Subgroups — #26188318

Order 12032.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253967	C₂	1	1	yes	maximal, characteristic
2	#20252454	C₂	1	2	yes	maximal, complemented
3	#20242303	C₂	1	2	yes	maximal, complemented
4	#20242219	C₂	1	4	yes	maximal, complemented
5	#20242213	C₂	1	2	yes	maximal, complemented
6	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241806	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689673	C₂²	1	2	yes	—
10	#15689528	C₄	1	1	yes	characteristic
11	#15689500	C₄	1	1	yes	characteristic
12	#15689470	C₄	1	2	yes	—
13	#15689359	C₄	1	2	yes	—
14	#15689359	C₄	1	2	yes	—
15	C₄×D₃₇₆	C₂²	1	2	yes	—
16	C₄×D₃₇₆	C₂²	1	2	yes	—
17	#15689202	C₂²	1	4	yes	—
18	#15689200	—	2	2	no	—
19	C₈×D₁₈₈	C₂²	1	4	yes	—
20	#15689188	—	2	2	no	—
21	#15689176	C₂²	1	2	yes	—
22	#15689088	—	2	4	no	—
23	#15688876	C₂²	1	2	yes	—
24	#15688874	C₂²	1	1	yes	characteristic
25	#15688874	C₂²	1	1	yes	characteristic
26	#15688852	C₂²	1	4	yes	—
27	#15688852	C₂²	1	4	yes	—
28	C₄₇×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#322	C₂²	1	2	yes	—
32	C₄₇×#322	C₂²	1	2	yes	—
33	C₂×#10625350	—	2	2	no	—
34	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
35	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
36	C₂×#10625324	—	2	4	no	—
37	C₂×C₈×D₄₇	—	2	4	no	—
38	#13094024	C₂×C₄	1	2	yes	—
39	#13094021	C₂×C₄	1	2	yes	—
40	#13094012	—	2	2	no	—
41	#13093998	—	2	2	no	—
42	#13093998	C₂×C₄	1	4	yes	—
43	C₄×D₁₈₈	—	2	2	no	—
44	C₄×D₁₈₈	C₂³	1	4	yes	—
45	#13093958	C₂×C₄	1	2	yes	—
46	#13093958	C₂×C₄	1	2	yes	—
47	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
49	#13093946	C₂×C₄	1	4	yes	—
50	#13093946	C₂×C₄	1	4	yes	—
51	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
52	(C₂×D₄₇)⋊C₈	—	2	4	no	—
53	(C₂×D₄₇)⋊C₈	—	2	4	no	—
54	C₄×C₄₇⋊C₈	—	2	2	no	—
55	C₂×M₁₆×C₄₇	—	2	1	no	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
60	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
61	#97×C₄₇	—	2	1	no	—
62	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂×C₂²⋊D₄₇	—	2	4	no	—
72	#10625350	—	8	2	no	—
73	#10625350	—	8	2	no	—
74	C₂×D₁₈₈	—	2	2	no	—
75	C₂×D₁₈₈	—	2	2	no	—
76	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
77	C₂×C₄×D₄₇	—	2	4	no	—
78	C₂×Dic₉₄	—	2	2	no	—
79	M₁₆×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	#41	1	1	yes	characteristic
91	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
92	(C₂×D₄₇)⋊C₄	—	2	4	no	—
93	(C₂×D₄₇)⋊C₄	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	D₃₇₆	—	2	4	no	—
99	D₃₇₆	—	4	2	no	—
100	#10625324	—	8	4	no	—
101	C₈×D₄₇	—	8	4	no	—
102	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
103	C₂²×D₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
114	C₂²⋊D₄₇	—	8	4	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	D₁₈₈	—	2	4	no	—
117	D₁₈₈	—	2	4	no	—
118	D₁₈₈	—	4	4	no	—
119	D₁₈₈	—	8	2	no	—
120	C₄×D₄₇	—	4	4	no	—
121	C₄×D₄₇	—	4	4	no	—
122	Dic₉₄	—	8	2	no	—
123	C₈×C₄₇	—	2	1	no	—
124	C₈×C₄₇	—	2	1	no	—
125	C₈×C₄₇	—	2	1	no	—
126	C₈×C₄₇	—	2	4	no	—
127	C₈×C₄₇	—	4	2	no	—
128	C₈×C₄₇	#131	1	2	yes	—
129	C₄₇⋊C₈	—	4	2	no	—
130	C₄₇⋊C₈	—	4	2	no	—
131	#18091	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×D₄₇	—	4	4	no	—
137	C₂×D₄₇	—	4	4	no	—
138	C₂×D₄₇	—	4	4	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	#444	1	2	yes	—
143	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
144	C₄×C₄₇	#568	1	1	yes	characteristic
145	C₄×C₄₇	#576	1	2	yes	—
146	Dic₄₇	—	4	4	no	—
147	#3169	—	47	1	no	—
148	#2979	—	47	2	no	—
149	#1594	—	47	2	no	—
150	#1581	—	47	4	no	—
151	#1580	—	47	2	no	—
152	#1564	—	47	1	no	—
153	#1484	—	47	2	no	—
154	#1451	D₄₇	1	1	yes	characteristic, complemented
155	C₂×C₄₇	—	4	1	no	—
156	C₂×C₄₇	#2935	1	1	yes	characteristic
157	C₂×C₄₇	#2948	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
159	D₄₇	—	8	4	no	—
160	#532	—	47	2	no	—
161	#500	—	47	1	no	—
162	#493	—	47	1	no	—
163	#486	—	47	2	no	—
164	#466	—	47	2	no	—
165	#466	—	47	2	no	—
166	C₄×D₈	—	47	2	no	—
167	C₄×D₈	—	47	2	no	—
168	#436	—	47	4	no	—
169	#436	—	94	2	no	—
170	C₈×D₄	—	47	4	no	—
171	C₈×D₄	—	94	2	no	—
172	#431	—	47	2	no	—
173	#431	C₂×D₄₇	1	1	yes	characteristic
174	#408	—	94	4	no	—
175	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	#335	—	47	2	no	—
178	#334	—	47	1	no	—
179	#334	—	47	1	no	—
180	D₄⋊C₈	—	47	4	no	—
181	D₄⋊C₈	—	47	4	no	—
182	#322	C₂×D₄₇	1	2	yes	—
183	#322	C₂×D₄₇	1	2	yes	—
184	C₄₇	#18091	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	94	2	no	—
186	C₂×D₈	—	47	2	no	—
187	C₂×D₈	—	47	2	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	94	4	no	—
190	C₂²×C₈	—	94	4	no	—
191	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
193	C₄⋊Q₈	—	47	2	no	—
194	#127	—	47	2	no	—
195	#124	—	94	2	no	—
196	#121	—	47	4	no	—
197	#121	—	94	2	no	—
198	C₄×D₄	—	47	4	no	—
199	C₄×D₄	—	94	2	no	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	#107	—	47	2	no	—
202	#107	—	47	2	no	—
203	C₄⋊C₈	—	47	2	no	—
204	C₄⋊C₈	—	47	2	no	—
205	C₄⋊C₈	D₁₈₈	1	2	yes	—
206	#102	—	47	4	no	—
207	#102	—	47	4	no	—
208	C₂²⋊C₈	—	94	2	no	—
209	C₂²⋊C₈	—	94	4	no	—
210	C₂²⋊C₈	—	94	4	no	—
211	C₂²⋊C₈	D₁₈₈	1	2	yes	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	94	2	no	—
214	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
215	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₄₇	1	2	yes	—
217	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₄₇	1	2	yes	—
220	#41	—	376	2	no	—
221	#41	—	376	2	no	—
222	C₂×Q₈	—	94	2	no	—
223	C₂×D₄	—	47	4	no	—
224	C₂×D₄	—	94	4	no	—
225	C₂×D₄	—	94	2	no	—
226	C₂×D₄	—	94	2	no	—
227	C₂²×C₄	—	94	4	no	—
228	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	#10625350	1	1	yes	characteristic
231	D₈	—	94	4	no	—
232	D₈	—	188	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	376	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	376	4	no	—
241	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
246	C₄⋊C₄	—	47	4	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₂²⋊C₄	—	94	4	no	—
249	C₂²⋊C₄	—	94	4	no	—
250	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
252	C₄²	#10625350	1	1	yes	characteristic
253	C₄²	C₂³×D₄₇	1	1	yes	characteristic
254	C₂³	—	94	4	no	—
255	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
256	Q₈	—	376	2	no	—
257	D₄	—	94	4	no	—
258	D₄	—	94	4	no	—
259	D₄	—	376	2	no	—
260	D₄	—	376	4	no	—
261	D₄	—	188	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	94	4	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	—	188	4	no	—
269	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
270	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
271	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
274	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	188	2	no	—
281	C₈	—	188	2	no	—
282	C₈	#13094032	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	188	4	no	—
287	C₂²	—	188	4	no	—
288	C₂²	—	188	4	no	—
289	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	188	4	no	—
294	C₄	#15689246	1	2	yes	—
295	C₄	C₂×#13094032	1	1	yes	characteristic
296	C₄	#15689889	1	1	yes	characteristic
297	C₄	#15689928	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	376	4	no	—
300	C₂	#20252134	1	1	yes	characteristic
301	C₂	#20252241	1	1	yes	characteristic
302	C₂	C₂×#15689246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1451	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#20253967	C₂
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1451	D₄₇
hypocenter	C₄₇	#18091
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #20252454	C₂	semidirect
3. #20242303	C₂	semidirect
4. #20242219	C₂	semidirect
5. #20242213	C₂	semidirect
6. #20242136	C₂	semidirect
7. #20241806	C₂	semidirect
8. C₄₇×#1451	C₂	semidirect
154. #1451	D₄₇	semidirect
184. C₄₇	#18091	semidirect

Maximal subgroups

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

subgroup	index
#20253967	2
#20252454	2
#20242303	2
#20242219	2
#20242213	2
#20242136	2
#20241806	2
C₄₇×#1451	2
#18091	47