	C₂²	—	376	1	no	—
308	C₂²	—	188	2	no	—
309	C₂²	—	188	1	no	—
310	C₂²	—	188	1	no	—
311	C₂²	#15688994	1	1	yes	characteristic
312	C₂²	#107×D₄₇	1	1	yes	characteristic
313	C₂²	#15689164	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	94	2	no	—
319	C₄	—	188	1	no	—
320	C₄	—	188	1	no	—
321	C₄	#15689255	1	1	yes	characteristic
322	C₄	#15689277	1	1	yes	characteristic
323	C₄	#15689438	1	1	yes	characteristic
324	C₄	#15689460	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	188	2	no	—
328	C₂	#20243574	1	1	yes	characteristic
329	C₂	#20243606	1	1	yes	characteristic
330	C₂	#20243672	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1571	C₂
socle	C₂²×C₄₇	#385
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1571	D₄₇
hypocenter	C₄₇	#8861
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1571

Complements

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

N	H ≅ G/N	type
3. #20242162	C₂	semidirect
5. #20241284	C₂	semidirect
6. #20241283	C₂	semidirect
7. C₄₇×#1571	C₂	semidirect
163. #1571	D₄₇	semidirect
190. C₄₇	#8861	semidirect

Maximal subgroups

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

subgroup	index
#20251950	2
#20243369	2
#20242162	2
#20241700	2
#20241284	2
#20241283	2
C₄₇×#1571	2
#8861	47