Subgroups — #26215082

Order 12032.

934 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20243258	C₂	1	2	yes	maximal, complemented
4	#20243253	C₂	1	2	yes	maximal, complemented
5	#20243247	C₂	1	4	yes	maximal, complemented
6	#20243164	C₂	1	2	yes	maximal
7	C₈×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
8	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689176	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13093934	C₂²	1	4	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
16	#15688991	C₄	1	2	yes	—
17	#15688989	C₄	1	2	yes	—
18	#15688986	C₄	1	2	yes	—
19	#15688983	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688969	C₂²	1	2	yes	—
23	#15688877	C₂²	1	4	yes	—
24	#15688877	C₂²	1	4	yes	—
25	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₇	C₄	1	8	yes	complemented
28	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
37	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
38	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
39	#13093934	—	4	4	no	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
41	C₈×Dic₄₇	—	4	4	no	—
42	C₄₇⋊#95	C₂×C₄	1	4	yes	—
43	C₄₇⋊#95	C₂×C₄	1	4	yes	—
44	C₄₇⋊#95	C₂×C₄	1	4	yes	—
45	C₄₇⋊#95	C₂×C₄	1	4	yes	—
46	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	—	4	4	no	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
60	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
61	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
62	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
68	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	4	8	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
83	C₄²×C₄₇	C₄²	1	4	yes	—
84	C₄²×C₄₇	C₄²	1	4	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂²⋊Dic₄₇	—	2	8	no	—
90	C₄⋊Dic₄₇	—	2	8	no	—
91	C₄×Dic₄₇	—	2	4	no	—
92	C₄×Dic₄₇	—	2	4	no	—
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	—	2	2	no	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂³×C₄₇	#131	1	4	yes	—
103	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₇	#131	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
123	C₂×Dic₄₇	—	2	8	no	—
124	C₂×Dic₄₇	—	2	8	no	—
125	C₂×Dic₄₇	—	2	8	no	—
126	C₈×C₄₇	—	2	4	no	—
127	C₈×C₄₇	—	2	4	no	—
128	C₄₇⋊C₈	—	2	4	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	#20545	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	8	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	#405	1	4	yes	—
135	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
136	C₂²×C₄₇	#520	1	2	yes	—
137	C₂²×C₄₇	C₂×#131	1	2	yes	—
138	C₂²×C₄₇	C₂×#131	1	2	yes	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	#405	1	4	yes	—
144	C₄×C₄₇	#518	1	1	yes	characteristic
145	C₄×C₄₇	#519	1	1	yes	characteristic
146	C₄×C₄₇	#520	1	2	yes	—
147	Dic₄₇	—	4	8	no	—
148	C₂×#431	—	47	1	no	—
149	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
150	#2283	—	47	1	no	—
151	#1770	—	47	2	no	—
152	C₈×C₄⋊C₄	—	47	2	no	—
153	#1768	—	47	4	no	—
154	#1752	—	47	2	no	—
155	C₈×C₂²⋊C₄	—	47	2	no	—
156	C₂×C₄₇	—	2	4	no	—
157	C₂×C₄₇	#2274	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
160	C₂×C₄₇	#2994	1	4	yes	—
161	C₂³×C₈	C₂×D₄₇	1	2	yes	—
162	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
163	#431	—	47	8	no	—
164	C₂×C₄⋊C₈	—	47	2	no	—
165	C₂×C₄⋊C₈	Dic₄₇	1	8	yes	complemented
166	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
167	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	—	47	1	no	—
170	C₂×C₂²⋊C₈	—	47	1	no	—
171	C₂×#97	—	47	4	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	C₂×C₄×C₈	—	47	4	no	—
174	#383	—	47	2	no	—
175	#382	—	47	2	no	—
176	#381	—	47	2	no	—
177	#380	—	47	2	no	—
178	C₄×C₄⋊C₄	—	47	2	no	—
179	C₄×C₂²⋊C₄	—	47	2	no	—
180	#376	—	47	2	no	—
181	#336	—	47	4	no	—
182	#336	—	47	4	no	—
183	C₄₇	#20545	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
187	C₂²×C₈	—	4	4	no	—
188	C₂²×C₈	—	47	2	no	—
189	C₂²×C₈	—	47	2	no	—
190	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
191	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
192	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
193	C₂×C₄⋊C₄	—	47	4	no	—
194	C₂×C₂²⋊C₄	—	47	4	no	—
195	C₂×C₄²	—	47	4	no	—
196	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₄₇	1	4	yes	—
199	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₈	—	47	8	no	—
203	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
204	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
205	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
206	C₂²⋊C₈	—	47	4	no	—
207	C₂²⋊C₈	—	47	4	no	—
208	#97	—	188	4	no	—
209	C₄×C₈	—	47	8	no	—
210	C₄×C₈	—	188	4	no	—
211	#95	—	47	4	no	—
212	#95	—	47	4	no	—
213	#95	—	47	4	no	—
214	#95	—	47	4	no	—
215	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	47	8	no	—
221	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
222	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
230	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	8	no	—
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	47	4	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
241	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
242	C₄⋊C₄	—	94	8	no	—
243	C₂²⋊C₄	—	94	8	no	—
244	C₄²	—	94	4	no	—
245	C₄²	—	94	4	no	—
246	C₄²	C₄×Dic₄₇	1	4	yes	—
247	C₄²	C₄×Dic₄₇	1	4	yes	—
248	C₄²	C₂×C₄×D₄₇	1	2	yes	—
249	C₄²	C₂×C₄×D₄₇	1	2	yes	—
250	C₄²	C₂²×Dic₄₇	1	2	yes	—
251	C₄²	C₂²×Dic₄₇	1	2	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
255	C₂³	#13094032	1	4	yes	—
256	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	94	8	no	—
268	C₂×C₄	—	94	8	no	—
269	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
270	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
272	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
273	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
274	C₂×C₄	#13094030	1	4	yes	—
275	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
277	C₂×C₄	#13094069	1	2	yes	—
278	C₂×C₄	#13094072	1	2	yes	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	—	94	4	no	—
282	C₈	—	94	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	#15689081	1	4	yes	—
287	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
288	C₂²	#15689604	1	2	yes	—
289	C₂²	C₂×#13094030	1	2	yes	—
290	C₂²	C₂×#13094032	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	188	8	no	—
296	C₄	#15689078	1	4	yes	—
297	C₄	#15689598	1	1	yes	characteristic
298	C₄	#15689601	1	1	yes	characteristic
299	C₄	#15689604	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	#20245745	1	1	yes	characteristic
302	C₂	C₂×#15689078	1	1	yes	characteristic
303	C₂	C₂×#15689081	1	1	yes	characteristic
304	C₂	#20252544	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#20545
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20243258	C₂	semidirect
4. #20243253	C₂	semidirect
5. #20243247	C₂	semidirect
9. #15689176	C₄	semidirect
27. C₂×C₄⋊C₈×C₄₇	C₄	semidirect
165. C₂×C₄⋊C₈	Dic₄₇	semidirect
183. C₄₇	#20545	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20245786	2
#20243258	2
#20243253	2
#20243247	2
#20243164	2
C₈×C₂²⋊Dic₄₇	2
C₂²×C₄⋊C₈×C₄₇	2
#20545	47