Subgroups — #25252025

Order 11008.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583335	C₂	1	4	yes	maximal, complemented
2	#19145023	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136859	C₂	1	4	yes	maximal, complemented
4	#19136850	C₂	1	2	yes	maximal, complemented
5	#19136844	C₂	1	2	yes	maximal, complemented
6	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	—	2	1	no	—
9	C₂×#13063347	C₂²	1	2	yes	—
10	#14583449	—	2	1	no	—
11	#14583335	—	2	16	no	—
12	C₂×#13063269	C₂²	1	8	yes	complemented
13	C₂×#13063266	C₂²	1	8	yes	complemented
14	C₂×#13063265	C₂²	1	2	yes	—
15	#14583283	—	2	1	no	—
16	#14583283	C₂²	1	2	yes	complemented
17	#14583279	C₂²	1	1	yes	characteristic, complemented
18	#14583276	C₂²	1	1	yes	characteristic, complemented
19	#14583272	—	2	1	no	—
20	#14582675	C₂²	1	2	yes	complemented
21	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
22	C₂²×D₈×C₄₃	—	2	1	no	—
23	C₂²×D₈×C₄₃	C₂²	1	2	yes	—
24	C₂×#127×C₄₃	—	2	1	no	—
25	C₄₃×#493	—	2	1	no	—
26	C₄₃×#493	—	2	1	no	—
27	C₄₃×#493	C₂²	1	2	yes	—
28	C₄₃×#493	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#10621898	—	2	2	no	—
31	#13063347	—	2	8	no	—
32	#13063347	—	4	4	no	—
33	C₂×Dic₁₇₂	—	2	1	no	—
34	C₂×Dic₁₇₂	C₂³	1	2	yes	—
35	C₂×#10621873	—	2	2	no	—
36	C₂×#10621873	—	2	2	no	—
37	C₂×D₃₄₄	—	2	1	no	—
38	C₂×D₃₄₄	C₂³	1	2	yes	—
39	#13063325	—	2	1	no	—
40	#13063322	—	2	1	no	—
41	#13063313	—	2	2	no	—
42	#13063299	—	4	2	no	—
43	C₄×D₁₇₂	—	4	2	no	—
44	#13063269	—	2	16	no	—
45	#13063266	—	2	16	no	—
46	#13063265	D₄	1	8	yes	complemented
47	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
48	C₂²×D₄×C₄₃	—	2	2	no	—
49	C₂²×D₄×C₄₃	—	4	2	no	—
50	C₂×D₈×C₄₃	—	2	8	no	—
51	C₂×D₈×C₄₃	—	2	2	no	—
52	C₂×D₈×C₄₃	—	2	2	no	—
53	C₂×D₈×C₄₃	—	2	2	no	—
54	C₂×D₈×C₄₃	—	4	4	no	—
55	C₂×D₈×C₄₃	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	#127×C₄₃	—	2	2	no	—
59	#127×C₄₃	—	2	2	no	—
60	#127×C₄₃	—	4	2	no	—
61	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
66	C₂⁴×C₄₃	—	4	2	no	—
67	C₂×D₄×C₄₃	—	2	4	no	—
68	C₂×D₄×C₄₃	—	4	4	no	—
69	C₂×D₄×C₄₃	—	4	4	no	—
70	C₂×D₄×C₄₃	—	4	2	no	—
71	C₂×D₄×C₄₃	—	4	4	no	—
72	C₂×D₄×C₄₃	—	4	4	no	—
73	C₂×D₄×C₄₃	—	8	8	no	—
74	C₂²×C₄×C₄₃	D₈	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₃	—	4	2	no	—
77	#10621898	—	4	4	no	—
78	#10621898	—	4	4	no	—
79	C₂×D₁₇₂	—	2	2	no	—
80	C₂×D₁₇₂	—	4	2	no	—
81	C₂×C₄×D₄₃	—	4	2	no	—
82	C₂×Dic₈₆	—	2	2	no	—
83	D₈×C₄₃	—	2	8	no	—
84	D₈×C₄₃	—	4	4	no	—
85	D₈×C₄₃	—	4	4	no	—
86	D₈×C₄₃	—	4	4	no	—
87	C₂×C₈×C₄₃	D₈	1	4	yes	—
88	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₄²×C₄₃	D₈	1	2	yes	—
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
96	(C₂×D₄₃)⋊C₄	—	4	4	no	—
97	C₄⋊Dic₄₃	—	4	2	no	—
98	Dic₁₇₂	—	2	4	no	—
99	Dic₁₇₂	—	4	2	no	—
100	D₃₄₄	—	2	4	no	—
101	D₃₄₄	—	4	2	no	—
102	#10621873	—	4	4	no	—
103	#10621873	—	4	4	no	—
104	C₄₃⋊C₁₆	C₂×D₄	1	8	yes	complemented
105	C₂³×C₄₃	—	4	4	no	—
106	C₂³×C₄₃	—	8	8	no	—
107	C₂³×C₄₃	—	8	4	no	—
108	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
109	C₂²×D₄₃	—	4	2	no	—
110	D₄×C₄₃	—	4	4	no	—
111	D₄×C₄₃	—	4	4	no	—
112	D₄×C₄₃	—	8	8	no	—
113	D₄×C₄₃	—	8	4	no	—
114	D₄×C₄₃	—	8	8	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	#127	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²⋊D₄₃	—	8	4	no	—
123	C₂×Dic₄₃	—	4	2	no	—
124	D₁₇₂	—	4	2	no	—
125	D₁₇₂	—	4	2	no	—
126	D₁₇₂	—	8	4	no	—
127	C₄×D₄₃	—	8	4	no	—
128	Dic₈₆	—	4	2	no	—
129	Dic₈₆	—	4	2	no	—
130	C₈×C₄₃	#127	1	2	yes	—
131	C₈×C₄₃	C₂×D₈	1	2	yes	—
132	C₈×C₄₃	C₂×D₈	1	2	yes	—
133	C₈×C₄₃	C₂²×D₄	1	2	yes	—
134	#33717	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	—	8	4	no	—
137	C₂²×C₄₃	—	8	8	no	—
138	C₂²×C₄₃	—	16	8	no	—
139	C₂²×C₄₃	#493	1	2	yes	—
140	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₄₃	—	8	2	no	—
142	C₂×D₄₃	—	8	4	no	—
143	C₄×C₄₃	—	2	2	no	—
144	C₄×C₄₃	#493	1	2	yes	—
145	C₄×C₄₃	#509	1	4	yes	complemented
146	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
147	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
148	Dic₄₃	—	8	2	no	—
149	C₂×#509	—	43	4	no	—
150	#3147	—	43	1	no	—
151	C₂×#493	D₄₃	1	1	yes	characteristic, complemented
152	#2250	—	43	4	no	—
153	#2248	—	43	2	no	—
154	#2246	—	43	2	no	—
155	C₄×M₃₂	—	43	1	no	—
156	C₂×C₄₃	—	2	2	no	—
157	C₂×C₄₃	—	16	4	no	—
158	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
159	C₂×C₄₃	C₂×#509	1	2	yes	—
160	D₄₃	—	16	2	no	—
161	C₂×#135	—	43	2	no	—
162	C₂×#135	—	86	1	no	—
163	C₂²×D₈	—	2	1	no	—
164	C₂²×D₈	C₂×D₄₃	1	2	yes	—
165	#532	—	86	1	no	—
166	C₂×#127	—	2	1	no	—
167	#509	—	86	16	no	—
168	C₂×SD₃₂	—	43	8	no	—
169	C₂×D₁₆	—	43	8	no	—
170	C₂×M₃₂	—	43	2	no	—
171	#495	—	43	2	no	—
172	#495	—	86	1	no	—
173	#494	—	43	1	no	—
174	#493	—	2	1	no	—
175	#493	—	2	1	no	—
176	#493	—	43	1	no	—
177	#493	C₂×D₄₃	1	2	yes	—
178	#493	C₂×D₄₃	1	2	yes	—
179	#492	—	86	1	no	—
180	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
181	#346	—	43	2	no	—
182	C₄×C₁₆	—	43	2	no	—
183	C₄₃	#33717	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	86	2	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	—	4	2	no	—
187	#135	—	86	8	no	—
188	#135	—	172	4	no	—
189	C₂×Q₁₆	—	43	2	no	—
190	C₂×Q₁₆	—	86	1	no	—
191	C₂×SD₁₆	—	86	2	no	—
192	C₂×SD₁₆	—	86	2	no	—
193	C₂×D₈	—	2	8	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	4	4	no	—
198	C₂×D₈	—	43	2	no	—
199	C₂×D₈	—	86	1	no	—
200	C₂×D₈	C₂²×D₄₃	1	4	yes	—
201	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	C₄⋊Q₈	—	86	1	no	—
204	#127	—	2	2	no	—
205	#127	—	2	2	no	—
206	#127	—	4	2	no	—
207	#127	—	86	1	no	—
208	#124	—	86	2	no	—
209	#121	—	172	2	no	—
210	C₄×D₄	—	172	2	no	—
211	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
212	SD₃₂	—	86	16	no	—
213	D₁₆	—	86	16	no	—
214	M₃₂	—	43	8	no	—
215	C₂×C₁₆	—	43	4	no	—
216	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
220	C₂⁴	—	4	2	no	—
221	#41	—	172	4	no	—
222	#41	—	172	4	no	—
223	C₂×Q₈	—	86	2	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	8	8	no	—
231	C₂×D₄	—	86	2	no	—
232	C₂×D₄	—	172	2	no	—
233	C₂×D₄	—	172	2	no	—
234	C₂²×C₄	—	172	2	no	—
235	C₂²×C₄	#10621882	1	2	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
237	Q₁₆	—	86	4	no	—
238	Q₁₆	—	172	2	no	—
239	SD₁₆	—	172	4	no	—
240	SD₁₆	—	172	4	no	—
241	D₈	—	2	8	no	—
242	D₈	—	4	4	no	—
243	D₈	—	4	4	no	—
244	D₈	—	4	4	no	—
245	D₈	—	86	4	no	—
246	D₈	—	172	2	no	—
247	C₂×C₈	#10621882	1	4	yes	—
248	C₂×C₈	D₄×D₄₃	1	4	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
253	C₄⋊C₄	—	172	2	no	—
254	C₂²⋊C₄	—	172	4	no	—
255	C₄²	#10621882	1	2	yes	—
256	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
258	C₁₆	—	43	8	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	8	8	no	—
261	C₂³	—	8	4	no	—
262	C₂³	—	172	2	no	—
263	C₂³	C₂×#10621882	1	1	yes	characteristic
264	Q₈	—	172	2	no	—
265	Q₈	—	172	2	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	8	8	no	—
269	D₄	—	8	8	no	—
270	D₄	—	8	4	no	—
271	D₄	—	344	4	no	—
272	D₄	—	344	4	no	—
273	D₄	—	172	2	no	—
274	D₄	—	172	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	344	4	no	—
277	C₂×C₄	—	172	2	no	—
278	C₂×C₄	#13063321	1	2	yes	—
279	C₂×C₄	C₂×#10621882	1	2	yes	—
280	C₂×C₄	C₂×#10621882	1	2	yes	—
281	C₂×C₄	C₂×#10621882	1	2	yes	—
282	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
284	C₈	#13063321	1	2	yes	—
285	C₈	C₂×#10621882	1	2	yes	—
286	C₈	C₂×#10621882	1	2	yes	—
287	C₈	C₂×D₄×D₄₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	8	no	—
291	C₂²	—	16	8	no	—
292	C₂²	—	344	2	no	—
293	C₂²	—	344	4	no	—
294	C₂²	#14583275	1	2	yes	—
295	C₂²	C₂²×#10621882	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	344	2	no	—
298	C₄	#14583275	1	2	yes	—
299	C₄	#14583335	1	4	yes	complemented
300	C₄	C₂×#13063321	1	1	yes	characteristic
301	C₄	C₂²×#10621882	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	4	no	—
304	C₂	—	688	2	no	—
305	C₂	C₂×#14583275	1	1	yes	characteristic
306	C₂	C₂×#14583335	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#493	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#33717
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#14583335	C₂	semidirect
2. #19145023	C₂	semidirect
3. #19136859	C₂	semidirect
4. #19136850	C₂	semidirect
5. #19136844	C₂	semidirect
6. C₄×#13063265	C₂	semidirect
7. C₂×C₄₃×#493	C₂	semidirect
12. C₂×#13063269	C₂²	semidirect
13. C₂×#13063266	C₂²	semidirect
16. #14583283	C₂²	semidirect
17. #14583279	C₂²	semidirect
18. #14583276	C₂²	semidirect
20. #14582675	C₂²	semidirect
21. C₄×C₄₃⋊C₁₆	C₂²	semidirect
46. #13063265	D₄	semidirect
104. C₄₃⋊C₁₆	C₂×D₄	semidirect
145. C₄×C₄₃	#509	semidirect
151. C₂×#493	D₄₃	semidirect
183. C₄₃	#33717	semidirect
299. C₄	#14583335	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583335	2
#19145023	2
#19136859	2
#19136850	2
#19136844	2
C₄×#13063265	2
C₂×C₄₃×#493	2
#33717	43