Subgroups — #25014848

Order 11008.

494 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145023	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582708	C₂	1	4	yes	maximal, complemented
3	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063347	—	2	1	no	—
6	C₂×#13063347	C₂²	1	2	yes	—
7	#14583449	—	2	1	no	—
8	C₂×#13063265	C₂²	1	2	yes	—
9	#14583283	—	2	1	no	—
10	#14583283	C₄	1	2	yes	complemented
11	#14583279	C₄	1	1	yes	characteristic, complemented
12	#14583276	C₄	1	1	yes	characteristic, complemented
13	#14583272	—	2	1	no	—
14	#14582708	—	2	8	no	—
15	#14582675	—	2	1	no	—
16	C₄×C₄₃⋊C₁₆	—	2	1	no	—
17	C₂×#107×C₄₃	—	2	1	no	—
18	C₂×#106×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#382	—	2	1	no	—
21	C₂×#10621898	—	2	2	no	—
22	#13063347	—	2	4	no	—
23	#13063347	—	2	4	no	—
24	#13063347	—	4	2	no	—
25	#13063347	—	4	2	no	—
26	C₂×Dic₁₇₂	—	2	1	no	—
27	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
28	C₂×#10621873	—	2	2	no	—
29	C₂×#10621873	—	2	2	no	—
30	C₂×D₃₄₄	—	2	1	no	—
31	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
32	#13063325	—	2	1	no	—
33	#13063322	—	2	1	no	—
34	#13063313	—	2	2	no	—
35	#13063299	—	4	2	no	—
36	C₄×D₁₇₂	—	4	2	no	—
37	#13063265	—	2	2	no	—
38	#13063265	D₄	1	4	yes	complemented
39	C₂×C₄₃⋊C₁₆	—	2	2	no	—
40	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	#107×C₄₃	—	4	2	no	—
45	#106×C₄₃	—	2	4	no	—
46	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
50	#95×C₄₃	—	4	2	no	—
51	C₂²×C₄×C₄₃	—	4	2	no	—
52	C₂²×C₄×C₄₃	D₈	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₃	—	4	2	no	—
55	#10621898	—	4	2	no	—
56	#10621898	—	4	2	no	—
57	#10621898	—	4	2	no	—
58	#10621898	—	4	2	no	—
59	C₂×D₁₇₂	—	2	2	no	—
60	C₂×D₁₇₂	—	4	2	no	—
61	C₂×C₄×D₄₃	—	4	2	no	—
62	C₂×Dic₈₆	—	2	2	no	—
63	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₃	D₈	1	2	yes	—
66	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	#41	1	2	yes	—
71	C₄⋊C₄×C₄₃	—	4	2	no	—
72	C₄⋊C₄×C₄₃	—	4	2	no	—
73	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₄₃	SD₁₆	1	2	yes	—
76	(C₂×D₄₃)⋊C₄	—	4	4	no	—
77	C₄⋊Dic₄₃	—	4	2	no	—
78	Dic₁₇₂	—	2	4	no	—
79	Dic₁₇₂	—	4	2	no	—
80	D₃₄₄	—	2	4	no	—
81	D₃₄₄	—	4	2	no	—
82	#10621873	—	4	4	no	—
83	#10621873	—	4	4	no	—
84	C₄₃⋊C₁₆	—	2	4	no	—
85	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₄₃	—	4	2	no	—
87	C₂×C₄×C₄₃	—	4	2	no	—
88	C₂×C₄×C₄₃	—	8	4	no	—
89	C₂×C₄×C₄₃	—	8	2	no	—
90	C₂×C₄×C₄₃	#102	1	2	yes	—
91	C₂×C₄×C₄₃	#102	1	2	yes	—
92	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂²⋊D₄₃	—	8	2	no	—
99	C₂²⋊D₄₃	—	8	2	no	—
100	C₂×Dic₄₃	—	4	2	no	—
101	D₁₇₂	—	4	2	no	—
102	D₁₇₂	—	4	2	no	—
103	D₁₇₂	—	8	4	no	—
104	C₄×D₄₃	—	8	2	no	—
105	C₄×D₄₃	—	8	2	no	—
106	Dic₈₆	—	4	2	no	—
107	Dic₈₆	—	4	2	no	—
108	C₈×C₄₃	#102	1	2	yes	—
109	C₈×C₄₃	#102	1	2	yes	—
110	C₈×C₄₃	C₄×D₄	1	2	yes	—
111	C₈×C₄₃	#121	1	2	yes	—
112	#13211	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
115	C₂²×C₄₃	#486	1	1	yes	characteristic
116	C₂²×C₄₃	#493	1	1	yes	characteristic
117	C₂×D₄₃	—	8	4	no	—
118	C₂×D₄₃	—	8	2	no	—
119	C₄×C₄₃	—	2	2	no	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	—	16	2	no	—
122	C₄×C₄₃	#390	1	1	yes	characteristic
123	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
124	C₄×C₄₃	#486	1	1	yes	characteristic
125	C₄×C₄₃	#492	1	1	yes	characteristic
126	Dic₄₃	—	8	2	no	—
127	#3147	—	43	1	no	—
128	C₂×#360	—	43	4	no	—
129	C₄×M₃₂	—	43	1	no	—
130	#1846	D₄₃	1	1	yes	characteristic, complemented
131	C₂×C₄₃	—	2	1	no	—
132	C₂×C₄₃	—	2	1	no	—
133	C₂×C₄₃	#1957	1	1	yes	characteristic
134	C₂×C₄₃	#2145	1	2	yes	—
135	D₄₃	—	16	2	no	—
136	C₂×#135	—	43	2	no	—
137	C₂×#135	—	86	1	no	—
138	#532	—	86	1	no	—
139	C₂×M₃₂	—	43	2	no	—
140	#495	—	43	2	no	—
141	#495	—	86	1	no	—
142	#494	—	43	1	no	—
143	#493	—	43	1	no	—
144	#492	—	86	1	no	—
145	C₂×#107	—	2	1	no	—
146	C₂×#106	C₂×D₄₃	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
148	#382	—	2	1	no	—
149	#360	—	86	8	no	—
150	#346	—	86	1	no	—
151	C₄×C₁₆	—	86	1	no	—
152	C₄₃	#13211	1	1	yes	characteristic, complemented, Sylow
153	C₂×#41	—	86	2	no	—
154	#135	—	86	4	no	—
155	#135	—	86	4	no	—
156	#135	—	172	2	no	—
157	#135	—	172	2	no	—
158	C₂×Q₁₆	—	43	2	no	—
159	C₂×Q₁₆	—	86	1	no	—
160	C₂×SD₁₆	—	86	2	no	—
161	C₂×SD₁₆	—	86	2	no	—
162	C₂×D₈	—	43	2	no	—
163	C₂×D₈	—	86	1	no	—
164	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
166	C₄⋊Q₈	—	86	1	no	—
167	#127	—	86	1	no	—
168	#124	—	86	2	no	—
169	#121	—	172	2	no	—
170	C₄×D₄	—	172	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
173	M₃₂	—	43	4	no	—
174	M₃₂	—	86	2	no	—
175	C₂×C₁₆	—	86	2	no	—
176	#107	—	4	2	no	—
177	#106	—	2	4	no	—
178	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
179	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
180	C₄×C₈	D₁₇₂	1	1	yes	characteristic
181	C₄×C₈	D₁₇₂	1	1	yes	characteristic
182	#95	—	4	2	no	—
183	#41	—	172	2	no	—
184	#41	—	172	2	no	—
185	#41	—	172	2	no	—
186	#41	—	172	2	no	—
187	C₂×Q₈	—	86	2	no	—
188	C₂×D₄	—	86	2	no	—
189	C₂×D₄	—	172	2	no	—
190	C₂×D₄	—	172	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	172	2	no	—
193	C₂²×C₄	#10621882	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	Q₁₆	—	86	4	no	—
196	Q₁₆	—	172	2	no	—
197	SD₁₆	—	172	4	no	—
198	SD₁₆	—	172	4	no	—
199	D₈	—	86	4	no	—
200	D₈	—	172	2	no	—
201	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	#10621882	1	2	yes	—
204	C₂×C₈	#10621884	1	2	yes	—
205	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
207	C₂×C₈	D₄×D₄₃	1	2	yes	—
208	C₂×C₈	#10621902	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	172	2	no	—
212	C₂²⋊C₄	—	172	4	no	—
213	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
214	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
215	C₄²	#10621884	1	2	yes	—
216	C₁₆	—	86	4	no	—
217	C₂³	—	172	2	no	—
218	C₂³	C₂×#10621882	1	1	yes	characteristic
219	Q₈	—	172	2	no	—
220	Q₈	—	172	2	no	—
221	D₄	—	344	4	no	—
222	D₄	—	344	2	no	—
223	D₄	—	344	2	no	—
224	D₄	—	172	2	no	—
225	D₄	—	172	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	4	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	344	2	no	—
230	C₂×C₄	—	344	2	no	—
231	C₂×C₄	—	172	2	no	—
232	C₂×C₄	#13063245	1	2	yes	—
233	C₂×C₄	#13063245	1	2	yes	—
234	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	#13063314	1	1	yes	characteristic
236	C₂×C₄	#13063321	1	1	yes	characteristic
237	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
238	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
239	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
240	C₈	#13063245	1	2	yes	—
241	C₈	#13063245	1	2	yes	—
242	C₈	#13063287	1	2	yes	—
243	C₈	#13063300	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	344	4	no	—
246	C₂²	—	344	2	no	—
247	C₂²	C₂×#13063245	1	1	yes	characteristic
248	C₂²	#14583247	1	1	yes	characteristic
249	C₂²	#14583275	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	16	2	no	—
253	C₄	—	344	2	no	—
254	C₄	#14582800	1	1	yes	characteristic
255	C₄	C₂×#13063245	1	1	yes	characteristic
256	C₄	#14583247	1	1	yes	characteristic
257	C₄	#14583273	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	688	2	no	—
261	C₂	#19135400	1	1	yes	characteristic
262	C₂	#19136330	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13063245
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1846	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	#19145023	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#13211
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
2. C₂×#14582708	C₂	semidirect
3. C₄×#13063265	C₂	semidirect
4. C₄₃×#1846	C₂	semidirect
10. #14583283	C₄	semidirect
11. #14583279	C₄	semidirect
12. #14583276	C₄	semidirect
38. #13063265	D₄	semidirect
130. #1846	D₄₃	semidirect
152. C₄₃	#13211	semidirect

Maximal subgroups

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

subgroup	index
#19145023	2
C₂×#14582708	2
C₄×#13063265	2
C₄₃×#1846	2
#13211	43