Subgroups — #26181992

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252016	C₂	1	2	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	#20241692	C₂	1	2	yes	maximal, complemented
4	#20241690	C₂	1	2	yes	maximal, complemented
5	#20241686	C₂	1	2	yes	maximal, complemented
6	#20241682	C₂	1	2	yes	maximal, complemented
7	#20241681	C₂	1	2	yes	maximal, complemented
8	#20241671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₄	1	2	yes	—
11	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
12	#15689179	C₄	1	4	yes	—
13	#15689177	C₄	1	4	yes	—
14	#15689176	C₂²	1	2	yes	—
15	#15689174	C₂²	1	4	yes	—
16	#15689145	C₄	1	4	yes	—
17	#15689085	C₄	1	4	yes	—
18	C₄×#10625324	C₄	1	2	yes	—
19	C₄×#10625324	C₄	1	2	yes	—
20	C₄×#10625324	C₂²	1	1	yes	characteristic
21	C₄×#10625324	C₂²	1	1	yes	characteristic
22	C₄×#10625324	C₂²	1	1	yes	characteristic
23	C₄×#10625324	C₂²	1	4	yes	—
24	C₄×#10625324	C₂²	1	1	yes	characteristic
25	C₂×#13093934	C₄	1	2	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	#15688845	C₂²	1	4	yes	complemented
28	#15688844	C₂²	1	2	yes	complemented
29	#15688844	C₂²	1	4	yes	complemented
30	C₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
31	C₄₇×#431	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#322	C₂²	1	2	yes	complemented
34	C₈²×C₄₇	C₂²	1	2	yes	complemented
35	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
37	C₂×#10625324	—	2	8	no	—
38	C₂×#10625324	C₂×C₄	1	2	yes	—
39	C₂×#10625324	C₂×C₄	1	2	yes	—
40	C₂×#10625324	C₂×C₄	1	2	yes	—
41	C₂×#10625324	C₂×C₄	1	2	yes	—
42	C₂×#10625324	C₂×C₄	1	8	yes	—
43	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
46	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₄₇	C₂×C₄	1	2	yes	—
49	C₄²×D₄₇	C₂×C₄	1	2	yes	—
50	C₄²×D₄₇	C₂³	1	1	yes	characteristic
51	C₄²×D₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
53	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
54	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
55	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
56	#13093934	C₂×C₄	1	2	yes	—
57	#13093934	C₂×C₄	1	2	yes	—
58	#13093934	C₂×C₄	1	2	yes	—
59	#13093934	C₂×C₄	1	2	yes	—
60	#13093934	C₂×C₄	1	8	yes	—
61	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
63	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
65	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₄₇	C₂³	1	4	yes	—
73	C₂³×D₄₇	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
77	C₂²×Dic₄₇	C₄²	1	2	yes	—
78	C₂×C₄×D₄₇	—	2	4	no	—
79	C₂×C₄×D₄₇	—	4	4	no	—
80	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
81	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
82	C₂×C₄×D₄₇	C₄²	1	2	yes	—
83	C₂×C₄×D₄₇	C₄²	1	2	yes	—
84	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
89	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄×Dic₄₇	C₄²	1	2	yes	—
96	C₄×Dic₄₇	C₄²	1	2	yes	—
97	C₄×Dic₄₇	C₄²	1	2	yes	—
98	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
101	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
106	#10625324	—	2	4	no	—
107	#10625324	—	2	4	no	—
108	#10625324	—	2	4	no	—
109	#10625324	—	2	4	no	—
110	#10625324	—	4	8	no	—
111	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
112	C₂²×D₄₇	—	2	2	no	—
113	C₂²×D₄₇	—	4	2	no	—
114	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
115	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
125	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
127	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
128	C₄×D₄₇	—	2	2	no	—
129	C₄×D₄₇	—	2	2	no	—
130	C₄×D₄₇	—	2	4	no	—
131	C₄×D₄₇	—	2	4	no	—
132	C₄×D₄₇	—	2	2	no	—
133	C₄×D₄₇	—	2	2	no	—
134	C₄×D₄₇	—	4	2	no	—
135	C₄×D₄₇	—	4	2	no	—
136	C₈×C₄₇	—	2	8	no	—
137	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
138	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
139	C₄₇⋊C₈	—	2	2	no	—
140	C₄₇⋊C₈	—	2	2	no	—
141	C₄₇⋊C₈	—	2	8	no	—
142	#17334	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
145	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
146	C₂×D₄₇	—	2	1	no	—
147	C₂×D₄₇	—	2	1	no	—
148	C₂×D₄₇	—	2	2	no	—
149	C₂×D₄₇	—	2	2	no	—
150	C₂×D₄₇	—	4	4	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
153	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
154	C₄×C₄₇	#568	1	4	yes	—
155	Dic₄₇	—	2	1	no	—
156	Dic₄₇	—	2	1	no	—
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	—	2	2	no	—
159	#2920	—	47	2	no	—
160	C₂×C₄×M₁₆	—	47	1	no	—
161	#1455	—	47	2	no	—
162	#1455	—	47	2	no	—
163	#1453	—	47	2	no	—
164	#1453	—	47	2	no	—
165	#1452	—	47	2	no	—
166	#1450	—	47	1	no	—
167	#1450	D₄₇	1	1	yes	characteristic, complemented
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
170	C₂×C₄₇	#2873	1	2	yes	—
171	D₄₇	—	4	1	no	—
172	D₄₇	—	4	1	no	—
173	C₂²×M₁₆	—	47	2	no	—
174	C₂²×C₄²	—	47	1	no	—
175	#432	—	47	4	no	—
176	#432	—	47	4	no	—
177	#431	—	47	2	no	—
178	#431	—	47	4	no	—
179	#431	C₂×D₄₇	1	2	yes	—
180	#423	—	47	4	no	—
181	#407	—	47	4	no	—
182	C₄×M₁₆	—	47	2	no	—
183	C₄×M₁₆	—	47	2	no	—
184	C₄×M₁₆	—	47	1	no	—
185	C₄×M₁₆	—	47	1	no	—
186	C₄×M₁₆	—	47	1	no	—
187	C₄×M₁₆	—	47	1	no	—
188	C₄×M₁₆	—	47	4	no	—
189	C₂×#97	—	47	2	no	—
190	C₂×C₄×C₈	—	47	1	no	—
191	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
192	#322	—	47	2	no	—
193	#322	—	47	4	no	—
194	#322	—	47	4	no	—
195	#322	C₂×D₄₇	1	2	yes	complemented
196	C₈²	—	47	2	no	—
197	C₈²	C₂×D₄₇	1	2	yes	complemented
198	C₄₇	#17334	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	—	47	1	no	—
200	C₂³×C₄	—	47	2	no	—
201	C₂×M₁₆	—	47	2	no	—
202	C₂×M₁₆	—	47	2	no	—
203	C₂×M₁₆	—	47	2	no	—
204	C₂×M₁₆	—	47	2	no	—
205	C₂×M₁₆	—	47	8	no	—
206	C₂×M₁₆	—	94	8	no	—
207	C₂²×C₈	—	47	2	no	—
208	C₂²×C₈	C₄×D₄₇	1	2	yes	—
209	C₂×C₄²	—	47	1	no	—
210	C₂×C₄²	—	47	2	no	—
211	C₂×C₄²	—	47	1	no	—
212	C₂×C₄²	—	47	1	no	—
213	C₂×C₄²	—	47	2	no	—
214	C₂×C₄²	—	47	2	no	—
215	C₂×C₄²	—	47	1	no	—
216	C₂×C₄²	—	47	1	no	—
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊C₈	—	47	8	no	—
219	C₄⋊C₈	—	47	4	no	—
220	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
221	C₂²⋊C₈	—	47	4	no	—
222	C₂²⋊C₈	—	47	8	no	—
223	C₂²⋊C₈	C₄×D₄₇	1	4	yes	—
224	#97	—	47	2	no	—
225	#97	—	47	2	no	—
226	#97	—	47	2	no	—
227	#97	—	47	2	no	—
228	#97	—	47	8	no	—
229	C₄×C₈	—	47	2	no	—
230	C₄×C₈	—	47	4	no	—
231	C₄×C₈	—	47	1	no	—
232	C₄×C₈	—	47	1	no	—
233	C₄×C₈	C₄×D₄₇	1	2	yes	—
234	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₄₇	1	4	yes	—
237	C₂⁴	—	47	1	no	—
238	C₂²×C₄	—	47	1	no	—
239	C₂²×C₄	—	47	2	no	—
240	C₂²×C₄	—	47	1	no	—
241	C₂²×C₄	—	47	1	no	—
242	C₂²×C₄	—	47	2	no	—
243	C₂²×C₄	—	47	1	no	—
244	C₂²×C₄	—	47	1	no	—
245	C₂²×C₄	—	47	2	no	—
246	C₂²×C₄	—	47	2	no	—
247	C₂²×C₄	—	47	2	no	—
248	C₂²×C₄	—	94	4	no	—
249	C₂²×C₄	—	188	4	no	—
250	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	M₁₆	—	94	4	no	—
253	M₁₆	—	94	4	no	—
254	M₁₆	—	94	4	no	—
255	M₁₆	—	94	4	no	—
256	M₁₆	—	188	8	no	—
257	C₂×C₈	—	47	2	no	—
258	C₂×C₈	—	47	2	no	—
259	C₂×C₈	—	47	8	no	—
260	C₂×C₈	—	94	4	no	—
261	C₂×C₈	#10625324	1	8	yes	complemented
262	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
263	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	2	no	—
267	C₄²	—	47	2	no	—
268	C₄²	—	47	1	no	—
269	C₄²	—	47	1	no	—
270	C₄²	—	47	2	no	—
271	C₄²	—	47	2	no	—
272	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₄₇	1	2	yes	—
274	C₄²	C₂³×D₄₇	1	1	yes	characteristic
275	C₂³	—	47	1	no	—
276	C₂³	—	47	1	no	—
277	C₂³	—	94	2	no	—
278	C₂³	—	188	2	no	—
279	C₂³	C₄²×D₄₇	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	47	1	no	—
282	C₂×C₄	—	47	1	no	—
283	C₂×C₄	—	47	2	no	—
284	C₂×C₄	—	47	2	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	—	94	4	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	C₄²×D₄₇	1	2	yes	—
296	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂×#10625324	1	4	yes	—
298	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
300	C₈	—	2	8	no	—
301	C₈	—	94	2	no	—
302	C₈	—	94	2	no	—
303	C₈	—	94	8	no	—
304	C₈	C₂×#10625324	1	4	yes	—
305	C₈	C₂×#10625324	1	4	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	—	94	1	no	—
308	C₂²	—	94	1	no	—
309	C₂²	—	94	2	no	—
310	C₂²	—	94	2	no	—
311	C₂²	—	188	4	no	—
312	C₂²	C₄×#10625324	1	2	yes	—
313	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	94	1	no	—
316	C₄	—	94	1	no	—
317	C₄	—	94	2	no	—
318	C₄	—	94	2	no	—
319	C₄	C₄×#10625324	1	2	yes	—
320	C₄	C₂²×#10625324	1	2	yes	—
321	C₄	#15689888	1	4	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	188	1	no	—
324	C₂	—	188	1	no	—
325	C₂	C₂×C₄×#10625324	1	1	yes	characteristic
326	C₂	#20251804	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#1450	C₂
prime radical	C₂³×D₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#17334
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20241692	C₂	semidirect
4. #20241690	C₂	semidirect
5. #20241686	C₂	semidirect
6. #20241682	C₂	semidirect
7. #20241681	C₂	semidirect
8. #20241671	C₂	semidirect
9. C₄₇×#1450	C₂	semidirect
27. #15688845	C₂²	semidirect
28. #15688844	C₂²	semidirect
29. #15688844	C₂²	semidirect
30. C₈×C₄₇⋊C₈	C₂²	semidirect
33. C₄₇×#322	C₂²	semidirect
34. C₈²×C₄₇	C₂²	semidirect
88. C₂×C₈×C₄₇	M₁₆	semidirect
167. #1450	D₄₇	semidirect
195. #322	C₂×D₄₇	semidirect
197. C₈²	C₂×D₄₇	semidirect
198. C₄₇	#17334	semidirect
261. C₂×C₈	#10625324	semidirect

Maximal subgroups

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

subgroup	index
#20252016	2
C₂×C₄×#10625324	2
#20241692	2
#20241690	2
#20241686	2
#20241682	2
#20241681	2
#20241671	2
C₄₇×#1450	2
#17334	47