Subgroups — #25105234

Order 11008.

1006 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145041	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583017	C₂	1	4	yes	maximal, complemented
3	#19143308	C₂	1	2	yes	maximal, complemented
4	#19135488	C₂	1	4	yes	maximal, complemented
5	#19135427	C₂	1	2	yes	maximal, complemented
6	#19134770	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	4	yes	—
9	C₂×#13063331	C₂²	1	4	yes	—
10	#14583449	C₂²	1	2	yes	—
11	#14583293	C₄	1	4	yes	complemented
12	#14583288	C₄	1	4	yes	complemented
13	#14583017	—	2	16	no	—
14	#14582978	—	2	4	no	—
15	C₈×D₁₇₂	—	2	4	no	—
16	C₂×#13063261	C₂²	1	4	yes	—
17	#14582923	C₂²	1	2	yes	—
18	C₂×#13063253	C₂²	1	8	yes	—
19	#14582865	C₂²	1	4	yes	—
20	C₂×#97×C₄₃	C₂²	1	2	yes	—
21	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
24	C₄₃×#376	C₂²	1	1	yes	characteristic
25	C₂×#10621898	C₂³	1	4	yes	—
26	#13063347	—	2	8	no	—
27	#13063347	—	2	8	no	—
28	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
29	C₂×#10621873	C₂×C₄	1	8	yes	—
30	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
31	#13063331	—	2	16	no	—
32	C₂×#10621877	C₂³	1	4	yes	—
33	C₂×#10621872	—	2	4	no	—
34	C₂×C₈×D₄₃	—	2	4	no	—
35	#13063325	C₂×C₄	1	2	yes	—
36	#13063322	C₂×C₄	1	2	yes	—
37	#13063313	C₂×C₄	1	4	yes	—
38	#13063299	—	2	4	no	—
39	C₄×D₁₇₂	—	2	4	no	—
40	#13063261	—	2	8	no	—
41	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
42	#13063253	—	2	16	no	—
43	(C₂×D₄₃)⋊C₈	—	2	8	no	—
44	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	#97×C₄₃	—	2	4	no	—
50	#97×C₄₃	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₄₃	D₄	1	2	yes	—
52	C₄×C₈×C₄₃	D₄	1	2	yes	—
53	C₄×C₈×C₄₃	D₄	1	8	yes	complemented
54	#95×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	#41	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
58	C₂×C₂²⋊D₄₃	—	2	4	no	—
59	#10621898	—	2	8	no	—
60	#10621898	—	2	8	no	—
61	C₂×D₁₇₂	—	2	4	no	—
62	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
63	C₂×C₄×D₄₃	—	2	4	no	—
64	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	#41	1	4	yes	—
73	C₄²×C₄₃	—	2	2	no	—
74	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₃	C₂×D₄	1	4	yes	—
76	C₄²×C₄₃	C₂×D₄	1	2	yes	—
77	(C₂×D₄₃)⋊C₄	—	2	8	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	#10621877	—	2	8	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	Dic₁₇₂	—	2	8	no	—
82	D₃₄₄	—	2	8	no	—
83	#10621873	—	2	16	no	—
84	#10621872	—	4	8	no	—
85	C₈×D₄₃	—	4	8	no	—
86	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
87	C₂²×D₄₃	—	2	4	no	—
88	C₂×C₄×C₄₃	—	2	4	no	—
89	C₂×C₄×C₄₃	—	4	2	no	—
90	C₂×C₄×C₄₃	—	4	4	no	—
91	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
96	C₂²⋊D₄₃	—	4	8	no	—
97	C₂×Dic₄₃	—	2	4	no	—
98	D₁₇₂	—	2	8	no	—
99	D₁₇₂	—	4	8	no	—
100	C₄×D₄₃	—	4	4	no	—
101	C₄×D₄₃	—	4	4	no	—
102	Dic₈₆	—	2	8	no	—
103	C₈×C₄₃	—	2	2	no	—
104	C₈×C₄₃	C₄×D₄	1	8	yes	—
105	C₈×C₄₃	#127	1	2	yes	—
106	C₈×C₄₃	#127	1	2	yes	—
107	C₄₃⋊C₈	—	4	4	no	—
108	#21708	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	—	2	2	no	—
110	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₄₃	#532	1	2	yes	—
112	C₂×D₄₃	—	4	4	no	—
113	C₂×D₄₃	—	4	8	no	—
114	C₄×C₄₃	—	2	4	no	—
115	C₄×C₄₃	—	4	4	no	—
116	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
117	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
118	C₄×C₄₃	#532	1	1	yes	characteristic
119	Dic₄₃	—	4	4	no	—
120	#3150	—	43	1	no	—
121	C₂×#443	—	43	4	no	—
122	#2931	—	43	2	no	—
123	#1975	—	43	4	no	—
124	#1963	—	43	2	no	—
125	#1849	—	43	1	no	—
126	#1726	D₄₃	1	1	yes	characteristic, complemented
127	C₂×C₄₃	—	2	2	no	—
128	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
129	C₂×C₄₃	#2955	1	2	yes	—
130	D₄₃	—	8	4	no	—
131	C₂×#135	—	43	4	no	—
132	C₂×#131	—	43	4	no	—
133	#532	—	43	2	no	—
134	#497	—	43	4	no	—
135	#496	—	43	4	no	—
136	#443	—	86	16	no	—
137	#436	—	86	4	no	—
138	C₈×D₄	—	86	4	no	—
139	C₂×#108	—	43	4	no	—
140	#423	—	43	2	no	—
141	C₂×#104	—	43	8	no	—
142	#408	—	43	4	no	—
143	C₂×#97	C₂×D₄₃	1	2	yes	—
144	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
147	#376	C₂×D₄₃	1	1	yes	characteristic
148	C₄₃	#21708	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	43	4	no	—
150	#135	—	86	8	no	—
151	#135	—	86	8	no	—
152	C₂×Q₁₆	—	43	4	no	—
153	C₂×SD₁₆	—	43	8	no	—
154	C₂×D₈	—	43	4	no	—
155	#131	—	86	16	no	—
156	C₂×M₁₆	—	43	4	no	—
157	C₂×M₁₆	—	86	4	no	—
158	C₂²×C₈	—	86	4	no	—
159	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
161	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	43	2	no	—
163	#127	—	43	2	no	—
164	#124	—	43	4	no	—
165	#121	—	86	4	no	—
166	C₄×D₄	—	86	4	no	—
167	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
168	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
169	#108	—	86	8	no	—
170	C₄⋊C₈	—	86	4	no	—
171	#104	—	86	16	no	—
172	C₂²⋊C₈	—	86	8	no	—
173	#97	—	2	4	no	—
174	#97	C₄×D₄₃	1	4	yes	—
175	C₄×C₈	D₁₇₂	1	2	yes	—
176	C₄×C₈	D₁₇₂	1	2	yes	—
177	C₄×C₈	D₁₇₂	1	8	yes	complemented
178	#95	—	2	2	no	—
179	#41	—	86	8	no	—
180	#41	—	86	8	no	—
181	C₂×Q₈	—	43	4	no	—
182	C₂×D₄	—	43	4	no	—
183	C₂×D₄	—	86	4	no	—
184	C₂×D₄	—	86	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	86	4	no	—
187	C₂²×C₄	#10621898	1	2	yes	—
188	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
189	Q₁₆	—	86	8	no	—
190	SD₁₆	—	86	16	no	—
191	D₈	—	86	8	no	—
192	M₁₆	—	86	8	no	—
193	M₁₆	—	172	8	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	86	4	no	—
196	C₂×C₈	—	172	8	no	—
197	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
198	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
199	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
200	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
201	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
202	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
203	C₂×C₈	#10621898	1	4	yes	—
204	C₄⋊C₄	—	86	4	no	—
205	C₂²⋊C₄	—	86	8	no	—
206	C₄²	—	2	2	no	—
207	C₄²	C₂×C₄×D₄₃	1	2	yes	—
208	C₄²	C₂×D₁₇₂	1	4	yes	—
209	C₄²	C₂×D₁₇₂	1	2	yes	—
210	C₂³	—	86	4	no	—
211	C₂³	C₂×#10621898	1	1	yes	characteristic
212	Q₈	—	86	8	no	—
213	D₄	—	86	8	no	—
214	D₄	—	172	8	no	—
215	D₄	—	172	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	86	4	no	—
220	C₂×C₄	—	172	4	no	—
221	C₂×C₄	—	172	4	no	—
222	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
223	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
225	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
226	C₂×C₄	C₂×#10621898	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	172	4	no	—
229	C₈	C₄×D₁₇₂	1	8	yes	—
230	C₈	#13063322	1	2	yes	—
231	C₈	#13063322	1	2	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	172	8	no	—
234	C₂²	—	172	4	no	—
235	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
236	C₂²	#14583449	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	4	4	no	—
239	C₄	—	172	4	no	—
240	C₄	C₂×C₄×D₁₇₂	1	2	yes	—
241	C₄	C₂×#13063322	1	1	yes	characteristic
242	C₄	#14583449	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	344	4	no	—
245	C₂	C₄×#13063322	1	1	yes	characteristic
246	C₂	#19143499	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1726	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#19145041	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1726	D₄₃
hypocenter	C₄₃	#21708
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. C₂×#14583017	C₂	semidirect
3. #19143308	C₂	semidirect
4. #19135488	C₂	semidirect
5. #19135427	C₂	semidirect
6. #19134770	C₂	semidirect
7. C₄₃×#1726	C₂	semidirect
11. #14583293	C₄	semidirect
12. #14583288	C₄	semidirect
53. C₄×C₈×C₄₃	D₄	semidirect
126. #1726	D₄₃	semidirect
148. C₄₃	#21708	semidirect
177. C₄×C₈	D₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
#19145041	2
C₂×#14583017	2
#19143308	2
#19135488	2
#19135427	2
#19134770	2
C₄₃×#1726	2
#21708	43