Subgroups — #25249245

Order 11008.

818 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583316	C₂	1	1	yes	maximal, characteristic
2	#131×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#19136194	C₂	1	3	yes	maximal, complemented
4	#19136191	C₂	1	3	yes	maximal, complemented
5	M₃₂×Dic₄₃	C₂	1	3	yes	maximal, complemented
6	C₂×#14582674	C₂	1	3	yes	maximal, complemented
7	C₂×C₄₃×#504	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063332	C₄	1	1	yes	characteristic
9	#41×Dic₄₃	C₄	1	1	yes	characteristic
10	#14583316	C₄	1	8	yes	complemented
11	C₂×#13063265	C₂²	1	3	yes	—
12	C₂²×C₄₃⋊C₁₆	C₂²	1	3	yes	—
13	#14582857	C₄	1	3	yes	—
14	#14582854	C₂²	1	3	yes	—
15	M₁₆×Dic₄₃	C₄	1	3	yes	—
16	C₂×C₈×Dic₄₃	C₂²	1	3	yes	—
17	#14582674	C₂²	1	1	yes	characteristic
18	#14582674	C₂²	1	3	yes	—
19	#14582674	C₂²	1	3	yes	—
20	#14582674	C₂²	1	3	yes	—
21	C₁₆×Dic₄₃	C₂²	1	6	yes	—
22	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#504	C₄	1	8	yes	complemented
24	C₂×M₃₂×C₄₃	C₂²	1	3	yes	—
25	C₂²×C₁₆×C₄₃	C₂²	1	3	yes	—
26	#13063332	C₈	1	8	yes	—
27	C₂×#10621877	C₂×C₄	1	3	yes	—
28	C₂²×C₄₃⋊C₈	C₂×C₄	1	3	yes	—
29	Q₈×Dic₄₃	C₈	1	2	yes	—
30	D₄×Dic₄₃	C₈	1	6	yes	—
31	#13063284	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₄₃	C₂×C₄	1	3	yes	—
33	#13063265	C₂×C₄	1	12	yes	complemented
34	C₂×C₄₃⋊C₁₆	C₂×C₄	1	12	yes	complemented
35	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
36	C₂×C₄₃⋊C₁₆	C₂³	1	3	yes	—
37	#13063235	C₂×C₄	1	6	yes	—
38	C₈×Dic₄₃	C₂×C₄	1	3	yes	—
39	C₈×Dic₄₃	C₂×C₄	1	3	yes	—
40	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
41	C₈×Dic₄₃	C₂³	1	3	yes	—
42	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
43	#131×C₄₃	C₂×C₄	1	4	yes	—
44	#131×C₄₃	C₂×C₄	1	4	yes	—
45	C₂×M₁₆×C₄₃	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
47	M₃₂×C₄₃	C₂×C₄	1	12	yes	complemented
48	C₂×C₁₆×C₄₃	C₂×C₄	1	12	yes	complemented
49	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₄₃	C₂³	1	3	yes	—
51	#41×C₄₃	C₄²	1	4	yes	—
52	#41×C₄₃	C₂×C₈	1	4	yes	—
53	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
54	C₂×D₄×C₄₃	C₂×C₈	1	3	yes	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
56	C₂²×Dic₄₃	C₂×C₈	1	6	yes	—
57	M₁₆×C₄₃	C₄²	1	12	yes	—
58	C₂×C₈×C₄₃	C₂²×C₄	1	3	yes	—
59	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
60	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
61	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
62	C₂²⋊Dic₄₃	C₂×C₈	1	6	yes	—
63	C₄⋊Dic₄₃	C₂×C₈	1	6	yes	—
64	C₄×Dic₄₃	C₂×C₈	1	6	yes	—
65	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₄₃	C₂²×C₄	1	3	yes	—
67	#10621877	C₂×C₈	1	12	yes	—
68	C₂×C₄₃⋊C₈	C₂×C₈	1	12	yes	—
69	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄₃⋊C₈	C₂²×C₄	1	3	yes	—
71	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
72	C₁₆×C₄₃	C₂²×C₄	1	6	yes	—
73	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
74	C₄₃⋊C₁₆	C₂²×C₄	1	6	yes	—
75	C₂³×C₄₃	C₂²×C₈	1	3	yes	—
76	Q₈×C₄₃	C₄×C₈	1	4	yes	—
77	D₄×C₄₃	C₄×C₈	1	12	yes	—
78	C₂×C₄×C₄₃	C₂×C₄²	1	6	yes	—
79	C₂×C₄×C₄₃	C₂²×C₈	1	6	yes	—
80	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
81	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×Dic₄₃	—	4	6	no	—
83	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
84	C₂×Dic₄₃	C₂²×C₈	1	6	yes	—
85	C₈×C₄₃	C₂×C₄²	1	6	yes	—
86	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
88	C₄₃⋊C₈	C₂²×C₈	1	2	yes	—
89	C₄₃⋊C₈	C₂²×C₈	1	6	yes	—
90	#33326	—	43	1	no	maximal, Sylow
91	C₂²×C₄₃	—	2	6	no	—
92	C₂²×C₄₃	C₂×C₄×C₈	1	6	yes	—
93	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₄₃	C₂×C₄×C₈	1	6	yes	—
95	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
96	C₄×C₄₃	C₂³×C₈	1	1	yes	characteristic
97	Dic₄₃	—	2	2	no	—
98	Dic₄₃	—	2	6	no	—
99	C₂×#504	—	43	1	no	—
100	C₂×#504	D₄₃	1	1	yes	characteristic
101	C₄×#131	—	43	1	no	—
102	#2108	—	43	3	no	—
103	#2108	—	43	3	no	—
104	C₄×M₃₂	—	43	3	no	—
105	C₂×#346	—	43	3	no	—
106	C₂×C₄₃	—	2	6	no	—
107	C₂×C₄₃	C₂²×C₄×C₈	1	1	yes	characteristic
108	C₂×C₄₃	#3407	1	2	yes	—
109	C₂×#131	—	43	1	no	—
110	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
111	C₄×#41	—	43	1	no	—
112	#504	—	43	8	no	—
113	#504	Dic₄₃	1	8	yes	complemented
114	C₂×M₃₂	—	43	3	no	—
115	C₂×M₃₂	C₂×D₄₃	1	3	yes	—
116	C₂²×C₁₆	—	43	3	no	—
117	C₂²×C₁₆	C₂×D₄₃	1	3	yes	—
118	#405	—	43	3	no	—
119	#405	—	43	3	no	—
120	C₄×M₁₆	—	43	3	no	—
121	C₂×C₄×C₈	—	43	3	no	—
122	#346	—	43	1	no	—
123	#346	—	43	3	no	—
124	#346	—	43	3	no	—
125	#346	—	43	3	no	—
126	C₄×C₁₆	—	43	6	no	—
127	C₄₃	#33326	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
129	#131	—	43	8	no	—
130	#131	C₄×D₄₃	1	4	yes	—
131	#131	C₂×Dic₄₃	1	4	yes	—
132	C₂×M₁₆	—	43	3	no	—
133	C₂×M₁₆	C₄×D₄₃	1	3	yes	—
134	C₂²×C₈	—	43	3	no	—
135	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
136	C₄×Q₈	—	43	2	no	—
137	C₄×D₄	—	43	6	no	—
138	#117	—	43	3	no	—
139	C₂×C₄²	—	43	3	no	—
140	M₃₂	—	43	12	no	—
141	M₃₂	C₂×Dic₄₃	1	12	yes	complemented
142	C₂×C₁₆	—	43	12	no	—
143	C₂×C₁₆	—	43	3	no	—
144	C₂×C₁₆	—	43	1	no	—
145	C₂×C₁₆	C₂×Dic₄₃	1	12	yes	complemented
146	C₂×C₁₆	C₂²×D₄₃	1	3	yes	—
147	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
148	#97	—	43	6	no	—
149	C₄×C₈	—	43	3	no	—
150	C₄×C₈	—	43	3	no	—
151	C₄×C₈	—	43	3	no	—
152	C₄×C₈	—	43	1	no	—
153	#41	C₈×D₄₃	1	4	yes	—
154	#41	C₄×Dic₄₃	1	4	yes	—
155	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
156	C₂×D₄	C₈×D₄₃	1	3	yes	—
157	C₂²×C₄	—	43	6	no	—
158	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
159	M₁₆	—	43	12	no	—
160	M₁₆	C₄×Dic₄₃	1	12	yes	—
161	C₂×C₈	—	43	12	no	—
162	C₂×C₈	—	43	1	no	—
163	C₂×C₈	—	43	3	no	—
164	C₂×C₈	C₂×C₄×D₄₃	1	3	yes	—
165	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
166	C₂×C₈	C₂²×Dic₄₃	1	6	yes	—
167	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
168	C₄⋊C₄	—	43	6	no	—
169	C₂²⋊C₄	—	43	6	no	—
170	C₄²	—	43	6	no	—
171	C₄²	—	43	1	no	—
172	C₄²	—	43	3	no	—
173	C₁₆	—	43	2	no	—
174	C₁₆	—	43	6	no	—
175	C₁₆	C₂²×Dic₄₃	1	2	yes	—
176	C₁₆	C₂²×Dic₄₃	1	6	yes	—
177	C₂³	C₂×C₈×D₄₃	1	3	yes	—
178	Q₈	C₈×Dic₄₃	1	4	yes	—
179	D₄	C₈×Dic₄₃	1	12	yes	—
180	C₂×C₄	—	43	2	no	—
181	C₂×C₄	—	43	6	no	—
182	C₂×C₄	—	172	6	no	—
183	C₂×C₄	C₂×C₄×Dic₄₃	1	6	yes	—
184	C₂×C₄	C₂×C₈×D₄₃	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₄₃	1	6	yes	—
186	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
187	C₈	—	43	2	no	—
188	C₈	—	43	6	no	—
189	C₈	C₂×C₄×Dic₄₃	1	6	yes	—
190	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
191	C₈	C₂³×Dic₄₃	1	1	yes	characteristic
192	C₂²	—	2	6	no	—
193	C₂²	C₂×C₈×Dic₄₃	1	6	yes	—
194	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
195	C₄	—	86	2	no	—
196	C₄	—	86	6	no	—
197	C₄	C₂×C₈×Dic₄₃	1	6	yes	—
198	C₄	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
199	C₄	C₂²×C₈×D₄₃	1	1	yes	characteristic
200	C₂	—	2	6	no	—
201	C₂	C₂²×C₈×Dic₄₃	1	1	yes	characteristic
202	C₂	#19147696	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator	C₂×C₄₃	C₂²×C₄×C₈
Fitting	C₂×C₄₃×#504	C₂
socle	C₂²×C₄₃	C₂³×C₈
prime radical	C₂×#41×C₄₃	C₂×C₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#504	D₄₃
hypocenter	C₄₃	#33326
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
3. #19136194	C₂	semidirect
4. #19136191	C₂	semidirect
5. M₃₂×Dic₄₃	C₂	semidirect
6. C₂×#14582674	C₂	semidirect
10. #14583316	C₄	semidirect
23. C₄₃×#504	C₄	semidirect
33. #13063265	C₂×C₄	semidirect
34. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
47. M₃₂×C₄₃	C₂×C₄	semidirect
48. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
113. #504	Dic₄₃	semidirect
127. C₄₃	#33326	semidirect
141. M₃₂	C₂×Dic₄₃	semidirect
145. C₂×C₁₆	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583316	2
#131×Dic₄₃	2
#19136194	2
#19136191	2
M₃₂×Dic₄₃	2
C₂×#14582674	2
C₂×C₄₃×#504	2
#33326	43