Subgroups — #26122016

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688887	C₂	1	2	yes	maximal
4	#20243081	C₂	1	2	yes	maximal
5	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂×#13093934	—	2	4	no	—
10	C₂×#13093934	C₄	1	4	yes	—
11	#15688888	C₄	1	4	yes	complemented
12	#15688887	—	2	4	no	—
13	#15688877	—	2	4	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#341	C₄	1	4	yes	complemented
17	C₂³×Dic₄₇	Q₈	1	2	yes	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
34	#13093934	—	2	8	no	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	#41	1	2	yes	—
55	C₂²×C₄×C₄₇	#41	1	2	yes	—
56	C₂²×C₄×C₄₇	#41	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	4	4	no	—
61	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
62	M₁₆×C₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	4	2	no	—
65	M₁₆×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	4	2	no	—
72	C₂×C₈×C₄₇	—	4	2	no	—
73	C₂×C₈×C₄₇	C₄²	1	4	yes	—
74	C₄×Dic₄₇	—	2	2	no	—
75	C₄×Dic₄₇	—	2	2	no	—
76	C₄×Dic₄₇	—	2	4	no	—
77	#10625329	—	2	4	no	—
78	#10625329	—	2	4	no	—
79	#10625329	—	4	2	no	—
80	#10625329	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	2	2	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
94	C₂³×C₄₇	#125	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#117	1	2	yes	—
101	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#117	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	2	yes	—
108	C₂×C₄×C₄₇	#123	1	2	yes	—
109	C₂×C₄×C₄₇	#123	1	2	yes	—
110	C₂×C₄×C₄₇	#124	1	2	yes	—
111	C₂×C₄×C₄₇	#126	1	2	yes	—
112	C₂×C₄×C₄₇	#126	1	2	yes	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	2	8	no	—
115	C₂×Dic₄₇	—	4	8	no	—
116	C₈×C₄₇	—	2	4	no	—
117	C₈×C₄₇	—	4	4	no	—
118	C₄₇⋊C₈	—	2	4	no	—
119	C₄₇⋊C₈	—	4	4	no	—
120	#10823	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₇	#380	1	1	yes	characteristic
127	C₂²×C₄₇	#382	1	1	yes	characteristic
128	C₂²×C₄₇	#387	1	2	yes	—
129	C₂²×C₄₇	#387	1	2	yes	—
130	C₂²×C₄₇	#430	1	4	yes	—
131	C₄×C₄₇	#376	1	1	yes	characteristic
132	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₄₇	#381	1	1	yes	characteristic
134	C₄×C₄₇	#383	1	1	yes	characteristic
135	C₄×C₄₇	#388	1	2	yes	—
136	C₄×C₄₇	#388	1	2	yes	—
137	Dic₄₇	—	4	4	no	—
138	C₂²×#97	—	47	1	no	—
139	C₂×#341	—	47	2	no	—
140	C₂×#341	—	47	1	no	—
141	C₂×#341	D₄₇	1	1	yes	characteristic
142	#1728	—	47	2	no	—
143	C₂×C₄₇	—	2	2	no	—
144	C₂×C₄₇	—	2	2	no	—
145	C₂×C₄₇	#1434	1	1	yes	characteristic
146	C₂×C₄₇	#1778	1	2	yes	—
147	C₂×C₄₇	#1811	1	2	yes	—
148	C₂×C₄₇	#1838	1	2	yes	—
149	C₂²×M₁₆	—	47	2	no	—
150	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
151	C₂³×C₈	—	47	1	no	—
152	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₂²×C₄²	—	47	1	no	—
154	C₂×#97	—	47	4	no	—
155	C₂×#97	—	94	4	no	—
156	#341	—	47	4	no	—
157	#341	—	94	4	no	—
158	#341	Dic₄₇	1	4	yes	complemented
159	#336	—	94	4	no	—
160	C₄₇	#10823	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	47	2	no	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	47	4	no	—
169	C₂×M₁₆	—	94	2	no	—
170	C₂×M₁₆	—	94	2	no	—
171	C₂×M₁₆	—	94	2	no	—
172	C₂×M₁₆	—	94	2	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	47	2	no	—
179	C₂²×C₈	—	47	4	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₂²×C₈	—	94	1	no	—
182	C₂²×C₈	—	94	1	no	—
183	C₂²×C₈	—	94	2	no	—
184	C₂²×C₈	C₄×D₄₇	1	2	yes	—
185	C₂²×C₈	C₄×D₄₇	1	2	yes	—
186	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
187	C₂×C₄²	—	47	2	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	94	2	no	—
190	C₂×C₄²	—	94	2	no	—
191	#97	—	94	8	no	—
192	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	—	47	4	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	—	188	4	no	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	#10625350	1	2	yes	—
199	C₂²×C₄	#10625350	1	2	yes	—
200	C₂²×C₄	D₄×D₄₇	1	2	yes	—
201	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	#10625352	1	2	yes	—
203	C₂²×C₄	#10625352	1	1	yes	characteristic
204	C₂²×C₄	#10625354	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	94	4	no	—
211	M₁₆	—	94	4	no	—
212	M₁₆	—	188	2	no	—
213	M₁₆	—	188	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	47	4	no	—
222	C₂×C₈	—	94	2	no	—
223	C₂×C₈	—	94	2	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	188	2	no	—
228	C₂×C₈	—	188	2	no	—
229	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
230	C₄²	—	94	2	no	—
231	C₄²	—	94	2	no	—
232	C₄²	—	94	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
235	C₂³	#13093989	1	1	yes	characteristic
236	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
237	C₂³	#13094002	1	2	yes	—
238	C₂³	#13094014	1	2	yes	—
239	C₂×C₄	—	94	4	no	—
240	C₂×C₄	—	94	8	no	—
241	C₂×C₄	—	188	8	no	—
242	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄	#13093980	1	1	yes	characteristic
245	C₂×C₄	#13093981	1	1	yes	characteristic
246	C₂×C₄	#13093981	1	1	yes	characteristic
247	C₂×C₄	#13093981	1	2	yes	—
248	C₂×C₄	#13093982	1	1	yes	characteristic
249	C₂×C₄	#13093983	1	2	yes	—
250	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
251	C₂×C₄	#13093986	1	1	yes	characteristic
252	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
253	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
254	C₂×C₄	#13093997	1	2	yes	—
255	C₂×C₄	#13094006	1	2	yes	—
256	C₂×C₄	#13094009	1	2	yes	—
257	C₂×C₄	#13094010	1	2	yes	—
258	C₂×C₄	#13094017	1	2	yes	—
259	C₂×C₄	#13094019	1	2	yes	—
260	C₈	—	2	4	no	—
261	C₈	—	4	4	no	—
262	C₈	—	94	4	no	—
263	C₈	—	188	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
269	C₂²	#15688982	1	1	yes	characteristic
270	C₂²	#15688987	1	1	yes	characteristic
271	C₂²	#15689005	1	2	yes	—
272	C₂²	#15689009	1	2	yes	—
273	C₂²	#15689170	1	4	yes	—
274	C₄	—	188	4	no	—
275	C₄	#15688969	1	1	yes	characteristic
276	C₄	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
277	C₄	#15688985	1	1	yes	characteristic
278	C₄	#15688990	1	1	yes	characteristic
279	C₄	#15689012	1	2	yes	—
280	C₄	#15689016	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	2	no	—
283	C₂	#20241613	1	1	yes	characteristic
284	C₂	#20243296	1	2	yes	—
285	C₂	#20243429	1	2	yes	—
286	C₂	#20243527	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#10823
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
11. #15688888	C₄	semidirect
16. C₄₇×#341	C₄	semidirect
158. #341	Dic₄₇	semidirect
160. C₄₇	#10823	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093934	2
C₂×#15688888	2
C₂×#15688887	2
#20243081	2
C₂×C₄₇×#341	2
#10823	47