Subgroups — #26132205

Order 12032.

486 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	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245143	C₂	1	2	yes	maximal, complemented
3	C₂×#15688905	C₂	1	1	yes	maximal, characteristic
4	#20241471	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#349	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
8	C₂×#13093964	C₂²	1	2	yes	—
9	M₁₆×Dic₄₇	—	2	4	no	—
10	C₂×#13093934	C₄	1	1	yes	characteristic
11	C₂×#13093934	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₁₆	—	2	4	no	—
15	#15688905	—	2	2	no	—
16	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#349	—	2	2	no	—
19	C₂³×Dic₄₇	C₈	1	2	yes	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
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	1	no	—
27	C₂×C₄×Dic₄₇	—	2	1	no	—
28	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
31	#13093964	—	2	2	no	—
32	#13093964	—	2	2	no	—
33	C₂×C₄₇⋊C₁₆	—	2	2	no	—
34	#13093934	—	2	2	no	—
35	#13093934	—	2	2	no	—
36	C₈×Dic₄₇	—	2	2	no	—
37	C₈×Dic₄₇	D₄	1	4	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
42	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	M₃₂×C₄₇	—	2	2	no	—
46	M₃₂×C₄₇	—	2	2	no	—
47	C₂×C₁₆×C₄₇	—	2	2	no	—
48	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
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	—	2	1	no	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	2	no	—
58	C₂²×Dic₄₇	—	2	2	no	—
59	C₂²×Dic₄₇	—	4	4	no	—
60	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
61	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
62	M₁₆×C₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	1	no	—
65	C₂×C₈×C₄₇	—	2	1	no	—
66	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₄₇	#41	1	2	yes	—
71	C₄×Dic₄₇	—	2	1	no	—
72	C₄×Dic₄₇	—	2	1	no	—
73	C₄×Dic₄₇	—	2	2	no	—
74	C₄×Dic₄₇	—	2	2	no	—
75	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
76	C₄×Dic₄₇	M₁₆	1	1	yes	characteristic
77	C₄×Dic₄₇	M₁₆	1	1	yes	characteristic
78	#10625329	—	2	8	no	—
79	C₂×C₄₇⋊C₈	—	2	1	no	—
80	C₂×C₄₇⋊C₈	—	2	1	no	—
81	C₂×C₄₇⋊C₈	—	2	1	no	—
82	C₂×C₄₇⋊C₈	—	2	1	no	—
83	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
85	C₁₆×C₄₇	—	4	2	no	—
86	C₄₇⋊C₁₆	—	4	2	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	4	2	no	—
90	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
91	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₂×Dic₄₇	—	4	2	no	—
110	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
111	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
112	C₈×C₄₇	—	2	2	no	—
113	C₈×C₄₇	#121	1	2	yes	—
114	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	C₄₇⋊C₈	—	2	2	no	—
116	C₄₇⋊C₈	—	2	2	no	—
117	#12785	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	4	2	no	—
121	C₂²×C₄₇	—	4	2	no	—
122	C₂²×C₄₇	—	4	2	no	—
123	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₄₇	#431	1	1	yes	characteristic
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	#380	1	1	yes	characteristic
128	C₄×C₄₇	#407	1	1	yes	characteristic
129	C₄×C₄₇	#423	1	1	yes	characteristic
130	C₄×C₄₇	#432	1	1	yes	characteristic
131	Dic₄₇	—	2	2	no	—
132	Dic₄₇	—	4	2	no	—
133	Dic₄₇	#349	1	4	yes	complemented
134	C₂×C₄×M₁₆	—	47	1	no	—
135	#2150	—	47	2	no	—
136	C₂×#349	—	47	1	no	—
137	C₂×#349	D₄₇	1	1	yes	characteristic
138	#1377	—	47	2	no	—
139	C₂×C₄₇	—	2	1	no	—
140	C₂×C₄₇	—	2	1	no	—
141	C₂×C₄₇	—	4	2	no	—
142	C₂×C₄₇	#1797	1	1	yes	characteristic
143	C₂×C₄₇	C₂×#349	1	2	yes	—
144	C₂²×M₁₆	—	47	1	no	—
145	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
146	C₂²×C₄²	—	47	1	no	—
147	C₂×M₃₂	—	47	2	no	—
148	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
149	C₄×M₁₆	—	94	4	no	—
150	C₂×#97	—	47	1	no	—
151	C₂×#97	—	47	1	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₂×C₄×C₈	—	47	1	no	—
154	C₄⋊C₁₆	—	94	4	no	—
155	#349	—	2	2	no	—
156	#349	—	94	2	no	—
157	C₄₇	#12785	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	47	2	no	—
159	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	94	2	no	—
163	C₂×M₁₆	—	94	2	no	—
164	C₂×M₁₆	—	94	2	no	—
165	C₂×M₁₆	Dic₉₄	1	2	yes	—
166	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
167	C₂²×C₈	—	47	1	no	—
168	C₂²×C₈	—	47	1	no	—
169	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
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	1	no	—
176	C₂×C₄²	—	94	1	no	—
177	M₃₂	—	2	2	no	—
178	M₃₂	—	2	2	no	—
179	M₃₂	—	94	2	no	—
180	M₃₂	—	94	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	94	2	no	—
183	#97	—	94	2	no	—
184	#97	—	94	2	no	—
185	C₄×C₈	—	47	4	no	—
186	C₄×C₈	—	94	2	no	—
187	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	47	2	no	—
193	C₂²×C₄	—	47	2	no	—
194	C₂²×C₄	—	94	2	no	—
195	C₂²×C₄	—	94	2	no	—
196	C₂²×C₄	—	188	4	no	—
197	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	#10625324	1	2	yes	—
199	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	94	8	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	47	2	no	—
207	C₂×C₈	—	47	2	no	—
208	C₂×C₈	—	94	1	no	—
209	C₂×C₈	—	94	1	no	—
210	C₂×C₈	—	94	1	no	—
211	C₂×C₈	—	94	1	no	—
212	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
213	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
214	C₂×C₈	D₄×D₄₇	1	2	yes	—
215	C₂×C₈	#10625352	1	2	yes	—
216	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₄²	—	47	2	no	—
218	C₄²	—	47	1	no	—
219	C₄²	—	47	1	no	—
220	C₄²	—	94	2	no	—
221	C₄²	—	94	2	no	—
222	C₄²	—	94	1	no	—
223	C₄²	—	94	1	no	—
224	C₁₆	—	4	2	no	—
225	C₁₆	—	188	2	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	2	no	—
229	C₂³	Dic₄₇⋊C₈	1	2	yes	—
230	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	47	2	no	—
238	C₂×C₄	—	47	2	no	—
239	C₂×C₄	—	94	4	no	—
240	C₂×C₄	—	94	2	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	2	no	—
244	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
245	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
247	C₂×C₄	#13093980	1	1	yes	characteristic
248	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
249	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	94	2	no	—
253	C₈	—	94	2	no	—
254	C₈	#13093996	1	2	yes	—
255	C₈	#13094001	1	2	yes	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
262	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
263	C₂²	#15689173	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	47	4	no	—
266	C₄	—	94	2	no	—
267	C₄	—	188	2	no	—
268	C₄	#15688981	1	1	yes	characteristic
269	C₄	#15689085	1	1	yes	characteristic
270	C₄	#15689146	1	1	yes	characteristic
271	C₄	#15689177	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	4	2	no	—
275	C₂	#20243366	1	1	yes	characteristic
276	C₂	#349×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#349	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#349	D₄₇
hypocenter	C₄₇	#12785
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. #20245143	C₂	semidirect
4. #20241471	C₂	semidirect
133. Dic₄₇	#349	semidirect
157. C₄₇	#12785	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₄₇	2
#20245143	2
C₂×#15688905	2
#20241471	2
C₂×C₄₇×#349	2
#12785	47