Subgroups — #26215041

Order 12032.

866 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20243265	C₂	1	4	yes	maximal, complemented
4	#20243260	C₂	1	4	yes	maximal, complemented
5	#20243156	C₂	1	4	yes	maximal
6	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15689180	C₄	1	8	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×#13093934	C₂²	1	4	yes	—
13	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
14	#15688993	C₄	1	2	yes	—
15	#15688989	C₄	1	2	yes	—
16	#15688983	C₄	1	2	yes	—
17	#15688983	C₄	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	#15688877	C₂²	1	8	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#432	C₄	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13093983	—	4	2	no	—
28	#13093983	—	4	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
35	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
36	#13093935	C₂×C₄	1	8	yes	—
37	#13093934	—	4	4	no	—
38	C₈×Dic₄₇	—	4	4	no	—
39	C₄₇⋊#95	C₂×C₄	1	4	yes	—
40	C₄₇⋊#95	C₂×C₄	1	4	yes	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
52	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
53	#97×C₄₇	C₂×C₄	1	8	yes	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
60	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
73	C₄²×C₄₇	C₄²	1	4	yes	—
74	C₄²×C₄₇	C₄²	1	4	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²⋊Dic₄₇	—	4	4	no	—
80	C₄⋊Dic₄₇	—	4	4	no	—
81	C₄⋊Dic₄₇	—	4	4	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
112	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	2	8	no	—
115	C₂×Dic₄₇	—	2	8	no	—
116	C₈×C₄₇	—	2	8	no	—
117	C₄₇⋊C₈	—	2	8	no	—
118	#20539	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	4	2	no	—
123	C₂²×C₄₇	—	4	2	no	—
124	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
125	C₂²×C₄₇	#518	1	1	yes	characteristic
126	C₂²×C₄₇	#519	1	1	yes	characteristic
127	C₂²×C₄₇	#568	1	2	yes	—
128	C₂²×C₄₇	#568	1	2	yes	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	#520	1	2	yes	—
135	C₄×C₄₇	#520	1	2	yes	—
136	Dic₄₇	—	4	8	no	—
137	C₂×#432	—	47	1	no	—
138	C₂×#432	D₄₇	1	1	yes	characteristic
139	#2285	—	47	1	no	—
140	#1771	—	47	4	no	—
141	#1771	—	47	4	no	—
142	#1750	—	47	4	no	—
143	C₂×C₄₇	—	4	2	no	—
144	C₂×C₄₇	#2274	1	1	yes	characteristic
145	C₂×C₄₇	#2873	1	1	yes	characteristic
146	C₂×C₄₇	#2873	1	1	yes	characteristic
147	C₂×C₄₇	#2980	1	2	yes	—
148	C₂×C₄₇	#2987	1	2	yes	—
149	C₂×#117	—	47	2	no	—
150	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
151	#432	—	47	8	no	—
152	#432	Dic₄₇	1	8	yes	complemented
153	C₂×C₄⋊C₈	—	47	2	no	—
154	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
155	C₂×C₂²⋊C₈	—	47	2	no	—
156	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
157	C₂×#97	—	47	2	no	—
158	C₂×#97	—	47	4	no	—
159	C₂×#97	C₂×D₄₇	1	2	yes	—
160	C₂×C₄×C₈	—	47	4	no	—
161	#384	—	47	2	no	—
162	#382	—	47	2	no	—
163	#380	—	47	2	no	—
164	#380	—	47	2	no	—
165	C₄×C₄⋊C₄	—	47	2	no	—
166	C₄×C₄⋊C₄	—	47	2	no	—
167	#336	—	47	8	no	—
168	C₄₇	#20539	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
170	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
171	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	—	47	4	no	—
173	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
174	#117	—	188	2	no	—
175	#117	—	188	2	no	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₄⋊C₄	—	47	4	no	—
178	C₂×C₄⋊C₄	—	47	4	no	—
179	C₂×C₂²⋊C₄	—	47	4	no	—
180	C₂×C₄²	—	47	4	no	—
181	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₄×D₄₇	1	4	yes	—
184	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₄⋊C₈	—	47	8	no	—
188	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
189	C₂²⋊C₈	—	47	8	no	—
190	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
191	#97	—	47	8	no	—
192	#97	—	188	4	no	—
193	#97	C₂×Dic₄₇	1	8	yes	—
194	C₄×C₈	—	188	4	no	—
195	#95	—	47	4	no	—
196	#95	—	47	4	no	—
197	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	47	8	no	—
203	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
204	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
205	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	47	8	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	94	4	no	—
218	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
219	C₄⋊C₄	—	188	4	no	—
220	C₄⋊C₄	—	188	4	no	—
221	C₄⋊C₄	—	188	4	no	—
222	C₂²⋊C₄	—	188	4	no	—
223	C₄²	—	94	4	no	—
224	C₄²	—	94	4	no	—
225	C₄²	C₄×Dic₄₇	1	4	yes	—
226	C₄²	C₄×Dic₄₇	1	4	yes	—
227	C₄²	C₂×C₄×D₄₇	1	2	yes	—
228	C₄²	C₂×C₄×D₄₇	1	2	yes	—
229	C₄²	C₂²×Dic₄₇	1	2	yes	—
230	C₄²	C₂²×Dic₄₇	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
236	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	94	8	no	—
248	C₂×C₄	—	94	8	no	—
249	C₂×C₄	—	94	8	no	—
250	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
251	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
252	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
253	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	#13094070	1	2	yes	—
258	C₂×C₄	#13094071	1	2	yes	—
259	C₈	—	2	8	no	—
260	C₈	—	94	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
267	C₂²	#15689597	1	1	yes	characteristic
268	C₂²	#15689600	1	1	yes	characteristic
269	C₂²	#15689889	1	2	yes	—
270	C₂²	#15689889	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	188	8	no	—
277	C₄	#15689603	1	2	yes	—
278	C₄	#15689606	1	2	yes	—
279	C₂	—	4	2	no	—
280	C₂	#20245744	1	1	yes	characteristic
281	C₂	#20251806	1	1	yes	characteristic
282	C₂	#20251806	1	1	yes	characteristic
283	C₂	#20252458	1	2	yes	—
284	C₂	#20252493	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#20539
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20243265	C₂	semidirect
4. #20243260	C₂	semidirect
8. #15689180	C₄	semidirect
22. C₄₇×#432	C₄	semidirect
152. #432	Dic₄₇	semidirect
168. C₄₇	#20539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
#20245795	2
#20243265	2
#20243260	2
#20243156	2
C₂×C₄₇×#432	2
#20539	47