Subgroups — #26107016

Order 12032.

516 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252354	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093950	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241768	C₂	1	1	yes	maximal, characteristic
5	#20241226	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241220	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10625350	C₂²	1	1	yes	characteristic
9	#15689253	C₄	1	2	yes	complemented
10	#15689202	C₄	1	2	yes	complemented
11	#15689194	—	2	2	no	—
12	#15689176	C₂²	1	1	yes	characteristic
13	#15689174	—	2	1	no	—
14	C₂×#13093950	C₄	1	2	yes	—
15	C₄×#10625324	—	2	1	no	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	—	2	2	no	—
18	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688884	C₄	1	2	yes	—
20	#15688853	C₄	1	2	yes	complemented
21	#15688850	C₄	1	2	yes	complemented
22	C₄₇×#431	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#322	—	2	1	no	—
26	C₄₇×#322	—	2	1	no	—
27	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
28	C₂×#10625324	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
33	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₇	—	2	1	no	—
35	C₄×C₂²⋊D₄₇	—	2	1	no	—
36	#13093983	C₂×C₄	1	1	yes	characteristic
37	#13093981	—	2	1	no	—
38	C₂×C₄×Dic₄₇	—	2	1	no	—
39	C₂×C₄×Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₄²×D₄₇	—	2	1	no	—
42	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
43	Dic₄₇⋊C₈	—	2	2	no	—
44	#13093950	—	2	4	no	—
45	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
46	(C₂×D₄₇)⋊C₈	—	2	2	no	—
47	#13093934	—	2	2	no	—
48	C₄×C₄₇⋊C₈	—	2	1	no	—
49	C₄×C₄₇⋊C₈	—	2	1	no	—
50	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
51	C₈×Dic₄₇	—	2	8	no	—
52	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₄₇	—	2	1	no	—
57	#97×C₄₇	—	2	1	no	—
58	C₄×C₈×C₄₇	—	2	1	no	—
59	C₄×C₈×C₄₇	—	2	1	no	—
60	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₄₇	—	2	1	no	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	#10625350	—	2	2	no	—
69	#10625350	—	2	2	no	—
70	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
71	C₂×C₄×D₄₇	—	2	1	no	—
72	C₂×C₄×D₄₇	—	2	1	no	—
73	C₂×C₄×D₄₇	—	2	1	no	—
74	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
75	M₁₆×C₄₇	—	2	2	no	—
76	M₁₆×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₇	#41	1	1	yes	characteristic
86	C₂²⋊Dic₄₇	—	2	1	no	—
87	(C₂×D₄₇)⋊C₄	—	2	1	no	—
88	(C₂×D₄₇)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
90	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
91	Dic₄₇⋊C₄	—	2	1	no	—
92	Dic₄₇⋊C₄	—	2	1	no	—
93	C₄×Dic₄₇	—	2	4	no	—
94	C₄×Dic₄₇	—	2	4	no	—
95	C₄×Dic₄₇	—	2	1	no	—
96	C₄×Dic₄₇	—	2	1	no	—
97	C₄×Dic₄₇	—	2	1	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	M₁₆	1	4	yes	complemented
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	#10625324	—	4	4	no	—
104	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₄₇	#131	1	2	yes	—
116	C₂²⋊D₄₇	—	4	2	no	—
117	C₂×Dic₄₇	—	2	4	no	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	1	no	—
121	C₂×Dic₄₇	—	2	1	no	—
122	C₂×Dic₄₇	—	2	1	no	—
123	D₁₈₈	—	2	1	no	—
124	D₁₈₈	—	2	1	no	—
125	C₄×D₄₇	—	4	2	no	—
126	C₄×D₄₇	—	4	2	no	—
127	C₄×D₄₇	—	4	2	no	—
128	Dic₉₄	—	2	1	no	—
129	Dic₉₄	—	2	1	no	—
130	C₈×C₄₇	—	2	2	no	—
131	C₈×C₄₇	—	4	2	no	—
132	C₈×C₄₇	—	4	2	no	—
133	C₈×C₄₇	#104	1	4	yes	complemented
134	C₄₇⋊C₈	—	2	4	no	—
135	C₄₇⋊C₈	—	4	2	no	—
136	#8194	—	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	#435	1	2	yes	—
141	C₂×D₄₇	—	4	1	no	—
142	C₂×D₄₇	—	4	1	no	—
143	C₂×D₄₇	—	4	1	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₄₇	#405	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#104	1	2	yes	—
149	C₄×C₄₇	#436	1	2	yes	—
150	C₄×C₄₇	#443	1	2	yes	—
151	Dic₄₇	—	2	8	no	—
152	Dic₄₇	—	4	1	no	—
153	Dic₄₇	—	4	1	no	—
154	Dic₄₇	—	4	1	no	—
155	#2965	—	47	1	no	—
156	C₄×#104	—	47	1	no	—
157	C₄²×C₈	—	47	1	no	—
158	#1476	—	47	1	no	—
159	#1451	D₄₇	1	1	yes	characteristic, complemented
160	#1278	—	47	1	no	—
161	#1276	—	47	1	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	#1753	1	1	yes	characteristic
164	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
165	C₂×C₄₇	#1764	1	1	yes	characteristic
166	D₄₇	—	8	1	no	—
167	C₄×#41	—	47	1	no	—
168	#446	—	47	2	no	—
169	#436	—	47	2	no	—
170	#435	—	94	2	no	—
171	#431	—	47	1	no	—
172	#431	—	94	1	no	—
173	#431	C₂×D₄₇	1	1	yes	characteristic
174	C₂×#104	—	47	2	no	—
175	C₄×M₁₆	—	94	1	no	—
176	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
177	C₂×C₄×C₈	—	47	1	no	—
178	C₂×C₄×C₈	—	94	2	no	—
179	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
180	C₄³	—	47	1	no	—
181	#339	—	47	2	no	—
182	Q₈⋊C₈	—	47	2	no	—
183	D₄⋊C₈	—	47	2	no	—
184	#322	—	2	1	no	—
185	#322	—	2	1	no	—
186	C₄₇	#8194	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	47	1	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
190	C₂²×C₈	—	94	1	no	—
191	C₂²×C₈	D₁₈₈	1	2	yes	—
192	C₄×Q₈	—	47	1	no	—
193	C₄×Q₈	—	47	1	no	—
194	C₄×D₄	—	47	1	no	—
195	C₄×D₄	—	47	1	no	—
196	C₄×D₄	—	94	1	no	—
197	C₄×D₄	—	94	1	no	—
198	#117	—	47	1	no	—
199	#117	—	94	1	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	94	1	no	—
202	C₂×C₄²	—	94	1	no	—
203	C₂×C₄²	—	94	1	no	—
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊C₈	—	47	2	no	—
206	C₄⋊C₈	—	94	2	no	—
207	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
208	#104	—	94	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	94	1	no	—
211	C₂²⋊C₈	—	94	2	no	—
212	#97	—	2	1	no	—
213	#97	—	94	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	47	2	no	—
217	C₄×C₈	—	94	8	no	—
218	C₄×C₈	—	94	1	no	—
219	C₄×C₈	—	94	1	no	—
220	C₄×C₈	C₄×D₄₇	1	2	yes	—
221	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
222	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
223	#41	—	94	2	no	—
224	#41	—	94	2	no	—
225	C₂×Q₈	—	47	1	no	—
226	C₂×D₄	—	47	1	no	—
227	C₂×D₄	—	94	1	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	94	1	no	—
230	C₂²×C₄	—	94	1	no	—
231	C₂²×C₄	—	94	1	no	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
234	C₂²×C₄	#10625350	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	188	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
245	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
246	C₄⋊C₄	—	47	1	no	—
247	C₄⋊C₄	—	47	1	no	—
248	C₄⋊C₄	—	94	1	no	—
249	C₄⋊C₄	—	94	1	no	—
250	C₂²⋊C₄	—	94	1	no	—
251	C₂²⋊C₄	—	94	1	no	—
252	C₂²⋊C₄	—	94	1	no	—
253	C₄²	—	47	4	no	—
254	C₄²	—	94	4	no	—
255	C₄²	—	94	4	no	—
256	C₄²	—	94	1	no	—
257	C₄²	—	94	1	no	—
258	C₄²	—	94	1	no	—
259	C₄²	—	94	2	no	—
260	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
262	C₄²	#10625350	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
264	C₂³	—	94	1	no	—
265	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
266	Q₈	—	94	1	no	—
267	Q₈	—	94	1	no	—
268	D₄	—	94	1	no	—
269	D₄	—	94	1	no	—
270	D₄	—	188	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	4	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	94	1	no	—
277	C₂×C₄	—	94	1	no	—
278	C₂×C₄	—	94	1	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#13093981	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
289	C₂×C₄	#13094032	1	2	yes	—
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	94	4	no	—
294	C₈	—	188	2	no	—
295	C₈	#13093950	1	4	yes	complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	188	1	no	—
299	C₂²	—	188	1	no	—
300	C₂²	—	188	1	no	—
301	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
302	C₂²	#15689196	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	94	8	no	—
306	C₄	—	188	1	no	—
307	C₄	—	188	1	no	—
308	C₄	—	188	1	no	—
309	C₄	#15689075	1	1	yes	characteristic
310	C₄	#15689080	1	1	yes	characteristic
311	C₄	C₂×#13093950	1	2	yes	—
312	C₄	#15689199	1	2	yes	—
313	C₄	#15689238	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	376	1	no	—
316	C₂	#20243173	1	1	yes	characteristic
317	C₂	C₄×#13093950	1	1	yes	characteristic
318	C₂	#20243227	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1451	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1451	D₄₇
hypocenter	C₄₇	#8194
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. C₄×C₈×Dic₄₇	C₂	semidirect
5. #20241226	C₂	semidirect
6. #20241220	C₂	semidirect
7. C₄₇×#1451	C₂	semidirect
9. #15689253	C₄	semidirect
10. #15689202	C₄	semidirect
20. #15688853	C₄	semidirect
21. #15688850	C₄	semidirect
99. C₄×Dic₄₇	M₁₆	semidirect
133. C₈×C₄₇	#104	semidirect
159. #1451	D₄₇	semidirect
186. C₄₇	#8194	semidirect
295. C₈	#13093950	semidirect

Maximal subgroups

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

subgroup	index
#20252354	2
C₄×#13093950	2
C₄×C₈×Dic₄₇	2
#20241768	2
#20241226	2
#20241220	2
C₄₇×#1451	2
#8194	47