Subgroups — #25196350

Order 11008.

710 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142958	C₂	1	1	yes	maximal, characteristic
2	#19136152	C₂	1	2	yes	maximal
3	#19135838	C₂	1	2	yes	maximal
4	#19135288	C₂	1	4	yes	maximal
5	#19134600	C₂	1	2	yes	maximal
6	#19134600	C₂	1	2	yes	maximal
7	#19134448	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063317	—	2	1	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×#13063249	C₂²	1	4	yes	—
12	C₂×#13063249	C₂²	1	4	yes	—
13	C₂×#13063236	C₂²	1	2	yes	—
14	#14582797	—	2	1	no	—
15	#14582797	—	2	1	no	—
16	#14582797	C₂²	1	2	yes	—
17	#14582797	C₂²	1	2	yes	—
18	#14582769	C₂²	1	1	yes	characteristic
19	#14582769	C₂²	1	1	yes	characteristic
20	#14582765	—	2	1	no	—
21	#14582661	C₂²	1	4	yes	—
22	#14582661	C₂²	1	4	yes	—
23	#14582654	C₂²	1	2	yes	—
24	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
26	C₄₃×#391	C₂²	1	2	yes	—
27	C₄₃×#389	—	2	1	no	—
28	C₄₃×#389	—	2	1	no	—
29	C₄₃×#384	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
31	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
32	C₄³×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
34	#13063317	—	4	4	no	—
35	C₂×C₄⋊Dic₄₃	—	2	1	no	—
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×C₄⋊Dic₄₃	—	2	1	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
41	C₂×Dic₄₃⋊C₄	—	2	2	no	—
42	C₂×Dic₄₃⋊C₄	—	2	2	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
44	#13063249	—	2	8	no	—
45	#13063249	—	4	4	no	—
46	#13063236	—	4	2	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	2	no	—
49	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₄₃	—	2	4	no	—
51	C₄×Q₈×C₄₃	—	2	4	no	—
52	C₄×Q₈×C₄₃	—	2	4	no	—
53	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₃	—	2	2	no	—
60	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	#95×C₄₃	—	2	2	no	—
68	#95×C₄₃	—	2	2	no	—
69	C₂×Q₈×C₄₃	—	2	2	no	—
70	C₂×Q₈×C₄₃	—	2	2	no	—
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	—	2	2	no	—
73	C₂×Q₈×C₄₃	—	2	2	no	—
74	C₂×Q₈×C₄₃	#41	1	4	yes	—
75	C₂²×C₄×C₄₃	—	2	2	no	—
76	C₂²×C₄×C₄₃	—	2	2	no	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₃	—	2	2	no	—
85	C₂²×Dic₄₃	—	2	2	no	—
86	C₄⋊C₄×C₄₃	—	2	4	no	—
87	C₄⋊C₄×C₄₃	—	2	4	no	—
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₄⋊C₄×C₄₃	—	2	2	no	—
90	C₄⋊C₄×C₄₃	—	4	4	no	—
91	C₄⋊C₄×C₄₃	—	4	2	no	—
92	C₄⋊C₄×C₄₃	—	4	2	no	—
93	C₄²×C₄₃	—	2	2	no	—
94	C₄²×C₄₃	—	2	2	no	—
95	C₄²×C₄₃	—	2	2	no	—
96	C₄²×C₄₃	—	2	2	no	—
97	C₄²×C₄₃	—	2	2	no	—
98	C₄²×C₄₃	—	2	1	no	—
99	C₄²×C₄₃	—	2	1	no	—
100	C₄²×C₄₃	—	4	4	no	—
101	C₄²×C₄₃	#41	1	4	yes	—
102	C₄⋊Dic₄₃	—	2	4	no	—
103	C₄⋊Dic₄₃	—	2	2	no	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	C₄⋊Dic₄₃	—	4	2	no	—
106	C₄⋊Dic₄₃	—	4	2	no	—
107	C₄⋊Dic₄₃	—	4	2	no	—
108	C₄⋊Dic₄₃	—	4	2	no	—
109	Dic₄₃⋊C₄	—	4	4	no	—
110	Dic₄₃⋊C₄	—	4	4	no	—
111	C₂×C₄₃⋊C₈	—	2	4	no	—
112	C₂×C₄₃⋊C₈	—	2	4	no	—
113	C₂×C₄₃⋊C₈	—	2	4	no	—
114	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₄₃	—	2	4	no	—
116	Q₈×C₄₃	—	2	4	no	—
117	Q₈×C₄₃	—	4	2	no	—
118	Q₈×C₄₃	—	4	2	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	—	2	4	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	#124	1	2	yes	—
138	C₂×C₄×C₄₃	#124	1	2	yes	—
139	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₄₃	#137	1	2	yes	—
141	C₂×C₄×C₄₃	#137	1	2	yes	—
142	C₂×C₄×C₄₃	#137	1	2	yes	—
143	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
144	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	—	4	2	no	—
153	C₄₃⋊C₈	—	4	4	no	—
154	#29132	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	#525	1	1	yes	characteristic
156	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
157	C₂²×C₄₃	#537	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	2	2	no	—
165	C₄×C₄₃	—	4	4	no	—
166	C₄×C₄₃	—	4	1	no	—
167	C₄×C₄₃	—	4	1	no	—
168	C₄×C₄₃	—	8	2	no	—
169	C₄×C₄₃	#491	1	4	yes	—
170	C₄×C₄₃	#552	1	2	yes	—
171	C₄×C₄₃	#553	1	2	yes	—
172	Dic₄₃	—	8	2	no	—
173	Dic₄₃	—	8	2	no	—
174	#2859	—	43	1	no	—
175	C₄×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
176	#2099	—	43	2	no	—
177	#2043	—	43	2	no	—
178	#1935	—	43	4	no	—
179	#1803	—	43	2	no	—
180	#1803	—	43	2	no	—
181	#1767	—	43	1	no	—
182	C₂×C₄₃	#2562	1	1	yes	characteristic
183	C₂×C₄₃	#3112	1	1	yes	characteristic
184	C₂×C₄₃	#3113	1	1	yes	characteristic
185	C₂×C₄₃	C₂×#491	1	1	yes	characteristic
186	C₂×C₄₃	#3141	1	1	yes	characteristic
187	C₂×C₄₃	#3254	1	1	yes	characteristic
188	C₂×C₄₃	#3255	1	1	yes	characteristic
189	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#125	—	86	1	no	—
191	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
192	C₂×C₄⋊C₈	—	43	2	no	—
193	C₂×#103	—	43	4	no	—
194	C₂×#103	—	43	4	no	—
195	C₂×#97	—	43	2	no	—
196	#391	C₂×D₄₃	1	2	yes	—
197	#389	—	2	1	no	—
198	#389	—	2	1	no	—
199	#389	—	43	2	no	—
200	#389	—	43	2	no	—
201	#389	—	86	1	no	—
202	#389	—	86	1	no	—
203	#384	—	43	1	no	—
204	#384	—	43	1	no	—
205	#384	C₂×D₄₃	1	1	yes	characteristic
206	#382	—	86	1	no	—
207	C₄×C₄⋊C₄	—	2	1	no	—
208	C₄×C₄⋊C₄	—	2	1	no	—
209	C₄³	C₂×D₄₃	1	1	yes	characteristic
210	#340	—	43	4	no	—
211	#340	—	43	4	no	—
212	#336	—	43	2	no	—
213	C₄₃	#29132	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
215	C₂²×C₈	—	43	4	no	—
216	C₄⋊Q₈	—	2	4	no	—
217	#125	—	172	4	no	—
218	C₄×Q₈	—	2	4	no	—
219	C₄×Q₈	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	43	2	no	—
226	C₂×C₄⋊C₄	—	43	2	no	—
227	C₂×C₄⋊C₄	—	86	2	no	—
228	C₂×C₄⋊C₄	—	86	2	no	—
229	C₂×C₄⋊C₄	—	86	1	no	—
230	C₂×C₄⋊C₄	—	86	1	no	—
231	C₂×C₄⋊C₄	—	86	1	no	—
232	C₂×C₄⋊C₄	—	86	1	no	—
233	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	C₂²⋊D₄₃	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	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
242	C₄⋊C₈	—	86	4	no	—
243	#103	—	86	8	no	—
244	#103	—	172	4	no	—
245	#97	—	172	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	86	2	no	—
249	#95	—	86	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	#10621898	1	4	yes	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	86	2	no	—
259	C₂²×C₄	—	86	2	no	—
260	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
267	C₂×C₈	—	86	4	no	—
268	C₂×C₈	—	86	4	no	—
269	C₂×C₈	—	86	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	86	4	no	—
278	C₄⋊C₄	—	86	2	no	—
279	C₄⋊C₄	—	86	2	no	—
280	C₄⋊C₄	—	172	4	no	—
281	C₄⋊C₄	—	172	2	no	—
282	C₄⋊C₄	—	172	2	no	—
283	C₄⋊C₄	—	172	4	no	—
284	C₄⋊C₄	—	172	2	no	—
285	C₄⋊C₄	—	172	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	4	4	no	—
294	C₄²	#10621900	1	4	yes	—
295	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	2	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	2	no	—
323	C₂×C₄	—	172	2	no	—
324	C₂×C₄	#13063312	1	2	yes	—
325	C₂×C₄	#13063312	1	2	yes	—
326	C₂×C₄	#13063350	1	2	yes	—
327	C₂×C₄	#13063355	1	2	yes	—
328	C₂×C₄	#13063356	1	2	yes	—
329	C₂×C₄	#13063356	1	2	yes	—
330	C₂×C₄	C₂×#10621900	1	2	yes	—
331	C₂×C₄	C₂×#10621898	1	2	yes	—
332	C₂×C₄	#13063371	1	1	yes	characteristic
333	C₂×C₄	#13063372	1	1	yes	characteristic
334	C₈	—	172	4	no	—
335	C₂²	#14583410	1	1	yes	characteristic
336	C₂²	C₂×#13063312	1	1	yes	characteristic
337	C₂²	#14583477	1	1	yes	characteristic
338	C₂²	C₂×#13063350	1	1	yes	characteristic
339	C₂²	C₂×#13063355	1	1	yes	characteristic
340	C₂²	C₂×#13063356	1	1	yes	characteristic
341	C₂²	C₂×#13063356	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	4	no	—
347	C₄	—	4	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	344	2	no	—
350	C₄	—	344	2	no	—
351	C₄	#14583267	1	4	yes	—
352	C₄	#14583589	1	2	yes	—
353	C₄	#14583595	1	2	yes	—
354	C₂	#19139761	1	1	yes	characteristic
355	C₂	#19144717	1	1	yes	characteristic
356	C₂	#19144725	1	1	yes	characteristic
357	C₂	C₂×#14583267	1	1	yes	characteristic
358	C₂	#19144988	1	1	yes	characteristic
359	C₂	#19145978	1	1	yes	characteristic
360	C₂	#19145986	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×C₄⋊Q₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#29132
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
213. C₄₃	#29132	semidirect

Maximal subgroups

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

subgroup	index
#19142958	2
#19136152	2
#19135838	2
#19135288	2
#19134600	2
#19134448	2
C₄×C₄⋊Q₈×C₄₃	2
#29132	43