Subgroups — #26121968

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688893	C₂	1	2	yes	maximal
2	#20243114	C₂	1	2	yes	maximal
3	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2873	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15688893	—	2	4	no	—
9	#15688887	C₄	1	8	yes	complemented
10	C₄₇⋊#338	—	2	2	no	—
11	C₄₇⋊#338	—	2	2	no	—
12	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#405	—	2	2	no	—
16	C₄₇×#405	C₄	1	4	yes	—
17	C₄×M₁₆×C₄₇	—	2	2	no	—
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	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₄₇	—	2	1	no	—
34	C₂×M₁₆×C₄₇	—	2	2	no	—
35	C₂×M₁₆×C₄₇	—	2	2	no	—
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
40	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
41	C₂²×C₈×C₄₇	—	2	1	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	#117×C₄₇	—	2	1	no	—
44	#117×C₄₇	—	2	1	no	—
45	#117×C₄₇	C₂×C₄	1	2	yes	—
46	#117×C₄₇	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
49	C₂×C₄²×C₄₇	—	2	1	no	—
50	#97×C₄₇	—	2	4	no	—
51	C₄×C₈×C₄₇	—	2	4	no	—
52	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	M₁₆×C₄₇	—	2	2	no	—
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	C₄²	1	4	yes	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₂²⋊C₄×C₄₇	—	2	2	no	—
77	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	#10625329	—	2	4	no	—
81	#10625329	—	2	4	no	—
82	#10625329	—	4	2	no	—
83	#10625329	—	4	2	no	—
84	#10625329	—	4	2	no	—
85	#10625329	—	4	2	no	—
86	#10625329	—	4	2	no	—
87	#10625329	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂³×C₄₇	—	8	1	no	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
97	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₄₇	#125	1	2	yes	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	8	2	no	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#117	1	2	yes	—
105	C₂×C₄×C₄₇	#117	1	2	yes	—
106	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#123	1	2	yes	—
116	C₂×C₄×C₄₇	#123	1	2	yes	—
117	C₂×C₄×C₄₇	#124	1	2	yes	—
118	C₂×C₄×C₄₇	#126	1	2	yes	—
119	C₂×C₄×C₄₇	#126	1	2	yes	—
120	C₈×C₄₇	—	2	4	no	—
121	C₈×C₄₇	—	2	4	no	—
122	C₄₇⋊C₈	—	4	4	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	#10812	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	4	1	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₄₇	#380	1	1	yes	characteristic
132	C₂²×C₄₇	#382	1	1	yes	characteristic
133	C₂²×C₄₇	#387	1	2	yes	—
134	C₂²×C₄₇	#387	1	2	yes	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	#376	1	1	yes	characteristic
137	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₄₇	#381	1	1	yes	characteristic
139	C₄×C₄₇	#383	1	1	yes	characteristic
140	C₄×C₄₇	#388	1	2	yes	—
141	C₄×C₄₇	#388	1	2	yes	—
142	#2873	D₄₇	1	1	yes	characteristic
143	C₂×#344	—	47	2	no	—
144	#1738	—	47	2	no	—
145	C₂×#338	—	47	1	no	—
146	C₂×#338	—	47	1	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	2	1	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	#1434	1	1	yes	characteristic
153	C₂²×M₁₆	—	47	2	no	—
154	C₂²×M₁₆	—	47	2	no	—
155	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
156	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
158	#405	—	2	2	no	—
159	#405	Dic₄₇	1	4	yes	—
160	C₄×M₁₆	—	2	2	no	—
161	#344	—	94	4	no	—
162	#341	—	47	8	no	—
163	#338	—	94	2	no	—
164	#338	—	94	2	no	—
165	C₄₇	#10812	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	47	4	no	—
172	C₂×M₁₆	—	94	2	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	2	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	—	94	2	no	—
178	C₂×M₁₆	—	94	2	no	—
179	C₂×M₁₆	—	94	2	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
184	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
185	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	94	2	no	—
190	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
191	#117	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	D₁₈₈	1	2	yes	—
194	#117	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
197	C₂×C₄²	—	2	1	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
203	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂²×C₄	#10625350	1	1	yes	characteristic
205	C₂²×C₄	#10625350	1	1	yes	characteristic
206	C₂²×C₄	#10625350	1	2	yes	—
207	C₂²×C₄	#10625350	1	2	yes	—
208	C₂²×C₄	#10625350	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	94	4	no	—
217	M₁₆	—	94	4	no	—
218	M₁₆	—	188	2	no	—
219	M₁₆	—	188	2	no	—
220	M₁₆	—	188	2	no	—
221	M₁₆	—	188	2	no	—
222	M₁₆	—	188	2	no	—
223	M₁₆	—	188	2	no	—
224	M₁₆	C₄×Dic₄₇	1	4	yes	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	94	4	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	188	2	no	—
234	C₂×C₈	—	188	2	no	—
235	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
244	C₂³	C₄×Dic₉₄	1	2	yes	—
245	C₂³	#13094003	1	1	yes	characteristic
246	C₂³	#13094003	1	1	yes	characteristic
247	C₂³	#13094016	1	2	yes	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
252	C₂×C₄	#13093981	1	2	yes	—
253	C₂×C₄	#13093981	1	2	yes	—
254	C₂×C₄	#13093983	1	1	yes	characteristic
255	C₂×C₄	#13093983	1	1	yes	characteristic
256	C₂×C₄	#13093983	1	1	yes	characteristic
257	C₂×C₄	#13093983	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
259	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
260	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
261	C₂×C₄	#13093998	1	1	yes	characteristic
262	C₂×C₄	#13093998	1	1	yes	characteristic
263	C₂×C₄	#13094007	1	2	yes	—
264	C₂×C₄	#13094007	1	2	yes	—
265	C₂×C₄	#13094012	1	2	yes	—
266	C₂×C₄	#13094018	1	2	yes	—
267	C₂×C₄	#13094018	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	188	4	no	—
271	C₈	—	188	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
278	C₂²	#15688983	1	1	yes	characteristic
279	C₂²	#15688989	1	1	yes	characteristic
280	C₂²	#15689007	1	2	yes	—
281	C₂²	#15689007	1	2	yes	—
282	C₄	—	4	4	no	—
283	C₄	#15688969	1	1	yes	characteristic
284	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
285	C₄	#15688986	1	1	yes	characteristic
286	C₄	#15688991	1	1	yes	characteristic
287	C₄	#15689014	1	2	yes	—
288	C₄	#15689014	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	#20241615	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2873	C₂
socle	C₂×C₄₇	#1434
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2873	D₄₇
hypocenter	C₄₇	#10812
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ #2873

Complements

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

N	H ≅ G/N	type
9. #15688887	C₄	semidirect
165. C₄₇	#10812	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688893	2
#20243114	2
C₂×C₄₇⋊#338	2
C₄₇×#2873	2
#10812	47