Subgroups — #26112281

Order 12032.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20249209	C₂	1	1	yes	maximal, characteristic
2	#20242070	C₂	1	2	yes	maximal, complemented
3	#20241702	C₂	1	1	yes	maximal, characteristic
4	#20241248	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	2	yes	—
8	#15689062	—	2	1	no	—
9	#15689059	—	2	1	no	—
10	#15689043	C₄	1	2	yes	—
11	#15689035	C₄	1	2	yes	—
12	#15689021	—	2	1	no	—
13	#15688989	C₂²	1	1	yes	characteristic
14	#15688983	—	2	1	no	—
15	#15688871	—	2	4	no	—
16	#15688849	—	2	1	no	—
17	#15688847	—	2	1	no	—
18	C₂²×M₁₆×C₄₇	—	2	1	no	—
19	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#424	—	2	4	no	—
21	C₄₇×#423	—	2	1	no	—
22	C₄₇×#423	C₂²	1	2	yes	—
23	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
24	#13094024	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×Dic₄₇⋊C₄	—	2	2	no	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₄×C₄₇⋊C₈	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	1	no	—
37	C₄₇⋊#95	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	4	2	no	—
42	C₂×M₁₆×C₄₇	—	4	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	#117×C₄₇	—	2	2	no	—
45	#117×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₇	—	2	4	no	—
53	C₄⋊C₈×C₄₇	—	2	2	no	—
54	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
55	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
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	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂×Dic₉₄	—	2	4	no	—
71	C₂×Dic₉₄	—	8	2	no	—
72	M₁₆×C₄₇	—	4	4	no	—
73	M₁₆×C₄₇	—	4	2	no	—
74	M₁₆×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	4	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₂²⋊C₄×C₄₇	—	2	2	no	—
80	C₂²⋊C₄×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	C₂×D₄	1	2	yes	—
84	C₄²×C₄₇	#41	1	2	yes	—
85	C₂²⋊Dic₄₇	—	8	2	no	—
86	C₄⋊Dic₄₇	—	4	1	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	1	no	—
89	C₄⋊Dic₄₇	—	4	1	no	—
90	C₄⋊Dic₄₇	—	4	1	no	—
91	Dic₄₇⋊C₄	—	8	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	4	1	no	—
96	C₂³×C₄₇	—	4	1	no	—
97	C₂³×C₄₇	#103	1	2	yes	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	#103	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#121	1	2	yes	—
115	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
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	1	yes	characteristic
119	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	Dic₉₄	—	8	4	no	—
127	C₈×C₄₇	—	4	4	no	—
128	C₄₇⋊C₈	—	8	2	no	—
129	#9191	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	4	2	no	—
134	C₂²×C₄₇	—	4	1	no	—
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	—	8	1	no	—
137	C₂²×C₄₇	#386	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
139	C₂²×C₄₇	#417	1	1	yes	characteristic
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	4	1	no	—
144	C₄×C₄₇	—	4	1	no	—
145	C₄×C₄₇	#450	1	1	yes	characteristic
146	C₄×C₄₇	#451	1	1	yes	characteristic
147	C₄×C₄₇	#483	1	1	yes	characteristic
148	C₄×C₄₇	#484	1	1	yes	characteristic
149	Dic₄₇	—	8	2	no	—
150	Dic₄₇	—	8	2	no	—
151	#2905	D₄₇	1	1	yes	characteristic
152	#2628	—	47	1	no	—
153	#1548	—	47	2	no	—
154	#1458	—	47	1	no	—
155	#1286	—	47	2	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	4	2	no	—
159	C₂×C₄₇	#1872	1	1	yes	characteristic
160	C₂×C₄₇	#1909	1	1	yes	characteristic
161	C₂×C₄₇	#1914	1	1	yes	characteristic
162	C₂²×M₁₆	—	2	1	no	—
163	C₂×C₄⋊Q₈	—	47	1	no	—
164	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
165	#431	—	47	2	no	—
166	#424	—	2	4	no	—
167	#423	—	2	1	no	—
168	#423	C₂×D₄₇	1	2	yes	—
169	#400	—	94	1	no	—
170	#399	—	94	1	no	—
171	#395	—	47	2	no	—
172	#393	—	47	2	no	—
173	#389	—	94	1	no	—
174	#382	—	47	1	no	—
175	#380	—	94	1	no	—
176	#333	—	94	4	no	—
177	#324	—	94	1	no	—
178	#323	—	94	1	no	—
179	C₄₇	#9191	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	47	2	no	—
181	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	4	2	no	—
185	C₂×M₁₆	—	4	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₄⋊Q₈	—	94	2	no	—
188	C₄⋊Q₈	—	94	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	—	94	2	no	—
192	C₂×C₄⋊C₄	—	94	1	no	—
193	C₂×C₄⋊C₄	—	94	1	no	—
194	C₂×C₄⋊C₄	—	94	1	no	—
195	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	94	2	no	—
198	C₂×C₂²⋊C₄	C₄×D₄₇	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	D₁₈₈	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	94	2	no	—
205	C₄⋊C₈	C₂²⋊D₄₇	1	4	yes	—
206	C₂²⋊C₈	—	94	2	no	—
207	C₄×C₈	—	94	2	no	—
208	#95	—	47	2	no	—
209	#95	—	94	2	no	—
210	#95	—	94	2	no	—
211	#95	—	94	1	no	—
212	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
213	C₂×Q₈	—	94	4	no	—
214	C₂×Q₈	—	376	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	94	2	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	#10625335	1	2	yes	—
225	C₂²×C₄	#10625337	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	M₁₆	—	4	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	188	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	376	2	no	—
238	C₄⋊C₄	—	188	1	no	—
239	C₄⋊C₄	—	188	2	no	—
240	C₄⋊C₄	—	188	1	no	—
241	C₄⋊C₄	—	188	1	no	—
242	C₄⋊C₄	—	188	1	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	376	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	#10625350	1	2	yes	—
249	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	#13093947	1	2	yes	—
255	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	Q₈	—	376	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	#13093947	1	2	yes	—
276	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
277	C₂×C₄	#13093993	1	1	yes	characteristic
278	C₂×C₄	#13093998	1	2	yes	—
279	C₂×C₄	#13094007	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
281	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
282	C₂×C₄	#13094050	1	1	yes	characteristic
283	C₂×C₄	#13094055	1	1	yes	characteristic
284	C₈	—	4	4	no	—
285	C₈	—	376	2	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	#15689002	1	1	yes	characteristic
294	C₂²	C₂×#13093947	1	1	yes	characteristic
295	C₂²	#15689122	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	376	2	no	—
302	C₄	—	376	2	no	—
303	C₄	#15689274	1	1	yes	characteristic
304	C₄	#15689279	1	1	yes	characteristic
305	C₄	#15689453	1	1	yes	characteristic
306	C₄	#15689458	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#20243695	1	1	yes	characteristic
311	C₂	#20243910	1	1	yes	characteristic
312	C₂	#20243935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689002
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2905	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2905	D₄₇
hypocenter	C₄₇	#9191
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #20242070	C₂	semidirect
4. #20241248	C₂	semidirect
179. C₄₇	#9191	semidirect

Maximal subgroups

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

subgroup	index
#20249209	2
#20242070	2
#20241702	2
#20241248	2
C₄₇×#2905	2
#9191	47