Subgroups — #25102015

Order 11008.

858 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135203	C₂	1	4	yes	maximal
2	#19135154	C₂	1	4	yes	maximal, complemented
3	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
5	#19134341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
13	#14582654	C₂²	1	4	yes	—
14	#14582654	C₂²	1	4	yes	—
15	#14582654	C₂²	1	4	yes	—
16	C₂×#117×C₄₃	C₄	1	1	yes	characteristic
17	C₂×#117×C₄₃	C₄	1	1	yes	characteristic
18	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₃	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
23	C₄₃×#376	C₄	1	1	yes	characteristic
24	C₄₃×#376	C₄	1	1	yes	characteristic
25	C₄³×C₄₃	C₂²	1	1	yes	characteristic
26	C₄³×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
30	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
31	C₄×C₄₃⋊C₈	—	2	4	no	—
32	C₄×C₄₃⋊C₈	—	2	4	no	—
33	C₄×C₄₃⋊C₈	—	2	8	no	—
34	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	#117×C₄₃	—	2	4	no	—
38	#117×C₄₃	C₈	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₃	—	2	4	no	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	#95×C₄₃	C₂×C₄	1	2	yes	—
57	#95×C₄₃	C₂×C₄	1	2	yes	—
58	#95×C₄₃	C₂×C₄	1	2	yes	—
59	#95×C₄₃	C₂×C₄	1	2	yes	—
60	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	2	4	no	—
67	C₂²×C₄×C₄₃	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	4	no	—
79	C₄⋊C₄×C₄₃	C₂×C₈	1	8	yes	—
80	C₂²⋊C₄×C₄₃	—	2	4	no	—
81	C₂²⋊C₄×C₄₃	C₂×C₈	1	8	yes	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	—	2	2	no	—
84	C₄²×C₄₃	—	2	2	no	—
85	C₄²×C₄₃	—	2	4	no	—
86	C₄²×C₄₃	—	2	4	no	—
87	C₄²×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	—	2	4	no	—
89	C₄²×C₄₃	—	2	4	no	—
90	C₄²×C₄₃	C₂×C₈	1	4	yes	—
91	C₄²×C₄₃	C₂×C₈	1	4	yes	—
92	C₄²×C₄₃	M₁₆	1	4	yes	—
93	C₄²×C₄₃	#41	1	4	yes	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	—	2	4	no	—
97	C₂×C₄₃⋊C₈	—	2	4	no	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	#41	1	8	yes	—
100	C₂³×C₄₃	—	2	1	no	—
101	C₂³×C₄₃	—	2	2	no	—
102	C₂³×C₄₃	—	2	1	no	—
103	C₂³×C₄₃	—	2	2	no	—
104	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
105	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	8	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	4	4	no	—
123	C₂×C₄×C₄₃	—	4	4	no	—
124	C₂×C₄×C₄₃	#117	1	2	yes	—
125	C₂×C₄×C₄₃	#117	1	2	yes	—
126	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
127	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
128	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₄₃	#131	1	4	yes	—
133	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
134	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
135	C₄₃⋊C₈	—	2	8	no	—
136	C₄₃⋊C₈	—	4	4	no	—
137	#21430	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	2	4	no	—
140	C₂²×C₄₃	—	4	1	no	—
141	C₂²×C₄₃	—	4	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
144	C₂²×C₄₃	C₄×#41	1	2	yes	—
145	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
146	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
147	C₂²×C₄₃	C₂×#131	1	2	yes	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	2	4	no	—
150	C₄×C₄₃	—	2	4	no	—
151	C₄×C₄₃	—	2	4	no	—
152	C₄×C₄₃	—	4	1	no	—
153	C₄×C₄₃	—	4	2	no	—
154	C₄×C₄₃	—	4	1	no	—
155	C₄×C₄₃	#431	1	4	yes	—
156	C₄×C₄₃	#433	1	4	yes	—
157	C₄×C₄₃	#529	1	4	yes	—
158	C₄×#117	D₄₃	1	1	yes	characteristic
159	#1923	—	43	4	no	—
160	#1916	—	43	4	no	—
161	C₄×C₄⋊C₈	—	43	2	no	—
162	C₄×C₂²⋊C₈	—	43	2	no	—
163	#1744	—	43	1	no	—
164	C₄²×C₈	—	43	1	no	—
165	C₂×C₄₃	—	4	2	no	—
166	C₂×C₄₃	#2298	1	1	yes	characteristic
167	C₂×C₄₃	C₂×#431	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#433	1	1	yes	characteristic
169	C₂×C₄₃	C₈×#41	1	2	yes	—
170	C₂×C₄₃	#2965	1	2	yes	—
171	C₂×#117	Dic₄₃	1	1	yes	characteristic
172	C₂×#117	Dic₄₃	1	1	yes	characteristic
173	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
174	C₂×C₄⋊C₈	—	43	4	no	—
175	C₂×C₂²⋊C₈	—	43	4	no	—
176	C₂×C₄×C₈	—	43	4	no	—
177	C₂×C₄×C₈	—	43	2	no	—
178	C₂×C₄×C₈	—	43	2	no	—
179	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
180	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
181	C₄×C₂²⋊C₄	Dic₄₃	1	2	yes	—
182	C₄×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
183	#376	Dic₄₃	1	1	yes	characteristic
184	#376	Dic₄₃	1	1	yes	characteristic
185	C₄³	C₂×D₄₃	1	1	yes	characteristic
186	C₄³	C₂×D₄₃	1	1	yes	characteristic
187	#336	—	43	4	no	—
188	#336	—	43	4	no	—
189	#336	—	43	4	no	—
190	C₄₃	#21430	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
192	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
193	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
194	C₂²×C₈	—	43	4	no	—
195	C₂²×C₈	—	43	4	no	—
196	#117	—	2	4	no	—
197	#117	C₄₃⋊C₈	1	8	yes	complemented
198	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
201	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
205	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
206	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
207	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
208	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
213	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
214	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
215	C₄⋊C₈	—	86	8	no	—
216	C₂²⋊C₈	—	86	8	no	—
217	C₄×C₈	—	86	8	no	—
218	C₄×C₈	—	86	4	no	—
219	C₄×C₈	—	86	4	no	—
220	#95	C₂×Dic₄₃	1	2	yes	—
221	#95	C₂×Dic₄₃	1	2	yes	—
222	#95	C₂×Dic₄₃	1	2	yes	—
223	#95	C₂×Dic₄₃	1	2	yes	—
224	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×C₄₃⋊C₈	1	4	yes	—
232	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
235	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
239	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
242	C₂×C₈	—	43	8	no	—
243	C₂×C₈	—	86	4	no	—
244	C₂×C₈	—	86	4	no	—
245	C₂×C₈	—	86	4	no	—
246	C₂×C₈	—	86	4	no	—
247	C₂×C₈	—	86	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	C₂×C₄₃⋊C₈	1	8	yes	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	8	yes	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	C₂×C₄₃⋊C₈	1	4	yes	—
261	C₄²	C₂×C₄₃⋊C₈	1	4	yes	—
262	C₄²	#10621877	1	4	yes	—
263	C₄²	#10621898	1	4	yes	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂²×C₄₃⋊C₈	1	2	yes	—
269	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	#13063284	1	2	yes	—
289	C₂×C₄	#13063284	1	2	yes	—
290	C₂×C₄	C₂²×C₄₃⋊C₈	1	4	yes	—
291	C₂×C₄	C₂²×C₄₃⋊C₈	1	4	yes	—
292	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
293	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
294	C₂×C₄	C₂×#10621877	1	4	yes	—
295	C₂×C₄	C₂×#10621877	1	2	yes	—
296	C₂×C₄	#13063332	1	4	yes	—
297	C₂×C₄	#41×D₄₃	1	4	yes	—
298	C₂×C₄	C₂×#10621898	1	2	yes	—
299	C₈	—	86	8	no	—
300	C₈	—	172	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂×#13063284	1	1	yes	characteristic
307	C₂²	#41×Dic₄₃	1	2	yes	—
308	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
309	C₂²	C₂²×#10621877	1	1	yes	characteristic
310	C₂²	C₂×#13063332	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	#14582952	1	4	yes	—
319	C₄	#14582960	1	4	yes	—
320	C₄	#14583427	1	4	yes	—
321	C₂	—	4	2	no	—
322	C₂	#19137093	1	1	yes	characteristic
323	C₂	C₂×#14582952	1	1	yes	characteristic
324	C₂	C₂×#14582960	1	1	yes	characteristic
325	C₂	#41×C₄₃⋊C₈	1	2	yes	—
326	C₂	#19143562	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#117	D₄₃
hypocenter	C₄₃	#21430
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #19135154	C₂	semidirect
3. C₄×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
5. #19134341	C₂	semidirect
6. C₄²×C₄₃⋊C₈	C₂	semidirect
38. #117×C₄₃	C₈	semidirect
190. C₄₃	#21430	semidirect
197. #117	C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#19135203	2
#19135154	2
C₄×C₄⋊(C₄₃⋊C₈)	2
C₄×C₂²⋊(C₄₃⋊C₈)	2
#19134341	2
C₄²×C₄₃⋊C₈	2
C₄×#117×C₄₃	2
#21430	43