Subgroups — #25251423

Order 11008.

632 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583135	C₂	1	1	yes	maximal, characteristic
2	#19136786	C₂	1	4	yes	maximal, complemented
3	#19136780	C₂	1	2	yes	maximal, complemented
4	#19136775	C₂	1	2	yes	maximal, complemented
5	#19136384	C₂	1	2	yes	maximal, complemented
6	#19136310	C₂	1	2	yes	maximal, complemented
7	#19136206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063299	—	2	1	no	—
11	C₂×#13063265	C₂²	1	2	yes	—
12	#14583135	—	2	1	no	—
13	#14583135	—	2	1	no	—
14	#14583135	C₂²	1	2	yes	—
15	#14583135	C₂²	1	2	yes	—
16	C₂×#13063259	C₂²	1	1	yes	characteristic
17	C₂×#13063247	—	2	1	no	—
18	#14582726	C₂²	1	4	yes	complemented
19	#14582724	C₂²	1	4	yes	complemented
20	#14582699	C₂²	1	4	yes	complemented
21	#14582699	C₂²	1	4	yes	complemented
22	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	complemented
23	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	complemented
24	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₃	—	2	1	no	—
26	C₄₃×#427	—	2	1	no	—
27	C₄₃×#427	—	2	1	no	—
28	C₄₃×#427	C₂²	1	2	yes	—
29	C₄₃×#427	C₂²	1	2	yes	complemented
30	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic, complemented
32	C₂×#106×C₄₃	—	2	1	no	—
33	C₂²×D₁₇₂	—	2	1	no	—
34	C₂²×C₂²⋊D₄₃	—	4	1	no	—
35	C₂×D₃₄₄	—	2	4	no	—
36	C₂×D₃₄₄	—	2	1	no	—
37	C₂×D₃₄₄	C₂³	1	2	yes	—
38	#13063299	—	2	2	no	—
39	#13063299	—	2	2	no	—
40	#13063299	—	4	1	no	—
41	#13063299	—	4	1	no	—
42	C₂×C₄⋊Dic₄₃	—	2	1	no	—
43	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
44	#13063265	—	2	4	no	—
45	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
46	#13063259	—	2	1	no	—
47	#13063259	C₂³	1	2	yes	—
48	#13063247	—	2	2	no	—
49	#13063247	—	2	2	no	—
50	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	—	2	4	no	—
52	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	#117×C₄₃	—	2	2	no	—
59	#117×C₄₃	—	2	2	no	—
60	#117×C₄₃	—	4	1	no	—
61	#117×C₄₃	—	4	1	no	—
62	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
65	C₂×C₄²×C₄₃	—	4	1	no	—
66	#107×C₄₃	—	2	1	no	—
67	#107×C₄₃	—	2	1	no	—
68	#107×C₄₃	C₂³	1	2	yes	—
69	#107×C₄₃	C₂³	1	2	yes	—
70	#106×C₄₃	—	2	2	no	—
71	#106×C₄₃	—	2	2	no	—
72	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
73	C₂³×D₄₃	—	4	1	no	—
74	C₂²×C₄×C₄₃	—	2	2	no	—
75	C₂²×C₄×C₄₃	—	2	2	no	—
76	C₂²×C₄×C₄₃	—	4	1	no	—
77	C₂²×C₄×C₄₃	—	4	1	no	—
78	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
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	1	yes	characteristic
84	C₂×C₂²⋊D₄₃	—	4	2	no	—
85	C₂×C₂²⋊D₄₃	—	4	2	no	—
86	C₂×C₂²⋊D₄₃	—	8	4	no	—
87	C₂²×Dic₄₃	—	4	1	no	—
88	C₂×D₁₇₂	—	2	2	no	—
89	C₂×D₁₇₂	—	4	1	no	—
90	C₂×D₁₇₂	—	4	2	no	—
91	C₂×D₁₇₂	—	4	2	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	#41	1	4	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄⋊C₄×C₄₃	—	4	1	no	—
106	C₂²⋊C₄×C₄₃	—	4	2	no	—
107	C₂²⋊C₄×C₄₃	—	4	2	no	—
108	C₄²×C₄₃	—	4	2	no	—
109	C₄²×C₄₃	—	4	2	no	—
110	(C₂×D₄₃)⋊C₄	—	4	2	no	—
111	(C₂×D₄₃)⋊C₄	—	4	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	4	1	no	—
114	D₃₄₄	—	2	4	no	—
115	D₃₄₄	—	4	2	no	—
116	C₄₃⋊C₁₆	—	2	4	no	—
117	C₂³×C₄₃	—	2	2	no	—
118	C₂³×C₄₃	—	2	2	no	—
119	C₂³×C₄₃	—	4	1	no	—
120	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
121	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
122	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₄₃	—	4	2	no	—
124	C₂²×D₄₃	—	8	2	no	—
125	C₂²×D₄₃	—	8	2	no	—
126	C₂²×D₄₃	—	8	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	1	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	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	8	2	no	—
137	C₂×C₄×C₄₃	—	8	2	no	—
138	C₂×C₄×C₄₃	#123	1	2	yes	—
139	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#137	1	2	yes	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₄₃	—	8	4	no	—
145	C₂²⋊D₄₃	—	8	4	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	8	2	no	—
148	D₁₇₂	—	4	2	no	—
149	D₁₇₂	—	4	2	no	—
150	D₁₇₂	—	8	2	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₈×C₄₃	—	2	1	no	—
153	C₈×C₄₃	#123	1	2	yes	—
154	C₈×C₄₃	C₂×#41	1	2	yes	—
155	#33648	—	43	1	no	maximal, Sylow
156	C₂²×C₄₃	—	2	1	no	—
157	C₂²×C₄₃	—	2	1	no	—
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	—	2	1	no	—
161	C₂²×C₄₃	—	2	1	no	—
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	4	4	no	—
164	C₂²×C₄₃	#480	1	2	yes	—
165	C₂²×C₄₃	#509	1	2	yes	—
166	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₂×D₄₃	—	8	2	no	—
170	C₂×D₄₃	—	16	4	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	2	1	no	—
173	C₄×C₄₃	—	8	2	no	—
174	C₄×C₄₃	—	8	2	no	—
175	C₄×C₄₃	#480	1	2	yes	—
176	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
177	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
178	Dic₄₃	—	8	2	no	—
179	C₂×#466	—	43	1	no	—
180	C₂×#427	D₄₃	1	1	yes	characteristic, complemented
181	#2234	—	43	4	no	—
182	#2233	—	43	2	no	—
183	#2232	—	43	2	no	—
184	#2159	—	43	2	no	—
185	#2140	—	43	2	no	—
186	#2112	—	43	1	no	—
187	C₂×C₄₃	—	2	2	no	—
188	C₂×C₄₃	—	2	2	no	—
189	C₂×C₄₃	—	4	1	no	—
190	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
191	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
192	C₂×C₄₃	#3414	1	1	yes	characteristic
193	D₄₃	—	16	2	no	—
194	C₂²×D₈	—	43	1	no	—
195	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
196	C₂×#121	—	86	1	no	—
197	C₂×#117	—	2	1	no	—
198	C₂×M₃₂	—	43	2	no	—
199	#466	—	43	2	no	—
200	#466	—	43	2	no	—
201	#466	—	86	1	no	—
202	#466	—	86	1	no	—
203	#427	—	2	1	no	—
204	#427	—	2	1	no	—
205	#427	C₂×D₄₃	1	2	yes	—
206	#427	C₂×D₄₃	1	2	yes	complemented
207	C₂×#107	—	43	1	no	—
208	C₂×#107	C₂×D₄₃	1	1	yes	characteristic, complemented
209	C₂×#107	C₂×D₄₃	1	1	yes	characteristic, complemented
210	C₂×#106	—	2	1	no	—
211	C₂×#102	—	86	1	no	—
212	#367	—	43	4	no	—
213	#366	—	43	4	no	—
214	#357	—	43	4	no	—
215	#357	—	43	4	no	—
216	C₂²⋊C₁₆	—	43	2	no	—
217	C₂²⋊C₁₆	—	43	2	no	—
218	C₄₃	#33648	1	1	yes	characteristic, complemented, Sylow
219	C₂²×D₄	—	86	1	no	—
220	C₂²×D₄	—	172	1	no	—
221	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂×D₈	—	43	2	no	—
223	C₂×D₈	—	86	4	no	—
224	C₂×D₈	—	86	1	no	—
225	C₂²×C₈	—	2	4	no	—
226	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
227	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
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	#121	—	86	2	no	—
233	#121	—	86	2	no	—
234	#121	—	172	1	no	—
235	#121	—	172	1	no	—
236	#117	—	2	2	no	—
237	#117	—	2	2	no	—
238	#117	—	4	1	no	—
239	#117	—	4	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	86	1	no	—
243	C₂×C₂²⋊C₄	—	4	1	no	—
244	C₂×C₂²⋊C₄	—	172	1	no	—
245	C₂×C₄²	—	4	1	no	—
246	M₃₂	—	86	4	no	—
247	C₂×C₁₆	—	43	4	no	—
248	#107	—	2	1	no	—
249	#107	—	2	1	no	—
250	#107	—	43	2	no	—
251	#107	—	86	1	no	—
252	#107	C₂²×D₄₃	1	2	yes	—
253	#107	C₂²×D₄₃	1	2	yes	—
254	#106	—	2	2	no	—
255	#106	—	2	2	no	—
256	#102	—	86	2	no	—
257	#102	—	86	2	no	—
258	C₂⁴	—	172	1	no	—
259	C₂⁴	#10621882	1	1	yes	characteristic
260	C₂×D₄	—	86	2	no	—
261	C₂×D₄	—	344	4	no	—
262	C₂×D₄	—	172	2	no	—
263	C₂×D₄	—	172	2	no	—
264	C₂×D₄	—	172	2	no	—
265	C₂×D₄	—	172	1	no	—
266	C₂×D₄	—	172	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	172	1	no	—
272	C₂²×C₄	#10621882	1	1	yes	characteristic
273	C₂²×C₄	#10621882	1	1	yes	characteristic
274	C₂²×C₄	#10621882	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
278	D₈	—	86	4	no	—
279	D₈	—	172	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	#10621898	1	4	yes	—
286	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	86	2	no	—
295	C₄⋊C₄	—	172	1	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	172	2	no	—
299	C₂²⋊C₄	—	172	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	2	no	—
302	C₁₆	—	86	4	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	—	344	2	no	—
307	C₂³	—	344	2	no	—
308	C₂³	—	344	2	no	—
309	C₂³	—	172	2	no	—
310	C₂³	C₂×#10621882	1	1	yes	characteristic
311	C₂³	C₂×#10621882	1	1	yes	characteristic
312	C₂³	C₂×#10621882	1	1	yes	characteristic
313	D₄	—	344	4	no	—
314	D₄	—	344	4	no	—
315	D₄	—	344	2	no	—
316	D₄	—	172	2	no	—
317	D₄	—	172	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	344	2	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	#13063308	1	2	yes	—
332	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
333	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
334	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
335	C₂×C₄	#13063356	1	2	yes	—
336	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	#13063308	1	2	yes	—
340	C₈	C₂×#10621898	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	4	4	no	—
349	C₂²	—	344	2	no	—
350	C₂²	—	344	2	no	—
351	C₂²	—	344	2	no	—
352	C₂²	—	688	4	no	—
353	C₂²	#14583212	1	2	yes	—
354	C₂²	#14583334	1	2	yes	—
355	C₂²	C₂²×#10621882	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	2	no	—
360	C₄	—	344	2	no	—
361	C₄	#14583212	1	2	yes	—
362	C₄	C₂×#13063308	1	1	yes	characteristic
363	C₄	C₂×#13063356	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	2	2	no	—
366	C₂	—	4	1	no	—
367	C₂	—	688	2	no	—
368	C₂	C₂×#14583212	1	1	yes	characteristic
369	C₂	C₂×#14583334	1	1	yes	characteristic
370	C₂	#19147729	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#427	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#14583135	C₂
hypercenter	C₂×#427	D₄₃
hypocenter	C₄₃	#33648
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #19136786	C₂	semidirect
3. #19136780	C₂	semidirect
4. #19136775	C₂	semidirect
5. #19136384	C₂	semidirect
6. #19136310	C₂	semidirect
7. #19136206	C₂	semidirect
8. C₂×C₄₃×#427	C₂	semidirect
18. #14582726	C₂²	semidirect
19. #14582724	C₂²	semidirect
20. #14582699	C₂²	semidirect
21. #14582699	C₂²	semidirect
22. C₂²⋊(C₄₃⋊C₁₆)	C₂²	semidirect
23. C₂²⋊(C₄₃⋊C₁₆)	C₂²	semidirect
29. C₄₃×#427	C₂²	semidirect
30. C₂×#107×C₄₃	C₂²	semidirect
31. C₂×#107×C₄₃	C₂²	semidirect
180. C₂×#427	D₄₃	semidirect
206. #427	C₂×D₄₃	semidirect
208. C₂×#107	C₂×D₄₃	semidirect
209. C₂×#107	C₂×D₄₃	semidirect
218. C₄₃	#33648	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583135	2
#19136786	2
#19136780	2
#19136775	2
#19136384	2
#19136310	2
#19136206	2
C₂×C₄₃×#427	2
#33648	43