Subgroups — #25107034

Order 11008.

662 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143230	C₂	1	2	yes	maximal
2	C₂×#14582925	C₂	1	1	yes	maximal, characteristic
3	#19135369	C₂	1	4	yes	maximal, complemented
4	#19135367	C₂	1	4	yes	maximal, complemented
5	#19134635	C₂	1	2	yes	maximal
6	#19134633	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1810	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10621877	C₂²	1	1	yes	characteristic
9	C₂×#13063284	C₂²	1	1	yes	characteristic
10	#14582957	—	2	8	no	—
11	C₂×#13063261	C₂²	1	2	yes	—
12	C₂×#13063260	C₂²	1	4	yes	—
13	#14582925	C₄	1	8	yes	—
14	#14582922	C₂²	1	4	yes	—
15	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
17	#14582863	C₂²	1	2	yes	—
18	M₁₆×Dic₄₃	C₂²	1	4	yes	—
19	#14582664	C₂²	1	2	yes	—
20	#14582663	C₂²	1	4	yes	—
21	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#108×C₄₃	C₂²	1	2	yes	—
23	C₄₃×#407	C₂²	1	2	yes	—
24	C₄₃×#341	C₂²	1	2	yes	—
25	C₂×#10621877	—	4	2	no	—
26	C₂×#10621877	C₂×C₄	1	4	yes	—
27	C₂×#10621877	C₂³	1	4	yes	—
28	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
29	#13063284	C₂×C₄	1	4	yes	—
30	#13063284	D₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₃	Q₈	1	2	yes	—
32	C₂×C₂²⋊Dic₄₃	Q₈	1	2	yes	—
33	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
34	#13063261	—	4	2	no	—
35	#13063260	—	4	4	no	—
36	Dic₄₃⋊C₈	—	2	8	no	—
37	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
38	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
39	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
40	#13063235	—	2	4	no	—
41	C₈×Dic₄₃	—	2	4	no	—
42	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
44	C₂²×C₈×C₄₃	—	4	2	no	—
45	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	#108×C₄₃	—	4	2	no	—
49	C₂²⋊C₈×C₄₃	—	2	4	no	—
50	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	—	2	2	no	—
52	C₂²×C₄×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₃	C₂×Q₈	1	4	yes	—
58	M₁₆×C₄₃	—	2	4	no	—
59	M₁₆×C₄₃	—	2	4	no	—
60	C₂×C₈×C₄₃	—	2	2	no	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	—	4	4	no	—
64	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
66	C₂²⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
67	C₄⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
68	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
69	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
70	#10621877	—	2	4	no	—
71	#10621877	—	2	4	no	—
72	#10621877	—	4	4	no	—
73	C₂×C₄₃⋊C₈	—	2	4	no	—
74	C₂×C₄₃⋊C₈	—	2	4	no	—
75	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
76	C₂³×C₄₃	—	2	2	no	—
77	C₂³×C₄₃	—	4	2	no	—
78	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	4	no	—
84	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
88	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
89	C₂×Dic₄₃	—	4	4	no	—
90	C₂×Dic₄₃	C₂×C₄⋊C₄	1	8	yes	—
91	C₈×C₄₃	—	2	2	no	—
92	C₈×C₄₃	—	2	2	no	—
93	C₈×C₄₃	—	4	4	no	—
94	C₄₃⋊C₈	—	2	4	no	—
95	C₄₃⋊C₈	—	4	4	no	—
96	#21971	—	43	1	no	maximal, Sylow
97	C₂²×C₄₃	—	2	2	no	—
98	C₂²×C₄₃	—	2	2	no	—
99	C₂²×C₄₃	—	2	4	no	—
100	C₂²×C₄₃	—	4	4	no	—
101	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄₃	#544	1	2	yes	—
103	C₄×C₄₃	—	2	2	no	—
104	C₄×C₄₃	#518	1	1	yes	characteristic
105	C₄×C₄₃	#519	1	1	yes	characteristic
106	C₄×C₄₃	#537	1	2	yes	—
107	Dic₄₃	—	2	8	no	—
108	#2921	—	43	2	no	—
109	C₂×#424	—	43	1	no	—
110	#1950	—	43	4	no	—
111	#1950	—	43	4	no	—
112	#1811	—	43	2	no	—
113	#1810	—	43	1	no	—
114	#1810	D₄₃	1	1	yes	characteristic
115	C₂×C₄₃	—	2	2	no	—
116	C₂×C₄₃	—	4	2	no	—
117	C₂×C₄₃	#2316	1	1	yes	characteristic
118	C₂×C₄₃	#2916	1	2	yes	—
119	C₂²×M₁₆	—	43	1	no	—
120	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
121	C₂×#117	—	43	1	no	—
122	#433	—	86	8	no	—
123	C₂×#108	—	43	4	no	—
124	C₂×#108	—	43	2	no	—
125	C₂×#108	C₂×D₄₃	1	2	yes	—
126	#424	—	43	8	no	—
127	#423	—	43	4	no	—
128	C₂×C₄⋊C₈	—	43	2	no	—
129	C₂×C₄⋊C₈	—	43	2	no	—
130	#407	—	43	2	no	—
131	#407	C₂×D₄₃	1	2	yes	—
132	C₄×M₁₆	—	43	4	no	—
133	#341	—	43	4	no	—
134	#341	—	43	2	no	—
135	#341	C₂×D₄₃	1	2	yes	—
136	C₄₃	#21971	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
138	C₂×M₁₆	—	43	4	no	—
139	C₂×M₁₆	—	43	4	no	—
140	C₂×M₁₆	—	172	2	no	—
141	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
142	C₂²×C₈	—	4	2	no	—
143	C₂²×C₈	—	43	2	no	—
144	C₂²×C₈	C₄×D₄₃	1	4	yes	—
145	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
147	#117	—	43	4	no	—
148	#117	—	43	4	no	—
149	C₂×C₄⋊C₄	—	43	2	no	—
150	C₂×C₂²⋊C₄	—	43	2	no	—
151	C₂×C₄²	—	43	2	no	—
152	#108	—	4	2	no	—
153	#108	—	172	2	no	—
154	#108	—	172	4	no	—
155	C₄⋊C₈	—	43	8	no	—
156	C₄⋊C₈	—	43	8	no	—
157	C₄⋊C₈	—	86	8	no	—
158	C₂²⋊C₈	—	2	4	no	—
159	C₂²⋊C₈	—	86	4	no	—
160	#97	—	86	4	no	—
161	C₄×C₈	—	86	4	no	—
162	C₂⁴	Q₈×D₄₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	43	4	no	—
166	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
167	C₂²×C₄	D₄×D₄₃	1	2	yes	—
168	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
170	M₁₆	—	2	4	no	—
171	M₁₆	—	2	4	no	—
172	M₁₆	—	86	4	no	—
173	M₁₆	—	86	4	no	—
174	M₁₆	—	172	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	43	4	no	—
180	C₂×C₈	—	86	4	no	—
181	C₂×C₈	—	86	4	no	—
182	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
183	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
184	C₄⋊C₄	—	43	8	no	—
185	C₂²⋊C₄	—	43	8	no	—
186	C₄²	—	43	4	no	—
187	C₄²	—	43	4	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₄⋊C₄×D₄₃	1	2	yes	—
191	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	43	8	no	—
197	C₂×C₄	—	172	4	no	—
198	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
199	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
201	C₂×C₄	#13063370	1	2	yes	—
202	C₂×C₄	#13063372	1	2	yes	—
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	86	4	no	—
207	C₈	—	172	4	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
213	C₂²	#14583528	1	2	yes	—
214	C₄	—	2	2	no	—
215	C₄	—	86	8	no	—
216	C₄	#14583372	1	1	yes	characteristic
217	C₄	#14583376	1	1	yes	characteristic
218	C₄	#14583476	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	—	4	2	no	—
221	C₂	#19137212	1	1	yes	characteristic
222	C₂	#19143203	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1810	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1810	D₄₃
hypocenter	C₄₃	#21971
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. #19135369	C₂	semidirect
4. #19135367	C₂	semidirect
136. C₄₃	#21971	semidirect

Maximal subgroups

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

subgroup	index
#19143230	2
C₂×#14582925	2
#19135369	2
#19135367	2
#19134635	2
#19134633	2
C₄₃×#1810	2
#21971	43