Subgroups — #25249587

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#19136355	C₂	1	3	yes	maximal, complemented
4	#19136353	C₂	1	3	yes	maximal, complemented
5	#19136350	C₂	1	3	yes	maximal, complemented
6	C₂×Dic₄₃⋊C₁₆	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	—	2	4	no	—
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	Dic₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	Dic₄₃⋊C₁₆	C₂²	1	3	yes	—
19	Dic₄₃⋊C₁₆	C₂²	1	3	yes	—
20	Dic₄₃⋊C₁₆	C₂²	1	3	yes	—
21	Dic₄₃⋊C₁₆	C₂²	1	6	yes	—
22	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#504	—	2	4	no	—
24	C₂×M₃₂×C₄₃	C₂²	1	3	yes	—
25	C₂²×C₁₆×C₄₃	C₂²	1	3	yes	—
26	#13063332	—	2	4	no	—
27	C₂×#10621877	C₂×C₄	1	3	yes	—
28	C₂²×C₄₃⋊C₈	C₂×C₄	1	3	yes	—
29	Q₈×Dic₄₃	C₈	1	1	yes	characteristic
30	Q₈×Dic₄₃	C₈	1	1	yes	characteristic
31	D₄×Dic₄₃	C₈	1	3	yes	—
32	D₄×Dic₄₃	C₈	1	3	yes	—
33	#13063284	C₂×C₄	1	3	yes	—
34	C₂×C₄×Dic₄₃	C₂×C₄	1	3	yes	—
35	#13063265	—	2	6	no	—
36	C₂×C₄₃⋊C₁₆	—	2	6	no	—
37	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₄₃⋊C₁₆	C₂³	1	3	yes	—
39	#13063235	C₂×C₄	1	6	yes	—
40	C₈×Dic₄₃	C₂×C₄	1	3	yes	—
41	C₈×Dic₄₃	C₂×C₄	1	3	yes	—
42	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
43	C₈×Dic₄₃	C₂³	1	3	yes	—
44	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
45	#131×C₄₃	D₄	1	4	yes	—
46	#131×C₄₃	Q₈	1	4	yes	—
47	C₂×M₁₆×C₄₃	C₂×C₄	1	3	yes	—
48	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
49	M₃₂×C₄₃	—	2	6	no	—
50	C₂×C₁₆×C₄₃	—	2	6	no	—
51	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₄₃	C₂³	1	3	yes	—
53	#41×C₄₃	C₄⋊C₄	1	4	yes	—
54	#41×C₄₃	M₁₆	1	4	yes	—
55	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₄₃	C₂×C₈	1	3	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
58	C₂²×Dic₄₃	C₂×C₈	1	3	yes	—
59	C₂²×Dic₄₃	C₂×C₈	1	3	yes	—
60	M₁₆×C₄₃	C₄⋊C₄	1	12	yes	—
61	C₂×C₈×C₄₃	C₂²×C₄	1	3	yes	—
62	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
63	C₂×C₈×C₄₃	C₂×Q₈	1	6	yes	—
64	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
65	C₂²⋊Dic₄₃	C₂×C₈	1	6	yes	—
66	C₄⋊Dic₄₃	C₂×C₈	1	3	yes	—
67	C₄⋊Dic₄₃	C₂×C₈	1	3	yes	—
68	C₄×Dic₄₃	C₂×C₈	1	3	yes	—
69	C₄×Dic₄₃	C₂×C₈	1	3	yes	—
70	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₄×Dic₄₃	C₂²×C₄	1	3	yes	—
72	#10621877	—	2	6	no	—
73	C₂×C₄₃⋊C₈	—	2	6	no	—
74	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄₃⋊C₈	C₂²×C₄	1	3	yes	—
76	C₁₆×C₄₃	—	2	1	no	—
77	C₁₆×C₄₃	—	2	3	no	—
78	C₄₃⋊C₁₆	—	2	1	no	—
79	C₄₃⋊C₁₆	—	2	3	no	—
80	C₂³×C₄₃	C₂²×C₈	1	3	yes	—
81	Q₈×C₄₃	C₄⋊C₈	1	4	yes	—
82	D₄×C₄₃	C₄⋊C₈	1	12	yes	—
83	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
85	C₂×C₄×C₄₃	C₂×M₁₆	1	6	yes	—
86	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₄₃	—	2	6	no	—
88	C₂×Dic₄₃	—	4	3	no	—
89	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×Dic₄₃	C₂²×C₈	1	3	yes	—
91	C₂×Dic₄₃	C₂²×C₈	1	3	yes	—
92	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
93	C₈×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
94	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
96	C₄₃⋊C₈	—	2	1	no	—
97	C₄₃⋊C₈	—	2	3	no	—
98	#33398	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	—	2	3	no	—
100	C₂²×C₄₃	—	2	3	no	—
101	C₂²×C₄₃	C₂×C₄⋊C₈	1	6	yes	—
102	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₄₃	C₂×C₄⋊C₈	1	6	yes	—
104	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
106	Dic₄₃	—	2	3	no	—
107	Dic₄₃	—	2	1	no	—
108	Dic₄₃	—	2	3	no	—
109	Dic₄₃	#504	1	2	yes	complemented
110	C₂×#504	—	43	1	no	—
111	C₂×#504	D₄₃	1	1	yes	characteristic
112	C₄×#131	—	43	1	no	—
113	#2151	—	43	3	no	—
114	#2151	—	43	3	no	—
115	#2150	—	43	3	no	—
116	C₂×C₄⋊C₁₆	—	43	3	no	—
117	C₂×C₄₃	—	2	6	no	—
118	C₂×C₄₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
119	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
120	C₂×C₄₃	#3407	1	1	yes	characteristic
121	C₂×#131	—	43	1	no	—
122	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
123	C₄×#41	—	43	1	no	—
124	#504	—	2	4	no	—
125	#504	—	86	4	no	—
126	C₂×M₃₂	—	43	3	no	—
127	C₂×M₃₂	C₂×D₄₃	1	3	yes	—
128	C₂²×C₁₆	—	43	3	no	—
129	C₂²×C₁₆	C₂×D₄₃	1	3	yes	—
130	#405	—	43	3	no	—
131	#405	—	43	3	no	—
132	C₄×M₁₆	—	43	3	no	—
133	C₂×C₄×C₈	—	43	3	no	—
134	C₄⋊C₁₆	—	43	1	no	—
135	C₄⋊C₁₆	—	43	3	no	—
136	C₄⋊C₁₆	—	43	3	no	—
137	C₄⋊C₁₆	—	43	3	no	—
138	C₄⋊C₁₆	—	43	6	no	—
139	C₄₃	#33398	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
141	#131	—	86	4	no	—
142	#131	Dic₈₆	1	4	yes	—
143	#131	C₂²⋊D₄₃	1	4	yes	—
144	C₂×M₁₆	—	43	3	no	—
145	C₂×M₁₆	C₄×D₄₃	1	3	yes	—
146	C₂²×C₈	—	43	3	no	—
147	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
148	C₄×Q₈	—	43	1	no	—
149	C₄×Q₈	—	43	1	no	—
150	C₄×D₄	—	43	3	no	—
151	C₄×D₄	—	43	3	no	—
152	#117	—	43	3	no	—
153	C₂×C₄²	—	43	3	no	—
154	M₃₂	—	2	6	no	—
155	M₃₂	—	86	6	no	—
156	C₂×C₁₆	—	2	6	no	—
157	C₂×C₁₆	—	43	1	no	—
158	C₂×C₁₆	—	43	3	no	—
159	C₂×C₁₆	—	86	6	no	—
160	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
161	C₂×C₁₆	C₂²×D₄₃	1	3	yes	—
162	#97	—	43	6	no	—
163	C₄×C₈	—	43	3	no	—
164	C₄×C₈	—	43	3	no	—
165	C₄×C₈	—	43	1	no	—
166	C₄×C₈	—	43	3	no	—
167	#41	#10621872	1	4	yes	—
168	#41	Dic₄₃⋊C₄	1	4	yes	—
169	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
170	C₂×D₄	C₈×D₄₃	1	3	yes	—
171	C₂²×C₄	—	43	3	no	—
172	C₂²×C₄	—	43	3	no	—
173	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
174	M₁₆	—	86	6	no	—
175	M₁₆	Dic₄₃⋊C₄	1	12	yes	—
176	C₂×C₈	—	43	1	no	—
177	C₂×C₈	—	43	3	no	—
178	C₂×C₈	—	86	6	no	—
179	C₂×C₈	C₂×Dic₈₆	1	6	yes	—
180	C₂×C₈	C₂×C₄×D₄₃	1	3	yes	—
181	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
182	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
183	C₄⋊C₄	—	43	3	no	—
184	C₄⋊C₄	—	43	3	no	—
185	C₂²⋊C₄	—	43	6	no	—
186	C₄²	—	43	3	no	—
187	C₄²	—	43	3	no	—
188	C₄²	—	43	1	no	—
189	C₄²	—	43	3	no	—
190	C₁₆	—	2	1	no	—
191	C₁₆	—	2	3	no	—
192	C₁₆	—	86	1	no	—
193	C₁₆	—	86	3	no	—
194	C₂³	C₂×C₈×D₄₃	1	3	yes	—
195	Q₈	Dic₄₃⋊C₈	1	4	yes	—
196	D₄	Dic₄₃⋊C₈	1	12	yes	—
197	C₂×C₄	—	43	1	no	—
198	C₂×C₄	—	43	1	no	—
199	C₂×C₄	—	43	3	no	—
200	C₂×C₄	—	43	3	no	—
201	C₂×C₄	—	86	6	no	—
202	C₂×C₄	—	172	3	no	—
203	C₂×C₄	C₂×Dic₄₃⋊C₄	1	6	yes	—
204	C₂×C₄	C₂×C₈×D₄₃	1	3	yes	—
205	C₂×C₄	C₂×#10621872	1	6	yes	—
206	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
207	C₈	—	86	1	no	—
208	C₈	—	86	3	no	—
209	C₈	C₂×Dic₄₃⋊C₄	1	6	yes	—
210	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
211	C₈	C₂²×Dic₈₆	1	1	yes	characteristic
212	C₂²	—	2	3	no	—
213	C₂²	—	2	3	no	—
214	C₂²	C₂×Dic₄₃⋊C₈	1	6	yes	—
215	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
216	C₄	—	43	2	no	—
217	C₄	—	86	3	no	—
218	C₄	—	86	1	no	—
219	C₄	—	86	3	no	—
220	C₄	C₂×Dic₄₃⋊C₈	1	6	yes	—
221	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
222	C₄	C₂²×#10621872	1	1	yes	characteristic
223	C₂	—	2	6	no	—
224	C₂	C₂²×Dic₄₃⋊C₈	1	1	yes	characteristic
225	C₂	#504×D₄₃	1	1	yes	characteristic
226	C₂	#19147696	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₂×C₄₃×#504	C₂
prime radical	C₂×#41×C₄₃	C₂×C₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#504	D₄₃
hypocenter	C₄₃	#33398
LCS	G ▷ C₂²×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. #19136355	C₂	semidirect
4. #19136353	C₂	semidirect
5. #19136350	C₂	semidirect
6. C₂×Dic₄₃⋊C₁₆	C₂	semidirect
109. Dic₄₃	#504	semidirect
139. C₄₃	#33398	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
#19136355	2
#19136353	2
#19136350	2
C₂×Dic₄₃⋊C₁₆	2
C₂×C₄₃×#504	2
#33398	43