Subgroups — #25084536

Order 11008.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145713	C₂	1	2	yes	maximal, complemented
2	C₂×#14583304	C₂	1	1	yes	maximal, characteristic, complemented
3	#19145055	C₂	1	2	yes	maximal, complemented
4	#19134268	C₂	1	4	yes	maximal, complemented
5	#19134212	C₂	1	2	yes	maximal, complemented
6	#19134184	C₂	1	2	yes	maximal, complemented
7	C₈×#10621877	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#3169	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063355	—	2	4	no	—
10	C₂×#13063325	C₂²	1	2	yes	—
11	#14583304	—	2	2	no	—
12	#14583304	C₂²	1	2	yes	—
13	#14583304	C₂²	1	2	yes	—
14	#14583302	C₂²	1	4	yes	—
15	#14583294	—	2	4	no	—
16	#14583278	C₂²	1	4	yes	—
17	#14583271	C₂²	1	4	yes	—
18	#14583245	C₂²	1	4	yes	—
19	#14583210	—	2	4	no	—
20	#14583179	—	2	4	no	—
21	#14582952	C₂²	1	1	yes	characteristic
22	C₂×#13063259	—	2	2	no	—
23	C₄×#10621877	C₂²	1	1	yes	characteristic
24	#14582621	C₂²	1	2	yes	—
25	C₈×C₄₃⋊C₈	C₂²	1	2	yes	—
26	C₄₃×#532	C₂²	1	2	yes	—
27	C₄₃×#500	C₂²	1	1	yes	characteristic
28	C₄₃×#493	C₂²	1	1	yes	characteristic
29	C₄₃×#486	C₂²	1	2	yes	—
30	C₄₃×#466	—	2	2	no	—
31	C₄×D₈×C₄₃	—	2	2	no	—
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₈₆	—	2	2	no	—
34	#13063355	—	4	16	no	—
35	C₂×#10621885	—	2	8	no	—
36	C₂×#10621883	—	2	8	no	—
37	C₂×#10621877	D₄	1	2	yes	—
38	#13063325	—	2	2	no	—
39	#13063325	—	2	2	no	—
40	#13063325	—	2	2	no	—
41	#13063325	C₂³	1	4	yes	—
42	C₂×C₄⋊Dic₄₃	—	2	2	no	—
43	C₂×C₄⋊Dic₄₃	—	2	2	no	—
44	#13063259	—	2	4	no	—
45	#13063259	—	2	4	no	—
46	#13063258	—	2	8	no	—
47	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
48	#13063246	—	2	8	no	—
49	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
50	#13063236	D₄	1	2	yes	—
51	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
52	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
53	C₂×#41×C₄₃	—	2	2	no	—
54	C₂×D₈×C₄₃	—	2	2	no	—
55	C₂²×C₈×C₄₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
57	#127×C₄₃	C₂³	1	2	yes	—
58	#124×C₄₃	—	2	2	no	—
59	#121×C₄₃	—	2	2	no	—
60	#121×C₄₃	—	2	2	no	—
61	C₄×D₄×C₄₃	—	2	2	no	—
62	C₄×D₄×C₄₃	—	2	2	no	—
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	#107×C₄₃	—	2	2	no	—
65	#102×C₄₃	—	2	4	no	—
66	C₄×C₈×C₄₃	D₄	1	2	yes	—
67	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	#41×C₄₃	—	4	8	no	—
70	C₂×Q₈×C₄₃	—	2	2	no	—
71	C₂×D₄×C₄₃	—	2	2	no	—
72	C₂×D₄×C₄₃	—	2	2	no	—
73	C₂×D₄×C₄₃	—	2	2	no	—
74	C₂×D₄×C₄₃	—	4	2	no	—
75	C₂²×C₄×C₄₃	—	4	2	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	4	2	no	—
80	C₂×Dic₈₆	—	2	4	no	—
81	C₂×Dic₈₆	—	4	8	no	—
82	C₂×Dic₈₆	—	4	2	no	—
83	D₈×C₄₃	—	4	4	no	—
84	C₂×C₈×C₄₃	—	2	4	no	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₄₃	—	2	2	no	—
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₂²⋊C₄×C₄₃	—	4	2	no	—
90	C₂²⋊C₄×C₄₃	—	4	2	no	—
91	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
95	#10621885	—	4	16	no	—
96	#10621883	—	4	16	no	—
97	C₄⋊Dic₄₃	—	2	4	no	—
98	C₄⋊Dic₄₃	—	2	4	no	—
99	C₄⋊Dic₄₃	—	4	2	no	—
100	C₄⋊Dic₄₃	—	4	2	no	—
101	#10621877	D₈	1	8	yes	complemented
102	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
103	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
104	C₂³×C₄₃	—	4	2	no	—
105	C₂³×C₄₃	#127	1	1	yes	characteristic
106	Q₈×C₄₃	—	4	4	no	—
107	D₄×C₄₃	—	4	4	no	—
108	D₄×C₄₃	—	4	4	no	—
109	D₄×C₄₃	—	4	4	no	—
110	D₄×C₄₃	—	8	4	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	8	4	no	—
115	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
119	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₄₃	—	4	4	no	—
123	C₂×Dic₄₃	—	8	4	no	—
124	Dic₈₆	—	4	8	no	—
125	Dic₈₆	—	8	4	no	—
126	C₈×C₄₃	—	2	2	no	—
127	C₈×C₄₃	#137	1	4	yes	complemented
128	C₄₃⋊C₈	—	2	4	no	—
129	C₄₃⋊C₈	C₂×D₈	1	8	yes	—
130	#19630	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	2	1	no	—
132	C₂²×C₄₃	—	2	1	no	—
133	C₂²×C₄₃	—	8	2	no	—
134	C₂²×C₄₃	—	8	2	no	—
135	C₂²×C₄₃	—	8	2	no	—
136	C₂²×C₄₃	#493	1	2	yes	—
137	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	2	1	no	—
140	C₄×C₄₃	—	8	2	no	—
141	C₄×C₄₃	#493	1	2	yes	—
142	C₄×C₄₃	C₂²×D₈	1	2	yes	—
143	C₄×C₄₃	C₂×#137	1	2	yes	—
144	C₄×C₄₃	#578	1	2	yes	—
145	Dic₄₃	—	8	4	no	—
146	#3226	—	43	2	no	—
147	#3169	D₄₃	1	1	yes	characteristic
148	C₂×#500	—	43	1	no	—
149	#3152	—	43	2	no	—
150	#1722	—	43	4	no	—
151	#1713	—	43	2	no	—
152	#1706	—	43	2	no	—
153	C₈×M₁₆	—	43	1	no	—
154	C₂×C₄₃	—	2	2	no	—
155	C₂×C₄₃	—	16	2	no	—
156	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
157	C₂×C₄₃	#3152	1	1	yes	characteristic
158	C₂×C₄₃	#3156	1	1	yes	characteristic
159	C₂×#137	—	86	4	no	—
160	#532	C₂×D₄₃	1	2	yes	—
161	C₂×C₄⋊Q₈	—	43	2	no	—
162	#500	—	43	4	no	—
163	#500	—	43	2	no	—
164	#500	—	43	2	no	—
165	#500	—	86	2	no	—
166	#500	C₂×D₄₃	1	1	yes	characteristic
167	#497	—	86	4	no	—
168	#494	—	43	4	no	—
169	#493	C₂×D₄₃	1	1	yes	characteristic
170	#492	—	43	4	no	—
171	#486	—	43	4	no	—
172	#486	C₂×D₄₃	1	2	yes	—
173	#480	—	86	4	no	—
174	#474	—	86	4	no	—
175	#466	—	2	2	no	—
176	C₄×D₈	—	2	2	no	—
177	#431	—	43	1	no	—
178	C₂×#107	—	86	2	no	—
179	C₄×M₁₆	—	43	1	no	—
180	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
181	#322	—	43	2	no	—
182	C₈²	—	43	2	no	—
183	C₄₃	#19630	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	2	2	no	—
185	C₂²×Q₈	—	86	2	no	—
186	#137	—	172	16	no	—
187	C₂×Q₁₆	—	86	8	no	—
188	C₂×SD₁₆	—	86	8	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×M₁₆	—	43	2	no	—
191	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
192	C₄⋊Q₈	—	43	4	no	—
193	C₄⋊Q₈	—	86	2	no	—
194	C₄⋊Q₈	—	86	2	no	—
195	C₄⋊Q₈	—	86	2	no	—
196	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
197	#127	C₂²×D₄₃	1	2	yes	—
198	#124	—	2	2	no	—
199	#121	—	2	2	no	—
200	#121	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	86	2	no	—
204	C₂×C₄⋊C₄	—	86	2	no	—
205	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
206	#107	—	2	2	no	—
207	#107	—	86	8	no	—
208	#107	—	86	4	no	—
209	#107	—	86	4	no	—
210	C₄⋊C₈	—	43	2	no	—
211	#102	—	2	4	no	—
212	#102	—	86	8	no	—
213	C₂²⋊C₈	—	43	2	no	—
214	#97	—	43	2	no	—
215	C₄×C₈	—	43	2	no	—
216	C₄×C₈	—	43	2	no	—
217	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
218	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
220	#41	—	4	8	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	86	4	no	—
223	C₂×Q₈	—	172	8	no	—
224	C₂×Q₈	—	172	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	172	2	no	—
231	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
234	Q₁₆	—	172	16	no	—
235	SD₁₆	—	172	16	no	—
236	D₈	—	4	4	no	—
237	M₁₆	—	43	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	43	4	no	—
240	C₂×C₈	—	43	4	no	—
241	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	86	4	no	—
246	C₄⋊C₄	—	86	4	no	—
247	C₄⋊C₄	—	172	2	no	—
248	C₄⋊C₄	—	172	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	D₄×D₄₃	1	1	yes	characteristic
252	C₄²	D₄×D₄₃	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
254	C₄²	C₂³×D₄₃	1	1	yes	characteristic
255	C₂³	—	4	2	no	—
256	C₂³	#13063321	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	344	4	no	—
259	Q₈	—	172	8	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	8	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	344	4	no	—
269	C₂×C₄	—	172	4	no	—
270	C₂×C₄	#13063321	1	1	yes	characteristic
271	C₂×C₄	#13063321	1	1	yes	characteristic
272	C₂×C₄	#13063321	1	1	yes	characteristic
273	C₂×C₄	D₈×D₄₃	1	4	yes	—
274	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	43	8	no	—
279	C₈	—	86	4	no	—
280	C₈	#13063355	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	#14583274	1	2	yes	—
287	C₂²	C₂×#13063321	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	—	344	4	no	—
292	C₄	#14583274	1	2	yes	—
293	C₄	C₂×D₈×D₄₃	1	2	yes	—
294	C₄	C₂×#13063355	1	2	yes	—
295	C₄	#14583713	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₂×#14583274	1	1	yes	characteristic
299	C₂	#19145055	1	1	yes	characteristic
300	C₂	#19145090	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063321
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3169	C₂
socle	C₂²×C₄₃	C₂×#127
hypercenter	#3169	D₄₃
hypocenter	C₄₃	#19630
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #19145713	C₂	semidirect
2. C₂×#14583304	C₂	semidirect
3. #19145055	C₂	semidirect
4. #19134268	C₂	semidirect
5. #19134212	C₂	semidirect
6. #19134184	C₂	semidirect
7. C₈×#10621877	C₂	semidirect
101. #10621877	D₈	semidirect
127. C₈×C₄₃	#137	semidirect
183. C₄₃	#19630	semidirect
280. C₈	#13063355	semidirect

Maximal subgroups

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

subgroup	index
#19145713	2
C₂×#14583304	2
#19145055	2
#19134268	2
#19134212	2
#19134184	2
C₈×#10621877	2
C₄₃×#3169	2
#19630	43