Subgroups — #26213541

Order 12032.

578 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	#20256204	C₂	1	2	yes	maximal, complemented
2	#20256200	C₂	1	2	yes	maximal, complemented
3	#20242994	C₂	1	1	yes	maximal, characteristic
4	#20242929	C₂	1	2	yes	maximal, complemented
5	#20242928	C₂	1	2	yes	maximal, complemented
6	#20242530	C₂	1	2	yes	maximal, complemented
7	#20242449	C₂	1	2	yes	maximal
8	#20242286	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689858	C₂²	1	2	yes	—
11	Q₈×Dic₉₄	C₂²	1	2	yes	—
12	#15689503	C₂²	1	1	yes	characteristic
13	#15689500	C₂²	1	1	yes	characteristic
14	#15689484	—	2	2	no	—
15	#15689483	—	2	2	no	—
16	#15689470	C₂²	1	2	yes	—
17	#15689470	C₂²	1	2	yes	—
18	#15689419	C₂²	1	2	yes	—
19	#15689418	C₂²	1	2	yes	—
20	#15689404	—	2	2	no	—
21	#15689402	—	2	2	no	—
22	#15689347	—	2	2	no	—
23	#15689346	—	2	2	no	—
24	#15689324	C₂²	1	2	yes	—
25	#15689323	C₂²	1	2	yes	—
26	#15689253	C₂²	1	2	yes	—
27	#15689226	—	2	2	no	—
28	#15689225	—	2	2	no	—
29	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
30	#15688876	C₂²	1	1	yes	characteristic
31	#15688871	C₂²	1	2	yes	—
32	#15688866	C₂²	1	2	yes	—
33	#15688855	C₂²	1	2	yes	—
34	C₄₇×#446	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#335	C₂²	1	1	yes	characteristic
37	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₈₈	—	4	1	no	—
40	C₂×Dic₁₈₈	D₄	1	1	yes	characteristic
41	C₂×#10625325	—	2	2	no	—
42	C₂×#10625336	—	2	2	no	—
43	C₂×D₃₇₆	—	4	1	no	—
44	#13094024	—	2	2	no	—
45	#13094024	C₂³	1	2	yes	—
46	#13094023	—	2	2	no	—
47	#13094023	—	4	2	no	—
48	#13094021	C₂³	1	2	yes	—
49	#13094013	—	2	2	no	—
50	#13094012	—	2	2	no	—
51	#13094009	—	4	2	no	—
52	Q₈×Dic₄₇	—	2	2	no	—
53	C₄×Dic₉₄	—	2	2	no	—
54	C₄×Dic₉₄	C₂³	1	2	yes	—
55	C₄×D₁₈₈	—	2	2	no	—
56	#13093986	—	2	2	no	—
57	#13093958	—	2	2	no	—
58	#13093958	D₄	1	1	yes	characteristic
59	#13093957	—	2	2	no	—
60	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	#13093949	—	2	2	no	—
63	#13093949	D₄	1	2	yes	—
64	#13093948	—	2	2	no	—
65	#13093946	—	2	2	no	—
66	#13093946	—	2	2	no	—
67	#13093946	—	2	2	no	—
68	#13093946	—	2	2	no	—
69	#13093946	—	2	2	no	—
70	#13093944	—	2	2	no	—
71	#13093934	—	2	2	no	—
72	C₂×Q₁₆×C₄₇	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
74	#107×C₄₇	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₄₇	—	2	2	no	—
76	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
77	#103×C₄₇	D₄	1	2	yes	—
78	#97×C₄₇	—	2	2	no	—
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×D₁₈₈	—	2	2	no	—
84	C₂×D₁₈₈	—	2	2	no	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	C₂×Dic₉₄	—	2	2	no	—
88	C₂×Dic₉₄	—	4	2	no	—
89	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
90	Q₁₆×C₄₇	—	4	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
99	C₄²×C₄₇	—	2	2	no	—
100	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
101	#10625336	—	4	4	no	—
102	(C₂×D₄₇)⋊C₄	—	4	2	no	—
103	(C₂×D₄₇)⋊C₄	—	4	2	no	—
104	(C₂×D₄₇)⋊C₄	—	4	2	no	—
105	C₄⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	4	2	no	—
108	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
109	Dic₄₇⋊C₄	—	2	2	no	—
110	Dic₄₇⋊C₄	—	4	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	Dic₁₈₈	—	4	1	no	—
115	Dic₁₈₈	—	8	2	no	—
116	D₃₇₆	—	8	2	no	—
117	#10625325	—	4	4	no	—
118	C₂²×D₄₇	—	4	2	no	—
119	Q₈×C₄₇	—	2	2	no	—
120	Q₈×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	2	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	D₁₈₈	—	4	2	no	—
130	D₁₈₈	—	4	2	no	—
131	D₁₈₈	—	4	2	no	—
132	D₁₈₈	—	4	2	no	—
133	D₁₈₈	—	8	2	no	—
134	C₄×D₄₇	—	16	2	no	—
135	Dic₉₄	—	2	2	no	—
136	Dic₉₄	—	2	2	no	—
137	Dic₉₄	—	8	2	no	—
138	Dic₉₄	—	16	2	no	—
139	C₈×C₄₇	—	2	2	no	—
140	C₈×C₄₇	—	2	1	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	—	2	1	no	—
143	C₈×C₄₇	—	2	1	no	—
144	C₈×C₄₇	—	4	2	no	—
145	C₄₇⋊C₈	—	4	2	no	—
146	#20319	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	D₄²	1	1	yes	characteristic
148	C₂×D₄₇	—	8	2	no	—
149	C₂×D₄₇	—	8	2	no	—
150	C₂×D₄₇	—	8	2	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	—	4	2	no	—
154	C₄×C₄₇	#575	1	1	yes	characteristic
155	C₄×C₄₇	#575	1	1	yes	characteristic
156	C₄×C₄₇	#578	1	1	yes	characteristic
157	C₄×C₄₇	#578	1	1	yes	characteristic
158	Dic₄₇	—	4	2	no	—
159	Dic₄₇	—	4	2	no	—
160	Dic₄₇	—	8	2	no	—
161	#3375	—	47	2	no	—
162	#3375	—	47	2	no	—
163	#1709	—	47	1	no	—
164	#1698	—	47	2	no	—
165	#1698	—	47	2	no	—
166	#1633	—	47	2	no	—
167	#1620	—	47	2	no	—
168	#1591	—	47	1	no	—
169	#1591	D₄₇	1	1	yes	characteristic, complemented
170	C₂×C₄₇	#3277	1	1	yes	characteristic
171	C₂×C₄₇	#3277	1	1	yes	characteristic
172	C₂×C₄₇	#3292	1	1	yes	characteristic
173	D₄₇	—	16	2	no	—
174	#559	—	47	2	no	—
175	Q₈²	—	47	2	no	—
176	#494	—	47	1	no	—
177	#493	—	47	1	no	—
178	#489	—	94	2	no	—
179	#489	—	94	2	no	—
180	#486	—	47	2	no	—
181	#486	—	47	2	no	—
182	#477	—	47	2	no	—
183	#477	—	47	2	no	—
184	#474	—	94	2	no	—
185	#474	—	94	2	no	—
186	#464	—	94	2	no	—
187	#464	—	94	2	no	—
188	#460	—	47	2	no	—
189	#460	—	47	2	no	—
190	#446	—	47	2	no	—
191	#446	C₂×D₄₇	1	2	yes	—
192	#440	—	94	2	no	—
193	#440	—	94	2	no	—
194	C₄×Q₁₆	—	47	1	no	—
195	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
196	#335	—	47	1	no	—
197	#335	C₂×D₄₇	1	1	yes	characteristic
198	#333	—	47	2	no	—
199	#331	—	47	2	no	—
200	Q₈⋊C₈	—	47	2	no	—
201	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
202	C₈²	C₂×D₄₇	1	1	yes	characteristic
203	C₄₇	#20319	1	1	yes	characteristic, complemented, Sylow
204	C₂×Q₁₆	—	47	1	no	—
205	C₂×Q₁₆	—	188	1	no	—
206	C₂×Q₁₆	D₁₈₈	1	1	yes	characteristic
207	C₂×SD₁₆	—	94	2	no	—
208	C₂×SD₁₆	—	94	2	no	—
209	C₂×D₈	—	188	1	no	—
210	C₄⋊Q₈	—	47	2	no	—
211	C₄⋊Q₈	—	94	2	no	—
212	C₄⋊Q₈	—	94	2	no	—
213	C₄⋊Q₈	—	188	2	no	—
214	#127	—	47	2	no	—
215	#124	—	94	2	no	—
216	#124	—	94	2	no	—
217	#123	—	188	2	no	—
218	C₄×Q₈	—	47	2	no	—
219	C₄×Q₈	—	94	2	no	—
220	C₄×Q₈	—	94	2	no	—
221	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
222	C₄×D₄	—	94	2	no	—
223	C₄×D₄	—	94	2	no	—
224	#107	—	47	1	no	—
225	#107	—	94	2	no	—
226	#107	—	94	2	no	—
227	#107	D₁₈₈	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	47	2	no	—
230	C₄⋊C₈	—	94	2	no	—
231	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
232	#103	—	47	2	no	—
233	#103	—	94	2	no	—
234	#103	—	94	2	no	—
235	#103	D₁₈₈	1	2	yes	—
236	#102	—	94	2	no	—
237	#102	—	94	2	no	—
238	#102	—	94	2	no	—
239	#102	—	94	2	no	—
240	#102	—	94	2	no	—
241	#102	—	94	2	no	—
242	#97	—	2	2	no	—
243	#97	—	94	2	no	—
244	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
247	C₂×Q₈	—	47	2	no	—
248	C₂×Q₈	—	94	2	no	—
249	C₂×Q₈	—	188	2	no	—
250	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
251	C₂×D₄	—	94	2	no	—
252	C₂×D₄	—	94	2	no	—
253	C₂×D₄	—	94	2	no	—
254	C₂²×C₄	—	188	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	376	2	no	—
257	Q₁₆	—	188	1	no	—
258	SD₁₆	—	188	4	no	—
259	SD₁₆	—	188	4	no	—
260	D₈	—	376	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
265	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
266	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
267	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	47	2	no	—
270	C₄⋊C₄	—	94	2	no	—
271	C₄⋊C₄	—	94	2	no	—
272	C₄⋊C₄	—	94	2	no	—
273	C₄⋊C₄	—	188	2	no	—
274	C₄⋊C₄	—	188	2	no	—
275	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
276	C₂²⋊C₄	—	188	2	no	—
277	C₂²⋊C₄	—	188	2	no	—
278	C₂²⋊C₄	—	188	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	94	2	no	—
281	C₄²	—	188	2	no	—
282	C₄²	C₂³×D₄₇	1	1	yes	characteristic
283	C₂³	—	188	2	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	94	2	no	—
287	Q₈	—	94	2	no	—
288	Q₈	—	376	2	no	—
289	Q₈	—	752	2	no	—
290	D₄	—	376	2	no	—
291	D₄	—	188	2	no	—
292	D₄	—	188	2	no	—
293	D₄	—	188	2	no	—
294	D₄	—	188	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	188	2	no	—
300	C₂×C₄	—	752	2	no	—
301	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
303	C₂×C₄	#13094070	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	188	2	no	—
311	C₂²	—	376	2	no	—
312	C₂²	—	376	2	no	—
313	C₂²	—	376	2	no	—
314	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	376	2	no	—
319	C₄	—	188	2	no	—
320	C₄	—	188	2	no	—
321	C₄	#15689920	1	1	yes	characteristic
322	C₄	#15689923	1	1	yes	characteristic
323	C₄	#15689936	1	1	yes	characteristic
324	C₄	#15689940	1	1	yes	characteristic
325	C₂	—	752	2	no	—
326	C₂	#20255044	1	1	yes	characteristic
327	C₂	#20255046	1	1	yes	characteristic
328	C₂	#20255188	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₈₈
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1591	C₂
socle	C₂²×C₄₇	D₄²
prime radical	#15689500	C₂²
hypercenter	#1591	D₄₇
hypocenter	C₄₇	#20319
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1591

Complements

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

N	H ≅ G/N	type
1. #20256204	C₂	semidirect
2. #20256200	C₂	semidirect
4. #20242929	C₂	semidirect
5. #20242928	C₂	semidirect
6. #20242530	C₂	semidirect
8. #20242286	C₂	semidirect
9. C₄₇×#1591	C₂	semidirect
169. #1591	D₄₇	semidirect
203. C₄₇	#20319	semidirect

Maximal subgroups

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

subgroup	index
#20256204	2
#20256200	2
#20242994	2
#20242929	2
#20242928	2
#20242530	2
#20242449	2
#20242286	2
C₄₇×#1591	2
#20319	47