Subgroups — #26226711

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252017	C₂	1	2	yes	maximal
2	C₂×#15689146	C₂	1	1	yes	maximal, characteristic
3	#20244162	C₂	1	4	yes	maximal, complemented
4	#20244160	C₂	1	4	yes	maximal, complemented
5	#20243430	C₂	1	1	yes	maximal, characteristic
6	#20243428	C₂	1	2	yes	maximal
7	C₄₇×#1811	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15689177	—	2	2	no	—
11	#15689177	—	2	2	no	—
12	#15689173	—	2	2	no	—
13	#15689173	—	2	2	no	—
14	C₂×#13093960	C₂²	1	2	yes	—
15	C₂×#13093959	C₂²	1	4	yes	—
16	#15689146	C₄	1	2	yes	—
17	#15689146	C₄	1	4	yes	—
18	#15689146	C₄	1	2	yes	—
19	#15689146	C₂²	1	4	yes	—
20	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
21	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
22	#15689087	C₂²	1	2	yes	—
23	M₁₆×Dic₄₇	C₂²	1	4	yes	—
24	#15688888	C₂²	1	2	yes	—
25	#15688887	C₂²	1	4	yes	—
26	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#407	C₂²	1	2	yes	—
29	C₄₇×#341	C₂²	1	2	yes	—
30	C₂³×Dic₄₇	Q₈	1	2	yes	—
31	C₂×#10625329	—	4	2	no	—
32	C₂×#10625329	C₂×C₄	1	2	yes	—
33	C₂×#10625329	C₂×C₄	1	2	yes	—
34	C₂×#10625329	C₂³	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
44	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
45	#13093960	—	4	2	no	—
46	#13093959	—	4	4	no	—
47	Dic₄₇⋊C₈	—	2	4	no	—
48	Dic₄₇⋊C₈	—	2	4	no	—
49	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
50	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
51	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
52	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
53	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
54	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
55	#13093934	—	2	4	no	—
56	C₈×Dic₄₇	—	2	4	no	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₄₇	—	4	2	no	—
59	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
61	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	#108×C₄₇	—	4	2	no	—
65	C₂²⋊C₈×C₄₇	—	2	2	no	—
66	C₂²⋊C₈×C₄₇	—	2	2	no	—
67	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
78	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
79	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
80	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
81	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
82	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
83	M₁₆×C₄₇	—	2	4	no	—
84	M₁₆×C₄₇	—	2	4	no	—
85	M₁₆×C₄₇	—	4	2	no	—
86	M₁₆×C₄₇	—	4	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	4	no	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
93	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
94	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
95	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
96	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
97	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
98	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
99	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
100	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
101	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
102	#10625329	—	2	4	no	—
103	#10625329	—	2	4	no	—
104	#10625329	—	4	2	no	—
105	#10625329	—	4	2	no	—
106	C₂×C₄₇⋊C₈	—	2	4	no	—
107	C₂×C₄₇⋊C₈	—	2	4	no	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
109	C₂³×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	—	4	2	no	—
111	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	—	2	2	no	—
139	C₈×C₄₇	—	4	4	no	—
140	C₄₇⋊C₈	—	2	4	no	—
141	C₄₇⋊C₈	—	4	4	no	—
142	#21970	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₄₇	#544	1	2	yes	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	#518	1	1	yes	characteristic
154	C₄×C₄₇	#519	1	1	yes	characteristic
155	C₄×C₄₇	#537	1	2	yes	—
156	Dic₄₇	—	2	2	no	—
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	—	2	2	no	—
159	Dic₄₇	#430	1	4	yes	—
160	#2920	—	47	2	no	—
161	C₂×#423	—	47	1	no	—
162	#1950	—	47	4	no	—
163	#1950	—	47	4	no	—
164	#1811	—	47	2	no	—
165	#1811	—	47	1	no	—
166	#1811	D₄₇	1	1	yes	characteristic
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	—	4	1	no	—
169	C₂×C₄₇	—	4	1	no	—
170	C₂×C₄₇	#2316	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#430	1	2	yes	—
172	C₂²×M₁₆	—	47	1	no	—
173	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
174	C₂²×C₄²	—	47	1	no	—
175	#432	—	94	2	no	—
176	#432	—	94	2	no	—
177	#431	—	94	2	no	—
178	#431	—	94	2	no	—
179	C₂×#108	—	47	4	no	—
180	C₂×#108	—	47	2	no	—
181	C₂×#108	C₂×D₄₇	1	2	yes	—
182	#423	—	47	2	no	—
183	#423	—	47	4	no	—
184	#423	—	47	2	no	—
185	#423	—	47	4	no	—
186	C₂×C₄⋊C₈	—	47	2	no	—
187	C₂×C₄⋊C₈	—	47	2	no	—
188	#407	—	47	2	no	—
189	#407	C₂×D₄₇	1	2	yes	—
190	C₄×M₁₆	—	47	4	no	—
191	#341	—	47	4	no	—
192	#341	—	47	2	no	—
193	#341	C₂×D₄₇	1	2	yes	—
194	C₄₇	#21970	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	47	2	no	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×M₁₆	—	4	2	no	—
198	C₂×M₁₆	—	47	2	no	—
199	C₂×M₁₆	—	47	2	no	—
200	C₂×M₁₆	—	47	4	no	—
201	C₂×M₁₆	—	188	2	no	—
202	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
203	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
204	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
205	C₂²×C₈	—	47	2	no	—
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×C₄²	—	47	1	no	—
209	C₂×C₄²	—	47	2	no	—
210	C₂×C₄²	—	47	1	no	—
211	C₂×C₄²	—	47	1	no	—
212	C₂×C₄²	—	47	2	no	—
213	C₂×C₄²	—	47	1	no	—
214	C₂×C₄²	—	47	1	no	—
215	C₂×C₄²	—	47	1	no	—
216	C₂×C₄²	—	47	2	no	—
217	#108	—	4	2	no	—
218	#108	—	188	2	no	—
219	#108	—	188	4	no	—
220	C₄⋊C₈	—	47	4	no	—
221	C₄⋊C₈	—	47	4	no	—
222	C₄⋊C₈	—	47	4	no	—
223	C₄⋊C₈	—	47	4	no	—
224	C₄⋊C₈	—	94	4	no	—
225	C₄⋊C₈	—	94	4	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	94	2	no	—
229	C₂²⋊C₈	—	94	2	no	—
230	#97	—	94	4	no	—
231	C₄×C₈	—	94	4	no	—
232	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	47	2	no	—
236	C₂²×C₄	—	47	2	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	47	2	no	—
239	C₂²×C₄	—	47	2	no	—
240	C₂²×C₄	—	47	2	no	—
241	C₂²×C₄	—	188	4	no	—
242	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
245	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
246	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	2	no	—
252	M₁₆	—	94	4	no	—
253	M₁₆	—	94	4	no	—
254	M₁₆	—	188	2	no	—
255	M₁₆	—	188	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	47	4	no	—
260	C₂×C₈	—	94	4	no	—
261	C₂×C₈	—	94	4	no	—
262	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
263	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
264	C₄²	—	47	1	no	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	1	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	—	47	2	no	—
273	C₄²	—	47	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
277	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
278	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	47	2	no	—
287	C₂×C₄	—	47	2	no	—
288	C₂×C₄	—	47	2	no	—
289	C₂×C₄	—	47	2	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	—	94	4	no	—
292	C₂×C₄	—	94	4	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	2	no	—
296	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
300	C₂×C₄	#13094069	1	2	yes	—
301	C₂×C₄	#13094071	1	2	yes	—
302	C₈	—	2	2	no	—
303	C₈	—	2	2	no	—
304	C₈	—	4	4	no	—
305	C₈	—	94	4	no	—
306	C₈	—	188	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	4	4	no	—
313	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
314	C₂²	#15689752	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	47	4	no	—
318	C₄	—	94	2	no	—
319	C₄	—	94	2	no	—
320	C₄	—	94	2	no	—
321	C₄	#15689596	1	1	yes	characteristic
322	C₄	#15689600	1	1	yes	characteristic
323	C₄	#15689700	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	4	1	no	—
326	C₂	—	4	1	no	—
327	C₂	#20246005	1	1	yes	characteristic
328	C₂	#430×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1811	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1811	D₄₇
hypocenter	C₄₇	#21970
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1811

Complements

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

N	H ≅ G/N	type
3. #20244162	C₂	semidirect
4. #20244160	C₂	semidirect
194. C₄₇	#21970	semidirect

Maximal subgroups

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

subgroup	index
#20252017	2
C₂×#15689146	2
#20244162	2
#20244160	2
#20243430	2
#20243428	2
C₄₇×#1811	2
#21970	47