Subgroups — #25079875

Order 11008.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145533	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583247	C₂	1	1	yes	maximal, characteristic
3	#19143530	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134137	C₂	1	4	yes	maximal, complemented
5	#19134013	C₂	1	4	yes	maximal, complemented
6	#19133341	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133276	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133262	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063322	C₂²	1	1	yes	characteristic
11	#14583273	—	2	1	no	—
12	#14583247	—	2	1	no	—
13	#14583247	—	2	1	no	—
14	#14583247	—	2	1	no	—
15	#14583247	C₂²	1	2	yes	—
16	#14583247	C₂²	1	2	yes	—
17	#14583099	C₂²	1	4	yes	—
18	#14583049	—	2	2	no	—
19	#14583030	—	2	1	no	—
20	#14583030	C₂²	1	2	yes	complemented
21	Q₈×C₄₃⋊C₈	—	2	1	no	—
22	Q₈×C₄₃⋊C₈	C₂²	1	2	yes	complemented
23	#14582952	C₂²	1	1	yes	characteristic
24	#14582926	—	2	1	no	—
25	#14582923	C₂²	1	1	yes	characteristic
26	C₂×#13063245	—	2	1	no	—
27	C₂×#13063245	—	2	1	no	—
28	#14582855	—	2	1	no	—
29	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
30	#14582651	C₂²	1	2	yes	complemented
31	#14582649	C₂²	1	2	yes	complemented
32	#14582648	C₂²	1	4	yes	complemented
33	#14582642	C₂²	1	4	yes	—
34	C₄₃×#533	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₄₃×#500	—	2	1	no	—
37	C₄₃×#487	—	2	1	no	—
38	C₄₃×#484	—	2	2	no	—
39	C₄₃×#477	C₂²	1	4	yes	complemented
40	C₄₃×#431	C₂²	1	1	yes	characteristic
41	C₂²×D₁₇₂	—	2	1	no	—
42	C₂²×D₁₇₂	—	2	1	no	—
43	C₂²×D₁₇₂	—	2	1	no	—
44	C₂×#10621884	—	2	4	no	—
45	C₂×#10621877	—	2	1	no	—
46	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
48	#13063322	—	2	1	no	—
49	#13063322	—	2	1	no	—
50	#13063322	—	2	1	no	—
51	#13063322	C₂³	1	2	yes	—
52	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
54	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
55	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
56	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
57	#13063245	—	2	4	no	—
58	#13063245	—	2	2	no	—
59	#13063245	—	2	2	no	—
60	#13063245	—	2	2	no	—
61	#13063245	—	2	2	no	—
62	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
63	#13063236	—	2	2	no	—
64	C₄×C₄₃⋊C₈	—	2	1	no	—
65	C₄×C₄₃⋊C₈	—	2	2	no	—
66	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₄₃	—	2	1	no	—
71	C₄⋊Q₈×C₄₃	—	2	1	no	—
72	C₄⋊Q₈×C₄₃	—	2	1	no	—
73	C₄⋊Q₈×C₄₃	—	2	1	no	—
74	C₄⋊Q₈×C₄₃	D₄	1	1	yes	characteristic
75	C₄⋊Q₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
77	#125×C₄₃	—	2	1	no	—
78	C₂²⋊Q₈×C₄₃	—	2	1	no	—
79	C₂²⋊Q₈×C₄₃	D₄	1	2	yes	—
80	C₄×Q₈×C₄₃	—	2	1	no	—
81	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
82	#117×C₄₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
86	#107×C₄₃	—	2	4	no	—
87	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
88	#103×C₄₃	—	2	4	no	—
89	C₂²⋊C₈×C₄₃	—	2	1	no	—
90	C₄×C₈×C₄₃	—	2	1	no	—
91	C₂³×D₄₃	—	4	1	no	—
92	C₂×Q₈×C₄₃	—	2	1	no	—
93	C₂×Q₈×C₄₃	—	2	2	no	—
94	C₂×Q₈×C₄₃	—	4	1	no	—
95	C₂×Q₈×C₄₃	—	8	2	no	—
96	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₄₃	—	4	1	no	—
98	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
101	C₂×D₁₇₂	—	2	2	no	—
102	C₂×D₁₇₂	—	2	2	no	—
103	C₂×D₁₇₂	—	2	2	no	—
104	C₂×D₁₇₂	—	4	1	no	—
105	C₂×D₁₇₂	—	4	1	no	—
106	C₂×D₁₇₂	—	4	1	no	—
107	C₂×D₁₇₂	—	4	2	no	—
108	C₂×D₁₇₂	—	4	2	no	—
109	C₂×D₁₇₂	—	8	2	no	—
110	C₂×D₁₇₂	—	8	2	no	—
111	C₂×C₈×C₄₃	—	2	2	no	—
112	C₄⋊C₄×C₄₃	—	2	1	no	—
113	C₄⋊C₄×C₄₃	—	2	1	no	—
114	C₄⋊C₄×C₄₃	—	2	1	no	—
115	C₄⋊C₄×C₄₃	—	2	2	no	—
116	C₄⋊C₄×C₄₃	—	2	2	no	—
117	C₄⋊C₄×C₄₃	—	2	2	no	—
118	C₄⋊C₄×C₄₃	—	2	2	no	—
119	C₄⋊C₄×C₄₃	—	4	1	no	—
120	C₄⋊C₄×C₄₃	—	4	1	no	—
121	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
122	C₂²⋊C₄×C₄₃	—	2	2	no	—
123	C₄²×C₄₃	—	2	2	no	—
124	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₄₃	#41	1	1	yes	characteristic
127	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
128	#10621884	—	8	4	no	—
129	#10621877	—	4	2	no	—
130	C₂×C₄₃⋊C₈	—	2	2	no	—
131	C₂×C₄₃⋊C₈	—	2	2	no	—
132	C₂×C₄₃⋊C₈	—	2	2	no	—
133	C₂×C₄₃⋊C₈	—	2	2	no	—
134	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
138	C₂³×C₄₃	#121	1	1	yes	characteristic
139	C₂²×D₄₃	—	4	2	no	—
140	C₂²×D₄₃	—	8	2	no	—
141	C₂²×D₄₃	—	8	2	no	—
142	C₂²×D₄₃	—	8	2	no	—
143	Q₈×C₄₃	—	2	4	no	—
144	Q₈×C₄₃	—	4	2	no	—
145	Q₈×C₄₃	—	8	2	no	—
146	Q₈×C₄₃	—	8	2	no	—
147	C₂×C₄×C₄₃	—	2	2	no	—
148	C₂×C₄×C₄₃	—	2	2	no	—
149	C₂×C₄×C₄₃	—	2	2	no	—
150	C₂×C₄×C₄₃	—	2	2	no	—
151	C₂×C₄×C₄₃	—	4	2	no	—
152	C₂×C₄×C₄₃	—	8	2	no	—
153	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
155	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
156	C₂×C₄×C₄₃	#135	1	2	yes	—
157	C₂×C₄×C₄₃	#137	1	2	yes	—
158	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
161	D₁₇₂	—	4	2	no	—
162	D₁₇₂	—	4	2	no	—
163	D₁₇₂	—	4	4	no	—
164	D₁₇₂	—	8	2	no	—
165	D₁₇₂	—	8	2	no	—
166	D₁₇₂	—	8	2	no	—
167	D₁₇₂	—	8	2	no	—
168	C₈×C₄₃	—	4	2	no	—
169	C₄₃⋊C₈	—	2	1	no	—
170	C₄₃⋊C₈	—	2	1	no	—
171	C₄₃⋊C₈	—	2	1	no	—
172	C₄₃⋊C₈	—	2	1	no	—
173	C₄₃⋊C₈	—	4	2	no	—
174	C₄₃⋊C₈	—	4	2	no	—
175	#19212	—	43	1	no	maximal, Sylow
176	C₂²×C₄₃	—	2	1	no	—
177	C₂²×C₄₃	—	2	1	no	—
178	C₂²×C₄₃	#464	1	2	yes	—
179	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
180	C₂×D₄₃	—	8	2	no	—
181	C₂×D₄₃	—	8	2	no	—
182	C₂×D₄₃	—	8	2	no	—
183	C₂×D₄₃	—	16	4	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	—	4	2	no	—
187	C₄×C₄₃	—	4	2	no	—
188	C₄×C₄₃	—	8	2	no	—
189	C₄×C₄₃	#464	1	2	yes	—
190	C₄×C₄₃	#472	1	2	yes	—
191	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
192	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
193	C₄×C₄₃	#577	1	2	yes	—
194	#3230	D₄₃	1	1	yes	characteristic, complemented
195	#3208	—	43	1	no	—
196	C₂×#486	—	43	1	no	—
197	#2960	—	43	1	no	—
198	#1698	—	43	4	no	—
199	#1678	—	43	4	no	—
200	#1564	—	43	1	no	—
201	#1548	—	43	1	no	—
202	#1545	—	43	1	no	—
203	C₂×C₄₃	—	2	2	no	—
204	C₂×C₄₃	C₂×#464	1	1	yes	characteristic
205	C₂×C₄₃	#3057	1	1	yes	characteristic
206	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
207	D₄₃	—	16	2	no	—
208	#533	C₂×D₄₃	1	1	yes	characteristic
209	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
210	C₂×#127	—	43	1	no	—
211	#500	—	2	1	no	—
212	#492	—	86	1	no	—
213	#487	—	2	1	no	—
214	#486	—	43	2	no	—
215	#486	—	43	2	no	—
216	#486	—	86	1	no	—
217	#486	—	86	1	no	—
218	#486	—	86	1	no	—
219	#484	—	2	2	no	—
220	#477	C₂×D₄₃	1	4	yes	complemented
221	#460	—	43	4	no	—
222	#450	—	86	2	no	—
223	#446	—	43	2	no	—
224	#446	—	86	1	no	—
225	C₈×Q₈	—	43	2	no	—
226	C₈×Q₈	—	86	1	no	—
227	#431	—	43	1	no	—
228	#431	C₂×D₄₃	1	1	yes	characteristic
229	#424	—	86	1	no	—
230	#423	—	43	1	no	—
231	C₂×#102	—	86	1	no	—
232	C₂×#102	—	86	1	no	—
233	#405	—	86	1	no	—
234	C₂×C₄×C₈	—	43	1	no	—
235	#335	—	43	2	no	—
236	#334	—	43	2	no	—
237	#333	—	43	4	no	—
238	#331	—	43	4	no	—
239	C₄₃	#19212	1	1	yes	characteristic, complemented, Sylow
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×D₄	—	86	1	no	—
242	C₂²×D₄	—	86	1	no	—
243	C₂²×D₄	—	86	1	no	—
244	C₂×SD₁₆	—	86	4	no	—
245	C₂×M₁₆	—	86	1	no	—
246	C₂²×C₈	—	43	1	no	—
247	C₂²×C₈	—	43	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
252	C₄⋊Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
253	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
254	#127	—	43	2	no	—
255	#127	—	86	1	no	—
256	#127	—	86	1	no	—
257	#127	—	86	1	no	—
258	#125	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₂²⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
263	#117	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
267	#107	—	2	4	no	—
268	C₄⋊C₈	—	43	2	no	—
269	C₄⋊C₈	—	43	2	no	—
270	C₄⋊C₈	—	86	2	no	—
271	C₄⋊C₈	—	86	2	no	—
272	C₄⋊C₈	—	86	1	no	—
273	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
274	#103	—	2	4	no	—
275	#102	—	86	2	no	—
276	#102	—	86	2	no	—
277	#102	—	86	2	no	—
278	#102	—	86	2	no	—
279	#102	—	86	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	86	1	no	—
282	#97	—	86	2	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	43	1	no	—
285	C₄×C₈	—	43	1	no	—
286	C₄×C₈	—	43	1	no	—
287	C₄×C₈	—	43	1	no	—
288	C₄×C₈	—	86	2	no	—
289	C₄×C₈	—	86	1	no	—
290	C₂⁴	—	172	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	8	2	no	—
295	C₂×Q₈	C₂×C₂²⋊D₄₃	1	2	yes	—
296	C₂×D₄	—	86	2	no	—
297	C₂×D₄	—	86	2	no	—
298	C₂×D₄	—	86	2	no	—
299	C₂×D₄	—	344	2	no	—
300	C₂×D₄	—	344	2	no	—
301	C₂×D₄	—	172	1	no	—
302	C₂×D₄	—	172	1	no	—
303	C₂×D₄	—	172	1	no	—
304	C₂×D₄	—	172	2	no	—
305	C₂×D₄	—	172	2	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
308	C₂²×C₄	#10621900	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
310	SD₁₆	—	344	4	no	—
311	M₁₆	—	172	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	43	1	no	—
314	C₂×C₈	—	43	1	no	—
315	C₂×C₈	—	43	1	no	—
316	C₂×C₈	—	43	1	no	—
317	C₂×C₈	—	86	2	no	—
318	C₂×C₈	—	86	2	no	—
319	C₂×C₈	—	86	2	no	—
320	C₂×C₈	—	86	2	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	D₄×D₄₃	1	1	yes	characteristic
334	C₄²	#10621900	1	1	yes	characteristic
335	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
336	C₄²	C₂³×D₄₃	1	1	yes	characteristic
337	C₂³	—	344	2	no	—
338	C₂³	—	344	2	no	—
339	C₂³	—	344	2	no	—
340	C₂³	—	172	2	no	—
341	C₂³	#13063297	1	1	yes	characteristic
342	Q₈	—	2	4	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	2	no	—
345	Q₈	—	8	2	no	—
346	D₄	—	344	2	no	—
347	D₄	—	344	2	no	—
348	D₄	—	344	2	no	—
349	D₄	—	344	2	no	—
350	D₄	—	172	2	no	—
351	D₄	—	172	2	no	—
352	D₄	—	172	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	#13063297	1	1	yes	characteristic
360	C₂×C₄	#13063297	1	1	yes	characteristic
361	C₂×C₄	#13063297	1	1	yes	characteristic
362	C₂×C₄	#13063348	1	2	yes	—
363	C₂×C₄	#13063357	1	2	yes	—
364	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
365	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
366	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
367	C₈	—	4	2	no	—
368	C₈	—	86	1	no	—
369	C₈	—	86	1	no	—
370	C₈	—	86	1	no	—
371	C₈	—	86	1	no	—
372	C₈	—	172	2	no	—
373	C₈	—	172	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	344	2	no	—
377	C₂²	—	344	2	no	—
378	C₂²	—	344	2	no	—
379	C₂²	—	688	4	no	—
380	C₂²	#14583123	1	2	yes	—
381	C₂²	C₂×#13063297	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	—	8	2	no	—
387	C₄	#14583123	1	2	yes	—
388	C₄	#14583167	1	2	yes	—
389	C₄	C₂×#13063348	1	1	yes	characteristic
390	C₄	C₂×#13063357	1	1	yes	characteristic
391	C₄	#14583707	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	688	2	no	—
394	C₂	C₂×#14583123	1	1	yes	characteristic
395	C₂	#19144236	1	1	yes	characteristic
396	C₂	C₂×#14583167	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063297
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#3230	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂×#13063322	C₂²
hypercenter	#3230	D₄₃
hypocenter	C₄₃	#19212
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3230

Complements

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

N	H ≅ G/N	type
3. #19143530	C₂	semidirect
4. #19134137	C₂	semidirect
5. #19134013	C₂	semidirect
6. #19133341	C₂	semidirect
7. #19133276	C₂	semidirect
9. C₄₃×#3230	C₂	semidirect
20. #14583030	C₂²	semidirect
22. Q₈×C₄₃⋊C₈	C₂²	semidirect
30. #14582651	C₂²	semidirect
31. #14582649	C₂²	semidirect
32. #14582648	C₂²	semidirect
39. C₄₃×#477	C₂²	semidirect
194. #3230	D₄₃	semidirect
220. #477	C₂×D₄₃	semidirect
239. C₄₃	#19212	semidirect

Maximal subgroups

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

subgroup	index
#19145533	2
C₂×#14583247	2
#19143530	2
#19134137	2
#19134013	2
#19133341	2
#19133276	2
#19133262	2
C₄₃×#3230	2
#19212	43