Subgroups — #26109096

Order 12032.

496 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×#13094024	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688868	C₂	1	2	yes	maximal
4	#20241272	C₂	1	2	yes	maximal
5	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₉₄	—	2	1	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15689029	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	C₄	1	2	yes	—
12	#15688993	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₄₇	—	2	1	no	—
14	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
15	#15688878	—	2	1	no	—
16	#15688878	—	2	1	no	—
17	#15688868	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₄³×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₉₄	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	#13094024	—	2	1	no	—
25	#13094024	—	2	1	no	—
26	#13094024	D₄	1	2	yes	—
27	#13094024	Q₈	1	2	yes	—
28	C₄×Dic₉₄	—	4	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
38	C₄₇⋊#95	—	2	2	no	—
39	C₄₇⋊#95	—	2	2	no	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂×C₄	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	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₇	—	2	2	no	—
49	C₄⋊C₈×C₄₇	—	2	2	no	—
50	C₄⋊C₈×C₄₇	—	4	2	no	—
51	C₄⋊C₈×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	2	no	—
60	C₂²×Dic₄₇	—	2	2	no	—
61	C₂×Dic₉₄	—	2	2	no	—
62	C₂×Dic₉₄	—	2	2	no	—
63	C₂×Dic₉₄	—	4	2	no	—
64	C₂×Dic₉₄	—	4	2	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	2	1	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	1	no	—
73	C₄²×C₄₇	—	4	1	no	—
74	C₄²×C₄₇	SD₁₆	1	2	yes	—
75	C₄²×C₄₇	Q₁₆	1	2	yes	—
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₄₇	#41	1	1	yes	characteristic
79	C₄²×C₄₇	#41	1	1	yes	characteristic
80	C₄⋊Dic₄₇	—	2	2	no	—
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	C₄⋊Dic₄₇	—	4	1	no	—
84	C₄⋊Dic₄₇	—	4	2	no	—
85	C₄⋊Dic₄₇	—	4	1	no	—
86	C₄⋊Dic₄₇	—	4	2	no	—
87	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
88	Dic₄₇⋊C₄	—	4	4	no	—
89	C₄×Dic₄₇	—	4	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	1	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	#101	1	2	yes	—
106	C₂×C₄×C₄₇	#103	1	2	yes	—
107	C₂×C₄×C₄₇	#103	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#135	1	2	yes	—
117	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
120	C₂×Dic₄₇	—	2	4	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	—	4	4	no	—
127	Dic₉₄	—	4	4	no	—
128	C₈×C₄₇	—	8	2	no	—
129	C₄₇⋊C₈	—	8	2	no	—
130	#8591	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	#380	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
133	C₂²×C₄₇	#413	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
135	C₂²×C₄₇	#416	1	1	yes	characteristic
136	C₂²×C₄₇	#417	1	1	yes	characteristic
137	C₂²×C₄₇	#418	1	1	yes	characteristic
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	4	1	no	—
144	C₄×C₄₇	#460	1	1	yes	characteristic
145	C₄×C₄₇	#462	1	1	yes	characteristic
146	C₄×C₄₇	#463	1	2	yes	—
147	C₄×C₄₇	#475	1	1	yes	characteristic
148	C₄×C₄₇	#477	1	1	yes	characteristic
149	C₄×C₄₇	#478	1	2	yes	—
150	Dic₄₇	—	4	4	no	—
151	Dic₄₇	—	8	2	no	—
152	C₄×C₄⋊Q₈	—	47	1	no	—
153	#1842	D₄₇	1	1	yes	characteristic
154	#1831	—	47	1	no	—
155	C₂×#332	—	47	2	no	—
156	#1296	—	47	2	no	—
157	C₂×C₄₇	#1542	1	1	yes	characteristic
158	C₂×C₄₇	#1549	1	1	yes	characteristic
159	C₂×C₄₇	#1552	1	1	yes	characteristic
160	C₂×C₄₇	#1557	1	1	yes	characteristic
161	C₂×C₄₇	#1801	1	1	yes	characteristic
162	C₂×C₄₇	#1803	1	1	yes	characteristic
163	C₂×C₄₇	#1804	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	—	47	1	no	—
165	C₂×C₄×Q₈	—	94	1	no	—
166	C₂×C₄⋊C₈	—	2	1	no	—
167	C₂×C₄⋊C₈	—	2	1	no	—
168	C₂×C₄⋊C₈	—	47	2	no	—
169	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
170	#391	—	94	1	no	—
171	#389	—	47	2	no	—
172	#389	—	94	1	no	—
173	#384	—	47	1	no	—
174	C₄×C₄⋊C₄	—	47	2	no	—
175	C₄×C₄⋊C₄	—	94	1	no	—
176	C₄³	C₂×D₄₇	1	1	yes	characteristic
177	#336	—	94	1	no	—
178	#336	—	94	1	no	—
179	#332	—	94	4	no	—
180	C₄₇	#8591	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	94	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	94	2	no	—
184	C₄⋊Q₈	—	47	2	no	—
185	C₄⋊Q₈	—	47	2	no	—
186	C₄⋊Q₈	—	94	1	no	—
187	C₄⋊Q₈	—	94	1	no	—
188	C₄×Q₈	—	188	4	no	—
189	C₂×C₄⋊C₄	—	47	2	no	—
190	C₂×C₄⋊C₄	—	47	2	no	—
191	C₂×C₄⋊C₄	—	94	1	no	—
192	C₂×C₄⋊C₄	—	94	2	no	—
193	C₂×C₄⋊C₄	—	94	1	no	—
194	C₂×C₄⋊C₄	—	94	1	no	—
195	C₂×C₄²	—	94	2	no	—
196	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
198	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
199	C₂×C₄²	D₁₈₈	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	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	4	2	no	—
206	C₄⋊C₈	—	4	2	no	—
207	C₄⋊C₈	—	94	2	no	—
208	C₄⋊C₈	—	94	2	no	—
209	#95	—	94	2	no	—
210	#95	—	94	2	no	—
211	C₂×Q₈	—	94	2	no	—
212	C₂×Q₈	—	94	2	no	—
213	C₂×Q₈	—	188	2	no	—
214	C₂×Q₈	—	188	2	no	—
215	C₂²×C₄	—	94	2	no	—
216	C₂²×C₄	—	94	2	no	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	188	2	no	—
228	C₂×C₈	—	188	2	no	—
229	C₂×C₈	—	188	2	no	—
230	C₄⋊C₄	—	47	4	no	—
231	C₄⋊C₄	—	94	2	no	—
232	C₄⋊C₄	—	94	2	no	—
233	C₄⋊C₄	—	94	2	no	—
234	C₄⋊C₄	—	188	2	no	—
235	C₄⋊C₄	—	188	4	no	—
236	C₄⋊C₄	—	188	1	no	—
237	C₄⋊C₄	—	188	2	no	—
238	C₄⋊C₄	—	188	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	188	4	no	—
246	C₄²	#10625335	1	2	yes	—
247	C₄²	#10625337	1	2	yes	—
248	C₄²	D₄×D₄₇	1	1	yes	characteristic
249	C₄²	#10625352	1	1	yes	characteristic
250	C₄²	Q₈×D₄₇	1	1	yes	characteristic
251	C₄²	#10625354	1	1	yes	characteristic
252	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
253	Q₈	—	188	4	no	—
254	Q₈	—	188	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	1	no	—
266	C₂×C₄	—	94	4	no	—
267	C₂×C₄	—	188	2	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	#13093942	1	2	yes	—
273	C₂×C₄	#13093947	1	2	yes	—
274	C₂×C₄	#13093947	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	#13093982	1	1	yes	characteristic
277	C₂×C₄	#13093995	1	1	yes	characteristic
278	C₂×C₄	#13093999	1	1	yes	characteristic
279	C₂×C₄	#13094000	1	1	yes	characteristic
280	C₂×C₄	#13094004	1	1	yes	characteristic
281	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
283	C₂×C₄	#13094046	1	2	yes	—
284	C₂×C₄	#13094050	1	1	yes	characteristic
285	C₂×C₄	#13094052	1	2	yes	—
286	C₂×C₄	#13094055	1	1	yes	characteristic
287	C₈	—	8	2	no	—
288	C₈	—	376	2	no	—
289	C₂²	#15688980	1	1	yes	characteristic
290	C₂²	C₂×#13093942	1	1	yes	characteristic
291	C₂²	#15689105	1	1	yes	characteristic
292	C₂²	C₂×#13093947	1	1	yes	characteristic
293	C₂²	#15689120	1	1	yes	characteristic
294	C₂²	#15689122	1	1	yes	characteristic
295	C₂²	#15689129	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	376	2	no	—
303	C₄	—	188	4	no	—
304	C₄	#15689320	1	1	yes	characteristic
305	C₄	#15689330	1	1	yes	characteristic
306	C₄	#15689339	1	2	yes	—
307	C₄	#15689405	1	1	yes	characteristic
308	C₄	#15689415	1	1	yes	characteristic
309	C₄	#15689424	1	2	yes	—
310	C₂	#20242046	1	1	yes	characteristic
311	C₂	#20242074	1	1	yes	characteristic
312	C₂	#20242086	1	1	yes	characteristic
313	C₂	#20242108	1	1	yes	characteristic
314	C₂	#20243381	1	1	yes	characteristic
315	C₂	#20243390	1	1	yes	characteristic
316	C₂	#20243399	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×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#8591
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
180. C₄₇	#8591	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094024	2
#20243505	2
C₂×#15688868	2
#20241272	2
C₄₇×#1842	2
#8591	47