Subgroups — #25080488

Order 11008.

870 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144071	C₂	1	2	yes	maximal, complemented
2	C₈×#10621898	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133952	C₂	1	4	yes	maximal, complemented
4	#19133947	C₂	1	4	yes	maximal, complemented
5	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133263	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063351	—	2	2	no	—
9	C₄×#10621898	C₂²	1	1	yes	characteristic
10	#14583096	—	2	2	no	—
11	#14583096	C₂²	1	4	yes	complemented
12	#14583091	—	2	2	no	—
13	#14583091	C₂²	1	4	yes	complemented
14	C₈×Dic₈₆	C₂²	1	2	yes	complemented
15	C₈×D₁₇₂	C₂²	1	2	yes	complemented
16	C₈×C₂²⋊D₄₃	—	2	2	no	—
17	#14582952	C₂²	1	1	yes	characteristic
18	#14582950	—	2	1	no	—
19	#14582923	C₂²	1	2	yes	—
20	#14582904	—	2	2	no	—
21	C₄×C₈×D₄₃	—	2	1	no	—
22	#14582652	C₂²	1	4	yes	complemented
23	#14582643	C₂²	1	8	yes	complemented
24	C₂²×D₈×C₄₃	—	2	2	no	—
25	C₂×#127×C₄₃	C₂²	1	2	yes	—
26	C₄₃×#493	—	2	1	no	—
27	C₄₃×#493	—	2	1	no	—
28	C₄₃×#493	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#10621898	D₄	1	1	yes	characteristic
31	#13063351	—	4	8	no	—
32	C₂×#10621883	—	2	4	no	—
33	C₂×#10621882	—	2	4	no	—
34	C₂×#10621877	—	2	2	no	—
35	C₂×C₈×D₄₃	—	2	2	no	—
36	C₄×Dic₈₆	D₄	1	1	yes	characteristic
37	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
38	C₄×D₁₇₂	D₄	1	1	yes	characteristic
39	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
40	C₄×C₂²⋊D₄₃	—	2	1	no	—
41	C₄×C₂²⋊D₄₃	—	2	1	no	—
42	#13063284	D₄	1	1	yes	characteristic
43	#13063282	—	2	1	no	—
44	C₄²×D₄₃	—	2	1	no	—
45	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
46	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
47	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
48	Dic₄₃⋊C₈	—	2	2	no	—
49	#13063246	—	2	4	no	—
50	#13063246	—	2	4	no	—
51	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
52	(C₂×D₄₃)⋊C₈	—	2	2	no	—
53	C₄×C₄₃⋊C₈	—	2	1	no	—
54	C₈×Dic₄₃	—	2	2	no	—
55	C₂²×D₄×C₄₃	—	2	2	no	—
56	C₂²×D₄×C₄₃	—	2	2	no	—
57	C₂²×D₄×C₄₃	—	2	2	no	—
58	C₂×D₈×C₄₃	—	2	4	no	—
59	C₂×D₈×C₄₃	—	2	4	no	—
60	C₂×D₈×C₄₃	—	4	8	no	—
61	C₂²×C₈×C₄₃	D₄	1	2	yes	—
62	#127×C₄₃	—	2	2	no	—
63	#127×C₄₃	—	2	2	no	—
64	#127×C₄₃	—	2	2	no	—
65	#127×C₄₃	C₂³	1	4	yes	—
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₃	C₂³	1	2	yes	—
70	C₂⁴×C₄₃	—	4	2	no	—
71	C₂×D₄×C₄₃	—	2	4	no	—
72	C₂×D₄×C₄₃	—	2	4	no	—
73	C₂×D₄×C₄₃	—	2	4	no	—
74	C₂×D₄×C₄₃	—	4	2	no	—
75	C₂×D₄×C₄₃	—	4	2	no	—
76	C₂×D₄×C₄₃	—	4	2	no	—
77	C₂×D₄×C₄₃	—	4	8	no	—
78	C₂×D₄×C₄₃	—	4	4	no	—
79	C₂×D₄×C₄₃	—	4	4	no	—
80	C₂×D₄×C₄₃	—	8	4	no	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
84	C₂×C₂²⋊D₄₃	—	2	1	no	—
85	#10621898	—	2	2	no	—
86	#10621898	D₈	1	4	yes	complemented
87	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×D₄₃	—	2	1	no	—
89	C₂×C₄×D₄₃	—	2	1	no	—
90	C₂×C₄×D₄₃	—	2	1	no	—
91	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
92	D₈×C₄₃	—	4	8	no	—
93	D₈×C₄₃	—	4	8	no	—
94	C₂×C₈×C₄₃	—	2	4	no	—
95	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	#41	1	1	yes	characteristic
100	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₄₃	—	2	1	no	—
102	#10621883	—	4	8	no	—
103	#10621882	—	4	8	no	—
104	(C₂×D₄₃)⋊C₄	—	2	1	no	—
105	(C₂×D₄₃)⋊C₄	—	2	1	no	—
106	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
108	Dic₄₃⋊C₄	—	2	1	no	—
109	Dic₄₃⋊C₄	—	2	1	no	—
110	C₄×Dic₄₃	—	2	1	no	—
111	C₄×Dic₄₃	—	2	1	no	—
112	C₄×Dic₄₃	—	2	1	no	—
113	#10621877	—	4	4	no	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	2	4	no	—
116	C₈×D₄₃	—	4	4	no	—
117	C₂³×C₄₃	—	4	4	no	—
118	C₂³×C₄₃	—	8	4	no	—
119	C₂³×C₄₃	—	8	4	no	—
120	C₂³×C₄₃	—	8	4	no	—
121	C₂³×C₄₃	#121	1	1	yes	characteristic
122	C₂²×D₄₃	—	2	1	no	—
123	D₄×C₄₃	—	4	4	no	—
124	D₄×C₄₃	—	4	4	no	—
125	D₄×C₄₃	—	4	8	no	—
126	D₄×C₄₃	—	8	4	no	—
127	D₄×C₄₃	—	8	4	no	—
128	D₄×C₄₃	—	8	4	no	—
129	D₄×C₄₃	—	8	4	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₄₃	#135	1	2	yes	—
137	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₂²⋊D₄₃	—	2	4	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	2	1	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	D₁₇₂	—	2	1	no	—
145	D₁₇₂	C₂×D₈	1	2	yes	complemented
146	C₄×D₄₃	—	2	4	no	—
147	C₄×D₄₃	—	4	2	no	—
148	C₄×D₄₃	—	4	2	no	—
149	Dic₈₆	—	2	1	no	—
150	Dic₈₆	C₂×D₈	1	2	yes	complemented
151	C₈×C₄₃	—	2	2	no	—
152	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	complemented
153	C₄₃⋊C₈	—	4	4	no	—
154	C₄₃⋊C₈	—	4	2	no	—
155	#19267	—	43	1	no	maximal, Sylow
156	C₂²×C₄₃	—	2	1	no	—
157	C₂²×C₄₃	—	2	1	no	—
158	C₂²×C₄₃	—	8	4	no	—
159	C₂²×C₄₃	—	8	4	no	—
160	C₂²×C₄₃	—	8	4	no	—
161	C₂²×C₄₃	—	16	8	no	—
162	C₂²×C₄₃	#466	1	2	yes	—
163	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
164	C₂×D₄₃	—	2	2	no	—
165	C₂×D₄₃	—	4	1	no	—
166	C₂×D₄₃	—	4	1	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	#466	1	2	yes	—
170	C₄×C₄₃	#472	1	2	yes	—
171	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
173	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
174	Dic₄₃	—	2	2	no	—
175	Dic₄₃	—	4	1	no	—
176	Dic₄₃	—	4	1	no	—
177	C₂×#493	D₄₃	1	1	yes	characteristic, complemented
178	#3038	—	43	2	no	—
179	C₈×#41	—	43	1	no	—
180	#1668	—	43	4	no	—
181	#1667	—	43	4	no	—
182	#1567	—	43	1	no	—
183	#1545	—	43	2	no	—
184	C₂×C₄₃	—	2	2	no	—
185	C₂×C₄₃	—	16	4	no	—
186	C₂×C₄₃	#3038	1	1	yes	characteristic
187	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
188	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
189	D₄₃	—	4	2	no	—
190	C₂×AGL(1,ℤ/8)	—	86	2	no	—
191	C₂²×D₈	—	2	2	no	—
192	C₂×#127	C₂×D₄₃	1	2	yes	—
193	C₄×#41	—	43	1	no	—
194	#493	—	2	1	no	—
195	#493	—	2	1	no	—
196	#493	C₂×D₄₃	1	4	yes	—
197	#460	—	43	4	no	—
198	#460	—	86	2	no	—
199	#459	—	43	4	no	—
200	#459	—	86	2	no	—
201	C₈×Q₈	—	43	2	no	—
202	C₈×D₄	—	43	2	no	—
203	C₈×D₄	—	86	2	no	—
204	#431	—	43	1	no	—
205	#431	—	86	1	no	—
206	#423	—	43	2	no	—
207	#418	—	86	2	no	—
208	C₂×C₄×C₈	—	86	1	no	—
209	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
210	#335	—	43	4	no	—
211	#331	—	43	8	no	—
212	C₄₃	#19267	1	1	yes	characteristic, complemented, Sylow
213	C₂×#41	—	43	1	no	—
214	C₂²×D₄	—	2	2	no	—
215	C₂²×D₄	—	2	2	no	—
216	C₂²×D₄	—	2	2	no	—
217	AGL(1,ℤ/8)	—	172	8	no	—
218	C₂×SD₁₆	—	86	4	no	—
219	C₂×D₈	—	2	4	no	—
220	C₂×D₈	—	2	4	no	—
221	C₂×D₈	—	4	8	no	—
222	C₂×D₈	—	86	4	no	—
223	C₂×M₁₆	—	86	2	no	—
224	C₂²×C₈	—	86	2	no	—
225	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
226	#127	—	2	2	no	—
227	#127	—	2	2	no	—
228	#127	—	2	2	no	—
229	#127	C₂²×D₄₃	1	4	yes	—
230	C₄×Q₈	—	43	1	no	—
231	C₄×Q₈	—	43	1	no	—
232	C₄×D₄	—	43	1	no	—
233	C₄×D₄	—	43	1	no	—
234	C₄×D₄	—	86	1	no	—
235	C₄×D₄	—	86	1	no	—
236	#117	—	43	1	no	—
237	#117	—	86	1	no	—
238	C₂×C₄²	—	86	1	no	—
239	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	C₄⋊C₈	—	43	2	no	—
241	C₄⋊C₈	—	43	4	no	—
242	C₄⋊C₈	—	86	2	no	—
243	C₄⋊C₈	—	86	2	no	—
244	#102	—	86	4	no	—
245	#102	—	86	4	no	—
246	C₂²⋊C₈	—	86	1	no	—
247	C₂²⋊C₈	—	86	2	no	—
248	C₄×C₈	—	86	2	no	—
249	C₄×C₈	—	86	1	no	—
250	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
251	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₄₃	1	2	yes	—
253	C₂⁴	—	4	2	no	—
254	#41	—	43	4	no	—
255	#41	—	86	2	no	—
256	C₂×Q₈	—	43	1	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	4	8	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	4	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	8	4	no	—
267	C₂×D₄	—	43	1	no	—
268	C₂×D₄	—	86	1	no	—
269	C₂²×C₄	—	86	1	no	—
270	C₂²×C₄	—	86	1	no	—
271	C₂²×C₄	—	86	1	no	—
272	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	#10621900	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
275	SD₁₆	—	172	8	no	—
276	D₈	—	4	8	no	—
277	D₈	—	4	8	no	—
278	D₈	—	172	8	no	—
279	M₁₆	—	172	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	86	4	no	—
282	C₂×C₈	—	86	2	no	—
283	C₂×C₈	—	172	4	no	—
284	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
285	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
286	C₄⋊C₄	—	43	1	no	—
287	C₄⋊C₄	—	43	1	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	86	1	no	—
290	C₂²⋊C₄	—	86	1	no	—
291	C₂²⋊C₄	—	86	1	no	—
292	C₂²⋊C₄	—	86	1	no	—
293	C₄²	—	86	1	no	—
294	C₄²	—	86	1	no	—
295	C₄²	—	86	1	no	—
296	C₄²	D₄×D₄₃	1	1	yes	characteristic
297	C₄²	#10621900	1	1	yes	characteristic
298	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
299	C₄²	C₂³×D₄₃	1	1	yes	characteristic
300	C₂³	—	4	4	no	—
301	C₂³	—	8	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	—	8	4	no	—
304	C₂³	—	86	1	no	—
305	C₂³	#13063296	1	1	yes	characteristic
306	Q₈	—	43	2	no	—
307	Q₈	—	86	1	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	8	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	4	no	—
315	D₄	—	43	2	no	—
316	D₄	—	86	4	no	—
317	D₄	—	86	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	4	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	#13063296	1	1	yes	characteristic
327	C₂×C₄	#13063296	1	1	yes	characteristic
328	C₂×C₄	#13063296	1	1	yes	characteristic
329	C₂×C₄	D₈×D₄₃	1	2	yes	—
330	C₂×C₄	#13063344	1	2	yes	—
331	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
333	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	172	4	no	—
336	C₈	—	172	2	no	—
337	C₈	#13063351	1	4	yes	complemented
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	8	no	—
344	C₂²	—	86	2	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	#14583132	1	2	yes	—
348	C₂²	C₂×#13063296	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	86	2	no	—
352	C₄	—	172	1	no	—
353	C₄	—	172	1	no	—
354	C₄	#14583132	1	2	yes	—
355	C₄	#14583166	1	2	yes	—
356	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
357	C₄	C₂×#13063344	1	1	yes	characteristic
358	C₄	C₂×#13063351	1	2	yes	—
359	C₂	—	2	2	no	—
360	C₂	—	16	4	no	—
361	C₂	—	172	2	no	—
362	C₂	#19144071	1	1	yes	characteristic
363	C₂	C₂×#14583132	1	1	yes	characteristic
364	C₂	C₂×#14583166	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#493	C₂
socle	C₂²×C₄₃	C₂×#121
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#19267
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #19144071	C₂	semidirect
2. C₈×#10621898	C₂	semidirect
3. #19133952	C₂	semidirect
4. #19133947	C₂	semidirect
5. #19133354	C₂	semidirect
6. #19133263	C₂	semidirect
7. C₂×C₄₃×#493	C₂	semidirect
11. #14583096	C₂²	semidirect
13. #14583091	C₂²	semidirect
14. C₈×Dic₈₆	C₂²	semidirect
15. C₈×D₁₇₂	C₂²	semidirect
22. #14582652	C₂²	semidirect
23. #14582643	C₂²	semidirect
86. #10621898	D₈	semidirect
145. D₁₇₂	C₂×D₈	semidirect
150. Dic₈₆	C₂×D₈	semidirect
152. C₈×C₄₃	AGL(1,ℤ/8)	semidirect
177. C₂×#493	D₄₃	semidirect
212. C₄₃	#19267	semidirect
337. C₈	#13063351	semidirect

Maximal subgroups

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

subgroup	index
#19144071	2
C₈×#10621898	2
#19133952	2
#19133947	2
#19133354	2
#19133263	2
C₂×C₄₃×#493	2
#19267	43