Subgroups — #25014600

Order 11008.

338 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₃⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×#14582675	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	#19132672	C₂	1	2	yes	maximal
5	C₂×C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
10	C₄×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
11	Dic₄₃⋊C₁₆	—	2	4	no	—
12	#14582675	—	2	2	no	—
13	#14582653	C₄	1	2	yes	—
14	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×C₄⋊C₁₆	—	2	2	no	—
17	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄⋊Dic₄₃	C₈	1	2	yes	—
19	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
20	C₂×C₄₃⋊C₁₆	—	2	1	no	—
21	C₂×C₄₃⋊C₁₆	—	2	1	no	—
22	C₂×C₄₃⋊C₁₆	—	2	1	no	—
23	C₂×C₄₃⋊C₁₆	—	2	1	no	—
24	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₄₃⋊C₁₆	Q₈	1	2	yes	—
26	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
27	C₈×Dic₄₃	—	2	2	no	—
28	C₈×Dic₄₃	—	2	2	no	—
29	C₄₃⋊#95	C₈	1	1	yes	characteristic
30	C₄₃⋊#95	C₈	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₃	—	2	2	no	—
35	C₂×C₁₆×C₄₃	—	2	2	no	—
36	C₂×C₁₆×C₄₃	—	2	2	no	—
37	C₄×C₈×C₄₃	D₄	1	2	yes	—
38	C₄×C₈×C₄₃	Q₈	1	2	yes	—
39	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
43	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₄₃	—	2	1	no	—
45	C₂×C₈×C₄₃	—	2	1	no	—
46	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
53	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₄₃	M₁₆	1	2	yes	—
55	C₄⋊Dic₄₃	—	2	4	no	—
56	C₄×Dic₄₃	—	2	2	no	—
57	C₄×Dic₄₃	—	2	2	no	—
58	C₂×C₄₃⋊C₈	—	2	2	no	—
59	C₂×C₄₃⋊C₈	—	2	2	no	—
60	C₂×C₄₃⋊C₈	—	2	2	no	—
61	C₁₆×C₄₃	—	4	2	no	—
62	C₄₃⋊C₁₆	—	2	2	no	—
63	C₄₃⋊C₁₆	—	2	2	no	—
64	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
68	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
69	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
74	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
75	C₂×Dic₄₃	—	2	2	no	—
76	C₂×Dic₄₃	—	2	2	no	—
77	C₂×Dic₄₃	—	2	2	no	—
78	C₂×Dic₄₃	—	2	2	no	—
79	C₂×Dic₄₃	—	2	2	no	—
80	C₂×Dic₄₃	M₃₂	1	4	yes	—
81	C₈×C₄₃	—	2	2	no	—
82	C₈×C₄₃	#121	1	1	yes	characteristic
83	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₈×C₄₃	#125	1	1	yes	characteristic
85	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₄₃⋊C₈	—	4	2	no	—
87	#13160	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
90	C₂²×C₄₃	C₈×Q₈	1	1	yes	characteristic
91	C₂²×C₄₃	C₂×M₃₂	1	2	yes	—
92	C₂²×C₄₃	#504	1	2	yes	—
93	C₄×C₄₃	—	2	2	no	—
94	C₄×C₄₃	—	2	2	no	—
95	C₄×C₄₃	#389	1	1	yes	characteristic
96	C₄×C₄₃	#424	1	1	yes	characteristic
97	C₄×C₄₃	#436	1	1	yes	characteristic
98	C₄×C₄₃	#446	1	1	yes	characteristic
99	Dic₄₃	—	2	4	no	—
100	Dic₄₃	—	4	2	no	—
101	C₂×C₄⋊C₁₆	—	43	2	no	—
102	C₂×C₄⋊C₁₆	D₄₃	1	1	yes	characteristic
103	C₂×#346	—	43	1	no	—
104	C₈×C₄⋊C₄	—	43	1	no	—
105	#1374	—	43	2	no	—
106	C₂×C₄₃	#1939	1	1	yes	characteristic
107	C₂×C₄₃	#2150	1	1	yes	characteristic
108	C₂×C₄₃	#2151	1	1	yes	characteristic
109	C₂×C₄₃	#2168	1	2	yes	—
110	C₂×C₄₃	#2183	1	2	yes	—
111	C₂²×C₁₆	—	43	1	no	—
112	C₂²×C₁₆	—	43	1	no	—
113	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
114	C₂×C₄⋊C₈	—	43	1	no	—
115	C₂×C₄×C₈	—	43	2	no	—
116	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	43	1	no	—
118	C₄⋊C₁₆	—	2	2	no	—
119	C₄⋊C₁₆	—	86	4	no	—
120	#346	—	86	2	no	—
121	#336	—	43	2	no	—
122	C₄₃	#13160	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	43	2	no	—
124	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	43	2	no	—
127	C₂×C₄²	—	43	2	no	—
128	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	43	2	no	—
133	C₂×C₁₆	—	43	2	no	—
134	C₂×C₁₆	—	86	1	no	—
135	C₂×C₁₆	—	86	1	no	—
136	C₂×C₁₆	—	86	1	no	—
137	C₂×C₁₆	—	86	1	no	—
138	C₄⋊C₈	—	86	2	no	—
139	C₄×C₈	—	86	2	no	—
140	C₄×C₈	—	86	2	no	—
141	C₄×C₈	Dic₈₆	1	2	yes	—
142	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
143	#95	—	43	1	no	—
144	#95	—	43	1	no	—
145	C₂²×C₄	—	43	2	no	—
146	C₂²×C₄	—	43	2	no	—
147	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
148	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	86	2	no	—
153	C₂×C₈	—	86	2	no	—
154	C₂×C₈	—	86	2	no	—
155	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
156	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
157	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
158	C₂×C₈	#10621900	1	1	yes	characteristic
159	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
160	C₂×C₈	#10621902	1	1	yes	characteristic
161	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
162	C₄⋊C₄	—	86	4	no	—
163	C₄²	—	86	2	no	—
164	C₄²	—	86	2	no	—
165	C₄²	#10621872	1	2	yes	—
166	C₄²	Dic₄₃⋊C₄	1	2	yes	—
167	C₁₆	—	4	2	no	—
168	C₁₆	—	86	2	no	—
169	C₁₆	—	86	2	no	—
170	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	43	4	no	—
174	C₂×C₄	—	86	2	no	—
175	C₂×C₄	—	86	2	no	—
176	C₂×C₄	—	86	2	no	—
177	C₂×C₄	—	86	2	no	—
178	C₂×C₄	—	86	2	no	—
179	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
180	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
181	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#13063287	1	1	yes	characteristic
183	C₂×C₄	#13063290	1	1	yes	characteristic
184	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
185	C₂×C₄	#13063333	1	1	yes	characteristic
186	C₂×C₄	#13063333	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	172	2	no	—
189	C₈	#13063296	1	1	yes	characteristic
190	C₈	#13063301	1	1	yes	characteristic
191	C₈	#13063316	1	1	yes	characteristic
192	C₈	#13063324	1	1	yes	characteristic
193	C₂²	C₂×Dic₄₃⋊C₈	1	1	yes	characteristic
194	C₂²	#14582964	1	1	yes	characteristic
195	C₂²	#14583023	1	1	yes	characteristic
196	C₂²	M₃₂×D₄₃	1	2	yes	—
197	C₂²	#14583315	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	86	4	no	—
201	C₄	—	172	2	no	—
202	C₄	#14582795	1	1	yes	characteristic
203	C₄	#14582927	1	1	yes	characteristic
204	C₄	#14582976	1	1	yes	characteristic
205	C₄	#14583026	1	1	yes	characteristic
206	C₂	#19135311	1	1	yes	characteristic
207	C₂	#19136350	1	1	yes	characteristic
208	C₂	#19136353	1	1	yes	characteristic
209	C₂	#19136419	1	2	yes	—
210	C₂	#19136502	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂²×C₈
Fitting	C₂×C₄₃×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#13160
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
122. C₄₃	#13160	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₄₃⋊C₁₆	2
C₂×#14582675	2
C₈×C₄⋊Dic₄₃	2
#19132672	2
C₂×C₄₃×C₄⋊C₁₆	2
#13160	43