Subgroups — #26121977

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	#20243104	C₂	1	2	yes	maximal
4	C₂×#15688879	C₂	1	2	yes	maximal
5	C₄₇×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689079	—	2	4	no	—
10	C₂×#13093935	C₄	1	2	yes	complemented
11	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	complemented
12	#15688890	—	2	2	no	—
13	#15688885	—	2	4	no	—
14	#15688879	C₄	1	8	yes	complemented
15	C₂×#117×C₄₇	C₂²	1	2	yes	—
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#342	—	2	2	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	1	no	—
20	C₂×#10625329	—	2	1	no	—
21	C₂×#10625329	C₂×C₄	1	4	yes	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
29	#13093983	—	2	2	no	—
30	#13093983	—	2	2	no	—
31	#13093983	—	2	2	no	—
32	#13093983	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	#13093935	—	2	4	no	—
37	C₄×C₄₇⋊C₈	—	2	4	no	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	—	2	2	no	—
41	#117×C₄₇	—	2	2	no	—
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	#117×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	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×Dic₄₇	—	4	2	no	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₄⋊C₄×C₄₇	—	2	4	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₂²⋊C₄×C₄₇	—	2	4	no	—
69	C₂²⋊C₄×C₄₇	—	2	4	no	—
70	C₂²⋊C₄×C₄₇	—	4	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	4	no	—
74	C₄²×C₄₇	—	4	2	no	—
75	C₂²⋊Dic₄₇	—	2	4	no	—
76	C₂²⋊Dic₄₇	—	4	2	no	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	4	2	no	—
79	C₄×Dic₄₇	—	2	4	no	—
80	C₄×Dic₄₇	—	4	2	no	—
81	#10625329	—	2	4	no	—
82	#10625329	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	2	2	no	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	C₂×C₄₇⋊C₈	—	2	1	no	—
86	C₂×C₄₇⋊C₈	—	2	1	no	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
92	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₄×D₄	1	2	yes	—
97	C₂³×C₄₇	#121	1	2	yes	—
98	C₂³×C₄₇	#124	1	1	yes	characteristic
99	C₂³×C₄₇	#124	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	8	4	no	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#117	1	2	yes	—
107	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#117	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	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	#123	1	2	yes	—
117	C₂×C₄×C₄₇	#123	1	2	yes	—
118	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#126	1	2	yes	—
121	C₂×C₄×C₄₇	#126	1	2	yes	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	8	4	no	—
124	C₄₇⋊C₈	—	2	4	no	—
125	C₄₇⋊C₈	—	2	2	no	—
126	C₄₇⋊C₈	—	2	2	no	—
127	#10814	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	1	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	#380	1	1	yes	characteristic
135	C₂²×C₄₇	#382	1	1	yes	characteristic
136	C₂²×C₄₇	#388	1	2	yes	—
137	C₂²×C₄₇	#388	1	2	yes	—
138	C₄×C₄₇	—	4	4	no	—
139	C₄×C₄₇	—	8	4	no	—
140	C₄×C₄₇	#376	1	1	yes	characteristic
141	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₄₇	#381	1	1	yes	characteristic
143	C₄×C₄₇	#383	1	1	yes	characteristic
144	C₄×C₄₇	#387	1	2	yes	—
145	C₄×C₄₇	#387	1	2	yes	—
146	Dic₄₇	—	8	4	no	—
147	C₂×#405	—	47	1	no	—
148	#1740	—	47	1	no	—
149	#1740	D₄₇	1	1	yes	characteristic
150	#1735	—	47	2	no	—
151	C₂×H(3,ℤ/4)	—	47	2	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	—	2	1	no	—
156	C₂×C₄₇	#1434	1	1	yes	characteristic
157	C₂×C₄₇	#1812	1	2	yes	—
158	C₂²×M₁₆	—	47	1	no	—
159	C₂³×C₈	—	47	1	no	—
160	C₂×#117	—	47	2	no	—
161	C₂×#117	C₂×D₄₇	1	2	yes	—
162	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
163	#405	—	94	4	no	—
164	C₂×#97	—	47	2	no	—
165	C₂×C₄×C₈	—	47	2	no	—
166	#342	—	2	2	no	—
167	#342	—	94	2	no	—
168	#340	—	94	4	no	—
169	H(3,ℤ/4)	—	47	8	no	—
170	C₄₇	#10814	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	1	no	—
175	C₂×M₁₆	—	94	1	no	—
176	C₂²×C₈	—	47	1	no	—
177	C₂²×C₈	—	47	1	no	—
178	C₂²×C₈	—	47	2	no	—
179	C₂²×C₈	—	47	4	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₂²×C₈	—	94	1	no	—
182	C₂²×C₈	—	94	1	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	47	4	no	—
189	#117	—	94	2	no	—
190	#117	—	94	2	no	—
191	#117	—	94	2	no	—
192	#117	C₄×D₄₇	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	94	2	no	—
195	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	94	2	no	—
198	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	94	2	no	—
201	C₂×C₄²	C₄×D₄₇	1	2	yes	—
202	#97	—	94	4	no	—
203	C₄×C₈	—	94	4	no	—
204	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	188	2	no	—
207	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
208	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
211	C₂²×C₄	#10625350	1	2	yes	—
212	C₂²×C₄	D₄×D₄₇	1	2	yes	—
213	C₂²×C₄	#10625352	1	1	yes	characteristic
214	C₂²×C₄	#10625352	1	2	yes	—
215	C₂²×C₄	#10625352	1	1	yes	characteristic
216	C₂²×C₄	#10625354	1	1	yes	characteristic
217	C₂²×C₄	#10625354	1	1	yes	characteristic
218	M₁₆	—	94	4	no	—
219	M₁₆	—	94	4	no	—
220	C₂×C₈	—	47	4	no	—
221	C₂×C₈	—	47	2	no	—
222	C₂×C₈	—	47	2	no	—
223	C₂×C₈	—	94	2	no	—
224	C₂×C₈	—	94	1	no	—
225	C₂×C₈	—	94	1	no	—
226	C₂×C₈	—	94	1	no	—
227	C₂×C₈	—	94	1	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	94	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₄⋊C₄	—	188	2	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	94	4	no	—
240	C₂²⋊C₄	—	188	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	94	4	no	—
246	C₄²	—	188	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
249	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
250	C₂³	#13093996	1	2	yes	—
251	C₂³	#13094011	1	1	yes	characteristic
252	C₂³	#13094013	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	376	4	no	—
258	C₂×C₄	—	188	4	no	—
259	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
261	C₂×C₄	#13093980	1	2	yes	—
262	C₂×C₄	#13093980	1	1	yes	characteristic
263	C₂×C₄	#13093981	1	1	yes	characteristic
264	C₂×C₄	#13093982	1	1	yes	characteristic
265	C₂×C₄	#13093982	1	2	yes	—
266	C₂×C₄	#13093982	1	1	yes	characteristic
267	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
268	C₂×C₄	#13093986	1	2	yes	—
269	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
270	C₂×C₄	#13094001	1	2	yes	—
271	C₂×C₄	#13094006	1	2	yes	—
272	C₂×C₄	#13094009	1	2	yes	—
273	C₂×C₄	#13094015	1	1	yes	characteristic
274	C₂×C₄	#13094015	1	1	yes	characteristic
275	C₂×C₄	#13094017	1	2	yes	—
276	C₂×C₄	#13094019	1	2	yes	—
277	C₈	—	94	4	no	—
278	C₈	—	94	2	no	—
279	C₈	—	94	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
286	C₂²	#15688981	1	1	yes	characteristic
287	C₂²	#15688988	1	1	yes	characteristic
288	C₂²	#15689013	1	2	yes	—
289	C₂²	#15689017	1	2	yes	—
290	C₄	—	4	4	no	—
291	C₄	—	8	4	no	—
292	C₄	—	376	4	no	—
293	C₄	#15688968	1	1	yes	characteristic
294	C₄	#15688970	1	1	yes	characteristic
295	C₄	#15688985	1	1	yes	characteristic
296	C₄	#15688990	1	1	yes	characteristic
297	C₄	#15689006	1	2	yes	—
298	C₄	#15689010	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	#20241614	1	1	yes	characteristic
304	C₂	#20243432	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093980
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1740	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1740	D₄₇
hypocenter	C₄₇	#10814
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
10. C₂×#13093935	C₄	semidirect
11. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
14. #15688879	C₄	semidirect
170. 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×#15689079	2
#20243121	2
#20243104	2
C₂×#15688879	2
C₄₇×#1740	2
#10814	47