Subgroups — #26121975

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
2	#20243120	C₂	1	2	yes	maximal
3	#20243105	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688879	C₂	1	2	yes	maximal
5	C₄₇×#1735	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	#15689078	—	2	2	no	—
10	#15689078	—	2	2	no	—
11	C₂×#13093934	C₄	1	2	yes	complemented
12	C₂×C₈×Dic₄₇	C₄	1	2	yes	complemented
13	#15688889	—	2	4	no	—
14	#15688886	—	2	2	no	—
15	#15688879	C₄	1	8	yes	complemented
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#340	—	2	2	no	—
19	C₂×#10625329	—	2	1	no	—
20	C₂×#10625329	—	2	1	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	C₂×C₄	1	4	yes	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	#13093983	—	2	1	no	—
25	#13093983	—	2	1	no	—
26	#13093983	—	2	1	no	—
27	#13093983	—	2	1	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	#13093983	—	2	2	no	—
31	#13093983	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
34	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
35	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	#13093934	—	2	4	no	—
39	C₈×Dic₄₇	—	2	4	no	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	#117×C₄₇	—	2	2	no	—
48	#117×C₄₇	—	2	2	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×Dic₄₇	—	4	2	no	—
67	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	—	2	1	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₂²⋊Dic₄₇	—	2	2	no	—
83	C₂²⋊Dic₄₇	—	2	2	no	—
84	C₂²⋊Dic₄₇	—	2	4	no	—
85	C₂²⋊Dic₄₇	—	4	2	no	—
86	C₄⋊Dic₄₇	—	2	2	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄×Dic₄₇	—	2	2	no	—
91	C₄×Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	2	4	no	—
93	C₄×Dic₄₇	—	4	2	no	—
94	#10625329	—	2	4	no	—
95	#10625329	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	C₄×D₄	1	2	yes	—
102	C₂³×C₄₇	#121	1	1	yes	characteristic
103	C₂³×C₄₇	#121	1	1	yes	characteristic
104	C₂³×C₄₇	#124	1	2	yes	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	8	4	no	—
107	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#117	1	2	yes	—
114	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#123	1	2	yes	—
121	C₂×C₄×C₄₇	#123	1	2	yes	—
122	C₂×C₄×C₄₇	#125	1	2	yes	—
123	C₂×C₄×C₄₇	#126	1	2	yes	—
124	C₂×C₄×C₄₇	#126	1	2	yes	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	8	4	no	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	—	2	2	no	—
131	C₄₇⋊C₈	—	2	4	no	—
132	#10814	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	4	2	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	#380	1	1	yes	characteristic
140	C₂²×C₄₇	#382	1	1	yes	characteristic
141	C₂²×C₄₇	#388	1	2	yes	—
142	C₂²×C₄₇	#388	1	2	yes	—
143	C₄×C₄₇	—	8	4	no	—
144	C₄×C₄₇	#376	1	1	yes	characteristic
145	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₄₇	#381	1	1	yes	characteristic
147	C₄×C₄₇	#383	1	1	yes	characteristic
148	C₄×C₄₇	#387	1	2	yes	—
149	C₄×C₄₇	#387	1	2	yes	—
150	Dic₄₇	—	4	4	no	—
151	Dic₄₇	—	8	4	no	—
152	C₂×#405	—	47	1	no	—
153	#1740	—	47	2	no	—
154	#1735	—	47	1	no	—
155	#1735	D₄₇	1	1	yes	characteristic
156	C₂×H(3,ℤ/4)	—	47	2	no	—
157	C₂×C₄₇	—	2	2	no	—
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	#1434	1	1	yes	characteristic
162	C₂×C₄₇	#1812	1	2	yes	—
163	C₂²×M₁₆	—	47	1	no	—
164	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂×#117	—	47	1	no	—
166	C₂×#117	—	47	2	no	—
167	C₂×#117	C₂×D₄₇	1	2	yes	—
168	#405	—	94	2	no	—
169	#405	—	94	2	no	—
170	C₂×#97	—	47	2	no	—
171	C₂×C₄×C₈	—	47	2	no	—
172	#342	—	94	4	no	—
173	#340	—	2	2	no	—
174	#340	—	94	2	no	—
175	H(3,ℤ/4)	—	47	8	no	—
176	C₄₇	#10814	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	94	1	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	47	2	no	—
185	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
187	C₂²×C₈	D₁₈₈	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	47	4	no	—
193	#117	—	94	1	no	—
194	#117	—	94	1	no	—
195	#117	—	94	1	no	—
196	#117	—	94	1	no	—
197	#117	—	94	2	no	—
198	#117	—	94	2	no	—
199	#117	—	94	2	no	—
200	#117	C₄×D₄₇	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	47	2	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	47	2	no	—
206	C₂×C₂²⋊C₄	—	94	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	—	94	2	no	—
210	#97	—	94	4	no	—
211	C₄×C₈	—	94	4	no	—
212	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	47	4	no	—
215	C₂²×C₄	—	188	2	no	—
216	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
218	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
219	C₂²×C₄	#10625350	1	2	yes	—
220	C₂²×C₄	#10625352	1	2	yes	—
221	C₂²×C₄	#10625352	1	1	yes	characteristic
222	C₂²×C₄	#10625352	1	2	yes	—
223	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	#10625354	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	M₁₆	—	94	4	no	—
227	M₁₆	—	94	4	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	94	2	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
238	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₄⋊C₄	—	94	2	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	94	2	no	—
248	C₂²⋊C₄	—	94	2	no	—
249	C₂²⋊C₄	—	94	4	no	—
250	C₂²⋊C₄	—	188	2	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	94	2	no	—
254	C₄²	—	94	2	no	—
255	C₄²	—	94	4	no	—
256	C₄²	—	188	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
259	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
260	C₂³	#13093995	1	1	yes	characteristic
261	C₂³	#13093999	1	1	yes	characteristic
262	C₂³	#13094010	1	2	yes	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	94	4	no	—
266	C₂×C₄	—	376	4	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	—	188	4	no	—
269	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
270	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	#13093980	1	1	yes	characteristic
272	C₂×C₄	#13093980	1	2	yes	—
273	C₂×C₄	#13093980	1	1	yes	characteristic
274	C₂×C₄	#13093981	1	1	yes	characteristic
275	C₂×C₄	#13093982	1	2	yes	—
276	C₂×C₄	#13093982	1	1	yes	characteristic
277	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
278	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
279	C₂×C₄	#13093989	1	2	yes	—
280	C₂×C₄	#13094000	1	1	yes	characteristic
281	C₂×C₄	#13094004	1	1	yes	characteristic
282	C₂×C₄	#13094005	1	2	yes	—
283	C₂×C₄	#13094008	1	2	yes	—
284	C₂×C₄	#13094014	1	2	yes	—
285	C₂×C₄	#13094017	1	2	yes	—
286	C₂×C₄	#13094019	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	94	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#15688974	1	1	yes	characteristic
296	C₂²	#15688980	1	1	yes	characteristic
297	C₂²	#15688987	1	1	yes	characteristic
298	C₂²	#15689011	1	2	yes	—
299	C₂²	#15689015	1	2	yes	—
300	C₄	—	8	4	no	—
301	C₄	—	376	4	no	—
302	C₄	—	188	4	no	—
303	C₄	#15688968	1	1	yes	characteristic
304	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
305	C₄	#15688984	1	1	yes	characteristic
306	C₄	#15688990	1	1	yes	characteristic
307	C₄	#15689004	1	2	yes	—
308	C₄	#15689008	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	2	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	#20241612	1	1	yes	characteristic
314	C₂	#20243431	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093982
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1735	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1735	D₄₇
hypocenter	C₄₇	#10814
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
11. C₂×#13093934	C₄	semidirect
12. C₂×C₈×Dic₄₇	C₄	semidirect
15. #15688879	C₄	semidirect
176. C₄₇	#10814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689078	2
#20243120	2
#20243105	2
C₂×#15688879	2
C₄₇×#1735	2
#10814	47