Subgroups — #26131311

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688898	C₂	1	2	yes	maximal, complemented
4	#20241465	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₈×Dic₄₇	—	2	4	no	—
10	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	complemented
15	#15688898	—	2	4	no	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
18	C₄₇×C₂²⋊C₁₆	C₄	1	4	yes	complemented
19	C₂³×Dic₄₇	C₈	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	C₈	1	4	yes	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄₇⋊C₁₆	—	2	2	no	—
31	C₂×C₄₇⋊C₁₆	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
33	C₈×Dic₄₇	—	2	2	no	—
34	C₈×Dic₄₇	—	2	2	no	—
35	C₈×Dic₄₇	—	2	2	no	—
36	C₈×Dic₄₇	D₄	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₄₇	—	2	2	no	—
45	C₂×C₁₆×C₄₇	—	2	2	no	—
46	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
47	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	2	4	no	—
57	C₂²×Dic₄₇	—	2	4	no	—
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
60	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₄₇	—	2	1	no	—
62	C₂×C₈×C₄₇	—	2	1	no	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	C₄²	1	2	yes	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₇	#41	1	2	yes	—
70	C₄×Dic₄₇	—	2	1	no	—
71	C₄×Dic₄₇	—	2	1	no	—
72	C₄×Dic₄₇	—	2	4	no	—
73	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
74	C₄×Dic₄₇	M₁₆	1	2	yes	—
75	C₂×C₄₇⋊C₈	—	2	1	no	—
76	C₂×C₄₇⋊C₈	—	2	1	no	—
77	C₂×C₄₇⋊C₈	—	2	8	no	—
78	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
80	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
81	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
82	C₁₆×C₄₇	—	2	4	no	—
83	C₄₇⋊C₁₆	—	2	4	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	—	2	1	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	C₄×C₈	1	2	yes	—
88	C₂³×C₄₇	M₃₂	1	4	yes	—
89	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₄₇	—	2	16	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	C₂²⋊C₈	1	4	yes	—
109	C₂×Dic₄₇	M₃₂	1	4	yes	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	C₄×D₄	1	2	yes	—
112	C₈×C₄₇	C₄×D₄	1	2	yes	—
113	C₄₇⋊C₈	—	2	2	no	—
114	C₄₇⋊C₈	C₂²⋊C₈	1	4	yes	—
115	#12599	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	#346	1	4	yes	complemented
121	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₄₇	#431	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
125	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
129	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₄₇	#431	1	1	yes	characteristic
131	Dic₄₇	—	2	4	no	—
132	Dic₄₇	—	2	4	no	—
133	C₂²×C₄×C₈	—	47	1	no	—
134	C₂×C₂²⋊C₁₆	—	47	1	no	—
135	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic
136	C₂×#346	—	47	2	no	—
137	#1374	—	47	2	no	—
138	C₂×C₄₇	—	2	4	no	—
139	C₂×C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#346	1	1	yes	characteristic
141	C₂×C₄₇	C₄×M₃₂	1	1	yes	characteristic
142	C₂×C₄₇	#2112	1	2	yes	—
143	C₂×C₄₇	#2163	1	2	yes	—
144	C₂³×C₈	—	47	1	no	—
145	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
146	C₂²×C₄²	—	47	1	no	—
147	C₂²×C₁₆	—	47	2	no	—
148	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
149	C₂×C₄×C₈	—	47	1	no	—
150	C₂×C₄×C₈	—	47	1	no	—
151	C₂×C₄×C₈	—	47	1	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₂×C₄×C₈	—	94	4	no	—
154	C₂²⋊C₁₆	—	47	4	no	—
155	C₂²⋊C₁₆	Dic₄₇	1	4	yes	complemented
156	#346	—	94	4	no	—
157	C₄₇	#12599	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	47	2	no	—
159	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	47	1	no	—
163	C₂²×C₈	—	47	4	no	—
164	C₂²×C₈	—	47	1	no	—
165	C₂²×C₈	—	94	2	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
168	C₂²×C₈	C₄×D₄₇	1	2	yes	—
169	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
170	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	—	47	1	no	—
172	C₂×C₄²	—	47	2	no	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	—	94	2	no	—
175	C₂×C₄²	—	94	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	47	4	no	—
179	C₂×C₁₆	—	94	2	no	—
180	C₂×C₁₆	—	94	2	no	—
181	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	complemented
182	C₄×C₈	—	47	4	no	—
183	C₄×C₈	—	94	2	no	—
184	C₄×C₈	—	94	2	no	—
185	C₄×C₈	—	94	2	no	—
186	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	—	47	2	no	—
193	C₂²×C₄	—	94	4	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	C₈×D₄₇	1	2	yes	—
198	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	47	2	no	—
205	C₂×C₈	—	47	2	no	—
206	C₂×C₈	—	47	2	no	—
207	C₂×C₈	—	47	2	no	—
208	C₂×C₈	—	94	8	no	—
209	C₂×C₈	—	94	1	no	—
210	C₂×C₈	—	94	1	no	—
211	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₈	D₄×D₄₇	1	2	yes	—
214	C₂×C₈	#10625352	1	2	yes	—
215	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₄²	—	47	2	no	—
217	C₄²	—	47	2	no	—
218	C₄²	—	94	4	no	—
219	C₄²	—	94	1	no	—
220	C₄²	—	94	1	no	—
221	C₁₆	—	2	4	no	—
222	C₁₆	—	94	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	C₈×Dic₄₇	1	2	yes	—
227	C₂³	#13093963	1	4	yes	—
228	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	47	4	no	—
236	C₂×C₄	—	47	4	no	—
237	C₂×C₄	—	94	16	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	2	no	—
241	C₂×C₄	—	94	2	no	—
242	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₄	#13093980	1	1	yes	characteristic
246	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
248	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	47	4	no	—
251	C₈	—	94	2	no	—
252	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
253	C₈	D₄×Dic₄₇	1	2	yes	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	#15688898	1	4	yes	complemented
259	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
260	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
261	C₂²	#15689173	1	1	yes	characteristic
262	C₂²	M₃₂×D₄₇	1	2	yes	—
263	C₂²	C₂×#13093963	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	94	4	no	—
266	C₄	—	94	4	no	—
267	C₄	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
268	C₄	M₁₆×Dic₄₇	1	1	yes	characteristic
269	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
270	C₄	#15689173	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	C₂²⋊C₈×Dic₄₇	1	1	yes	characteristic
273	C₂	C₂×#15688898	1	1	yes	characteristic
274	C₂	M₃₂×Dic₄₇	1	1	yes	characteristic
275	C₂	#20244997	1	2	yes	—
276	C₂	#20245188	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂×C₄₇×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	#12599
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#15688898	C₂	semidirect
4. #20241465	C₂	semidirect
14. C₂²⋊(C₄₇⋊C₁₆)	C₄	semidirect
18. C₄₇×C₂²⋊C₁₆	C₄	semidirect
32. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
46. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
120. C₂²×C₄₇	#346	semidirect
155. C₂²⋊C₁₆	Dic₄₇	semidirect
157. C₄₇	#12599	semidirect
181. C₂×C₁₆	C₂×Dic₄₇	semidirect
258. C₂²	#15688898	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂×#15688898	2
#20241465	2
C₂×C₄₇×C₂²⋊C₁₆	2
#12599	47