Subgroups — C₄₃×#8231 #24928248

Order 11008.

684 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#127×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1767	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₃×#331	C₂	1	2	yes	maximal
4	C₄₃×#1296	C₂	1	2	yes	maximal, complemented
5	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×D₄×C₄₃	—	2	2	no	—
7	C₂×C₄×D₄×C₄₃	C₄	1	2	yes	—
8	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
9	C₂×#97×C₄₃	—	2	2	no	—
10	C₄₃×#390	—	2	2	no	—
11	C₄₃×#390	C₄	1	2	yes	—
12	C₄₃×#384	C₂²	1	1	yes	characteristic
13	C₄³×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#336	—	2	2	no	—
15	C₄₃×#331	C₄	1	8	yes	complemented
16	C₂²×D₄×C₄₃	—	2	2	no	—
17	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₄₃	—	2	2	no	—
19	C₂²×C₈×C₄₃	—	2	4	no	—
20	#127×C₄₃	—	2	2	no	—
21	#127×C₄₃	C₂×C₄	1	4	yes	—
22	C₄×D₄×C₄₃	—	4	8	no	—
23	C₄×D₄×C₄₃	—	4	4	no	—
24	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
26	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
27	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
28	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₃	D₄	1	2	yes	—
31	C₂×C₄²×C₄₃	D₄	1	2	yes	—
32	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₃	—	2	4	no	—
34	C₄⋊C₈×C₄₃	C₂×C₄	1	8	yes	complemented
35	#97×C₄₃	—	4	4	no	—
36	C₂⁴×C₄₃	—	2	2	no	—
37	C₂×D₄×C₄₃	—	2	2	no	—
38	C₂×D₄×C₄₃	—	2	8	no	—
39	C₂×D₄×C₄₃	—	4	4	no	—
40	C₂×D₄×C₄₃	—	4	2	no	—
41	C₂×D₄×C₄₃	—	4	2	no	—
42	C₂×D₄×C₄₃	—	4	2	no	—
43	C₂×D₄×C₄₃	—	4	2	no	—
44	C₂×D₄×C₄₃	C₄²	1	4	yes	—
45	C₂²×C₄×C₄₃	—	2	2	no	—
46	C₂²×C₄×C₄₃	—	4	8	no	—
47	C₂²×C₄×C₄₃	—	4	4	no	—
48	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₄₃	—	2	8	no	—
54	C₂×C₈×C₄₃	—	4	4	no	—
55	C₂×C₈×C₄₃	—	4	4	no	—
56	C₄⋊C₄×C₄₃	—	4	2	no	—
57	C₄⋊C₄×C₄₃	—	4	4	no	—
58	C₂²⋊C₄×C₄₃	—	4	4	no	—
59	C₂²⋊C₄×C₄₃	—	4	8	no	—
60	C₄²×C₄₃	—	2	2	no	—
61	C₄²×C₄₃	—	2	2	no	—
62	C₄²×C₄₃	—	2	4	no	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	—	4	1	no	—
65	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
66	C₄²×C₄₃	#41	1	2	yes	—
67	C₂³×C₄₃	—	2	4	no	—
68	C₂³×C₄₃	—	4	2	no	—
69	C₂³×C₄₃	—	4	2	no	—
70	C₂³×C₄₃	—	4	4	no	—
71	C₂³×C₄₃	—	4	4	no	—
72	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	D₄×C₄₃	—	4	4	no	—
74	D₄×C₄₃	—	4	4	no	—
75	D₄×C₄₃	—	4	8	no	—
76	D₄×C₄₃	—	4	8	no	—
77	C₂×C₄×C₄₃	—	2	1	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	2	2	no	—
80	C₂×C₄×C₄₃	—	2	1	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	4	2	no	—
86	C₂×C₄×C₄₃	—	4	4	no	—
87	C₂×C₄×C₄₃	—	8	8	no	—
88	C₂×C₄×C₄₃	#100	1	2	yes	—
89	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
93	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×C₄×C₄₃	#137	1	2	yes	—
95	C₈×C₄₃	—	4	8	no	—
96	#8231	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₄₃	—	4	4	no	—
98	C₂²×C₄₃	—	4	8	no	—
99	C₂²×C₄₃	—	8	2	no	—
100	C₂²×C₄₃	—	8	4	no	—
101	C₂²×C₄₃	—	8	2	no	—
102	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
104	C₂²×C₄₃	#413	1	1	yes	characteristic
105	C₂²×C₄₃	#417	1	2	yes	—
106	C₂²×C₄₃	#418	1	2	yes	—
107	C₄×C₄₃	—	2	2	no	—
108	C₄×C₄₃	—	2	1	no	—
109	C₄×C₄₃	—	2	1	no	—
110	C₄×C₄₃	—	4	2	no	—
111	C₄×C₄₃	—	4	1	no	—
112	C₄×C₄₃	—	8	2	no	—
113	C₄×C₄₃	#440	1	4	yes	—
114	C₄×C₄₃	#442	1	4	yes	—
115	C₄×#127	C₂×C₄₃	1	1	yes	characteristic
116	#1767	C₂×C₄₃	1	2	yes	complemented
117	C₂×#331	C₂×C₄₃	1	2	yes	—
118	#1296	C₂×C₄₃	1	2	yes	complemented
119	C₂×C₄₃	—	8	4	no	—
120	C₂×C₄₃	#1555	1	1	yes	characteristic
121	C₂×C₄₃	#1557	1	2	yes	—
122	C₂×C₄₃	#1560	1	1	yes	characteristic
123	C₂×C₄₃	C₄×#100	1	1	yes	characteristic
124	C₂×C₄₃	#1762	1	2	yes	—
125	C₂×#127	C₂²×C₄₃	1	1	yes	characteristic
126	C₂×C₄×D₄	—	2	2	no	—
127	C₂×C₄×D₄	C₄×C₄₃	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₄₃	1	4	yes	—
129	C₂×#97	—	2	2	no	—
130	#390	—	2	2	no	—
131	#390	C₄×C₄₃	1	2	yes	—
132	#384	C₂²×C₄₃	1	1	yes	characteristic
133	C₄³	C₂²×C₄₃	1	1	yes	characteristic
134	#336	—	2	2	no	—
135	#331	C₄×C₄₃	1	8	yes	complemented
136	C₄₃	#8231	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	C₂×C₄×C₄₃	1	2	yes	—
139	C₂³×C₄	—	2	2	no	—
140	C₂²×C₈	—	2	4	no	—
141	#127	—	2	2	no	—
142	#127	C₂×C₄×C₄₃	1	4	yes	—
143	C₄×D₄	—	4	8	no	—
144	C₄×D₄	—	4	4	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	2	yes	—
147	C₂×C₂²⋊C₄	—	2	4	no	—
148	C₂×C₂²⋊C₄	—	2	2	no	—
149	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
151	C₂×C₄²	D₄×C₄₃	1	2	yes	—
152	C₂×C₄²	D₄×C₄₃	1	2	yes	—
153	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	C₂×C₄×C₄₃	1	8	yes	complemented
156	#97	—	4	4	no	—
157	C₂⁴	—	2	2	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	C₄²×C₄₃	1	4	yes	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	8	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
171	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	4	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₂²⋊C₄	—	4	4	no	—
180	C₂²⋊C₄	—	4	8	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	1	no	—
186	C₄²	C₂²×C₄×C₄₃	1	2	yes	—
187	C₄²	#41×C₄₃	1	2	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	—	4	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
194	D₄	—	4	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	8	no	—
197	D₄	—	4	8	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	#100×C₄₃	1	2	yes	—
210	C₂×C₄	C₂×C₄²×C₄₃	1	1	yes	characteristic
211	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
212	C₂×C₄	C₄×D₄×C₄₃	1	4	yes	—
213	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	4	yes	—
214	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	2	yes	—
215	C₂×C₄	#137×C₄₃	1	2	yes	—
216	C₈	—	4	8	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	8	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	4	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₄×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
223	C₂²	C₂×#100×C₄₃	1	1	yes	characteristic
224	C₂²	C₄₃×#413	1	1	yes	characteristic
225	C₂²	C₄₃×#417	1	2	yes	—
226	C₂²	C₄₃×#418	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	8	2	no	—
233	C₄	C₄₃×#440	1	4	yes	—
234	C₄	C₄₃×#442	1	4	yes	—
235	C₂	—	8	4	no	—
236	C₂	C₄₃×#1555	1	1	yes	characteristic
237	C₂	C₄₃×#1557	1	2	yes	—
238	C₂	C₄₃×#1560	1	1	yes	characteristic
239	C₂	C₄×#100×C₄₃	1	1	yes	characteristic
240	C₂	C₄₃×#1762	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
prime radical	C₂×#127×C₄₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₄³×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1767	C₂	semidirect
4. C₄₃×#1296	C₂	semidirect
15. C₄₃×#331	C₄	semidirect
34. C₄⋊C₈×C₄₃	C₂×C₄	semidirect
96. #8231	C₄₃	direct (N × H)
116. #1767	C₂×C₄₃	semidirect
118. #1296	C₂×C₄₃	semidirect
135. #331	C₄×C₄₃	semidirect
136. C₄₃	#8231	direct (N × H)
155. C₄⋊C₈	C₂×C₄×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#127×C₄₃	2
C₄₃×#1767	2
C₂×C₄₃×#331	2
C₄₃×#1296	2
#8231	43

Subgroups — C43×#8231 #24928248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8231 #24928248