Subgroups — #26225140

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253900	C₂	1	1	yes	maximal, characteristic
2	#20252321	C₂	1	2	yes	maximal
3	#20244363	C₂	1	2	yes	maximal
4	#20244141	C₂	1	1	yes	maximal, characteristic
5	#20244138	C₂	1	2	yes	maximal
6	#20243558	C₂	1	1	yes	maximal, characteristic
7	#20243275	C₂	1	1	yes	maximal, characteristic
8	#20243274	C₂	1	1	yes	maximal, characteristic
9	#20243271	C₂	1	2	yes	maximal
10	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1946	C₂	1	1	yes	maximal, characteristic
12	#15689676	C₂²	1	2	yes	—
13	#15689527	C₄	1	2	yes	complemented
14	#15689524	C₄	1	2	yes	complemented
15	#15689523	C₄	1	2	yes	complemented
16	#15689520	C₄	1	2	yes	complemented
17	#15689252	—	2	4	no	—
18	#15689247	—	2	4	no	—
19	C₂×#13093960	C₂²	1	1	yes	characteristic
20	C₂×#13093959	C₂²	1	2	yes	—
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689159	C₂²	1	2	yes	—
23	#15689149	C₂²	1	2	yes	—
24	#15689146	C₂²	1	2	yes	—
25	#15689081	C₂²	1	2	yes	—
26	#15689079	C₂²	1	2	yes	—
27	C₂×#13093935	C₂²	1	1	yes	characteristic
28	C₂×#13093934	C₂²	1	1	yes	characteristic
29	C₂×#13093934	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
33	#15688884	C₂²	1	2	yes	—
34	#15688883	C₂²	1	2	yes	—
35	#15688882	C₂²	1	2	yes	—
36	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#427	C₂²	1	1	yes	characteristic
38	C₄₇×#424	C₂²	1	2	yes	—
39	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#339	C₂²	1	2	yes	—
41	C₂×#10625329	C₂³	1	2	yes	—
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	#13094023	C₂×C₄	1	4	yes	—
45	#13094015	C₂×C₄	1	4	yes	—
46	#13094001	—	2	4	no	—
47	C₄×Dic₉₄	—	2	4	no	—
48	#13093983	C₂³	1	2	yes	—
49	C₂×C₄×Dic₄₇	—	2	2	no	—
50	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	#13093960	—	2	2	no	—
53	#13093959	—	2	4	no	—
54	#13093958	C₂×C₄	1	2	yes	—
55	#13093957	C₂×C₄	1	4	yes	—
56	#13093956	C₂×C₄	1	2	yes	—
57	#13093955	C₂×C₄	1	4	yes	—
58	Dic₄₇⋊C₈	—	2	4	no	—
59	Dic₄₇⋊C₈	—	2	4	no	—
60	#13093935	—	2	2	no	—
61	#13093935	Q₈	1	4	yes	complemented
62	#13093934	—	2	2	no	—
63	#13093934	—	2	2	no	—
64	#13093934	Q₈	1	4	yes	complemented
65	C₄×C₄₇⋊C₈	—	2	2	no	—
66	C₈×Dic₄₇	—	2	2	no	—
67	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₄₇	D₄	1	4	yes	—
72	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	#117×C₄₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	#108×C₄₇	—	2	2	no	—
78	#107×C₄₇	C₂×C₄	1	2	yes	—
79	#106×C₄₇	C₂×C₄	1	2	yes	—
80	C₄⋊C₈×C₄₇	—	2	4	no	—
81	#97×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₇	—	2	1	no	—
85	C₂²×Dic₄₇	#41	1	2	yes	—
86	C₂×Dic₉₄	—	2	4	no	—
87	M₁₆×C₄₇	—	2	2	no	—
88	M₁₆×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₄₇	#41	1	2	yes	—
97	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₄₇	—	2	2	no	—
99	C₄²×C₄₇	—	2	2	no	—
100	C₄²×C₄₇	—	2	1	no	—
101	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
102	C₂²⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
105	Dic₄₇⋊C₄	—	2	4	no	—
106	Dic₄₇⋊C₄	—	2	4	no	—
107	C₄×Dic₄₇	—	2	2	no	—
108	C₄×Dic₄₇	—	2	1	no	—
109	C₄×Dic₄₇	—	4	4	no	—
110	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
111	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
113	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
114	#10625329	—	2	2	no	—
115	#10625329	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
120	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
121	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
122	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
123	C₂×C₄₇⋊C₈	#41	1	2	yes	—
124	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₄₇	—	2	4	no	—
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	C₄×D₄	1	2	yes	—
139	C₂×Dic₄₇	C₄×D₄	1	2	yes	—
140	Dic₉₄	—	4	8	no	—
141	C₈×C₄₇	—	2	2	no	—
142	C₈×C₄₇	—	4	2	no	—
143	C₈×C₄₇	C₄×Q₈	1	2	yes	—
144	C₈×C₄₇	C₄×Q₈	1	2	yes	—
145	C₄₇⋊C₈	—	4	2	no	—
146	C₄₇⋊C₈	C₄×Q₈	1	2	yes	—
147	C₄₇⋊C₈	C₄×Q₈	1	2	yes	—
148	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
149	#21734	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄₇	#533	1	1	yes	characteristic
153	C₂²×C₄₇	#533	1	1	yes	characteristic
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	—	4	4	no	—
156	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₄×C₄₇	#532	1	1	yes	characteristic
160	Dic₄₇	—	2	2	no	—
161	Dic₄₇	—	4	2	no	—
162	Dic₄₇	—	4	4	no	—
163	Dic₄₇	#444	1	4	yes	—
164	#3160	—	47	1	no	—
165	#2960	—	47	2	no	—
166	#1994	—	47	2	no	—
167	#1946	—	47	2	no	—
168	#1946	—	47	1	no	—
169	#1946	D₄₇	1	1	yes	characteristic
170	#1848	—	47	1	no	—
171	#1773	—	47	1	no	—
172	#1773	—	47	1	no	—
173	#1773	—	47	2	no	—
174	C₄×#97	—	47	1	no	—
175	C₂×C₄₇	—	2	1	no	—
176	C₂×C₄₇	—	2	1	no	—
177	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#444	1	2	yes	—
179	#533	—	47	2	no	—
180	#500	—	47	2	no	—
181	#499	—	47	2	no	—
182	#499	—	47	2	no	—
183	#498	—	47	2	no	—
184	#446	—	94	4	no	—
185	C₈×Q₈	—	94	4	no	—
186	C₂×#108	—	47	2	no	—
187	C₂×#108	—	47	1	no	—
188	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
189	#427	—	47	1	no	—
190	#427	—	47	2	no	—
191	#427	C₂×D₄₇	1	1	yes	characteristic
192	#424	—	47	2	no	—
193	#424	C₂×D₄₇	1	2	yes	—
194	#423	—	47	2	no	—
195	#405	—	47	2	no	—
196	#405	—	47	2	no	—
197	C₂×#97	—	47	1	no	—
198	C₂×#97	—	47	1	no	—
199	C₂×#97	—	47	1	no	—
200	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
201	C₂×C₄×C₈	—	47	1	no	—
202	C₂×C₄×C₈	—	47	1	no	—
203	C₄³	—	47	1	no	—
204	#339	—	47	2	no	—
205	#339	—	47	2	no	—
206	#339	—	47	2	no	—
207	#339	C₂×D₄₇	1	2	yes	—
208	C₄₇	#21734	1	1	yes	characteristic, complemented, Sylow
209	C₂×M₁₆	—	47	2	no	—
210	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
211	C₂²×C₈	—	47	1	no	—
212	C₂²×C₈	—	47	1	no	—
213	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	47	4	no	—
216	#125	—	47	4	no	—
217	C₂²⋊Q₈	—	94	4	no	—
218	C₄×Q₈	—	94	4	no	—
219	#117	—	47	2	no	—
220	#117	C₂²×D₄₇	1	2	yes	—
221	C₂×C₄²	—	47	1	no	—
222	C₂×C₄²	—	47	1	no	—
223	C₂×C₄²	—	94	2	no	—
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	#108	—	2	2	no	—
226	#108	—	94	4	no	—
227	#108	—	94	2	no	—
228	#107	—	47	2	no	—
229	#107	—	47	4	no	—
230	#107	C₄×D₄₇	1	2	yes	—
231	#106	—	47	2	no	—
232	#106	—	47	4	no	—
233	#106	C₄×D₄₇	1	2	yes	—
234	C₄⋊C₈	—	2	4	no	—
235	C₄⋊C₈	—	94	4	no	—
236	C₄⋊C₈	—	94	4	no	—
237	#97	—	2	2	no	—
238	#97	—	47	4	no	—
239	#97	—	47	4	no	—
240	#97	—	94	2	no	—
241	#97	—	94	2	no	—
242	#97	—	94	2	no	—
243	C₄×C₈	—	47	1	no	—
244	C₄×C₈	—	47	1	no	—
245	C₄×C₈	—	47	1	no	—
246	C₄×C₈	—	47	1	no	—
247	C₄×C₈	—	47	4	no	—
248	C₄×C₈	—	94	2	no	—
249	C₄×C₈	—	94	2	no	—
250	C₂×Q₈	—	94	4	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	47	2	no	—
253	C₂²×C₄	—	94	1	no	—
254	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	94	2	no	—
258	M₁₆	—	94	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	47	1	no	—
263	C₂×C₈	—	47	1	no	—
264	C₂×C₈	—	47	2	no	—
265	C₂×C₈	—	47	2	no	—
266	C₂×C₈	—	47	2	no	—
267	C₂×C₈	—	47	2	no	—
268	C₂×C₈	—	47	2	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₈	D₄×D₄₇	1	2	yes	—
273	C₂×C₈	#10625352	1	2	yes	—
274	C₂×C₈	Q₈×D₄₇	1	2	yes	—
275	C₄⋊C₄	—	47	4	no	—
276	C₄⋊C₄	—	94	4	no	—
277	C₄⋊C₄	—	94	4	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	94	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	47	1	no	—
285	C₄²	—	47	1	no	—
286	C₄²	—	47	2	no	—
287	C₄²	—	47	2	no	—
288	C₄²	—	94	2	no	—
289	C₄²	—	94	1	no	—
290	C₄²	—	188	4	no	—
291	C₄²	Q₈×D₄₇	1	2	yes	—
292	C₂³	#41×D₄₇	1	1	yes	characteristic
293	Q₈	—	188	8	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	47	2	no	—
298	C₂×C₄	—	47	2	no	—
299	C₂×C₄	—	94	4	no	—
300	C₂×C₄	—	94	4	no	—
301	C₂×C₄	—	94	2	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
307	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	47	2	no	—
312	C₈	—	47	2	no	—
313	C₈	—	47	4	no	—
314	C₈	—	188	2	no	—
315	C₈	#13093989	1	2	yes	—
316	C₈	#13093989	1	2	yes	—
317	C₂²	—	2	2	no	—
318	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
319	C₂²	#15689675	1	1	yes	characteristic
320	C₂²	#15689679	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	4	no	—
323	C₄	—	47	4	no	—
324	C₄	—	94	2	no	—
325	C₄	—	188	2	no	—
326	C₄	—	188	4	no	—
327	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
328	C₄	C₂×#13093989	1	1	yes	characteristic
329	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
330	C₄	#15689668	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	#20245959	1	1	yes	characteristic
334	C₂	#444×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1946	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1946	D₄₇
hypocenter	C₄₇	#21734
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1946

Complements

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

N	H ≅ G/N	type
13. #15689527	C₄	semidirect
14. #15689524	C₄	semidirect
15. #15689523	C₄	semidirect
16. #15689520	C₄	semidirect
61. #13093935	Q₈	semidirect
64. #13093934	Q₈	semidirect
208. C₄₇	#21734	semidirect

Maximal subgroups

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

subgroup	index
#20253900	2
#20252321	2
#20244363	2
#20244141	2
#20244138	2
#20243558	2
#20243275	2
#20243274	2
#20243271	2
#97×Dic₄₇	2
C₄₇×#1946	2
#21734	47