Subgroups — #25250171

Order 11008.

552 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	#19136504	C₂	1	2	yes	maximal, complemented
2	#19136421	C₂	1	4	yes	maximal, complemented
3	D₄×C₄₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#19136398	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136398	C₂	1	2	yes	maximal, complemented
6	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582675	C₂	1	2	yes	maximal, complemented
8	C₈×#41×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063265	C₂²	1	2	yes	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	4	yes	—
11	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
12	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
13	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
14	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
15	#14582675	C₂²	1	2	yes	—
16	#14582675	C₂²	1	1	yes	characteristic
17	#14582675	C₂²	1	4	yes	—
18	#14582675	C₂²	1	1	yes	characteristic
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	2	yes	—
24	C₈×D₄×C₄₃	C₄	1	1	yes	characteristic
25	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#431	C₄	1	1	yes	characteristic
28	C₄₃×#431	C₄	1	2	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	#13063265	—	2	4	no	—
32	C₂×C₄₃⋊C₁₆	—	2	8	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	2	yes	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
48	#117×C₄₃	C₈	1	2	yes	—
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	1	yes	characteristic
54	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
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	2	yes	—
59	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₄₃	C₂³	1	2	yes	—
65	#41×C₄₃	—	2	4	no	—
66	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₄₃	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
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	2	yes	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₃	—	2	4	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
86	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
87	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
89	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
92	C₄²×C₄₃	C₂×C₈	1	2	yes	—
93	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×C₈	1	2	yes	—
95	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₄₃⋊C₁₆	—	2	4	no	—
100	C₄₃⋊C₁₆	—	2	2	no	—
101	C₄₃⋊C₁₆	#41	1	4	yes	complemented
102	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
104	Q₈×C₄₃	—	2	2	no	—
105	D₄×C₄₃	—	2	4	no	—
106	D₄×C₄₃	M₃₂	1	4	yes	complemented
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×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	—	4	2	no	—
114	C₂×C₄×C₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₄₃	—	2	2	no	—
125	C₈×C₄₃	—	2	1	no	—
126	C₈×C₄₃	—	2	1	no	—
127	C₈×C₄₃	—	2	2	no	—
128	C₈×C₄₃	#131	1	2	yes	—
129	C₈×C₄₃	C₂×#41	1	2	yes	—
130	#33484	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	1	no	—
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	C₂×M₃₂	1	4	yes	—
137	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	—	2	1	no	—
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	—	2	2	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	C₂×M₃₂	1	2	yes	—
146	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
147	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
148	C₈×#41	D₄₃	1	1	yes	characteristic
149	#2183	—	43	2	no	—
150	#2168	—	43	4	no	—
151	D₄×C₁₆	—	43	2	no	—
152	#2163	—	43	1	no	—
153	#2163	—	43	2	no	—
154	C₄×M₃₂	—	43	1	no	—
155	C₂×#346	—	43	2	no	—
156	C₂×C₄₃	—	2	4	no	—
157	C₂×C₄₃	—	4	1	no	—
158	C₂×C₄₃	C₈×#41	1	1	yes	characteristic
159	C₂×C₄₃	C₂²×M₃₂	1	1	yes	characteristic
160	C₂×C₄₃	#3407	1	1	yes	characteristic
161	C₄×#41	Dic₄₃	1	1	yes	characteristic
162	C₂×M₃₂	—	43	2	no	—
163	C₂²×C₁₆	—	43	4	no	—
164	C₈×Q₈	Dic₄₃	1	1	yes	characteristic
165	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
166	C₈×D₄	Dic₄₃	1	2	yes	—
167	C₈×D₄	Dic₄₃	1	1	yes	characteristic
168	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
169	C₈×D₄	C₂×D₄₃	1	2	yes	—
170	#431	Dic₄₃	1	1	yes	characteristic
171	#431	Dic₄₃	1	2	yes	—
172	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
173	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
174	C₄⋊C₁₆	—	43	2	no	—
175	C₄⋊C₁₆	—	43	4	no	—
176	C₂²⋊C₁₆	—	43	4	no	—
177	C₂²⋊C₁₆	—	43	2	no	—
178	#346	—	43	1	no	—
179	#346	—	43	2	no	—
180	#346	—	43	4	no	—
181	#346	—	43	1	no	—
182	C₄×C₁₆	—	43	2	no	—
183	C₄₃	#33484	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	C₄₃⋊C₈	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
186	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
188	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	C₄×Q₈	C₄₃⋊C₈	1	1	yes	characteristic
190	C₄×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
191	C₄×D₄	C₄₃⋊C₈	1	2	yes	—
192	C₄×D₄	C₄₃⋊C₈	1	1	yes	characteristic
193	C₄×D₄	C₂×Dic₄₃	1	2	yes	—
194	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
195	#117	C₄₃⋊C₈	1	1	yes	characteristic
196	#117	C₄₃⋊C₈	1	2	yes	—
197	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
198	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
199	M₃₂	—	86	4	no	—
200	C₂×C₁₆	—	43	2	no	—
201	C₂×C₁₆	—	43	2	no	—
202	C₂×C₁₆	—	43	4	no	—
203	C₂×C₁₆	—	86	8	no	—
204	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
205	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
206	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
208	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
209	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
210	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
211	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
212	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
213	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
214	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₄₃	1	2	yes	—
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
220	C₂×D₄	C₂×C₄₃⋊C₈	1	2	yes	—
221	C₂×D₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
223	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
234	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
235	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
239	C₄⋊C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
240	C₄⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
241	C₄⋊C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
243	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
244	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
245	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
246	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
247	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
248	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
249	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
250	C₄²	C₂²×Dic₄₃	1	2	yes	—
251	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
252	C₁₆	—	43	4	no	—
253	C₁₆	—	86	4	no	—
254	C₁₆	—	86	2	no	—
255	C₂³	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
256	C₂³	C₂²×C₄₃⋊C₈	1	2	yes	—
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	#13063265	1	4	yes	complemented
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
271	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
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	C₂³×Dic₄₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	#13063332	1	2	yes	—
282	C₈	#41×D₄₃	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	C₂×#13063265	1	4	yes	—
289	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	C₂×#13063265	1	2	yes	—
298	C₄	#41×Dic₄₃	1	1	yes	characteristic
299	C₄	C₂×#13063332	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	4	1	no	—
302	C₂	#41×C₄₃⋊C₈	1	1	yes	characteristic
303	C₂	C₂²×#13063265	1	1	yes	characteristic
304	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	#33484
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. #19136421	C₂	semidirect
3. D₄×C₄₃⋊C₁₆	C₂	semidirect
4. #19136398	C₂	semidirect
5. #19136398	C₂	semidirect
6. C₄×#13063265	C₂	semidirect
7. C₂×#14582675	C₂	semidirect
101. C₄₃⋊C₁₆	#41	semidirect
106. D₄×C₄₃	M₃₂	semidirect
183. C₄₃	#33484	semidirect
259. D₄	#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
#19136421	2
D₄×C₄₃⋊C₁₆	2
#19136398	2
C₄×#13063265	2
C₂×#14582675	2
C₈×#41×C₄₃	2
#33484	43