Subgroups — #25067468

Order 11008.

576 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	#19145165	C₂	1	1	yes	maximal, characteristic
2	#19143746	C₂	1	2	yes	maximal, complemented
3	#19133475	C₂	1	2	yes	maximal, complemented
4	#19133454	C₂	1	2	yes	maximal, complemented
5	#19133447	C₂	1	2	yes	maximal, complemented
6	#19133426	C₂	1	2	yes	maximal, complemented
7	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133061	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583433	C₂²	1	2	yes	—
11	#14583301	C₄	1	1	yes	characteristic
12	#14583283	C₄	1	1	yes	characteristic
13	#14583254	C₄	1	2	yes	—
14	#14583140	C₄	1	1	yes	characteristic
15	#14583135	C₄	1	1	yes	characteristic
16	#14583128	C₄	1	2	yes	—
17	#14583029	C₂²	1	2	yes	—
18	C₈×Dic₈₆	C₂²	1	2	yes	—
19	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
20	C₄×#10621873	C₂²	1	2	yes	—
21	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
22	#14582978	C₂²	1	2	yes	—
23	#14582974	—	2	2	no	—
24	#14582973	—	2	2	no	—
25	C₈×D₁₇₂	C₂²	1	2	yes	—
26	#14582959	—	2	2	no	—
27	#14582924	—	2	2	no	—
28	#14582923	C₂²	1	2	yes	—
29	#14582652	C₂²	1	2	yes	—
30	#14582652	C₂²	1	1	yes	characteristic
31	#14582652	C₂²	1	1	yes	characteristic
32	#14582631	C₂²	1	2	yes	—
33	#14582631	C₂²	1	2	yes	—
34	#14582628	C₂²	1	2	yes	—
35	#14582628	C₂²	1	2	yes	—
36	C₄₃×#431	C₂²	1	1	yes	characteristic
37	C₄₃×#431	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#322	C₂²	1	2	yes	—
40	C₄₃×#322	C₂²	1	2	yes	—
41	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
42	C₂×#10621873	C₂×C₄	1	2	yes	—
43	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
44	C₂×#10621877	—	2	2	no	—
45	C₂×#10621872	—	2	2	no	—
46	C₂×C₈×D₄₃	—	2	2	no	—
47	#13063319	—	2	2	no	—
48	#13063317	C₂×C₄	1	2	yes	—
49	#13063313	C₂×C₄	1	2	yes	—
50	#13063308	—	2	2	no	—
51	#13063304	C₂×C₄	1	2	yes	—
52	#13063299	C₂×C₄	1	2	yes	—
53	C₄×Dic₈₆	C₂³	1	2	yes	—
54	C₄×D₁₇₂	C₂³	1	2	yes	—
55	C₄×C₂²⋊D₄₃	—	2	2	no	—
56	#13063282	—	2	2	no	—
57	#13063259	C₂×C₄	1	1	yes	characteristic
58	#13063259	C₂×C₄	1	1	yes	characteristic
59	#13063257	C₂×C₄	1	2	yes	—
60	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
61	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
63	Dic₄₃⋊C₈	—	2	2	no	—
64	Dic₄₃⋊C₈	—	2	2	no	—
65	#13063250	C₂×C₄	1	2	yes	—
66	#13063250	C₂×C₄	1	2	yes	—
67	#13063247	C₂×C₄	1	2	yes	—
68	#13063247	C₂×C₄	1	2	yes	—
69	(C₂×D₄₃)⋊C₈	—	2	2	no	—
70	(C₂×D₄₃)⋊C₈	—	2	2	no	—
71	#13063235	—	2	2	no	—
72	C₈×Dic₄₃	—	2	2	no	—
73	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	—	2	1	no	—
77	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
78	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
79	C₂²⋊C₈×C₄₃	—	2	1	no	—
80	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
81	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₄₃	C₂³	1	2	yes	—
84	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₃	C₂³	1	2	yes	—
87	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₃	—	2	2	no	—
91	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₄₃	—	2	2	no	—
93	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
103	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₃	#41	1	1	yes	characteristic
106	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₄₃	—	2	2	no	—
108	(C₂×D₄₃)⋊C₄	—	2	2	no	—
109	(C₂×D₄₃)⋊C₄	—	2	2	no	—
110	(C₂×D₄₃)⋊C₄	—	2	2	no	—
111	(C₂×D₄₃)⋊C₄	—	2	2	no	—
112	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
113	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
114	Dic₄₃⋊C₄	—	2	2	no	—
115	Dic₄₃⋊C₄	—	2	2	no	—
116	Dic₄₃⋊C₄	—	2	2	no	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	#10621877	—	8	2	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	Dic₁₇₂	—	4	1	no	—
123	D₃₄₄	—	4	1	no	—
124	#10621873	—	4	2	no	—
125	#10621872	—	8	2	no	—
126	C₈×D₄₃	—	8	2	no	—
127	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂²×D₄₃	—	2	2	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	—	2	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂³×C₄	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	—	8	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	D₁₇₂	—	2	2	no	—
145	D₁₇₂	—	2	2	no	—
146	C₄×D₄₃	—	4	2	no	—
147	C₄×D₄₃	—	4	2	no	—
148	Dic₈₆	—	2	2	no	—
149	Dic₈₆	—	2	2	no	—
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₈×C₄₃	—	2	1	no	—
153	C₈×C₄₃	—	2	1	no	—
154	C₈×C₄₃	—	2	2	no	—
155	C₈×C₄₃	—	2	2	no	—
156	C₈×C₄₃	—	4	2	no	—
157	C₄₃⋊C₈	—	4	2	no	—
158	C₄₃⋊C₈	—	4	2	no	—
159	#17992	—	43	1	no	maximal, Sylow
160	C₂²×C₄₃	—	2	1	no	—
161	C₂²×C₄₃	—	2	1	no	—
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₄₃	—	4	2	no	—
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	4	2	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	#568	1	1	yes	characteristic
172	C₄×C₄₃	#568	1	1	yes	characteristic
173	C₄×C₄₃	#576	1	1	yes	characteristic
174	C₄×C₄₃	#578	1	1	yes	characteristic
175	Dic₄₃	—	4	2	no	—
176	Dic₄₃	—	4	2	no	—
177	Dic₄₃	—	4	2	no	—
178	#3168	—	43	1	no	—
179	#2993	—	43	2	no	—
180	#1588	—	43	2	no	—
181	#1585	—	43	2	no	—
182	#1584	—	43	2	no	—
183	#1581	—	43	2	no	—
184	#1567	—	43	1	no	—
185	#1495	—	43	2	no	—
186	#1452	D₄₃	1	1	yes	characteristic, complemented
187	C₂×C₄₃	—	4	1	no	—
188	C₂×C₄₃	#2932	1	1	yes	characteristic
189	C₂×C₄₃	#2951	1	1	yes	characteristic
190	C₂×C₄₃	#2957	1	1	yes	characteristic
191	D₄₃	—	8	2	no	—
192	#529	—	43	2	no	—
193	#499	—	43	1	no	—
194	#495	—	43	1	no	—
195	#488	—	43	2	no	—
196	#467	—	43	1	no	—
197	#466	—	43	1	no	—
198	#465	—	43	2	no	—
199	#446	—	43	2	no	—
200	C₈×Q₈	—	43	2	no	—
201	C₄×Q₁₆	—	43	1	no	—
202	C₄×SD₁₆	—	43	2	no	—
203	C₄×D₈	—	43	1	no	—
204	#436	—	43	2	no	—
205	#436	—	86	2	no	—
206	#435	—	86	2	no	—
207	C₈×D₄	—	43	2	no	—
208	#433	—	86	2	no	—
209	#431	C₂×D₄₃	1	1	yes	characteristic
210	#431	C₂×D₄₃	1	1	yes	characteristic
211	#424	—	86	2	no	—
212	#423	—	43	2	no	—
213	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
214	#335	—	43	2	no	—
215	#335	—	43	1	no	—
216	#335	—	43	1	no	—
217	Q₈⋊C₈	—	43	2	no	—
218	Q₈⋊C₈	—	43	2	no	—
219	D₄⋊C₈	—	43	2	no	—
220	D₄⋊C₈	—	43	2	no	—
221	#322	C₂×D₄₃	1	2	yes	—
222	#322	C₂×D₄₃	1	2	yes	—
223	C₄₃	#17992	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₁₆	—	43	1	no	—
225	C₂×SD₁₆	—	43	2	no	—
226	C₂×D₈	—	43	1	no	—
227	C₂×M₁₆	—	86	2	no	—
228	C₂×M₁₆	—	86	2	no	—
229	C₂²×C₈	—	86	2	no	—
230	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
231	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
232	#126	—	86	2	no	—
233	#125	—	43	2	no	—
234	#124	—	43	2	no	—
235	#123	—	86	2	no	—
236	C₂²⋊Q₈	—	43	2	no	—
237	#121	—	43	2	no	—
238	C₄×Q₈	—	43	2	no	—
239	C₄×D₄	—	43	2	no	—
240	C₄×D₄	—	86	2	no	—
241	#117	—	86	2	no	—
242	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
243	#107	—	43	1	no	—
244	#107	—	43	1	no	—
245	#106	—	43	2	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	43	2	no	—
248	C₄⋊C₈	—	43	2	no	—
249	C₄⋊C₈	—	86	2	no	—
250	C₄⋊C₈	—	86	2	no	—
251	C₄⋊C₈	—	86	2	no	—
252	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
253	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
254	#103	—	43	2	no	—
255	#103	—	43	2	no	—
256	#102	—	43	2	no	—
257	#102	—	43	2	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	86	2	no	—
260	C₂²⋊C₈	—	86	2	no	—
261	C₂²⋊C₈	D₁₇₂	1	2	yes	—
262	#97	—	86	2	no	—
263	C₄×C₈	—	86	2	no	—
264	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
265	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₄₃	1	2	yes	—
267	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₄₃	1	2	yes	—
270	C₂×Q₈	—	43	2	no	—
271	C₂×D₄	—	43	2	no	—
272	C₂×D₄	—	86	2	no	—
273	C₂²×C₄	—	86	2	no	—
274	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
276	C₂²×C₄	#10621898	1	1	yes	characteristic
277	Q₁₆	—	172	1	no	—
278	SD₁₆	—	172	2	no	—
279	D₈	—	172	1	no	—
280	M₁₆	—	344	2	no	—
281	M₁₆	—	344	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	86	2	no	—
286	C₂×C₈	—	86	2	no	—
287	C₂×C₈	—	344	2	no	—
288	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
292	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
293	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
294	C₄⋊C₄	—	43	2	no	—
295	C₄⋊C₄	—	43	2	no	—
296	C₄⋊C₄	—	86	2	no	—
297	C₄⋊C₄	—	86	2	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₂²⋊C₄	—	86	2	no	—
301	C₂²⋊C₄	—	86	2	no	—
302	C₂²⋊C₄	—	86	2	no	—
303	C₂²⋊C₄	—	86	2	no	—
304	C₂²⋊C₄	—	86	2	no	—
305	C₄²	—	86	2	no	—
306	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
307	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
308	C₄²	#10621898	1	1	yes	characteristic
309	C₄²	C₂³×D₄₃	1	1	yes	characteristic
310	C₂³	—	86	2	no	—
311	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
312	Q₈	—	86	2	no	—
313	Q₈	—	86	2	no	—
314	D₄	—	86	2	no	—
315	D₄	—	86	2	no	—
316	D₄	—	344	2	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	—	4	1	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	86	2	no	—
324	C₂×C₄	—	86	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
328	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
329	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
332	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	172	2	no	—
341	C₈	—	172	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	172	2	no	—
346	C₂²	—	172	2	no	—
347	C₂²	—	172	2	no	—
348	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	172	2	no	—
354	C₄	—	172	2	no	—
355	C₄	—	172	2	no	—
356	C₄	#14583665	1	1	yes	characteristic
357	C₄	#14583665	1	1	yes	characteristic
358	C₄	#14583704	1	1	yes	characteristic
359	C₄	#14583716	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	344	2	no	—
362	C₂	#19143314	1	1	yes	characteristic
363	C₂	#19143472	1	1	yes	characteristic
364	C₂	#19143513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1452	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583135	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1452	D₄₃
hypocenter	C₄₃	#17992
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #19143746	C₂	semidirect
3. #19133475	C₂	semidirect
4. #19133454	C₂	semidirect
5. #19133447	C₂	semidirect
6. #19133426	C₂	semidirect
7. #19133354	C₂	semidirect
8. #19133061	C₂	semidirect
9. C₄₃×#1452	C₂	semidirect
186. #1452	D₄₃	semidirect
223. C₄₃	#17992	semidirect

Maximal subgroups

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

subgroup	index
#19145165	2
#19143746	2
#19133475	2
#19133454	2
#19133447	2
#19133426	2
#19133354	2
#19133061	2
C₄₃×#1452	2
#17992	43