Subgroups — #26122319

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251807	C₂	1	1	yes	maximal, characteristic
2	#20246008	C₂	1	1	yes	maximal, characteristic
3	#20243109	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	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689079	—	2	4	no	—
10	C₄×#10625329	—	2	2	no	—
11	#15689021	—	2	2	no	—
12	#15688995	—	2	2	no	—
13	#15688992	C₄	1	2	yes	complemented
14	#15688988	C₄	1	2	yes	complemented
15	#15688885	—	2	4	no	—
16	#15688879	C₄	1	8	yes	complemented
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#342	—	2	2	no	—
20	C₂³×Dic₄₇	—	2	1	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	C₂×C₄	1	4	yes	—
26	C₂×#10625329	Q₈	1	4	yes	—
27	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
28	#13093983	—	2	2	no	—
29	#13093983	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	4	no	—
36	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	#13093935	—	2	4	no	—
39	C₄×C₄₇⋊C₈	—	2	4	no	—
40	C₄₇⋊#95	—	2	4	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	#117×C₄₇	—	2	2	no	—
45	#117×C₄₇	—	2	2	no	—
46	#117×C₄₇	—	2	2	no	—
47	#117×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	8	2	no	—
71	C₄⋊C₄×C₄₇	—	2	2	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	2	4	no	—
76	C₂²⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₂²⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	4	1	no	—
86	C₄⋊Dic₄₇	—	4	1	no	—
87	Dic₄₇⋊C₄	—	8	4	no	—
88	C₄×Dic₄₇	—	2	2	no	—
89	C₄×Dic₄₇	C₄²	1	4	yes	—
90	#10625329	—	2	4	no	—
91	#10625329	—	2	4	no	—
92	#10625329	—	2	4	no	—
93	#10625329	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	complemented
97	C₂³×C₄₇	—	4	1	no	—
98	C₂³×C₄₇	—	4	1	no	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₄×D₄	1	2	yes	—
101	C₂³×C₄₇	#121	1	2	yes	—
102	C₂³×C₄₇	#124	1	1	yes	characteristic
103	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	8	2	no	—
110	C₂×C₄×C₄₇	—	8	2	no	—
111	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
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×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	#123	1	2	yes	—
124	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#125	1	2	yes	—
126	C₂×C₄×C₄₇	#126	1	2	yes	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	8	4	no	—
133	C₄₇⋊C₈	—	2	8	no	—
134	#10889	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	4	2	no	—
140	C₂²×C₄₇	—	4	1	no	—
141	C₂²×C₄₇	—	4	1	no	—
142	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄₇	#386	1	1	yes	characteristic
144	C₂²×C₄₇	#388	1	2	yes	—
145	C₂²×C₄₇	#389	1	2	yes	—
146	C₂²×C₄₇	#391	1	1	yes	characteristic
147	C₄×C₄₇	—	4	2	no	—
148	C₄×C₄₇	—	4	2	no	—
149	C₄×C₄₇	—	8	4	no	—
150	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
151	C₄×C₄₇	#385	1	2	yes	—
152	C₄×C₄₇	#387	1	2	yes	—
153	C₄×C₄₇	#387	1	2	yes	—
154	Dic₄₇	—	4	4	no	—
155	Dic₄₇	—	8	2	no	—
156	#2873	—	47	1	no	—
157	#2317	—	47	1	no	—
158	#1740	D₄₇	1	1	yes	characteristic
159	#1736	—	47	2	no	—
160	C₂×H(3,ℤ/4)	—	47	2	no	—
161	C₂×C₄₇	—	2	1	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	—	2	1	no	—
165	C₂×C₄₇	#1436	1	1	yes	characteristic
166	C₂×C₄₇	#1909	1	1	yes	characteristic
167	C₂×C₄₇	#1910	1	1	yes	characteristic
168	C₂²×M₁₆	—	47	2	no	—
169	C₂×#117	—	47	1	no	—
170	C₂×#117	C₂×D₄₇	1	2	yes	—
171	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	—	47	1	no	—
173	#405	—	94	4	no	—
174	C₄×M₁₆	—	94	2	no	—
175	#389	—	94	2	no	—
176	#385	—	94	2	no	—
177	#384	—	47	2	no	—
178	#382	—	47	2	no	—
179	#342	—	2	2	no	—
180	#340	—	94	4	no	—
181	H(3,ℤ/4)	—	47	8	no	—
182	C₄₇	#10889	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	—	94	1	no	—
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×M₁₆	—	47	4	no	—
186	C₂×M₁₆	—	47	4	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂²×C₈	—	47	4	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	47	4	no	—
198	#117	—	94	2	no	—
199	#117	C₄×D₄₇	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	47	2	no	—
202	C₂×C₄⋊C₄	—	94	1	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₄⋊C₄	—	94	2	no	—
205	C₂×C₄⋊C₄	—	94	4	no	—
206	C₂×C₄⋊C₄	—	94	1	no	—
207	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	94	1	no	—
211	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	47	2	no	—
214	C₂×C₄²	D₁₈₈	1	2	yes	—
215	#97	—	94	4	no	—
216	C₄×C₈	—	94	4	no	—
217	#95	—	94	4	no	—
218	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	—	94	2	no	—
222	C₂²×C₄	—	376	2	no	—
223	C₂²×C₄	—	188	2	no	—
224	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	#10625350	1	2	yes	—
230	C₂²×C₄	D₄×D₄₇	1	2	yes	—
231	C₂²×C₄	#10625352	1	2	yes	—
232	C₂²×C₄	#10625352	1	2	yes	—
233	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
234	M₁₆	—	94	4	no	—
235	M₁₆	—	94	4	no	—
236	M₁₆	—	94	4	no	—
237	M₁₆	—	94	4	no	—
238	C₂×C₈	—	47	8	no	—
239	C₂×C₈	—	94	4	no	—
240	C₂×C₈	—	94	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	94	4	no	—
246	C₄⋊C₄	—	94	4	no	—
247	C₄⋊C₄	—	376	4	no	—
248	C₄⋊C₄	—	188	1	no	—
249	C₄⋊C₄	—	188	1	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	94	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	47	4	no	—
259	C₄²	—	94	2	no	—
260	C₂³	—	4	1	no	—
261	C₂³	—	4	1	no	—
262	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
264	C₂³	#13093996	1	2	yes	—
265	C₂³	#13094011	1	1	yes	characteristic
266	C₂³	#13094022	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	94	4	no	—
275	C₂×C₄	—	376	4	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
281	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
282	C₂×C₄	#13093980	1	2	yes	—
283	C₂×C₄	#13093981	1	1	yes	characteristic
284	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
285	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
286	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
287	C₂×C₄	#13093989	1	2	yes	—
288	C₂×C₄	#13093989	1	2	yes	—
289	C₂×C₄	#13093994	1	1	yes	characteristic
290	C₂×C₄	#13094002	1	2	yes	—
291	C₂×C₄	#13094004	1	2	yes	—
292	C₂×C₄	#13094006	1	2	yes	—
293	C₂×C₄	#13094008	1	1	yes	characteristic
294	C₂×C₄	#13094014	1	2	yes	—
295	C₂×C₄	#13094017	1	2	yes	—
296	C₈	—	94	8	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
305	C₂²	#15689000	1	1	yes	characteristic
306	C₂²	#15689017	1	2	yes	—
307	C₂²	#15689018	1	2	yes	—
308	C₂²	#15689028	1	1	yes	characteristic
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	8	4	no	—
312	C₄	—	376	2	no	—
313	C₄	—	188	4	no	—
314	C₄	#15688974	1	2	yes	—
315	C₄	#15688997	1	2	yes	—
316	C₄	#15689005	1	2	yes	—
317	C₄	#15689008	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	#20241623	1	1	yes	characteristic
323	C₂	#20243908	1	1	yes	characteristic
324	C₂	#20243913	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689000
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1740	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1740	D₄₇
hypocenter	C₄₇	#10889
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
13. #15688992	C₄	semidirect
14. #15688988	C₄	semidirect
16. #15688879	C₄	semidirect
96. C₂×C₄₇⋊C₈	C₄⋊C₄	semidirect
182. C₄₇	#10889	semidirect

Maximal subgroups

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

subgroup	index
#20251807	2
#20246008	2
#20243109	2
C₂×#15688879	2
C₄₇×#1740	2
#10889	47