Subgroups — #26215073

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20243260	C₂	1	4	yes	maximal, complemented
4	#20243247	C₂	1	4	yes	maximal, complemented
5	#20243167	C₂	1	2	yes	maximal
6	#20243162	C₂	1	2	yes	maximal
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×#13093934	C₂²	1	4	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	#15688991	C₄	1	2	yes	—
16	#15688989	C₄	1	2	yes	—
17	#15688986	C₄	1	2	yes	—
18	#15688983	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	#15688969	C₂²	1	2	yes	—
22	#15688877	C₂²	1	4	yes	—
23	#15688877	C₂²	1	4	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#432	C₄	1	8	yes	complemented
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	2	yes	—
29	C₂²×C₄₇⋊C₈	—	4	2	no	—
30	C₂²×C₄₇⋊C₈	—	4	2	no	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
38	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
40	#13093934	—	4	4	no	—
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	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
58	#97×C₄₇	C₂×C₄	1	8	yes	—
59	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
65	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
78	C₄²×C₄₇	C₄²	1	4	yes	—
79	C₄²×C₄₇	C₄²	1	4	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂²⋊Dic₄₇	—	2	8	no	—
85	C₄⋊Dic₄₇	—	4	4	no	—
86	C₄×Dic₄₇	—	2	4	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	4	no	—
91	C₂×C₄₇⋊C₈	—	4	4	no	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
97	C₂³×C₄₇	#131	1	4	yes	—
98	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
117	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
118	C₂×Dic₄₇	—	2	8	no	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₈×C₄₇	—	2	8	no	—
122	C₄₇⋊C₈	—	2	4	no	—
123	C₄₇⋊C₈	—	2	4	no	—
124	#20544	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	8	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	#405	1	4	yes	—
129	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₄₇	#520	1	2	yes	—
131	C₂²×C₄₇	C₂×#131	1	2	yes	—
132	C₂²×C₄₇	#568	1	2	yes	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	#518	1	1	yes	characteristic
139	C₄×C₄₇	#519	1	1	yes	characteristic
140	C₄×C₄₇	#520	1	2	yes	—
141	Dic₄₇	—	4	8	no	—
142	C₂×#432	D₄₇	1	1	yes	characteristic
143	C₂²×C₄⋊C₈	—	47	1	no	—
144	#2283	—	47	1	no	—
145	#1771	—	47	4	no	—
146	#1768	—	47	4	no	—
147	#1752	—	47	2	no	—
148	C₈×C₂²⋊C₄	—	47	2	no	—
149	C₂×C₄₇	—	2	4	no	—
150	C₂×C₄₇	#2274	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
152	C₂×C₄₇	#2873	1	1	yes	characteristic
153	C₂×C₄₇	#2992	1	4	yes	—
154	C₂³×C₈	—	47	1	no	—
155	C₂³×C₈	—	47	1	no	—
156	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
157	#432	Dic₄₇	1	8	yes	complemented
158	C₂×C₄⋊C₈	—	47	8	no	—
159	C₂×C₄⋊C₈	—	47	2	no	—
160	C₂×C₄⋊C₈	—	47	2	no	—
161	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
162	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
163	C₂×#97	—	47	4	no	—
164	C₂×#97	C₂×D₄₇	1	2	yes	—
165	C₂×C₄×C₈	—	47	4	no	—
166	#383	—	47	2	no	—
167	#382	—	47	2	no	—
168	#381	—	47	2	no	—
169	#380	—	47	2	no	—
170	C₄×C₄⋊C₄	—	47	2	no	—
171	C₄×C₂²⋊C₄	—	47	2	no	—
172	#376	—	47	2	no	—
173	#336	—	47	4	no	—
174	#336	—	47	4	no	—
175	C₄₇	#20544	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
177	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	—	47	2	no	—
181	C₂²×C₈	—	47	4	no	—
182	C₂²×C₈	—	47	4	no	—
183	C₂²×C₈	—	188	2	no	—
184	C₂²×C₈	—	188	2	no	—
185	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
186	C₂×C₄⋊C₄	—	47	4	no	—
187	C₂×C₂²⋊C₄	—	47	4	no	—
188	C₂×C₄²	—	47	4	no	—
189	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
191	C₂×C₄²	C₄×D₄₇	1	4	yes	—
192	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
193	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
195	C₄⋊C₈	—	47	8	no	—
196	C₄⋊C₈	—	47	8	no	—
197	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
198	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
199	#97	—	188	4	no	—
200	#97	C₂×Dic₄₇	1	8	yes	—
201	C₄×C₈	—	188	4	no	—
202	#95	—	47	4	no	—
203	#95	—	47	4	no	—
204	#95	—	47	4	no	—
205	#95	—	47	4	no	—
206	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	47	8	no	—
212	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
213	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
221	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	47	4	no	—
225	C₂×C₈	—	47	4	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	4	no	—
228	C₂×C₈	—	188	4	no	—
229	C₂×C₈	—	188	4	no	—
230	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
231	C₄⋊C₄	—	188	4	no	—
232	C₂²⋊C₄	—	94	8	no	—
233	C₄²	—	94	4	no	—
234	C₄²	—	94	4	no	—
235	C₄²	C₄×Dic₄₇	1	4	yes	—
236	C₄²	C₄×Dic₄₇	1	4	yes	—
237	C₄²	C₂×C₄×D₄₇	1	2	yes	—
238	C₄²	C₂×C₄×D₄₇	1	2	yes	—
239	C₄²	C₂²×Dic₄₇	1	2	yes	—
240	C₄²	C₂²×Dic₄₇	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
244	C₂³	#13094032	1	4	yes	—
245	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	94	8	no	—
257	C₂×C₄	—	94	8	no	—
258	C₂×C₄	—	94	8	no	—
259	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
260	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
261	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
262	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
263	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
264	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	#13094070	1	2	yes	—
267	C₂×C₄	#13094071	1	2	yes	—
268	C₈	—	2	8	no	—
269	C₈	—	94	4	no	—
270	C₈	—	94	4	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	#15689081	1	4	yes	—
275	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
276	C₂²	#15689603	1	2	yes	—
277	C₂²	C₂×#13094032	1	2	yes	—
278	C₂²	#15689889	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	188	8	no	—
285	C₄	#15689597	1	1	yes	characteristic
286	C₄	#15689600	1	1	yes	characteristic
287	C₄	#15689606	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	#20245744	1	1	yes	characteristic
290	C₂	C₂×#15689081	1	1	yes	characteristic
291	C₂	#20251806	1	1	yes	characteristic
292	C₂	#20252529	1	4	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	#20544
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. #20243260	C₂	semidirect
4. #20243247	C₂	semidirect
10. C₂×C₄⋊(C₄₇⋊C₈)	C₄	semidirect
25. C₄₇×#432	C₄	semidirect
157. #432	Dic₄₇	semidirect
175. C₄₇	#20544	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₇⋊C₈)	2
#20245786	2
#20243260	2
#20243247	2
#20243167	2
#20243162	2
C₂×C₄₇×#432	2
#20544	47