Subgroups — #26106857

Order 12032.

494 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689200	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093940	C₂	1	1	yes	maximal, characteristic
3	#20241708	C₂	1	1	yes	maximal, characteristic
4	#20241675	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241669	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241238	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	C₄	1	2	yes	—
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	#15689200	—	2	1	no	—
11	#15689200	—	2	1	no	—
12	#15689200	—	2	1	no	—
13	#15689200	—	2	1	no	—
14	#15689176	C₂²	1	1	yes	characteristic
15	C₂×#13093940	C₄	1	2	yes	—
16	C₂×(C₂×D₄₇)⋊C₈	C₄	1	2	yes	—
17	C₄×#10625329	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	#15688891	C₄	1	2	yes	—
20	#15688846	C₄	1	4	yes	—
21	#15688845	—	2	1	no	—
22	#15688845	—	2	1	no	—
23	#15688845	—	2	1	no	—
24	C₈×C₄₇⋊C₈	—	2	1	no	—
25	C₄₇×#431	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
30	C₂×#10625329	C₂×C₄	1	2	yes	—
31	C₂×#10625324	—	2	2	no	—
32	C₂×#10625324	—	2	2	no	—
33	C₂×#10625324	—	4	4	no	—
34	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×D₁₈₈	—	2	1	no	—
37	C₄×D₁₈₈	—	2	1	no	—
38	C₄×D₁₈₈	—	2	1	no	—
39	C₄×D₁₈₈	—	2	1	no	—
40	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
42	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
43	#13093940	—	2	4	no	—
44	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
45	(C₂×D₄₇)⋊C₈	—	2	2	no	—
46	(C₂×D₄₇)⋊C₈	—	2	2	no	—
47	#13093935	—	2	1	no	—
48	C₄×C₄₇⋊C₈	—	2	1	no	—
49	C₄×C₄₇⋊C₈	—	2	1	no	—
50	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
52	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
53	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₇	—	2	1	no	—
58	C₄⋊C₈×C₄₇	—	2	1	no	—
59	C₄⋊C₈×C₄₇	—	2	1	no	—
60	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
61	#97×C₄₇	—	2	1	no	—
62	C₄×C₈×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	—	2	1	no	—
64	C₂³×D₄₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₄₇	C₄²	1	2	yes	—
69	C₂×D₁₈₈	—	2	1	no	—
70	C₂×D₁₈₈	—	2	1	no	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×C₄×D₄₇	—	2	2	no	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	—	4	2	no	—
76	C₂×C₄×D₄₇	—	4	2	no	—
77	M₁₆×C₄₇	—	2	2	no	—
78	M₁₆×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₇	#41	1	1	yes	characteristic
86	C₄²×C₄₇	#41	1	1	yes	characteristic
87	(C₂×D₄₇)⋊C₄	—	2	2	no	—
88	(C₂×D₄₇)⋊C₄	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	1	no	—
90	C₄⋊Dic₄₇	—	2	1	no	—
91	#10625329	—	2	2	no	—
92	#10625329	M₁₆	1	4	yes	complemented
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	4	1	no	—
97	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	#10625324	—	4	4	no	—
102	#10625324	—	4	4	no	—
103	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
104	C₂²×D₄₇	—	2	2	no	—
105	C₂²×D₄₇	—	2	2	no	—
106	C₂²×D₄₇	—	2	2	no	—
107	C₂²×D₄₇	—	4	2	no	—
108	C₂²×D₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₄₇	#131	1	2	yes	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	D₁₈₈	—	4	2	no	—
123	D₁₈₈	—	4	2	no	—
124	C₄×D₄₇	—	4	4	no	—
125	C₄×D₄₇	—	4	2	no	—
126	C₄×D₄₇	—	4	2	no	—
127	C₈×C₄₇	—	4	2	no	—
128	C₈×C₄₇	—	4	2	no	—
129	C₈×C₄₇	—	4	2	no	—
130	C₄₇⋊C₈	—	2	1	no	—
131	C₄₇⋊C₈	—	2	1	no	—
132	C₄₇⋊C₈	—	2	1	no	—
133	C₄₇⋊C₈	—	2	4	no	—
134	C₄₇⋊C₈	—	4	2	no	—
135	C₄₇⋊C₈	#100	1	2	yes	complemented
136	#8166	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
141	C₂²×C₄₇	#405	1	1	yes	characteristic
142	C₂×D₄₇	—	4	2	no	—
143	C₂×D₄₇	—	4	2	no	—
144	C₂×D₄₇	—	4	2	no	—
145	C₂×D₄₇	—	8	2	no	—
146	C₂×D₄₇	—	8	2	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
151	C₄×C₄₇	#413	1	1	yes	characteristic
152	C₄×C₄₇	#435	1	2	yes	—
153	C₄×C₄₇	#436	1	2	yes	—
154	Dic₄₇	—	4	2	no	—
155	C₂×#436	—	47	1	no	—
156	C₄×#100	—	47	1	no	—
157	#1460	—	47	1	no	—
158	#1451	—	47	1	no	—
159	C₈×M₁₆	—	47	1	no	—
160	#1283	—	47	1	no	—
161	#1283	D₄₇	1	1	yes	characteristic, complemented
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	—	2	1	no	—
164	C₂×C₄₇	#1753	1	1	yes	characteristic
165	C₂×C₄₇	C₄×#100	1	1	yes	characteristic
166	C₂×C₄₇	#1757	1	1	yes	characteristic
167	D₄₇	—	8	2	no	—
168	C₂²×M₁₆	—	47	2	no	—
169	C₂×C₄×D₄	—	47	1	no	—
170	#436	—	94	1	no	—
171	#436	—	94	1	no	—
172	#436	—	94	1	no	—
173	#436	—	94	1	no	—
174	#431	—	47	1	no	—
175	#431	C₂×D₄₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
177	C₂×#100	—	47	2	no	—
178	C₂×C₂²⋊C₈	—	47	2	no	—
179	C₄×M₁₆	—	47	1	no	—
180	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
181	C₂×C₄×C₈	—	47	1	no	—
182	#343	—	47	2	no	—
183	#323	—	47	4	no	—
184	#322	—	94	1	no	—
185	#322	—	94	1	no	—
186	#322	—	94	1	no	—
187	C₈²	—	94	1	no	—
188	C₄₇	#8166	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₄	—	47	1	no	—
190	C₂³×C₄	—	47	2	no	—
191	C₂×M₁₆	—	47	2	no	—
192	C₂×M₁₆	—	94	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	—	188	4	no	—
195	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
196	C₂²×C₈	—	47	1	no	—
197	C₂²×C₈	—	47	1	no	—
198	C₂²×C₈	C₄×D₄₇	1	2	yes	—
199	C₄×D₄	—	94	1	no	—
200	C₄×D₄	—	94	1	no	—
201	C₄×D₄	—	94	1	no	—
202	C₄×D₄	—	94	1	no	—
203	C₂×C₄⋊C₄	—	47	1	no	—
204	C₂×C₂²⋊C₄	—	47	2	no	—
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	94	1	no	—
210	#100	—	94	4	no	—
211	C₂²⋊C₈	—	47	2	no	—
212	C₂²⋊C₈	—	94	2	no	—
213	C₂²⋊C₈	—	94	2	no	—
214	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
215	#97	—	2	1	no	—
216	#97	—	94	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	47	1	no	—
220	C₄×C₈	—	47	1	no	—
221	C₄×C₈	—	47	1	no	—
222	C₄×C₈	—	47	1	no	—
223	C₄×C₈	—	94	1	no	—
224	C₄×C₈	—	94	1	no	—
225	C₂⁴	—	47	2	no	—
226	C₂×D₄	—	94	1	no	—
227	C₂×D₄	—	94	1	no	—
228	C₂×D₄	—	94	2	no	—
229	C₂×D₄	—	94	2	no	—
230	C₂²×C₄	—	47	2	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	—	188	2	no	—
234	C₂²×C₄	—	188	2	no	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	47	4	no	—
241	M₁₆	—	94	2	no	—
242	M₁₆	—	188	4	no	—
243	M₁₆	—	188	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	47	1	no	—
249	C₂×C₈	—	47	1	no	—
250	C₂×C₈	—	47	1	no	—
251	C₂×C₈	—	47	1	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	—	94	2	no	—
255	C₂×C₈	—	188	1	no	—
256	C₄⋊C₄	—	94	1	no	—
257	C₄⋊C₄	—	94	1	no	—
258	C₂²⋊C₄	—	94	2	no	—
259	C₂²⋊C₄	—	94	2	no	—
260	C₄²	D₄×D₄₇	1	1	yes	characteristic
261	C₄²	D₄×D₄₇	1	1	yes	characteristic
262	C₄²	#10625352	1	1	yes	characteristic
263	C₄²	#10625354	1	1	yes	characteristic
264	C₂³	—	94	2	no	—
265	C₂³	—	94	2	no	—
266	C₂³	—	94	2	no	—
267	C₂³	—	188	2	no	—
268	C₂³	—	188	2	no	—
269	C₂³	C₄²×D₄₇	1	1	yes	characteristic
270	D₄	—	188	2	no	—
271	D₄	—	188	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	94	2	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	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
281	C₂×C₄	#13093982	1	1	yes	characteristic
282	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#13093986	1	1	yes	characteristic
285	C₂×C₄	#13093986	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
287	C₂×C₄	#13094030	1	2	yes	—
288	C₈	—	4	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	47	2	no	—
292	C₈	—	94	1	no	—
293	C₈	—	94	1	no	—
294	C₈	—	94	1	no	—
295	C₈	—	94	4	no	—
296	C₈	—	188	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	376	2	no	—
300	C₂²	—	376	2	no	—
301	C₂²	—	188	2	no	—
302	C₂²	—	188	2	no	—
303	C₂²	—	188	2	no	—
304	C₂²	#15688970	1	1	yes	characteristic
305	C₂²	#15689075	1	1	yes	characteristic
306	C₂²	#15689077	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	188	2	no	—
311	C₄	#100×D₄₇	1	1	yes	characteristic
312	C₄	#15689106	1	1	yes	characteristic
313	C₄	#15689193	1	2	yes	—
314	C₄	#15689198	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	376	2	no	—
318	C₂	#20243170	1	1	yes	characteristic
319	C₂	#20243179	1	1	yes	characteristic
320	C₂	#20243187	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1283	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1283	D₄₇
hypocenter	C₄₇	#8166
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
4. #20241675	C₂	semidirect
5. #20241669	C₂	semidirect
6. #20241238	C₂	semidirect
7. C₄₇×#1283	C₂	semidirect
92. #10625329	M₁₆	semidirect
135. C₄₇⋊C₈	#100	semidirect
161. #1283	D₄₇	semidirect
188. C₄₇	#8166	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689200	2
C₄×#13093940	2
#20241708	2
#20241675	2
#20241669	2
#20241238	2
C₄₇×#1283	2
#8166	47