Subgroups — #24986652

Order 11008.

746 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×#13063236	C₂	1	1	yes	maximal, characteristic
2	#19136993	C₂	1	1	yes	maximal, characteristic
3	#19132477	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₃⋊C₈	—	2	1	no	—
6	C₂×#13063236	—	2	1	no	—
7	C₂×#13063236	—	2	2	no	—
8	C₂×#13063236	C₄	1	4	yes	complemented
9	C₂×#13063236	C₂²	1	2	yes	—
10	#14582767	—	2	1	no	—
11	#14582765	C₂²	1	2	yes	—
12	#14582762	—	2	1	no	—
13	#14582759	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₄₃	C₄	1	2	yes	—
15	C₄×C₂²⋊Dic₄₃	—	2	1	no	—
16	#14582745	—	2	1	no	—
17	C₂²×M₁₆×C₄₃	—	2	1	no	—
18	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₄₃	—	2	2	no	—
20	C₄×M₁₆×C₄₃	C₄	1	4	yes	complemented
21	C₂×#97×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	—	2	1	no	—
23	C₂²×C₄₃⋊C₈	—	2	8	no	—
24	C₂²×C₄₃⋊C₈	—	2	4	no	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄×Dic₄₃	—	2	2	no	—
29	#13063236	—	2	4	no	—
30	#13063236	—	2	2	no	—
31	#13063236	C₂×C₄	1	4	yes	—
32	C₄₃⋊#95	—	2	2	no	—
33	C₄₃⋊#95	—	2	2	no	—
34	C₄₃⋊#95	—	2	2	no	—
35	C₄₃⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₄₃	—	2	8	no	—
40	C₂×M₁₆×C₄₃	—	2	4	no	—
41	C₂²×C₈×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	2	yes	—
48	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	#97×C₄₃	—	2	2	no	—
51	#97×C₄₃	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₄₃	—	2	4	no	—
53	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	—	2	4	no	—
56	C₂²×C₄×C₄₃	—	4	2	no	—
57	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	2	16	no	—
71	C₂×C₈×C₄₃	—	2	8	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₄²×C₄₃	C₄²	1	2	yes	—
74	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²⋊Dic₄₃	—	4	4	no	—
84	C₄⋊Dic₄₃	—	4	4	no	—
85	C₄×Dic₄₃	—	4	4	no	—
86	C₂×C₄₃⋊C₈	—	2	8	no	—
87	C₂×C₄₃⋊C₈	—	2	16	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂³×C₄₃	—	2	2	no	—
90	C₂³×C₄₃	—	2	2	no	—
91	C₂³×C₄₃	#95	1	2	yes	—
92	C₂³×C₄₃	#104	1	4	yes	—
93	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	—	2	4	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	#95	1	2	yes	—
106	C₂×C₄×C₄₃	#95	1	2	yes	—
107	C₂×C₄×C₄₃	#95	1	2	yes	—
108	C₂×C₄×C₄₃	#104	1	4	yes	—
109	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×Dic₄₃	—	4	4	no	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₈×C₄₃	—	2	8	no	—
119	C₄₃⋊C₈	—	2	8	no	—
120	#8029	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	—	2	2	no	—
122	C₂²×C₄₃	—	2	2	no	—
123	C₂²×C₄₃	—	2	8	no	—
124	C₂²×C₄₃	#339	1	4	yes	—
125	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
126	C₂²×C₄₃	#413	1	2	yes	—
127	C₂²×C₄₃	C₂×#104	1	2	yes	—
128	C₂²×C₄₃	C₂×#104	1	2	yes	—
129	C₄×C₄₃	—	2	1	no	—
130	C₄×C₄₃	—	2	1	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	—	2	1	no	—
133	C₄×C₄₃	—	2	1	no	—
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	#339	1	4	yes	—
138	Dic₄₃	—	8	4	no	—
139	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
140	C₂²×#97	—	43	1	no	—
141	#2283	—	43	1	no	—
142	#1295	—	43	4	no	—
143	C₂×C₄₃	—	2	4	no	—
144	C₂×C₄₃	#1524	1	4	yes	—
145	C₂×C₄₃	C₂×#339	1	1	yes	characteristic
146	C₂×C₄₃	C₂×#339	1	1	yes	characteristic
147	C₂×C₄₃	#1742	1	1	yes	characteristic
148	C₂²×M₁₆	—	2	1	no	—
149	C₂³×C₈	—	86	1	no	—
150	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
151	C₄×M₁₆	—	2	2	no	—
152	C₄×M₁₆	Dic₄₃	1	4	yes	complemented
153	C₂×#97	—	43	2	no	—
154	C₂×#97	—	43	4	no	—
155	C₂×#97	—	86	1	no	—
156	C₂×#97	—	86	2	no	—
157	C₂×#97	C₂×D₄₃	1	2	yes	—
158	C₂×C₄×C₈	—	2	1	no	—
159	#383	—	86	1	no	—
160	#382	—	43	2	no	—
161	#381	—	86	1	no	—
162	#380	—	43	2	no	—
163	C₄×C₄⋊C₄	—	43	2	no	—
164	C₄×C₂²⋊C₄	—	86	1	no	—
165	#376	—	86	1	no	—
166	C₄₃	#8029	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
168	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	86	8	no	—
174	C₂²×C₈	—	86	4	no	—
175	C₂²×C₈	—	86	2	no	—
176	C₂×C₄⋊C₄	—	43	4	no	—
177	C₂×C₂²⋊C₄	—	86	2	no	—
178	C₂×C₄²	—	86	2	no	—
179	C₂×C₄²	Dic₈₆	1	2	yes	—
180	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₄₃	1	2	yes	—
182	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
186	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
187	#97	—	2	2	no	—
188	#97	—	43	4	no	—
189	#97	—	86	4	no	—
190	#97	—	86	2	no	—
191	#97	C₂×Dic₄₃	1	4	yes	—
192	C₄×C₈	—	2	4	no	—
193	#95	—	86	2	no	—
194	#95	—	86	2	no	—
195	#95	—	86	2	no	—
196	#95	—	86	2	no	—
197	C₂⁴	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	86	4	no	—
202	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
203	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
204	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
206	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
209	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	86	8	no	—
218	C₂×C₈	—	86	16	no	—
219	C₂×C₈	—	86	4	no	—
220	C₄⋊C₄	—	172	4	no	—
221	C₂²⋊C₄	—	172	4	no	—
222	C₄²	—	172	4	no	—
223	C₄²	C₄×Dic₄₃	1	2	yes	—
224	C₄²	Dic₄₃⋊C₄	1	2	yes	—
225	C₄²	Dic₄₃⋊C₄	1	2	yes	—
226	C₄²	C₄⋊Dic₄₃	1	2	yes	—
227	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
228	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
229	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
230	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
231	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₄₃⋊#95	1	2	yes	—
236	C₂³	#13063253	1	4	yes	—
237	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	172	4	no	—
250	C₂×C₄	—	172	4	no	—
251	C₂×C₄	—	172	4	no	—
252	C₂×C₄	C₄₃⋊#95	1	2	yes	—
253	C₂×C₄	C₄₃⋊#95	1	2	yes	—
254	C₂×C₄	C₄₃⋊#95	1	2	yes	—
255	C₂×C₄	#13063251	1	4	yes	—
256	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
257	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
262	C₈	—	2	8	no	—
263	C₈	—	86	8	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	#14582659	1	4	yes	—
268	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
269	C₂²	#14582881	1	2	yes	—
270	C₂²	C₂×#13063251	1	2	yes	—
271	C₂²	C₂×#13063253	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	344	4	no	—
281	C₄	#14582660	1	4	yes	—
282	C₂	—	2	4	no	—
283	C₂	#19133184	1	4	yes	—
284	C₂	C₂×#14582659	1	1	yes	characteristic
285	C₂	C₂×#14582660	1	1	yes	characteristic
286	C₂	#19134333	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	#8029
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #19132477	C₂	semidirect
8. C₂×#13063236	C₄	semidirect
20. C₄×M₁₆×C₄₃	C₄	semidirect
152. C₄×M₁₆	Dic₄₃	semidirect
166. C₄₃	#8029	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063236	2
#19136993	2
#19132477	2
C₂×C₄×M₁₆×C₄₃	2
#8029	43