Subgroups — #26122015

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 300 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	2	no	—
10	C₂×#13093934	—	2	2	no	—
11	C₂×#13093934	C₄	1	4	yes	—
12	#15688888	—	2	2	no	—
13	#15688887	C₄	1	8	yes	complemented
14	#15688877	—	2	4	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#341	—	2	2	no	—
18	C₂³×Dic₄₇	Q₈	1	2	yes	—
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	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
37	#13093934	—	2	8	no	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	—	2	2	no	—
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₄₇	D₄	1	2	yes	—
50	C₂²×C₈×C₄₇	D₄	1	2	yes	—
51	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×Dic₄₇	—	2	2	no	—
63	C₂²×Dic₄₇	—	2	2	no	—
64	C₂²×Dic₄₇	—	4	4	no	—
65	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	4	2	no	—
69	M₁₆×C₄₇	—	4	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	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
78	C₄×Dic₄₇	—	2	2	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₄×Dic₄₇	—	2	2	no	—
81	C₄×Dic₄₇	—	2	2	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	4	2	no	—
85	#10625329	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	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	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
94	C₂³×C₄₇	—	2	4	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	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#117	1	2	yes	—
104	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#117	1	2	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#123	1	2	yes	—
114	C₂×C₄×C₄₇	#123	1	2	yes	—
115	C₂×C₄×C₄₇	#124	1	2	yes	—
116	C₂×C₄×C₄₇	#126	1	2	yes	—
117	C₂×C₄×C₄₇	#126	1	2	yes	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₈×C₄₇	—	2	4	no	—
123	C₈×C₄₇	—	4	4	no	—
124	C₄₇⋊C₈	—	2	4	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	#10823	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₄₇	#380	1	1	yes	characteristic
133	C₂²×C₄₇	#382	1	1	yes	characteristic
134	C₂²×C₄₇	#387	1	2	yes	—
135	C₂²×C₄₇	#387	1	2	yes	—
136	C₂²×C₄₇	#430	1	4	yes	—
137	C₄×C₄₇	#376	1	1	yes	characteristic
138	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₄₇	#381	1	1	yes	characteristic
140	C₄×C₄₇	#383	1	1	yes	characteristic
141	C₄×C₄₇	#388	1	2	yes	—
142	C₄×C₄₇	#388	1	2	yes	—
143	Dic₄₇	—	4	4	no	—
144	C₂²×#97	—	47	1	no	—
145	C₂×#341	—	47	2	no	—
146	C₂×#341	—	47	1	no	—
147	C₂×#341	D₄₇	1	1	yes	characteristic
148	#1728	—	47	2	no	—
149	C₂×C₄₇	—	2	2	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	#1434	1	1	yes	characteristic
152	C₂×C₄₇	#1778	1	2	yes	—
153	C₂×C₄₇	#1811	1	2	yes	—
154	C₂×C₄₇	#1838	1	2	yes	—
155	C₂²×M₁₆	—	47	2	no	—
156	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
157	C₂³×C₈	—	47	1	no	—
158	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂²×C₄²	—	47	1	no	—
160	C₂×#97	—	47	4	no	—
161	C₂×#97	—	94	2	no	—
162	C₂×#97	—	94	2	no	—
163	#341	—	2	2	no	—
164	#341	—	47	8	no	—
165	#341	—	94	2	no	—
166	#336	—	94	4	no	—
167	C₄₇	#10823	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	—	47	2	no	—
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	47	4	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	C₄×D₄₇	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	47	4	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	94	1	no	—
188	C₂²×C₈	—	94	2	no	—
189	C₂²×C₈	—	94	1	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₂²×C₈	C₄×D₄₇	1	2	yes	—
192	C₂²×C₈	D₁₈₈	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
194	C₂×C₄²	—	47	2	no	—
195	C₂×C₄²	—	47	2	no	—
196	C₂×C₄²	—	94	1	no	—
197	C₂×C₄²	—	94	1	no	—
198	C₂×C₄²	—	94	1	no	—
199	C₂×C₄²	—	94	1	no	—
200	#97	—	94	8	no	—
201	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	—	47	4	no	—
203	C₂²×C₄	—	94	2	no	—
204	C₂²×C₄	—	94	2	no	—
205	C₂²×C₄	—	188	4	no	—
206	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
208	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
209	C₂²×C₄	#10625350	1	2	yes	—
210	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	#10625352	1	1	yes	characteristic
212	C₂²×C₄	#10625352	1	2	yes	—
213	C₂²×C₄	#10625352	1	2	yes	—
214	C₂²×C₄	#10625354	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	94	4	no	—
221	M₁₆	—	94	4	no	—
222	M₁₆	—	188	2	no	—
223	M₁₆	—	188	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	4	no	—
237	C₂×C₈	—	188	2	no	—
238	C₂×C₈	—	188	2	no	—
239	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
240	C₄²	—	94	2	no	—
241	C₄²	—	94	2	no	—
242	C₄²	—	94	2	no	—
243	C₄²	—	94	2	no	—
244	C₂³	—	2	4	no	—
245	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
246	C₂³	#13093989	1	2	yes	—
247	C₂³	#13094000	1	1	yes	characteristic
248	C₂³	#13094004	1	1	yes	characteristic
249	C₂³	#13094014	1	2	yes	—
250	C₂×C₄	—	94	4	no	—
251	C₂×C₄	—	94	8	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
255	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#13093980	1	1	yes	characteristic
257	C₂×C₄	#13093980	1	2	yes	—
258	C₂×C₄	#13093980	1	1	yes	characteristic
259	C₂×C₄	#13093981	1	1	yes	characteristic
260	C₂×C₄	#13093982	1	1	yes	characteristic
261	C₂×C₄	#13093982	1	2	yes	—
262	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
263	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
264	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
265	C₂×C₄	#13093995	1	1	yes	characteristic
266	C₂×C₄	#13093999	1	1	yes	characteristic
267	C₂×C₄	#13094005	1	2	yes	—
268	C₂×C₄	#13094008	1	2	yes	—
269	C₂×C₄	#13094010	1	2	yes	—
270	C₂×C₄	#13094017	1	2	yes	—
271	C₂×C₄	#13094019	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	4	4	no	—
274	C₈	—	94	4	no	—
275	C₈	—	188	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	#15688974	1	1	yes	characteristic
281	C₂²	#15688980	1	1	yes	characteristic
282	C₂²	#15688987	1	1	yes	characteristic
283	C₂²	#15689004	1	2	yes	—
284	C₂²	#15689008	1	2	yes	—
285	C₂²	#15689170	1	4	yes	—
286	C₄	—	188	4	no	—
287	C₄	#15688968	1	1	yes	characteristic
288	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
289	C₄	#15688984	1	1	yes	characteristic
290	C₄	#15688990	1	1	yes	characteristic
291	C₄	#15689011	1	2	yes	—
292	C₄	#15689015	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	#20241612	1	1	yes	characteristic
296	C₂	#20243293	1	2	yes	—
297	C₂	#20243427	1	2	yes	—
298	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
13. #15688887	C₄	semidirect
167. 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