Subgroups — #24989396

Order 11008.

664 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
2	#19134747	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582642	C₂	1	2	yes	maximal
4	#19132479	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063322	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₇₂	—	2	1	no	—
8	C₂×C₄×D₁₇₂	—	2	1	no	—
9	C₂×C₄×D₁₇₂	C₄	1	2	yes	—
10	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
13	#14582801	—	2	1	no	—
14	#14582801	—	2	1	no	—
15	#14582801	C₄	1	2	yes	—
16	#14582769	C₂²	1	1	yes	characteristic
17	#14582642	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₄³×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#336	—	2	1	no	—
21	C₄₃×#336	—	2	1	no	—
22	C₂²×D₁₇₂	—	2	1	no	—
23	C₂²×D₁₇₂	—	2	1	no	—
24	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₄₃	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	#13063322	—	2	1	no	—
28	#13063322	—	2	1	no	—
29	#13063322	D₄	1	2	yes	—
30	#13063322	Q₈	1	2	yes	—
31	C₄×D₁₇₂	—	4	4	no	—
32	C₄×D₁₇₂	—	4	2	no	—
33	C₄×D₁₇₂	—	4	2	no	—
34	C₄×D₁₇₂	—	4	4	no	—
35	C₂×C₄⋊Dic₄₃	—	2	1	no	—
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
38	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
40	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
41	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
43	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
44	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
45	C₂²×C₈×C₄₃	—	2	2	no	—
46	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	—	2	2	no	—
54	C₄⋊C₈×C₄₃	—	2	2	no	—
55	C₂³×D₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₄₃	—	2	2	no	—
64	C₂×D₁₇₂	—	2	2	no	—
65	C₂×D₁₇₂	—	2	2	no	—
66	C₂×D₁₇₂	—	2	2	no	—
67	C₂×D₁₇₂	—	2	2	no	—
68	C₂×D₁₇₂	—	2	2	no	—
69	C₂×D₁₇₂	—	4	1	no	—
70	C₂×D₁₇₂	—	4	1	no	—
71	C₂×D₁₇₂	—	4	2	no	—
72	C₂×D₁₇₂	—	4	2	no	—
73	C₂×D₁₇₂	—	4	1	no	—
74	C₂×D₁₇₂	—	4	1	no	—
75	C₂×D₁₇₂	—	4	2	no	—
76	C₂×D₁₇₂	—	4	2	no	—
77	C₂×D₁₇₂	C₄⋊C₄	1	4	yes	—
78	C₂×C₄×D₄₃	—	4	4	no	—
79	C₂×C₄×D₄₃	—	4	4	no	—
80	C₂×C₄×D₄₃	—	4	4	no	—
81	C₂×C₈×C₄₃	—	4	2	no	—
82	C₂×C₈×C₄₃	—	4	2	no	—
83	C₂×C₈×C₄₃	—	4	2	no	—
84	C₄²×C₄₃	—	2	2	no	—
85	C₄²×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	—	2	1	no	—
87	C₄²×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	—	2	1	no	—
89	C₄²×C₄₃	—	4	1	no	—
90	C₄²×C₄₃	D₈	1	2	yes	—
91	C₄²×C₄₃	SD₁₆	1	2	yes	—
92	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
94	C₄²×C₄₃	#41	1	1	yes	characteristic
95	C₄²×C₄₃	#41	1	1	yes	characteristic
96	(C₂×D₄₃)⋊C₄	—	4	4	no	—
97	(C₂×D₄₃)⋊C₄	—	4	4	no	—
98	(C₂×D₄₃)⋊C₄	—	4	4	no	—
99	C₄⋊Dic₄₃	—	4	1	no	—
100	C₄⋊Dic₄₃	—	4	1	no	—
101	C₄⋊Dic₄₃	—	4	2	no	—
102	C₄⋊Dic₄₃	—	4	2	no	—
103	C₂×C₄₃⋊C₈	—	4	2	no	—
104	C₂×C₄₃⋊C₈	—	4	2	no	—
105	C₂×C₄₃⋊C₈	—	4	2	no	—
106	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×D₄₃	—	2	4	no	—
108	C₂²×D₄₃	—	4	2	no	—
109	C₂²×D₄₃	—	4	2	no	—
110	C₂²×D₄₃	—	4	2	no	—
111	C₂²×D₄₃	—	4	2	no	—
112	C₂²×D₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	#100	1	2	yes	—
126	C₂×C₄×C₄₃	#102	1	2	yes	—
127	C₂×C₄×C₄₃	#102	1	2	yes	—
128	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#135	1	2	yes	—
137	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	4	2	no	—
143	D₁₇₂	—	4	4	no	—
144	D₁₇₂	—	4	4	no	—
145	D₁₇₂	—	4	2	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	—	4	4	no	—
148	D₁₇₂	—	4	2	no	—
149	D₁₇₂	—	4	2	no	—
150	D₁₇₂	—	4	4	no	—
151	C₄×D₄₃	—	8	8	no	—
152	C₈×C₄₃	—	8	2	no	—
153	C₄₃⋊C₈	—	8	2	no	—
154	#8588	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	#380	1	1	yes	characteristic
156	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
157	C₂²×C₄₃	#413	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
159	C₂²×C₄₃	#416	1	1	yes	characteristic
160	C₂²×C₄₃	#417	1	1	yes	characteristic
161	C₂²×C₄₃	#418	1	1	yes	characteristic
162	C₂×D₄₃	—	4	4	no	—
163	C₂×D₄₃	—	4	4	no	—
164	C₂×D₄₃	—	4	4	no	—
165	C₂×D₄₃	—	8	2	no	—
166	C₂×D₄₃	—	8	2	no	—
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	2	2	no	—
174	C₄×C₄₃	—	4	1	no	—
175	C₄×C₄₃	#459	1	1	yes	characteristic
176	C₄×C₄₃	#461	1	1	yes	characteristic
177	C₄×C₄₃	#463	1	2	yes	—
178	C₄×C₄₃	#474	1	1	yes	characteristic
179	C₄×C₄₃	#476	1	1	yes	characteristic
180	C₄×C₄₃	#478	1	2	yes	—
181	Dic₄₃	—	8	2	no	—
182	C₄×#127	—	43	1	no	—
183	#1842	—	43	1	no	—
184	#1831	D₄₃	1	1	yes	characteristic, complemented
185	C₂×#331	—	43	2	no	—
186	#1296	—	43	2	no	—
187	C₂×C₄₃	#1540	1	1	yes	characteristic
188	C₂×C₄₃	#1545	1	1	yes	characteristic
189	C₂×C₄₃	#1551	1	1	yes	characteristic
190	C₂×C₄₃	#1557	1	1	yes	characteristic
191	C₂×C₄₃	#1800	1	1	yes	characteristic
192	C₂×C₄₃	#1802	1	1	yes	characteristic
193	C₂×C₄₃	#1804	1	1	yes	characteristic
194	D₄₃	—	8	4	no	—
195	C₂×#127	—	43	1	no	—
196	C₂×C₄×D₄	—	43	2	no	—
197	C₂×C₄×D₄	—	86	1	no	—
198	C₂×C₄×D₄	—	86	1	no	—
199	C₂×C₄⋊C₈	—	43	2	no	—
200	C₂×C₄⋊C₈	—	86	1	no	—
201	C₂×C₄⋊C₈	—	86	1	no	—
202	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
203	#390	—	43	2	no	—
204	#390	—	86	1	no	—
205	#390	—	86	1	no	—
206	#384	—	43	1	no	—
207	C₄³	C₂×D₄₃	1	1	yes	characteristic
208	#336	—	2	1	no	—
209	#336	—	2	1	no	—
210	#331	—	86	4	no	—
211	C₄₃	#8588	1	1	yes	characteristic, complemented, Sylow
212	C₂²×D₄	—	43	2	no	—
213	C₂²×D₄	—	86	1	no	—
214	C₂²×D₄	—	86	1	no	—
215	C₂³×C₄	—	86	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	86	2	no	—
218	#127	—	43	2	no	—
219	#127	—	43	2	no	—
220	#127	—	86	1	no	—
221	#127	—	86	1	no	—
222	C₄×D₄	—	172	4	no	—
223	C₄×D₄	—	172	4	no	—
224	C₄×D₄	—	172	2	no	—
225	C₄×D₄	—	172	2	no	—
226	C₂×C₄⋊C₄	—	43	2	no	—
227	C₂×C₄⋊C₄	—	86	1	no	—
228	C₂×C₄⋊C₄	—	86	1	no	—
229	C₂×C₂²⋊C₄	—	86	2	no	—
230	C₂×C₂²⋊C₄	—	86	2	no	—
231	C₂×C₂²⋊C₄	—	86	2	no	—
232	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
233	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
234	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
235	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	86	2	no	—
242	C₄⋊C₈	—	86	2	no	—
243	C₄⋊C₈	—	172	2	no	—
244	C₄⋊C₈	—	172	2	no	—
245	C₂⁴	—	86	2	no	—
246	C₂×D₄	—	43	4	no	—
247	C₂×D₄	—	86	2	no	—
248	C₂×D₄	—	86	2	no	—
249	C₂×D₄	—	86	2	no	—
250	C₂×D₄	—	86	2	no	—
251	C₂×D₄	—	86	2	no	—
252	C₂×D₄	—	172	2	no	—
253	C₂×D₄	—	172	2	no	—
254	C₂×D₄	—	172	1	no	—
255	C₂×D₄	—	172	1	no	—
256	C₂×D₄	—	172	2	no	—
257	C₂×D₄	—	172	2	no	—
258	C₂×D₄	—	172	1	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂²×C₄	—	86	2	no	—
261	C₂²×C₄	—	172	4	no	—
262	C₂²×C₄	—	172	4	no	—
263	C₂²×C₄	—	172	4	no	—
264	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
266	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
267	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	172	2	no	—
275	C₂×C₈	—	172	2	no	—
276	C₂×C₈	—	172	2	no	—
277	C₄⋊C₄	—	172	2	no	—
278	C₄⋊C₄	—	172	1	no	—
279	C₄⋊C₄	—	172	1	no	—
280	C₄⋊C₄	—	172	2	no	—
281	C₂²⋊C₄	—	172	4	no	—
282	C₂²⋊C₄	—	172	4	no	—
283	C₂²⋊C₄	—	172	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	#10621873	1	2	yes	—
291	C₄²	D₃₄₄	1	2	yes	—
292	C₄²	D₄×D₄₃	1	1	yes	characteristic
293	C₄²	#10621900	1	1	yes	characteristic
294	C₄²	Q₈×D₄₃	1	1	yes	characteristic
295	C₄²	#10621902	1	1	yes	characteristic
296	C₂³	—	86	4	no	—
297	C₂³	—	172	2	no	—
298	C₂³	—	172	2	no	—
299	C₂³	—	172	2	no	—
300	C₂³	—	172	2	no	—
301	C₂³	—	172	2	no	—
302	C₂³	—	172	2	no	—
303	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
304	D₄	—	172	4	no	—
305	D₄	—	172	4	no	—
306	D₄	—	172	2	no	—
307	D₄	—	172	2	no	—
308	D₄	—	172	4	no	—
309	D₄	—	172	2	no	—
310	D₄	—	172	2	no	—
311	D₄	—	172	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	344	8	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	#13063241	1	2	yes	—
328	C₂×C₄	#13063247	1	2	yes	—
329	C₂×C₄	#13063247	1	2	yes	—
330	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
331	C₂×C₄	#13063283	1	1	yes	characteristic
332	C₂×C₄	#13063296	1	1	yes	characteristic
333	C₂×C₄	#13063300	1	1	yes	characteristic
334	C₂×C₄	#13063301	1	1	yes	characteristic
335	C₂×C₄	#13063305	1	1	yes	characteristic
336	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
337	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
338	C₂×C₄	#13063345	1	2	yes	—
339	C₂×C₄	#13063350	1	2	yes	—
340	C₂×C₄	#13063353	1	1	yes	characteristic
341	C₂×C₄	#13063358	1	1	yes	characteristic
342	C₈	—	8	2	no	—
343	C₈	—	344	2	no	—
344	C₂²	—	344	2	no	—
345	C₂²	—	344	2	no	—
346	C₂²	—	344	2	no	—
347	C₂²	—	344	2	no	—
348	C₂²	—	172	4	no	—
349	C₂²	—	172	4	no	—
350	C₂²	—	172	4	no	—
351	C₂²	#14582756	1	1	yes	characteristic
352	C₂²	C₂×#13063241	1	1	yes	characteristic
353	C₂²	#14582881	1	1	yes	characteristic
354	C₂²	C₂×#13063247	1	1	yes	characteristic
355	C₂²	#14582893	1	1	yes	characteristic
356	C₂²	#14582901	1	1	yes	characteristic
357	C₂²	#14582903	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	344	2	no	—
365	C₄	#14583095	1	1	yes	characteristic
366	C₄	#14583105	1	1	yes	characteristic
367	C₄	#14583111	1	2	yes	—
368	C₄	#14583180	1	1	yes	characteristic
369	C₄	#14583190	1	1	yes	characteristic
370	C₄	#14583196	1	2	yes	—
371	C₂	—	344	4	no	—
372	C₂	#19133242	1	1	yes	characteristic
373	C₂	#19133262	1	1	yes	characteristic
374	C₂	#19133291	1	1	yes	characteristic
375	C₂	#19133318	1	1	yes	characteristic
376	C₂	#19134584	1	1	yes	characteristic
377	C₂	#19134596	1	1	yes	characteristic
378	C₂	#19134602	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1831	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1831	D₄₃
hypocenter	C₄₃	#8588
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
2. #19134747	C₂	semidirect
4. #19132479	C₂	semidirect
5. C₄₃×#1831	C₂	semidirect
184. #1831	D₄₃	semidirect
211. C₄₃	#8588	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063322	2
#19134747	2
C₂×#14582642	2
#19132479	2
C₄₃×#1831	2
#8588	43