Subgroups — #25104701

Order 11008.

770 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583268	C₂	1	1	yes	maximal, characteristic
2	#19137336	C₂	1	1	yes	maximal, characteristic
3	#19137314	C₂	1	1	yes	maximal, characteristic
4	#19135560	C₂	1	1	yes	maximal, characteristic
5	#19135397	C₂	1	1	yes	maximal, characteristic
6	#19135285	C₂	1	2	yes	maximal
7	#19135271	C₂	1	2	yes	maximal
8	#19134729	C₂	1	1	yes	maximal, characteristic
9	#19134426	C₂	1	2	yes	maximal
10	C₄×#13063250	C₂	1	2	yes	maximal
11	C₄₃×#1726	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063325	C₂²	1	1	yes	characteristic
13	C₂×#13063317	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
15	#14583268	C₄	1	8	yes	complemented
16	C₂×#13063250	C₂²	1	2	yes	—
17	C₂×#13063250	C₂²	1	2	yes	—
18	C₂×#13063250	C₂²	1	2	yes	—
19	C₂×#13063250	C₂²	1	2	yes	—
20	#14582805	C₂²	1	2	yes	—
21	#14582797	C₂²	1	2	yes	—
22	#14582796	—	2	2	no	—
23	#14582786	—	2	2	no	—
24	#14582784	—	2	2	no	—
25	#14582784	—	2	2	no	—
26	#14582769	C₂²	1	1	yes	characteristic
27	#14582765	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
29	#14582662	C₂²	1	2	yes	—
30	#14582662	C₂²	1	2	yes	—
31	#14582662	C₂²	1	2	yes	—
32	#14582662	C₂²	1	2	yes	—
33	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#376	C₂²	1	1	yes	characteristic
40	C₂²×Dic₈₆	C₂³	1	2	yes	—
41	#13063325	—	2	2	no	—
42	#13063325	C₂×C₄	1	4	yes	—
43	#13063317	—	2	2	no	—
44	#13063317	C₂×C₄	1	4	yes	—
45	C₄×Dic₈₆	—	2	4	no	—
46	C₄×Dic₈₆	—	2	4	no	—
47	C₂×C₄⋊Dic₄₃	—	2	2	no	—
48	C₂×C₄⋊Dic₄₃	—	2	2	no	—
49	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
52	C₂×Dic₄₃⋊C₄	—	2	2	no	—
53	C₂×Dic₄₃⋊C₄	—	2	2	no	—
54	C₂×Dic₄₃⋊C₄	—	2	2	no	—
55	C₂×C₄×Dic₄₃	—	2	2	no	—
56	C₂×C₄×Dic₄₃	—	2	2	no	—
57	#13063250	—	2	4	no	—
58	#13063250	—	2	4	no	—
59	#13063250	C₂×C₄	1	8	yes	—
60	#13063250	C₂×C₄	1	8	yes	—
61	C₄₃⋊#95	—	2	2	no	—
62	C₄₃⋊#95	—	2	2	no	—
63	C₄₃⋊#95	—	2	2	no	—
64	C₄₃⋊#95	—	2	2	no	—
65	C₄₃⋊#95	—	2	2	no	—
66	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
73	#97×C₄₃	—	2	2	no	—
74	#97×C₄₃	—	2	2	no	—
75	#97×C₄₃	C₂×C₄	1	4	yes	—
76	C₄×C₈×C₄₃	—	2	2	no	—
77	C₄×C₈×C₄₃	—	2	2	no	—
78	C₄×C₈×C₄₃	—	2	2	no	—
79	#95×C₄₃	D₄	1	2	yes	—
80	#95×C₄₃	D₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	—	2	2	no	—
88	C₂²×Dic₄₃	—	2	2	no	—
89	C₂²×Dic₄₃	—	2	2	no	—
90	C₂×Dic₈₆	—	2	2	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	—	2	2	no	—
93	C₂×Dic₈₆	—	2	2	no	—
94	C₂×Dic₈₆	—	2	2	no	—
95	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	1	no	—
98	C₂×C₈×C₄₃	—	2	1	no	—
99	C₂×C₈×C₄₃	—	2	1	no	—
100	C₂×C₈×C₄₃	—	2	1	no	—
101	C₂×C₈×C₄₃	—	2	1	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	—	2	1	no	—
104	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₄₃	#41	1	2	yes	—
107	C₂×C₈×C₄₃	#41	1	2	yes	—
108	C₄²×C₄₃	—	2	1	no	—
109	C₄²×C₄₃	—	2	1	no	—
110	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₄₃	#41	1	2	yes	—
112	C₄²×C₄₃	#41	1	2	yes	—
113	C₄²×C₄₃	#41	1	2	yes	—
114	C₄⋊Dic₄₃	—	2	8	no	—
115	C₄⋊Dic₄₃	—	2	2	no	—
116	C₄⋊Dic₄₃	—	2	2	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	2	2	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄⋊Dic₄₃	—	8	2	no	—
121	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
122	Dic₄₃⋊C₄	—	2	8	no	—
123	Dic₄₃⋊C₄	—	4	4	no	—
124	Dic₄₃⋊C₄	—	8	2	no	—
125	C₄×Dic₄₃	—	4	4	no	—
126	C₄×Dic₄₃	—	8	2	no	—
127	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	2	1	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
137	C₂×C₄×C₄₃	#124	1	4	yes	—
138	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₄₃	—	2	4	no	—
145	C₂×Dic₄₃	—	2	4	no	—
146	C₂×Dic₄₃	—	4	2	no	—
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	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	Dic₈₆	—	2	4	no	—
157	Dic₈₆	—	2	4	no	—
158	Dic₈₆	—	4	2	no	—
159	Dic₈₆	—	4	2	no	—
160	C₈×C₄₃	—	2	2	no	—
161	C₈×C₄₃	—	2	2	no	—
162	C₈×C₄₃	—	2	2	no	—
163	C₈×C₄₃	—	2	2	no	—
164	#21653	—	43	1	no	maximal, Sylow
165	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
167	C₂²×C₄₃	#532	1	1	yes	characteristic
168	C₂²×C₄₃	#577	1	2	yes	—
169	C₂²×C₄₃	#578	1	2	yes	—
170	C₄×C₄₃	—	2	4	no	—
171	C₄×C₄₃	—	2	4	no	—
172	C₄×C₄₃	—	4	1	no	—
173	C₄×C₄₃	—	4	1	no	—
174	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
175	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
176	C₄×C₄₃	#529	1	1	yes	characteristic
177	C₄×C₄₃	#529	1	1	yes	characteristic
178	Dic₄₃	—	4	4	no	—
179	Dic₄₃	—	4	4	no	—
180	Dic₄₃	—	8	2	no	—
181	Dic₄₃	—	8	2	no	—
182	C₂×#491	—	43	1	no	—
183	#2330	—	43	1	no	—
184	#2327	—	43	1	no	—
185	#1991	—	43	1	no	—
186	#1956	—	43	1	no	—
187	#1935	—	43	2	no	—
188	#1933	—	43	2	no	—
189	#1836	—	43	1	no	—
190	#1763	—	43	2	no	—
191	C₄×#103	—	43	2	no	—
192	#1726	D₄₃	1	1	yes	characteristic
193	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
194	C₂×C₄₃	#2947	1	2	yes	—
195	C₂×C₄₃	#3137	1	2	yes	—
196	C₂×C₄₃	#3174	1	1	yes	characteristic
197	C₂×C₄₃	#3177	1	1	yes	characteristic
198	C₂×C₄⋊Q₈	—	43	1	no	—
199	C₂×#125	—	43	1	no	—
200	C₂×C₄×Q₈	—	43	2	no	—
201	#491	—	43	8	no	—
202	C₂×#103	—	43	2	no	—
203	C₂×#103	—	43	2	no	—
204	C₂×#103	—	43	2	no	—
205	C₂×#103	—	43	2	no	—
206	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
207	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
208	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
212	#391	—	43	2	no	—
213	#389	—	43	2	no	—
214	#389	—	86	2	no	—
215	#387	—	86	2	no	—
216	#387	—	86	2	no	—
217	#387	—	86	2	no	—
218	#384	—	43	1	no	—
219	#382	—	43	1	no	—
220	C₄×C₄⋊C₄	—	43	2	no	—
221	#376	C₂×D₄₃	1	1	yes	characteristic
222	#340	—	43	2	no	—
223	#340	—	43	2	no	—
224	#340	—	43	2	no	—
225	#340	—	43	2	no	—
226	C₄₃	#21653	1	1	yes	characteristic, complemented, Sylow
227	C₂²×Q₈	—	43	2	no	—
228	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
230	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₄⋊Q₈	—	43	4	no	—
233	C₄⋊Q₈	—	86	2	no	—
234	#125	—	43	4	no	—
235	#125	—	86	2	no	—
236	C₄×Q₈	—	86	4	no	—
237	C₄×Q₈	—	86	4	no	—
238	C₂×C₄⋊C₄	—	43	2	no	—
239	C₂×C₄⋊C₄	—	43	2	no	—
240	C₂×C₄⋊C₄	—	43	2	no	—
241	C₂×C₄⋊C₄	—	86	2	no	—
242	C₂×C₄⋊C₄	—	86	2	no	—
243	C₂×C₄⋊C₄	—	86	2	no	—
244	C₂×C₄⋊C₄	—	86	2	no	—
245	C₂×C₄⋊C₄	—	86	2	no	—
246	C₂×C₄²	—	86	2	no	—
247	C₂×C₄²	—	86	2	no	—
248	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#103	—	43	8	no	—
252	#103	—	43	8	no	—
253	#103	—	86	4	no	—
254	#103	—	86	4	no	—
255	#97	—	2	2	no	—
256	#97	—	2	2	no	—
257	#97	C₄×D₄₃	1	4	yes	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	—	2	2	no	—
261	#95	—	86	2	no	—
262	#95	—	86	2	no	—
263	#95	—	86	2	no	—
264	#95	—	86	2	no	—
265	#95	—	86	2	no	—
266	#95	D₁₇₂	1	2	yes	—
267	#95	D₁₇₂	1	2	yes	—
268	C₂×Q₈	—	43	4	no	—
269	C₂×Q₈	—	86	2	no	—
270	C₂×Q₈	—	86	2	no	—
271	C₂×Q₈	—	86	2	no	—
272	C₂×Q₈	—	86	2	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂²×C₄	—	86	2	no	—
275	C₂²×C₄	—	86	2	no	—
276	C₂²×C₄	—	86	2	no	—
277	C₂²×C₄	—	86	2	no	—
278	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
281	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
282	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
292	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
293	C₂×C₈	#10621898	1	2	yes	—
294	C₂×C₈	#10621898	1	2	yes	—
295	C₄⋊C₄	—	43	4	no	—
296	C₄⋊C₄	—	86	8	no	—
297	C₄⋊C₄	—	86	8	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	86	2	no	—
302	C₄⋊C₄	—	86	2	no	—
303	C₄⋊C₄	—	344	2	no	—
304	C₄⋊C₄	—	344	2	no	—
305	C₄⋊C₄	—	172	4	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	2	1	no	—
308	C₄²	—	344	2	no	—
309	C₄²	—	172	4	no	—
310	C₄²	C₂×C₄×D₄₃	1	2	yes	—
311	C₄²	#10621898	1	2	yes	—
312	C₄²	#10621898	1	2	yes	—
313	C₄²	#10621898	1	2	yes	—
314	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
315	Q₈	—	86	4	no	—
316	Q₈	—	86	4	no	—
317	Q₈	—	172	2	no	—
318	Q₈	—	172	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	86	4	no	—
328	C₂×C₄	—	86	4	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	—	172	2	no	—
333	C₂×C₄	—	172	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	—	172	2	no	—
338	C₂×C₄	—	172	2	no	—
339	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
340	C₂×C₄	#13063313	1	4	yes	—
341	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
343	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
344	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
345	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
346	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
352	C₂²	C₂×#13063313	1	1	yes	characteristic
353	C₂²	#14583449	1	1	yes	characteristic
354	C₂²	#14583711	1	2	yes	—
355	C₂²	#14583716	1	2	yes	—
356	C₄	—	2	4	no	—
357	C₄	—	2	4	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	344	2	no	—
361	C₄	—	344	2	no	—
362	C₄	—	172	4	no	—
363	C₄	—	172	4	no	—
364	C₄	C₄×#10621898	1	1	yes	characteristic
365	C₄	C₄×#10621898	1	1	yes	characteristic
366	C₄	#14583433	1	1	yes	characteristic
367	C₄	#14583433	1	1	yes	characteristic
368	C₂	C₄×#13063313	1	1	yes	characteristic
369	C₂	#19143440	1	2	yes	—
370	C₂	#19144962	1	2	yes	—
371	C₂	#19145223	1	1	yes	characteristic
372	C₂	#19145254	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1726	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1726	D₄₃
hypocenter	C₄₃	#21653
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
15. #14583268	C₄	semidirect
226. C₄₃	#21653	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583268	2
#19137336	2
#19137314	2
#19135560	2
#19135397	2
#19135285	2
#19135271	2
#19134729	2
#19134426	2
C₄×#13063250	2
C₄₃×#1726	2
#21653	43