Subgroups — #25067299

Order 11008.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 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	#19133510	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133505	C₂	1	1	yes	maximal, characteristic, complemented
5	#19133487	C₂	1	2	yes	maximal, complemented
6	#19133420	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133415	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#10621873	C₂	1	2	yes	maximal, complemented
9	#19133351	C₂	1	1	yes	maximal, characteristic, complemented
10	#19133061	C₂	1	2	yes	maximal, complemented
11	C₄₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583433	C₂²	1	2	yes	—
13	#14583301	C₄	1	1	yes	characteristic
14	#14583283	C₄	1	1	yes	characteristic
15	#14583254	C₄	1	2	yes	—
16	#14583140	C₄	1	1	yes	characteristic
17	#14583135	C₄	1	1	yes	characteristic
18	#14583128	C₄	1	2	yes	—
19	#14583029	C₂²	1	2	yes	—
20	C₈×Dic₈₆	C₂²	1	2	yes	—
21	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
22	C₄×#10621873	C₂²	1	2	yes	—
23	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
24	#14582978	C₂²	1	2	yes	—
25	#14582974	—	2	2	no	—
26	#14582973	—	2	2	no	—
27	C₈×D₁₇₂	C₂²	1	2	yes	—
28	#14582959	—	2	2	no	—
29	#14582924	—	2	2	no	—
30	#14582923	C₂²	1	2	yes	—
31	#14582650	C₂²	1	2	yes	—
32	#14582650	C₂²	1	1	yes	characteristic
33	#14582650	C₂²	1	1	yes	characteristic
34	#14582631	C₂²	1	2	yes	—
35	#14582631	C₂²	1	2	yes	—
36	#14582628	C₂²	1	2	yes	—
37	#14582628	C₂²	1	2	yes	—
38	C₄₃×#431	C₂²	1	1	yes	characteristic
39	C₄₃×#431	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×#322	C₂²	1	1	yes	characteristic
42	C₄₃×#322	C₂²	1	1	yes	characteristic
43	C₈²×C₄₃	C₂²	1	1	yes	characteristic
44	C₈²×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#10621873	C₂×C₄	1	2	yes	—
47	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#10621877	—	2	2	no	—
49	C₂×#10621872	—	2	2	no	—
50	C₂×C₈×D₄₃	—	2	2	no	—
51	#13063319	—	2	2	no	—
52	#13063317	C₂×C₄	1	2	yes	—
53	#13063313	C₂×C₄	1	2	yes	—
54	#13063308	—	2	2	no	—
55	#13063304	C₂×C₄	1	2	yes	—
56	#13063299	C₂×C₄	1	2	yes	—
57	C₄×Dic₈₆	C₂³	1	2	yes	—
58	C₄×D₁₇₂	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₄₃	—	2	2	no	—
60	#13063282	—	2	2	no	—
61	#13063259	C₂×C₄	1	1	yes	characteristic
62	#13063259	C₂×C₄	1	1	yes	characteristic
63	#13063257	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
65	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
67	Dic₄₃⋊C₈	—	2	2	no	—
68	Dic₄₃⋊C₈	—	2	2	no	—
69	#13063250	C₂×C₄	1	2	yes	—
70	#13063250	C₂×C₄	1	2	yes	—
71	#13063247	C₂×C₄	1	2	yes	—
72	#13063247	C₂×C₄	1	2	yes	—
73	(C₂×D₄₃)⋊C₈	—	2	2	no	—
74	(C₂×D₄₃)⋊C₈	—	2	2	no	—
75	#13063235	—	2	2	no	—
76	C₈×Dic₄₃	—	2	2	no	—
77	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₄₃	—	2	1	no	—
81	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
82	C₂²⋊C₈×C₄₃	—	2	1	no	—
83	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
84	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₃	—	2	2	no	—
96	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₄₃	—	2	2	no	—
98	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	1	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
109	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₃	#41	1	1	yes	characteristic
112	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₄₃	—	2	2	no	—
114	(C₂×D₄₃)⋊C₄	—	2	2	no	—
115	(C₂×D₄₃)⋊C₄	—	2	2	no	—
116	(C₂×D₄₃)⋊C₄	—	2	2	no	—
117	(C₂×D₄₃)⋊C₄	—	2	2	no	—
118	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
119	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
120	Dic₄₃⋊C₄	—	2	2	no	—
121	Dic₄₃⋊C₄	—	2	2	no	—
122	Dic₄₃⋊C₄	—	2	2	no	—
123	Dic₄₃⋊C₄	—	2	2	no	—
124	C₄×Dic₄₃	—	2	2	no	—
125	#10621877	—	8	2	no	—
126	C₂×C₄₃⋊C₈	—	2	2	no	—
127	C₂×C₄₃⋊C₈	—	2	2	no	—
128	Dic₁₇₂	—	2	2	no	—
129	D₃₄₄	—	2	2	no	—
130	#10621873	—	2	4	no	—
131	#10621872	—	8	2	no	—
132	C₈×D₄₃	—	8	2	no	—
133	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
134	C₂²×D₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	1	no	—
136	C₂×C₄×C₄₃	—	2	1	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₂²⋊D₄₃	—	8	2	no	—
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	D₁₇₂	—	2	2	no	—
150	D₁₇₂	—	2	2	no	—
151	C₄×D₄₃	—	4	2	no	—
152	C₄×D₄₃	—	4	2	no	—
153	Dic₈₆	—	2	2	no	—
154	Dic₈₆	—	2	2	no	—
155	C₈×C₄₃	—	2	1	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₈×C₄₃	—	2	2	no	—
158	C₈×C₄₃	—	2	2	no	—
159	C₈×C₄₃	—	4	1	no	—
160	C₈×C₄₃	—	4	1	no	—
161	C₈×C₄₃	#131	1	2	yes	—
162	C₈×C₄₃	#131	1	2	yes	—
163	C₄₃⋊C₈	—	4	2	no	—
164	C₄₃⋊C₈	—	4	2	no	—
165	#17974	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	—	2	1	no	—
168	C₂²×C₄₃	—	2	1	no	—
169	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₄₃	—	4	2	no	—
171	C₂×D₄₃	—	4	2	no	—
172	C₂×D₄₃	—	4	2	no	—
173	C₄×C₄₃	—	2	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
178	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
179	C₄×C₄₃	#577	1	2	yes	—
180	Dic₄₃	—	4	2	no	—
181	Dic₄₃	—	4	2	no	—
182	Dic₄₃	—	4	2	no	—
183	#3168	—	43	1	no	—
184	#2993	—	43	2	no	—
185	#1594	—	43	1	no	—
186	#1593	—	43	1	no	—
187	#1590	—	43	2	no	—
188	#1580	—	43	1	no	—
189	#1579	—	43	1	no	—
190	C₈×SD₁₆	—	43	2	no	—
191	#1566	—	43	1	no	—
192	#1495	—	43	2	no	—
193	#1450	D₄₃	1	1	yes	characteristic, complemented
194	C₂×C₄₃	—	4	1	no	—
195	C₂×C₄₃	#2931	1	1	yes	characteristic
196	C₂×C₄₃	#2952	1	1	yes	characteristic
197	C₂×C₄₃	#2957	1	1	yes	characteristic
198	D₄₃	—	8	2	no	—
199	#529	—	43	2	no	—
200	#499	—	43	1	no	—
201	#495	—	43	1	no	—
202	#488	—	43	2	no	—
203	#467	—	43	1	no	—
204	#466	—	43	1	no	—
205	#465	—	43	2	no	—
206	#446	—	43	2	no	—
207	C₈×Q₈	—	43	2	no	—
208	C₄×Q₁₆	—	43	1	no	—
209	C₄×SD₁₆	—	43	2	no	—
210	C₄×D₈	—	43	1	no	—
211	#436	—	43	2	no	—
212	#436	—	86	2	no	—
213	#435	—	86	2	no	—
214	C₈×D₄	—	43	2	no	—
215	#433	—	86	2	no	—
216	#431	C₂×D₄₃	1	1	yes	characteristic
217	#431	C₂×D₄₃	1	1	yes	characteristic
218	#424	—	86	2	no	—
219	#423	—	43	2	no	—
220	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
221	#334	—	43	2	no	—
222	#334	—	43	1	no	—
223	#334	—	43	1	no	—
224	Q₈⋊C₈	—	43	2	no	—
225	Q₈⋊C₈	—	43	2	no	—
226	D₄⋊C₈	—	43	2	no	—
227	D₄⋊C₈	—	43	2	no	—
228	#322	C₂×D₄₃	1	1	yes	characteristic
229	#322	C₂×D₄₃	1	1	yes	characteristic
230	C₈²	C₂×D₄₃	1	1	yes	characteristic
231	C₈²	C₂×D₄₃	1	1	yes	characteristic
232	C₄₃	#17974	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₁₆	—	43	1	no	—
234	C₂×SD₁₆	—	43	2	no	—
235	C₂×D₈	—	43	1	no	—
236	C₂×M₁₆	—	86	2	no	—
237	C₂×M₁₆	—	86	2	no	—
238	C₂²×C₈	—	86	2	no	—
239	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
240	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
241	#126	—	86	2	no	—
242	#125	—	43	2	no	—
243	#124	—	43	2	no	—
244	#123	—	86	2	no	—
245	C₂²⋊Q₈	—	43	2	no	—
246	#121	—	43	2	no	—
247	C₄×Q₈	—	43	2	no	—
248	C₄×D₄	—	43	2	no	—
249	C₄×D₄	—	86	2	no	—
250	#117	—	86	2	no	—
251	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
252	#107	—	43	1	no	—
253	#107	—	43	1	no	—
254	#106	—	43	2	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	43	2	no	—
257	C₄⋊C₈	—	43	2	no	—
258	C₄⋊C₈	—	86	2	no	—
259	C₄⋊C₈	—	86	2	no	—
260	C₄⋊C₈	—	86	2	no	—
261	C₄⋊C₈	D₁₇₂	1	2	yes	—
262	#103	—	43	2	no	—
263	#103	—	43	2	no	—
264	#102	—	43	2	no	—
265	#102	—	43	2	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	86	2	no	—
268	C₂²⋊C₈	—	86	2	no	—
269	C₂²⋊C₈	D₁₇₂	1	2	yes	—
270	#97	—	86	2	no	—
271	C₄×C₈	—	86	2	no	—
272	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
273	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
280	C₂×Q₈	—	43	2	no	—
281	C₂×D₄	—	43	2	no	—
282	C₂×D₄	—	86	2	no	—
283	C₂²×C₄	—	86	2	no	—
284	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
286	C₂²×C₄	#10621898	1	1	yes	characteristic
287	Q₁₆	—	86	2	no	—
288	SD₁₆	—	86	4	no	—
289	D₈	—	86	2	no	—
290	M₁₆	—	344	2	no	—
291	M₁₆	—	344	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	86	2	no	—
297	C₂×C₈	—	86	2	no	—
298	C₂×C₈	—	344	2	no	—
299	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
303	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
304	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
305	C₄⋊C₄	—	43	2	no	—
306	C₄⋊C₄	—	43	2	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	86	2	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₂²⋊C₄	—	86	2	no	—
312	C₂²⋊C₄	—	86	2	no	—
313	C₂²⋊C₄	—	86	2	no	—
314	C₂²⋊C₄	—	86	2	no	—
315	C₂²⋊C₄	—	86	2	no	—
316	C₄²	—	86	2	no	—
317	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
318	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
319	C₄²	#10621898	1	1	yes	characteristic
320	C₄²	C₂³×D₄₃	1	1	yes	characteristic
321	C₂³	—	86	2	no	—
322	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
323	Q₈	—	86	2	no	—
324	Q₈	—	86	2	no	—
325	D₄	—	86	2	no	—
326	D₄	—	86	2	no	—
327	D₄	—	344	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	86	2	no	—
333	C₂×C₄	—	86	2	no	—
334	C₂×C₄	—	86	2	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
338	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
339	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
342	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	2	no	—
346	C₈	—	2	2	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	1	no	—
349	C₈	—	172	2	no	—
350	C₈	—	172	2	no	—
351	C₈	#13063331	1	2	yes	—
352	C₈	#13063331	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	172	2	no	—
357	C₂²	—	172	2	no	—
358	C₂²	—	172	2	no	—
359	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	172	2	no	—
365	C₄	—	172	2	no	—
366	C₄	—	172	2	no	—
367	C₄	C₂×#13063331	1	1	yes	characteristic
368	C₄	C₂×#13063331	1	1	yes	characteristic
369	C₄	#14583711	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	344	2	no	—
372	C₂	#19143308	1	1	yes	characteristic
373	C₂	#19143480	1	1	yes	characteristic
374	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×#1450	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583135	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1450	D₄₃
hypocenter	C₄₃	#17974
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #19143746	C₂	semidirect
3. #19133510	C₂	semidirect
4. #19133505	C₂	semidirect
5. #19133487	C₂	semidirect
6. #19133420	C₂	semidirect
7. #19133415	C₂	semidirect
8. C₈×#10621873	C₂	semidirect
9. #19133351	C₂	semidirect
10. #19133061	C₂	semidirect
11. C₄₃×#1450	C₂	semidirect
193. #1450	D₄₃	semidirect
232. C₄₃	#17974	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
#19133510	2
#19133505	2
#19133487	2
#19133420	2
#19133415	2
C₈×#10621873	2
#19133351	2
#19133061	2
C₄₃×#1450	2
#17974	43