Subgroups — #26121514

Order 12032.

1374 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245771	C₂	1	2	yes	maximal
2	C₂×#15688880	C₂	1	4	yes	maximal, complemented
3	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093983	C₂²	1	4	yes	—
5	#15688986	—	2	2	no	—
6	#15688983	—	2	4	no	—
7	#15688979	—	2	2	no	—
8	C₄×C₂²⋊Dic₄₇	C₄	1	8	yes	complemented
9	#15688880	—	2	8	no	—
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₄₇	—	2	4	no	—
12	C₂²×M₁₆×C₄₇	—	2	8	no	—
13	C₂²×M₁₆×C₄₇	C₄	1	2	yes	complemented
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₄₇	C₄	1	2	yes	complemented
16	#13093983	—	2	8	no	—
17	#13093983	—	2	8	no	—
18	C₂×C₄⋊Dic₄₇	—	2	4	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
20	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
21	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
22	C₄₇⋊#95	—	2	8	no	—
23	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	—	2	4	no	—
25	C₂³×C₄×C₄₇	—	2	4	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	D₄	1	4	yes	—
31	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₄₇	—	2	32	no	—
35	C₂×M₁₆×C₄₇	—	2	4	no	—
36	C₂×M₁₆×C₄₇	—	2	4	no	—
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	—	2	8	no	—
39	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
40	C₂²×C₈×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	—	2	8	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
43	C₂⁴×C₄₇	—	2	4	no	—
44	C₂⁴×C₄₇	—	4	4	no	—
45	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
46	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	16	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₂²×Dic₄₇	—	2	8	no	—
75	M₁₆×C₄₇	—	2	16	no	—
76	M₁₆×C₄₇	—	2	16	no	—
77	C₂×C₈×C₄₇	—	2	16	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	complemented
81	C₂²⋊Dic₄₇	—	4	16	no	—
82	C₂²⋊Dic₄₇	—	4	8	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	4	4	no	—
85	C₄×Dic₄₇	—	2	16	no	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	2	8	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	8	no	—
91	C₂³×C₄₇	—	4	4	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	—	4	16	no	—
95	C₂³×C₄₇	#95	1	2	yes	—
96	C₂³×C₄₇	#95	1	2	yes	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₄₇	#121	1	4	yes	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	#95	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	#117	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
119	C₂×C₄×C₄₇	#120	1	4	yes	—
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₄₇	#123	1	4	yes	—
122	C₂×Dic₄₇	—	4	16	no	—
123	C₂×Dic₄₇	—	4	8	no	—
124	C₈×C₄₇	—	2	8	no	—
125	#10717	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	8	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	—	4	16	no	—
135	C₂²×C₄₇	—	4	4	no	—
136	C₂²×C₄₇	H(3,ℤ/4)	1	4	yes	complemented
137	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
138	C₂²×C₄₇	#380	1	2	yes	—
139	C₂²×C₄₇	#386	1	4	yes	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
143	C₄×C₄₇	#379	1	1	yes	characteristic
144	C₄×C₄₇	#381	1	1	yes	characteristic
145	C₄×C₄₇	#385	1	4	yes	—
146	Dic₄₇	—	8	8	no	—
147	C₂³×M₁₆	D₄₇	1	1	yes	characteristic
148	#2280	—	47	2	no	—
149	C₂×H(3,ℤ/4)	—	47	4	no	—
150	C₂×C₄₇	—	2	4	no	—
151	C₂×C₄₇	—	2	4	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	—	4	2	no	—
154	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
155	C₂×C₄₇	C₂×H(3,ℤ/4)	1	2	yes	—
156	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
157	C₂²×M₁₆	—	2	4	no	—
158	C₂²×M₁₆	—	2	8	no	—
159	C₂²×M₁₆	Dic₄₇	1	2	yes	complemented
160	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
161	C₂³×C₈	Dic₄₇	1	2	yes	complemented
162	C₂×#117	—	47	4	no	—
163	#381	—	94	2	no	—
164	#380	—	94	4	no	—
165	#379	—	94	2	no	—
166	C₄×C₂²⋊C₄	—	47	8	no	—
167	H(3,ℤ/4)	—	94	8	no	—
168	C₄₇	#10717	1	1	yes	characteristic, complemented, Sylow
169	C₂⁵	D₁₈₈	1	1	yes	characteristic
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
173	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
174	C₂³×C₄	C₄×D₄₇	1	4	yes	—
175	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂²⋊D₄₇	1	4	yes	—
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	32	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	2	8	no	—
185	C₂×M₁₆	C₂²⋊D₄₇	1	8	yes	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	8	no	—
188	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
189	#117	—	94	8	no	—
190	#117	—	94	8	no	—
191	C₂×C₄⋊C₄	—	94	4	no	—
192	C₂×C₂²⋊C₄	—	94	8	no	—
193	C₂×C₂²⋊C₄	—	94	4	no	—
194	C₂×C₄²	—	47	8	no	—
195	#95	—	94	8	no	—
196	C₂⁴	—	2	4	no	—
197	C₂⁴	—	4	4	no	—
198	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
199	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
200	C₂⁴	(C₂×D₄₇)⋊C₄	1	4	yes	—
201	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	16	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	94	8	no	—
213	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
214	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
215	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
216	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
217	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
220	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
221	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
224	C₂²×C₄	#10625350	1	4	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
228	M₁₆	—	2	16	no	—
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	16	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	complemented
234	C₄⋊C₄	—	188	4	no	—
235	C₄⋊C₄	—	188	4	no	—
236	C₂²⋊C₄	—	188	16	no	—
237	C₂²⋊C₄	—	188	8	no	—
238	C₄²	—	94	16	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	16	no	—
248	C₂³	C₄₇⋊#95	1	2	yes	—
249	C₂³	C₄₇⋊#95	1	2	yes	—
250	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
251	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
252	C₂³	#13093998	1	4	yes	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	188	16	no	—
265	C₂×C₄	—	188	8	no	—
266	C₂×C₄	C₄₇⋊#95	1	2	yes	—
267	C₂×C₄	C₄₇⋊#95	1	2	yes	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
270	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
272	C₂×C₄	#13093983	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
274	C₂×C₄	#13093993	1	4	yes	—
275	C₂×C₄	#13094003	1	4	yes	—
276	C₂×C₄	#13094007	1	4	yes	—
277	C₈	—	2	8	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	16	no	—
287	C₂²	—	4	4	no	—
288	C₂²	#15688880	1	4	yes	complemented
289	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
290	C₂²	#15688983	1	2	yes	—
291	C₂²	#15689002	1	4	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	376	8	no	—
295	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
296	C₄	#15688979	1	1	yes	characteristic
297	C₄	#15688986	1	1	yes	characteristic
298	C₄	#15688996	1	4	yes	—
299	C₂	—	2	4	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	4	no	—
302	C₂	—	4	2	no	—
303	C₂	#20241611	1	1	yes	characteristic
304	C₂	C₂×#15688880	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	#10717
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#15688880	C₂	semidirect
8. C₄×C₂²⋊Dic₄₇	C₄	semidirect
13. C₂²×M₁₆×C₄₇	C₄	semidirect
15. C₂³×C₈×C₄₇	C₄	semidirect
80. C₂×C₈×C₄₇	C₂²⋊C₄	semidirect
136. C₂²×C₄₇	H(3,ℤ/4)	semidirect
159. C₂²×M₁₆	Dic₄₇	semidirect
161. C₂³×C₈	Dic₄₇	semidirect
168. C₄₇	#10717	semidirect
233. C₂×C₈	C₂²⋊Dic₄₇	semidirect
288. C₂²	#15688880	semidirect

Maximal subgroups

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

subgroup	index
#20245771	2
C₂×#15688880	2
C₂³×M₁₆×C₄₇	2
#10717	47