Subgroups — #26198643

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253423	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
3	#20242789	C₂	1	2	yes	maximal, complemented
4	#20242779	C₂	1	2	yes	maximal, complemented
5	#20242767	C₂	1	4	yes	maximal, complemented
6	#20242136	C₂	1	1	yes	maximal, characteristic
7	#20242045	C₂	1	2	yes	maximal
8	C₄₇×#3160	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
10	#15689456	—	2	2	no	—
11	#15689415	C₂²	1	4	yes	—
12	#15689330	C₂²	1	4	yes	—
13	#15689278	—	2	2	no	—
14	C₈×Dic₉₄	—	2	1	no	—
15	C₈×Dic₉₄	—	2	1	no	—
16	C₈×Dic₉₄	C₂²	1	2	yes	—
17	C₈×Dic₉₄	C₂²	1	2	yes	—
18	C₄×#10625337	—	2	4	no	—
19	#15689176	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₇⋊C₈	—	2	1	no	—
22	C₂×C₈×Dic₄₇	—	2	1	no	—
23	#15688876	C₂²	1	2	yes	—
24	#15688874	C₂²	1	2	yes	—
25	#15688872	C₂²	1	4	yes	—
26	#15688869	C₂²	1	4	yes	—
27	C₄₇×#533	C₂²	1	2	yes	—
28	C₄₇×#500	C₂²	1	1	yes	characteristic
29	C₄₇×#500	C₂²	1	1	yes	characteristic
30	C₄₇×#499	—	2	1	no	—
31	C₄₇×#499	—	2	1	no	—
32	C₄₇×#498	C₂²	1	2	yes	—
33	C₄₇×#427	—	2	1	no	—
34	C₄₇×#427	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
37	C₂×#10625337	—	2	4	no	—
38	C₂²×C₄₇⋊C₈	—	2	1	no	—
39	C₄×Dic₉₄	—	2	1	no	—
40	C₄×Dic₉₄	—	2	1	no	—
41	C₄×Dic₉₄	D₄	1	2	yes	—
42	C₄×Dic₉₄	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
44	C₂×Dic₄₇⋊C₄	—	2	1	no	—
45	C₂×C₄×Dic₄₇	—	2	1	no	—
46	#13093957	—	2	4	no	—
47	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
49	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
50	Dic₄₇⋊C₈	—	2	2	no	—
51	Dic₄₇⋊C₈	—	2	2	no	—
52	#13093948	—	2	4	no	—
53	#13093947	—	2	4	no	—
54	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
55	C₄×C₄₇⋊C₈	—	2	2	no	—
56	C₈×Dic₄₇	—	2	2	no	—
57	C₈×Dic₄₇	—	2	2	no	—
58	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
62	#125×C₄₇	—	2	2	no	—
63	C₂²⋊Q₈×C₄₇	—	2	2	no	—
64	C₂²⋊Q₈×C₄₇	—	2	2	no	—
65	C₄×Q₈×C₄₇	—	2	2	no	—
66	C₄×Q₈×C₄₇	—	2	2	no	—
67	#117×C₄₇	—	2	2	no	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	#107×C₄₇	—	2	1	no	—
70	#107×C₄₇	—	2	1	no	—
71	#107×C₄₇	—	2	1	no	—
72	#107×C₄₇	—	2	1	no	—
73	#106×C₄₇	—	2	2	no	—
74	#106×C₄₇	—	2	2	no	—
75	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
84	C₂²×Dic₄₇	—	2	1	no	—
85	C₂²×Dic₄₇	—	2	1	no	—
86	C₂×Dic₉₄	—	2	1	no	—
87	C₂×Dic₉₄	—	2	4	no	—
88	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	Q₁₆	1	4	yes	complemented
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	4	2	no	—
100	C₄⋊C₄×C₄₇	—	4	2	no	—
101	C₂²⋊C₄×C₄₇	—	4	2	no	—
102	C₄²×C₄₇	—	4	2	no	—
103	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₇	#41	1	1	yes	characteristic
106	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
107	#10625337	—	4	8	no	—
108	C₄⋊Dic₄₇	—	2	1	no	—
109	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
110	Dic₄₇⋊C₄	—	2	2	no	—
111	Dic₄₇⋊C₄	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	4	no	—
116	C₂×C₄₇⋊C₈	—	4	2	no	—
117	C₂³×C₄₇	#121	1	1	yes	characteristic
118	Q₈×C₄₇	—	4	4	no	—
119	Q₈×C₄₇	—	8	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₄₇	#137	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	Dic₉₄	—	2	4	no	—
137	Dic₉₄	—	2	2	no	—
138	Dic₉₄	#135	1	4	yes	—
139	C₈×C₄₇	—	2	1	no	—
140	C₈×C₄₇	—	2	1	no	—
141	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
142	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
143	C₄₇⋊C₈	—	4	4	no	—
144	C₄₇⋊C₈	—	4	2	no	—
145	#19118	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	#462	1	2	yes	—
149	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	8	2	no	—
154	C₄×C₄₇	—	8	2	no	—
155	C₄×C₄₇	#462	1	2	yes	—
156	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
157	C₄×C₄₇	C₂×#135	1	2	yes	—
158	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
159	Dic₄₇	—	2	4	no	—
160	Dic₄₇	—	4	2	no	—
161	#3160	D₄₇	1	1	yes	characteristic
162	#3104	—	47	2	no	—
163	C₂×C₈×Q₈	—	47	1	no	—
164	#1676	—	47	2	no	—
165	#1674	—	47	2	no	—
166	#1672	—	47	4	no	—
167	#1564	—	47	1	no	—
168	#1542	—	47	2	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
171	C₂×C₄₇	#3050	1	1	yes	characteristic
172	C₂×C₄₇	#3067	1	1	yes	characteristic
173	#533	C₂×D₄₇	1	2	yes	—
174	C₂×C₄×Q₈	—	47	1	no	—
175	#500	C₂×D₄₇	1	1	yes	characteristic
176	#500	C₂×D₄₇	1	1	yes	characteristic
177	#499	—	2	1	no	—
178	#499	—	2	1	no	—
179	#498	C₂×D₄₇	1	2	yes	—
180	#484	—	94	2	no	—
181	#477	—	47	4	no	—
182	#462	—	47	4	no	—
183	#451	—	94	2	no	—
184	C₈×Q₈	—	47	2	no	—
185	C₈×Q₈	—	47	2	no	—
186	C₈×Q₈	—	94	1	no	—
187	C₈×Q₈	—	94	1	no	—
188	C₄×Q₁₆	—	94	4	no	—
189	#431	—	47	2	no	—
190	#427	—	2	1	no	—
191	#427	—	2	1	no	—
192	C₂×C₄⋊C₈	—	47	1	no	—
193	C₂×C₄⋊C₈	—	94	1	no	—
194	C₂×C₄×C₈	—	94	1	no	—
195	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
196	#335	—	47	2	no	—
197	#334	—	47	2	no	—
198	#333	—	47	4	no	—
199	#332	—	47	4	no	—
200	C₄₇	#19118	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	47	1	no	—
202	C₂×Q₁₆	—	94	4	no	—
203	C₂²×C₈	—	94	1	no	—
204	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
207	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
208	#125	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	47	2	no	—
214	C₄×Q₈	—	47	2	no	—
215	C₄×Q₈	—	94	1	no	—
216	C₄×Q₈	—	94	1	no	—
217	#117	—	2	2	no	—
218	C₂×C₄⋊C₄	—	47	1	no	—
219	C₂×C₄⋊C₄	—	94	1	no	—
220	C₂×C₄²	—	94	1	no	—
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#107	—	94	4	no	—
227	#106	—	2	2	no	—
228	#106	—	2	2	no	—
229	C₄⋊C₈	—	47	2	no	—
230	C₄⋊C₈	—	47	4	no	—
231	C₄⋊C₈	—	94	2	no	—
232	C₄⋊C₈	—	94	2	no	—
233	C₄⋊C₈	—	94	1	no	—
234	#103	—	94	4	no	—
235	#103	—	94	4	no	—
236	C₂²⋊C₈	—	94	2	no	—
237	C₄×C₈	—	94	2	no	—
238	C₄×C₈	—	94	2	no	—
239	C₄×C₈	—	94	2	no	—
240	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
241	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	47	2	no	—
247	C₂×Q₈	—	94	4	no	—
248	C₂×Q₈	—	94	1	no	—
249	C₂²×C₄	—	94	1	no	—
250	C₂²×C₄	—	94	1	no	—
251	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	#10625352	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
254	Q₁₆	—	188	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	94	4	no	—
257	C₂×C₈	—	94	2	no	—
258	C₂×C₈	—	188	2	no	—
259	C₂×C₈	#10625337	1	4	yes	complemented
260	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	47	2	no	—
271	C₄⋊C₄	—	94	2	no	—
272	C₄⋊C₄	—	94	1	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	94	2	no	—
277	C₄²	—	94	2	no	—
278	C₄²	D₄×D₄₇	1	1	yes	characteristic
279	C₄²	#10625352	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
281	C₄²	C₂³×D₄₇	1	1	yes	characteristic
282	C₂³	#13093995	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	47	4	no	—
286	Q₈	—	94	4	no	—
287	Q₈	—	94	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	94	4	no	—
293	C₂×C₄	—	94	2	no	—
294	C₂×C₄	—	94	2	no	—
295	C₂×C₄	—	188	2	no	—
296	C₂×C₄	#13093995	1	1	yes	characteristic
297	C₂×C₄	#13093995	1	1	yes	characteristic
298	C₂×C₄	#13093995	1	1	yes	characteristic
299	C₂×C₄	C₂×#10625337	1	2	yes	—
300	C₂×C₄	#13094055	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	188	4	no	—
307	C₈	—	188	2	no	—
308	C₈	C₂×#10625337	1	2	yes	—
309	C₈	C₂×#10625337	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#15689330	1	2	yes	—
313	C₂²	C₂×#13093995	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	94	4	no	—
320	C₄	—	188	2	no	—
321	C₄	#15689330	1	2	yes	—
322	C₄	C₂²×#10625337	1	1	yes	characteristic
323	C₄	#135×D₄₇	1	2	yes	—
324	C₄	C₂×#13094055	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×#15689330	1	1	yes	characteristic
327	C₂	#20252965	1	1	yes	characteristic
328	C₂	#20253104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3160	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂³×C₄₇	#121
hypercenter	#3160	D₄₇
hypocenter	C₄₇	#19118
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #20242789	C₂	semidirect
4. #20242779	C₂	semidirect
5. #20242767	C₂	semidirect
90. C₂×C₈×C₄₇	Q₁₆	semidirect
200. C₄₇	#19118	semidirect
259. C₂×C₈	#10625337	semidirect

Maximal subgroups

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

subgroup	index
#20253423	2
C₂×C₈×Dic₉₄	2
#20242789	2
#20242779	2
#20242767	2
#20242136	2
#20242045	2
C₄₇×#3160	2
#19118	47