Subgroups — #24986704

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582956	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19132478	C₂	1	4	yes	maximal, complemented
4	C₄₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₃⋊C₈	—	2	1	no	—
6	#14582956	—	2	2	no	—
7	#14582956	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂×#13063236	—	2	1	no	—
15	C₂×#117×C₄₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#384	C₂²	1	2	yes	—
18	C₄₃×#382	C₄	1	2	yes	—
19	C₄₃×#380	—	2	1	no	—
20	C₄₃×#380	—	2	1	no	—
21	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
23	C₂²×C₄₃⋊C₈	—	2	4	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	4	4	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
28	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
31	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
32	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
33	#13063236	—	2	4	no	—
34	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
37	#117×C₄₃	—	4	4	no	—
38	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	—	2	2	no	—
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	2	yes	—
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	#95×C₄₃	—	2	2	no	—
52	#95×C₄₃	—	2	2	no	—
53	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₄₃	—	4	4	no	—
73	C₄⋊C₄×C₄₃	—	4	4	no	—
74	C₄⋊C₄×C₄₃	—	4	2	no	—
75	C₄⋊C₄×C₄₃	—	4	2	no	—
76	C₂²⋊C₄×C₄₃	—	4	4	no	—
77	C₄²×C₄₃	—	4	4	no	—
78	C₄²×C₄₃	C₄²	1	2	yes	—
79	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	4	8	no	—
91	C₂×C₄₃⋊C₈	—	4	4	no	—
92	C₂×C₄₃⋊C₈	—	4	4	no	—
93	C₂³×C₄₃	—	2	2	no	—
94	C₂³×C₄₃	—	2	2	no	—
95	C₂³×C₄₃	#95	1	2	yes	—
96	C₂³×C₄₃	#100	1	2	yes	—
97	C₂³×C₄₃	#101	1	2	yes	—
98	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	2	4	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	#95	1	2	yes	—
116	C₂×C₄×C₄₃	#95	1	2	yes	—
117	C₂×C₄×C₄₃	#95	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#135	1	4	yes	—
125	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₄₃	#137	1	2	yes	—
127	C₄₃⋊C₈	—	4	4	no	—
128	C₄₃⋊C₈	—	4	4	no	—
129	#8044	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	#343	1	4	yes	—
135	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
136	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
137	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
138	C₂²×C₄₃	#416	1	2	yes	—
139	C₂²×C₄₃	#417	1	2	yes	—
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	2	2	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	—	8	4	no	—
146	C₄×C₄₃	#416	1	2	yes	—
147	C₄×C₄₃	#418	1	1	yes	characteristic
148	C₄×C₄₃	#419	1	1	yes	characteristic
149	C₄×C₄₃	#427	1	2	yes	—
150	C₄×C₄₃	#428	1	2	yes	—
151	C₂×#432	—	43	1	no	—
152	C₂²×C₄⋊C₈	—	43	1	no	—
153	#2285	D₄₃	1	1	yes	characteristic
154	#1296	—	43	4	no	—
155	C₂×C₄₃	—	2	4	no	—
156	C₂×C₄₃	#1551	1	2	yes	—
157	C₂×C₄₃	#1552	1	2	yes	—
158	C₂×C₄₃	#1735	1	1	yes	characteristic
159	C₂×C₄₃	#1736	1	1	yes	characteristic
160	C₂×C₄₃	C₂×#343	1	1	yes	characteristic
161	C₂³×C₈	—	86	1	no	—
162	C₂×#117	Dic₄₃	1	2	yes	—
163	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
164	#432	—	43	4	no	—
165	#432	—	86	2	no	—
166	C₂×C₄⋊C₈	—	43	2	no	—
167	C₂×C₄⋊C₈	—	43	2	no	—
168	C₂×C₄⋊C₈	—	43	4	no	—
169	C₂×C₄⋊C₈	—	86	2	no	—
170	C₂×C₄⋊C₈	—	86	1	no	—
171	C₂×C₂²⋊C₈	—	86	1	no	—
172	C₂×#97	—	86	1	no	—
173	#384	C₂×D₄₃	1	2	yes	—
174	#382	Dic₄₃	1	2	yes	—
175	#380	—	2	1	no	—
176	#380	—	2	1	no	—
177	C₄×C₄⋊C₄	—	2	1	no	—
178	C₄×C₄⋊C₄	—	2	1	no	—
179	C₄₃	#8044	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
182	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
183	C₂²×C₈	—	86	4	no	—
184	C₂²×C₈	—	86	2	no	—
185	C₂²×C₈	—	86	2	no	—
186	C₂²×C₈	—	172	4	no	—
187	#117	—	4	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	Dic₈₆	1	2	yes	—
194	C₂×C₄²	C₄×D₄₃	1	2	yes	—
195	C₂×C₄²	C₄×D₄₃	1	2	yes	—
196	C₂×C₄²	D₁₇₂	1	2	yes	—
197	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
201	C₄⋊C₈	—	43	4	no	—
202	C₄⋊C₈	—	43	4	no	—
203	C₄⋊C₈	—	86	4	no	—
204	C₄⋊C₈	—	86	2	no	—
205	C₄⋊C₈	—	86	2	no	—
206	C₂²⋊C₈	—	86	4	no	—
207	#97	—	86	4	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
218	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
219	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
221	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
222	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
229	C₂×C₈	—	86	4	no	—
230	C₂×C₈	—	86	4	no	—
231	C₂×C₈	—	172	8	no	—
232	C₂×C₈	—	172	4	no	—
233	C₂×C₈	—	172	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	—	4	4	no	—
240	C₄²	C₄×Dic₄₃	1	2	yes	—
241	C₄²	Dic₄₃⋊C₄	1	2	yes	—
242	C₄²	Dic₄₃⋊C₄	1	2	yes	—
243	C₄²	C₄⋊Dic₄₃	1	2	yes	—
244	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
245	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
246	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
249	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₄₃⋊#95	1	2	yes	—
253	C₂³	#13063242	1	2	yes	—
254	C₂³	#13063244	1	2	yes	—
255	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₄₃⋊#95	1	2	yes	—
273	C₂×C₄	C₄₃⋊#95	1	2	yes	—
274	C₂×C₄	C₄₃⋊#95	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
276	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
281	C₂×C₄	#13063345	1	4	yes	—
282	C₂×C₄	#13063350	1	2	yes	—
283	C₂×C₄	#13063355	1	2	yes	—
284	C₈	—	172	4	no	—
285	C₈	—	172	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	#14582668	1	4	yes	—
291	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
292	C₂²	C₂×#13063242	1	1	yes	characteristic
293	C₂²	C₂×#13063244	1	1	yes	characteristic
294	C₂²	#14582895	1	2	yes	—
295	C₂²	#14582900	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	8	4	no	—
302	C₄	#14582893	1	2	yes	—
303	C₄	#14582903	1	1	yes	characteristic
304	C₄	#14582907	1	1	yes	characteristic
305	C₄	#14582935	1	2	yes	—
306	C₄	#14582942	1	2	yes	—
307	C₂	—	2	4	no	—
308	C₂	#19133290	1	2	yes	—
309	C₂	#19133295	1	2	yes	—
310	C₂	#19134311	1	1	yes	characteristic
311	C₂	#19134316	1	1	yes	characteristic
312	C₂	C₂×#14582668	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2285	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2285	D₄₃
hypocenter	C₄₃	#8044
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #19132478	C₂	semidirect
7. #14582956	C₄	semidirect
10. C₂×C₄⋊(C₄₃⋊C₈)	C₄	semidirect
179. C₄₃	#8044	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582956	2
C₂²×C₄⋊(C₄₃⋊C₈)	2
#19132478	2
C₄₃×#2285	2
#8044	43