Subgroups — #25104940

Order 11008.

658 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137900	C₂	1	1	yes	maximal, characteristic
2	#19135318	C₂	1	2	yes	maximal
3	#19135159	C₂	1	4	yes	maximal
4	#19134715	C₂	1	1	yes	maximal, characteristic
5	C₈×Dic₄₃⋊C₄	C₂	1	4	yes	maximal
6	C₈×C₄⋊Dic₄₃	C₂	1	2	yes	maximal
7	C₄₃×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₄₃⋊C₈	C₂²	1	4	yes	—
10	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
11	#14582841	C₄	1	4	yes	—
12	#14582833	C₄	1	4	yes	—
13	C₄×Dic₄₃⋊C₄	C₂²	1	4	yes	—
14	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	#14582654	C₂²	1	4	yes	—
16	#14582653	C₂²	1	4	yes	—
17	#14582653	C₂²	1	4	yes	—
18	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#97×C₄₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#376	C₂²	1	1	yes	characteristic
23	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
24	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
25	C₂×Dic₄₃⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×Dic₄₃⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
28	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
29	Dic₄₃⋊C₈	—	2	8	no	—
30	C₈×Dic₄₃	—	2	8	no	—
31	C₄₃⋊#95	C₂×C₄	1	2	yes	—
32	C₄₃⋊#95	C₂×C₄	1	4	yes	—
33	C₄₃⋊#95	C₂×C₄	1	2	yes	—
34	C₄₃⋊#95	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	#97×C₄₃	—	2	4	no	—
41	#97×C₄₃	—	2	2	no	—
42	C₄×C₈×C₄₃	—	2	4	no	—
43	C₄×C₈×C₄₃	Q₈	1	4	yes	—
44	#95×C₄₃	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
49	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₄₃	—	2	2	no	—
51	C₂×C₈×C₄₃	—	2	2	no	—
52	C₂×C₈×C₄₃	—	2	2	no	—
53	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₄₃	#41	1	4	yes	—
56	C₂×C₈×C₄₃	#41	1	4	yes	—
57	C₄²×C₄₃	—	2	2	no	—
58	C₄²×C₄₃	—	2	2	no	—
59	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
60	C₄²×C₄₃	#41	1	2	yes	—
61	C₄⋊Dic₄₃	—	2	8	no	—
62	Dic₄₃⋊C₄	—	2	8	no	—
63	Dic₄₃⋊C₄	—	2	8	no	—
64	C₄×Dic₄₃	—	2	4	no	—
65	C₄×Dic₄₃	—	2	4	no	—
66	C₂×C₄₃⋊C₈	—	2	4	no	—
67	C₂×C₄₃⋊C₈	—	2	4	no	—
68	C₂×C₄₃⋊C₈	—	2	4	no	—
69	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	—	2	4	no	—
73	C₂×C₄×C₄₃	—	2	4	no	—
74	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
75	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
77	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
79	C₂×Dic₄₃	—	2	4	no	—
80	C₂×Dic₄₃	—	2	4	no	—
81	C₂×Dic₄₃	—	2	4	no	—
82	C₂×Dic₄₃	—	2	4	no	—
83	C₂×Dic₄₃	—	2	4	no	—
84	C₂×Dic₄₃	#131	1	8	yes	—
85	C₈×C₄₃	—	2	4	no	—
86	C₈×C₄₃	—	2	2	no	—
87	C₈×C₄₃	#125	1	4	yes	—
88	C₄₃⋊C₈	—	4	4	no	—
89	#21676	—	43	1	no	maximal, Sylow
90	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂²×C₄₃	C₄×#41	1	2	yes	—
92	C₂²×C₄₃	C₂×#131	1	4	yes	—
93	C₄×C₄₃	—	2	4	no	—
94	C₄×C₄₃	—	4	4	no	—
95	C₄×C₄₃	C₂×#125	1	1	yes	characteristic
96	C₄×C₄₃	#529	1	2	yes	—
97	C₄×C₄₃	#533	1	1	yes	characteristic
98	Dic₄₃	—	2	8	no	—
99	Dic₄₃	—	4	4	no	—
100	#2402	—	43	1	no	—
101	#1940	—	43	2	no	—
102	#1917	—	43	4	no	—
103	#1832	—	43	1	no	—
104	C₈×C₄⋊C₄	—	43	2	no	—
105	C₈×C₄⋊C₄	—	43	4	no	—
106	#1726	D₄₃	1	1	yes	characteristic
107	C₂×C₄₃	C₄×#125	1	1	yes	characteristic
108	C₂×C₄₃	#2960	1	2	yes	—
109	C₂×C₄₃	#2966	1	4	yes	—
110	C₂×C₄⋊C₈	—	43	4	no	—
111	C₂×C₄⋊C₈	—	43	2	no	—
112	C₂×#97	C₂×D₄₃	1	2	yes	—
113	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
114	C₂×C₄×C₈	—	43	4	no	—
115	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
117	#400	—	43	4	no	—
118	#398	—	43	4	no	—
119	C₄×C₄⋊C₄	—	43	4	no	—
120	C₄×C₄⋊C₄	—	43	2	no	—
121	#376	C₂×D₄₃	1	1	yes	characteristic
122	#336	—	43	4	no	—
123	#336	—	43	4	no	—
124	#336	—	43	4	no	—
125	C₄₃	#21676	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	43	4	no	—
127	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
129	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	43	4	no	—
131	C₂×C₄⋊C₄	—	43	4	no	—
132	C₂×C₄⋊C₄	—	43	4	no	—
133	C₂×C₄²	—	43	4	no	—
134	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
135	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
136	C₄⋊C₈	—	86	8	no	—
137	C₄⋊C₈	—	86	4	no	—
138	#97	—	2	4	no	—
139	#97	—	2	2	no	—
140	C₄×C₈	—	2	4	no	—
141	C₄×C₈	—	86	8	no	—
142	C₄×C₈	Dic₈₆	1	4	yes	—
143	#95	—	43	2	no	—
144	#95	—	43	4	no	—
145	#95	—	43	4	no	—
146	#95	—	43	2	no	—
147	#95	C₄×D₄₃	1	4	yes	—
148	C₂²×C₄	—	43	4	no	—
149	C₂²×C₄	—	43	4	no	—
150	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
152	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	86	4	no	—
157	C₂×C₈	—	86	4	no	—
158	C₂×C₈	—	86	4	no	—
159	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
160	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
161	C₂×C₈	#10621898	1	4	yes	—
162	C₂×C₈	#10621898	1	4	yes	—
163	C₄⋊C₄	—	86	8	no	—
164	C₄⋊C₄	—	86	8	no	—
165	C₄⋊C₄	—	86	8	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	86	4	no	—
169	C₄²	—	86	4	no	—
170	C₄²	C₂×Dic₈₆	1	2	yes	—
171	C₄²	#10621898	1	2	yes	—
172	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	43	8	no	—
178	C₂×C₄	—	86	4	no	—
179	C₂×C₄	—	86	4	no	—
180	C₂×C₄	—	86	4	no	—
181	C₂×C₄	—	86	4	no	—
182	C₂×C₄	—	86	4	no	—
183	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
184	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
185	C₂×C₄	C₂×#10621898	1	2	yes	—
186	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
187	C₂×C₄	C₂×#10621898	1	2	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	2	2	no	—
190	C₈	—	172	4	no	—
191	C₈	#13063317	1	4	yes	—
192	C₂²	C₂×C₄×Dic₈₆	1	1	yes	characteristic
193	C₂²	C₄×#10621898	1	2	yes	—
194	C₂²	#131×D₄₃	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	86	8	no	—
198	C₄	—	172	4	no	—
199	C₄	C₂×#13063317	1	1	yes	characteristic
200	C₄	#14583433	1	2	yes	—
201	C₄	#14583456	1	1	yes	characteristic
202	C₂	C₄×#13063317	1	1	yes	characteristic
203	C₂	#19143527	1	2	yes	—
204	C₂	#19143566	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1726	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1726	D₄₃
hypocenter	C₄₃	#21676
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
125. C₄₃	#21676	semidirect

Maximal subgroups

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

subgroup	index
#19137900	2
#19135318	2
#19135159	2
#19134715	2
C₈×Dic₄₃⋊C₄	2
C₈×C₄⋊Dic₄₃	2
C₄₃×#1726	2
#21676	43