Subgroups — #25250274

Order 11008.

526 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136504	C₂	1	2	yes	maximal, complemented
2	#19136427	C₂	1	2	yes	maximal, complemented
3	#19136421	C₂	1	4	yes	maximal, complemented
4	#19136398	C₂	1	2	yes	maximal, complemented
5	#19136394	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136351	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#41×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063265	C₂²	1	4	yes	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
11	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
12	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
13	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
15	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
17	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
18	#14582675	C₂²	1	4	yes	—
19	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
20	C₄×#41×C₄₃	C₄	1	1	yes	characteristic
21	C₈×Q₈×C₄₃	C₄	1	1	yes	characteristic
22	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₄₃	C₄	1	1	yes	characteristic
24	C₈×D₄×C₄₃	C₄	1	2	yes	—
25	C₈×D₄×C₄₃	C₂²	1	2	yes	—
26	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#431	C₄	1	2	yes	—
28	C₄₃×#431	C₄	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
31	#13063265	—	4	4	no	—
32	C₂×C₄₃⋊C₁₆	—	4	2	no	—
33	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
34	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
35	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
36	C₂×#41×C₄₃	C₈	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
41	C₄×Q₈×C₄₃	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₄₃	C₈	1	2	yes	—
44	C₄×D₄×C₄₃	C₈	1	1	yes	characteristic
45	C₄×D₄×C₄₃	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
47	#117×C₄₃	C₈	1	2	yes	—
48	#117×C₄₃	C₈	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
54	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₃	C₂³	1	2	yes	—
64	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	#41×C₄₃	—	2	4	no	—
66	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
68	C₂×D₄×C₄₃	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	—	2	1	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	1	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₂×C₈×C₄₃	—	4	2	no	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
89	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
90	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
91	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
92	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
94	C₄²×C₄₃	C₂×C₈	1	2	yes	—
95	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₄₃	C₂×C₈	1	2	yes	—
97	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
98	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₄₃⋊C₁₆	—	2	4	no	—
102	C₄₃⋊C₁₆	—	2	2	no	—
103	C₄₃⋊C₁₆	—	2	2	no	—
104	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
105	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
106	Q₈×C₄₃	—	2	2	no	—
107	D₄×C₄₃	—	2	4	no	—
108	D₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	M₃₂	1	4	yes	complemented
117	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₄₃	—	2	1	no	—
127	C₈×C₄₃	—	2	2	no	—
128	C₈×C₄₃	—	2	1	no	—
129	C₈×C₄₃	—	2	1	no	—
130	C₈×C₄₃	—	2	2	no	—
131	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
133	#33503	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	—	2	2	no	—
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	C₂×M₃₂	1	2	yes	—
140	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	2	1	no	—
146	C₄×C₄₃	—	2	2	no	—
147	C₄×C₄₃	C₂×M₃₂	1	2	yes	—
148	C₄×C₄₃	C₂×M₃₂	1	2	yes	—
149	C₄×C₄₃	#520	1	1	yes	characteristic
150	C₄×C₄₃	#568	1	1	yes	characteristic
151	C₈×#41	D₄₃	1	1	yes	characteristic
152	#2183	—	43	2	no	—
153	#2169	—	43	2	no	—
154	#2168	—	43	4	no	—
155	#2163	—	43	2	no	—
156	#2162	—	43	1	no	—
157	#2150	—	43	2	no	—
158	C₂×C₄⋊C₁₆	—	43	1	no	—
159	C₂×C₄₃	—	2	2	no	—
160	C₂×C₄₃	—	4	2	no	—
161	C₂×C₄₃	#2988	1	1	yes	characteristic
162	C₂×C₄₃	C₂²×M₃₂	1	1	yes	characteristic
163	C₂×C₄₃	#3407	1	1	yes	characteristic
164	C₄×#41	Dic₄₃	1	1	yes	characteristic
165	C₂×M₃₂	—	43	4	no	—
166	C₂²×C₁₆	—	43	2	no	—
167	C₈×Q₈	Dic₄₃	1	1	yes	characteristic
168	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
169	C₈×D₄	Dic₄₃	1	1	yes	characteristic
170	C₈×D₄	Dic₄₃	1	2	yes	—
171	C₈×D₄	C₂×D₄₃	1	2	yes	—
172	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
173	#431	Dic₄₃	1	2	yes	—
174	#431	Dic₄₃	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
177	C₄⋊C₁₆	—	43	2	no	—
178	C₄⋊C₁₆	—	43	2	no	—
179	C₄⋊C₁₆	—	43	1	no	—
180	C₄⋊C₁₆	—	43	4	no	—
181	C₄⋊C₁₆	—	43	1	no	—
182	C₂²⋊C₁₆	—	43	2	no	—
183	C₂²⋊C₁₆	—	43	4	no	—
184	#346	—	43	4	no	—
185	C₄×C₁₆	—	43	2	no	—
186	C₄₃	#33503	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	C₄₃⋊C₈	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
190	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
192	C₄×Q₈	C₄₃⋊C₈	1	1	yes	characteristic
193	C₄×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
194	C₄×D₄	C₄₃⋊C₈	1	2	yes	—
195	C₄×D₄	C₄₃⋊C₈	1	1	yes	characteristic
196	C₄×D₄	C₂×Dic₄₃	1	2	yes	—
197	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
198	#117	C₄₃⋊C₈	1	2	yes	—
199	#117	C₄₃⋊C₈	1	1	yes	characteristic
200	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
201	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
202	M₃₂	—	172	4	no	—
203	C₂×C₁₆	—	43	2	no	—
204	C₂×C₁₆	—	43	2	no	—
205	C₂×C₁₆	—	43	4	no	—
206	C₂×C₁₆	—	172	2	no	—
207	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
208	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
209	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
210	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
211	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
212	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
213	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
214	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
215	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
216	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
217	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
218	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₄₃	1	2	yes	—
220	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
221	#41	—	2	4	no	—
222	C₂×Q₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
223	C₂×D₄	C₂×C₄₃⋊C₈	1	2	yes	—
224	C₂×D₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
226	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
227	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
230	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
240	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
242	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
243	C₄⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
244	C₄⋊C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
245	C₄⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
246	C₄⋊C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
247	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
248	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
249	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
250	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
251	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
252	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
253	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
254	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
255	C₄²	C₂²×Dic₄₃	1	2	yes	—
256	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
257	C₁₆	—	86	4	no	—
258	C₁₆	—	86	2	no	—
259	C₁₆	—	86	2	no	—
260	C₂³	C₂²×C₄₃⋊C₈	1	2	yes	—
261	C₂³	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	D₄	—	2	4	no	—
264	D₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	#13063265	1	4	yes	complemented
273	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
275	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
277	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
279	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
281	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	2	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	2	no	—
287	C₈	#13063371	1	1	yes	characteristic
288	C₈	#13063372	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	C₂×#13063265	1	2	yes	—
295	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	C₂×#13063265	1	2	yes	—
303	C₄	C₂×#13063265	1	2	yes	—
304	C₄	#14583382	1	1	yes	characteristic
305	C₄	#14583666	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#19143706	1	1	yes	characteristic
309	C₂	C₂²×#13063265	1	1	yes	characteristic
310	C₂	#19147698	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₈×#41×C₄₃	C₂
prime radical	C₂×#41×C₄₃	C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₈×#41	D₄₃
hypocenter	C₄₃	#33503
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #19136504	C₂	semidirect
2. #19136427	C₂	semidirect
3. #19136421	C₂	semidirect
4. #19136398	C₂	semidirect
5. #19136394	C₂	semidirect
6. #19136351	C₂	semidirect
7. C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
116. C₂×C₄×C₄₃	M₃₂	semidirect
186. C₄₃	#33503	semidirect
272. C₂×C₄	#13063265	semidirect

Maximal subgroups

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

subgroup	index
#19136504	2
#19136427	2
#19136421	2
#19136398	2
#19136394	2
#19136351	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₈×#41×C₄₃	2
#33503	43