Subgroups — C₄₃×#28702 #24948719

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#3166	C₂	1	2	yes	maximal
2	C₂×C₄₃×#499	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2960	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2090	C₂	1	4	yes	maximal
5	C₄₃×#1998	C₂	1	2	yes	maximal
6	C₄₃×#1952	C₂	1	4	yes	maximal
7	C₄×#108×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#533	C₂²	1	1	yes	characteristic
9	C₂×#125×C₄₃	C₂²	1	2	yes	—
10	C₄₃×#501	—	2	4	no	—
11	C₄₃×#499	—	2	4	no	—
12	C₄₃×#499	—	2	2	no	—
13	C₄₃×#499	—	2	2	no	—
14	C₄₃×#446	—	2	2	no	—
15	C₈×Q₈×C₄₃	—	2	2	no	—
16	C₂×#108×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#428	C₂²	1	8	yes	—
18	C₂×#107×C₄₃	C₂²	1	2	yes	—
19	C₂×#106×C₄₃	C₂²	1	2	yes	—
20	C₄₃×#424	C₂²	1	2	yes	—
21	C₄₃×#423	C₂²	1	1	yes	characteristic
22	C₄₃×#405	C₂²	1	2	yes	—
23	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#343	C₂²	1	8	yes	—
26	C₄₃×#341	C₂²	1	2	yes	—
27	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
28	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
29	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
32	#125×C₄₃	—	2	2	no	—
33	#125×C₄₃	—	2	4	no	—
34	#125×C₄₃	—	2	2	no	—
35	#125×C₄₃	D₄	1	2	yes	—
36	C₂²⋊Q₈×C₄₃	—	2	2	no	—
37	C₄×Q₈×C₄₃	—	2	2	no	—
38	#117×C₄₃	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	#108×C₄₃	Q₈	1	8	yes	complemented
43	#107×C₄₃	—	2	2	no	—
44	#107×C₄₃	—	2	2	no	—
45	#107×C₄₃	—	2	8	no	—
46	#106×C₄₃	—	2	4	no	—
47	#106×C₄₃	—	2	8	no	—
48	C₄⋊C₈×C₄₃	—	2	4	no	—
49	C₄⋊C₈×C₄₃	—	2	2	no	—
50	#97×C₄₃	—	2	2	no	—
51	#97×C₄₃	—	2	2	no	—
52	C₄×C₈×C₄₃	—	2	2	no	—
53	C₄×C₈×C₄₃	—	2	2	no	—
54	C₄×C₈×C₄₃	D₄	1	2	yes	—
55	C₄×C₈×C₄₃	D₄	1	2	yes	—
56	C₂×Q₈×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	#41	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₄₃	—	2	4	no	—
61	M₁₆×C₄₃	—	2	4	no	—
62	M₁₆×C₄₃	C₂×Q₈	1	8	yes	—
63	C₂×C₈×C₄₃	—	2	4	no	—
64	C₂×C₈×C₄₃	—	2	2	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
70	C₄⋊C₄×C₄₃	—	2	4	no	—
71	C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₄⋊C₄×C₄₃	—	2	4	no	—
73	C₄⋊C₄×C₄₃	—	2	4	no	—
74	C₄⋊C₄×C₄₃	—	4	4	no	—
75	C₄⋊C₄×C₄₃	—	4	4	no	—
76	C₄⋊C₄×C₄₃	C₂×D₄	1	4	yes	—
77	C₂²⋊C₄×C₄₃	—	2	2	no	—
78	C₄²×C₄₃	—	2	2	no	—
79	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂×D₄	1	2	yes	—
81	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
83	Q₈×C₄₃	—	4	4	no	—
84	C₂×C₄×C₄₃	—	2	4	no	—
85	C₂×C₄×C₄₃	—	4	2	no	—
86	C₂×C₄×C₄₃	—	4	4	no	—
87	C₂×C₄×C₄₃	—	8	4	no	—
88	C₂×C₄×C₄₃	#121	1	2	yes	—
89	C₂×C₄×C₄₃	#121	1	2	yes	—
90	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
94	C₈×C₄₃	—	2	2	no	—
95	C₈×C₄₃	—	2	8	no	—
96	C₈×C₄₃	—	4	2	no	—
97	C₈×C₄₃	C₄⋊Q₈	1	2	yes	—
98	C₈×C₄₃	C₄⋊Q₈	1	2	yes	—
99	#28702	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₄₃	—	2	2	no	—
101	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
102	C₂²×C₄₃	#557	1	2	yes	—
103	C₄×C₄₃	—	2	2	no	—
104	C₄×C₄₃	—	2	2	no	—
105	C₄×C₄₃	—	4	4	no	—
106	C₄×C₄₃	—	8	4	no	—
107	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₄×C₄₃	#536	1	1	yes	characteristic
109	C₄×C₄₃	D₄×Q₈	1	2	yes	—
110	#3166	C₂×C₄₃	1	2	yes	—
111	C₂×#499	C₂×C₄₃	1	1	yes	characteristic
112	#2960	C₂×C₄₃	1	1	yes	characteristic
113	#2090	C₂×C₄₃	1	4	yes	—
114	#1998	C₂×C₄₃	1	2	yes	—
115	#1952	C₂×C₄₃	1	4	yes	—
116	C₄×#108	C₂×C₄₃	1	1	yes	characteristic
117	C₂×C₄₃	—	2	2	no	—
118	C₂×C₄₃	#2546	1	1	yes	characteristic
119	C₂×C₄₃	#3067	1	2	yes	—
120	#533	C₂²×C₄₃	1	1	yes	characteristic
121	C₂×#125	C₂²×C₄₃	1	2	yes	—
122	#501	—	2	4	no	—
123	#499	—	2	4	no	—
124	#499	—	2	2	no	—
125	#499	—	2	2	no	—
126	#446	—	2	2	no	—
127	C₈×Q₈	—	2	2	no	—
128	C₂×#108	C₂²×C₄₃	1	2	yes	—
129	#428	C₂²×C₄₃	1	8	yes	—
130	C₂×#107	C₂²×C₄₃	1	2	yes	—
131	C₂×#106	C₂²×C₄₃	1	2	yes	—
132	#424	C₂²×C₄₃	1	2	yes	—
133	#423	C₂²×C₄₃	1	1	yes	characteristic
134	#405	C₂²×C₄₃	1	2	yes	—
135	C₄×M₁₆	C₂²×C₄₃	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
137	#343	C₂²×C₄₃	1	8	yes	—
138	#341	C₂²×C₄₃	1	2	yes	—
139	C₄₃	#28702	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×M₁₆	C₂³×C₄₃	1	2	yes	—
141	C₂×M₁₆	C₂³×C₄₃	1	4	yes	—
142	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
144	C₄⋊Q₈	D₄×C₄₃	1	2	yes	—
145	#125	—	2	2	no	—
146	#125	—	2	4	no	—
147	#125	—	2	2	no	—
148	#125	D₄×C₄₃	1	2	yes	—
149	C₂²⋊Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	#117	C₂³×C₄₃	1	2	yes	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	C₂³×C₄₃	1	4	yes	—
154	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
155	#108	Q₈×C₄₃	1	8	yes	complemented
156	#107	—	2	2	no	—
157	#107	—	2	2	no	—
158	#107	—	2	8	no	—
159	#106	—	2	4	no	—
160	#106	—	2	8	no	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	D₄×C₄₃	1	2	yes	—
168	C₄×C₈	D₄×C₄₃	1	2	yes	—
169	C₂×Q₈	—	2	2	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	#41×C₄₃	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
173	M₁₆	—	2	4	no	—
174	M₁₆	—	2	4	no	—
175	M₁₆	C₂×Q₈×C₄₃	1	8	yes	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂×D₄×C₄₃	1	2	yes	—
180	C₂×C₈	C₂×D₄×C₄₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₄₃	1	1	yes	characteristic
182	C₂×C₈	C₂×Q₈×C₄₃	1	4	yes	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	C₂×D₄×C₄₃	1	4	yes	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₄₃	1	2	yes	—
194	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
195	C₂³	C₂×#41×C₄₃	1	1	yes	characteristic
196	Q₈	—	4	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	4	no	—
201	C₂×C₄	#121×C₄₃	1	2	yes	—
202	C₂×C₄	#121×C₄₃	1	2	yes	—
203	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
205	C₂×C₄	C₂²×Q₈×C₄₃	1	2	yes	—
206	C₂×C₄	2₋¹⁺⁴×C₄₃	1	2	yes	—
207	C₈	—	2	2	no	—
208	C₈	—	2	8	no	—
209	C₈	—	4	2	no	—
210	C₈	C₄⋊Q₈×C₄₃	1	2	yes	—
211	C₈	C₄⋊Q₈×C₄₃	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	C₂×#121×C₄₃	1	1	yes	characteristic
214	C₂²	C₄₃×#557	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	4	no	—
219	C₄	C₂×C₄⋊Q₈×C₄₃	1	1	yes	characteristic
220	C₄	C₄₃×#536	1	1	yes	characteristic
221	C₄	D₄×Q₈×C₄₃	1	2	yes	—
222	C₂	—	2	2	no	—
223	C₂	C₄₃×#2546	1	1	yes	characteristic
224	C₂	C₄₃×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₃	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂³×C₄₃	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₃ ◁ C₂×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
42. #108×C₄₃	Q₈	semidirect
99. #28702	C₄₃	direct (N × H)
139. C₄₃	#28702	direct (N × H)
155. #108	Q₈×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄₃×#3166	2
C₂×C₄₃×#499	2
C₄₃×#2960	2
C₄₃×#2090	2
C₄₃×#1998	2
C₄₃×#1952	2
C₄×#108×C₄₃	2
#28702	43

Subgroups — C43×#28702 #24948719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#28702 #24948719