Subgroups — #25096829

Order 11008.

778 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143158	C₂	1	2	yes	maximal
2	C₂×#14582857	C₂	1	1	yes	maximal, characteristic
3	#19134734	C₂	1	4	yes	maximal, complemented
4	#19134731	C₂	1	4	yes	maximal, complemented
5	#19134322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14582663	C₂	1	2	yes	maximal
7	C₄₃×#1738	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10621877	C₂²	1	2	yes	—
9	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#13063284	C₂²	1	1	yes	characteristic
11	#14582927	—	2	4	no	—
12	#14582925	—	2	4	no	—
13	#14582922	C₂²	1	8	yes	—
14	#14582857	C₄	1	8	yes	—
15	C₂×#13063235	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
17	#14582664	C₂²	1	4	yes	—
18	#14582663	C₂²	1	4	yes	—
19	#14582663	C₂²	1	4	yes	—
20	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#341	C₂²	1	4	yes	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	4	no	—
25	C₂×#10621877	—	2	4	no	—
26	C₂×#10621877	C₂³	1	4	yes	—
27	C₂²×C₄₃⋊C₈	—	2	4	no	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
32	#13063284	C₂×C₄	1	4	yes	—
33	#13063284	D₄	1	4	yes	—
34	C₂×C₄⋊Dic₄₃	Q₈	1	2	yes	—
35	C₂×C₂²⋊Dic₄₃	Q₈	1	2	yes	—
36	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
37	Dic₄₃⋊C₈	—	2	16	no	—
38	#13063235	C₂×C₄	1	8	yes	—
39	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₃	—	2	4	no	—
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂×M₁₆×C₄₃	—	2	4	no	—
44	C₂×M₁₆×C₄₃	—	2	4	no	—
45	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
47	C₂²×C₈×C₄₃	—	2	2	no	—
48	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
49	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	#41	1	4	yes	—
54	C₂²×C₄×C₄₃	#41	1	4	yes	—
55	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₄₃	C₂×Q₈	1	4	yes	—
57	M₁₆×C₄₃	—	2	8	no	—
58	M₁₆×C₄₃	—	2	4	no	—
59	M₁₆×C₄₃	—	2	4	no	—
60	M₁₆×C₄₃	—	4	4	no	—
61	C₂×C₈×C₄₃	—	2	4	no	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	—	4	4	no	—
64	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
65	C₂²⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
66	C₄⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
67	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
68	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
69	#10621877	—	2	4	no	—
70	#10621877	—	2	4	no	—
71	#10621877	—	4	4	no	—
72	C₂×C₄₃⋊C₈	—	2	8	no	—
73	C₂×C₄₃⋊C₈	—	2	2	no	—
74	C₂×C₄₃⋊C₈	—	2	2	no	—
75	C₂×C₄₃⋊C₈	—	2	4	no	—
76	C₂×C₄₃⋊C₈	—	4	4	no	—
77	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
79	C₂³×C₄₃	—	4	2	no	—
80	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₄₃	#125	1	4	yes	—
82	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₄₃	#117	1	4	yes	—
85	C₂×C₄×C₄₃	#117	1	4	yes	—
86	C₂×C₄×C₄₃	#124	1	4	yes	—
87	C₂×C₄×C₄₃	#124	1	4	yes	—
88	C₂×C₄×C₄₃	#125	1	4	yes	—
89	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
92	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
93	C₂×Dic₄₃	—	4	4	no	—
94	C₂×Dic₄₃	C₂×C₄⋊C₄	1	8	yes	—
95	C₈×C₄₃	—	2	4	no	—
96	C₈×C₄₃	—	4	4	no	—
97	C₄₃⋊C₈	—	2	2	no	—
98	C₄₃⋊C₈	—	2	2	no	—
99	C₄₃⋊C₈	—	4	4	no	—
100	#20768	—	43	1	no	maximal, Sylow
101	C₂²×C₄₃	—	2	4	no	—
102	C₂²×C₄₃	—	2	2	no	—
103	C₂²×C₄₃	—	4	4	no	—
104	C₂²×C₄₃	#382	1	4	yes	—
105	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₄₃	C₂×#125	1	2	yes	—
107	C₄×C₄₃	#382	1	4	yes	—
108	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
109	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
110	C₄×C₄₃	C₂×#124	1	2	yes	—
111	Dic₄₃	—	2	8	no	—
112	#2905	—	43	2	no	—
113	C₂×#405	—	43	1	no	—
114	#1838	—	43	4	no	—
115	#1837	—	43	4	no	—
116	#1738	—	43	1	no	—
117	#1738	D₄₃	1	1	yes	characteristic
118	C₂×#341	—	43	2	no	—
119	C₂×C₄₃	—	2	2	no	—
120	C₂×C₄₃	—	2	4	no	—
121	C₂×C₄₃	C₂×#382	1	1	yes	characteristic
122	C₂×C₄₃	#2916	1	2	yes	—
123	C₂²×M₁₆	—	43	2	no	—
124	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
125	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
126	C₂³×C₈	—	43	1	no	—
127	C₂×#117	—	43	1	no	—
128	#424	—	86	4	no	—
129	#424	—	86	4	no	—
130	#423	—	43	8	no	—
131	#405	—	43	8	no	—
132	C₂×#97	—	43	2	no	—
133	C₂×C₄×C₈	—	43	2	no	—
134	#341	—	43	4	no	—
135	#341	—	43	4	no	—
136	#341	—	43	4	no	—
137	#341	C₂×D₄₃	1	4	yes	—
138	C₄₃	#20768	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	43	4	no	—
145	C₂×M₁₆	—	86	2	no	—
146	C₂×M₁₆	—	86	4	no	—
147	C₂×M₁₆	—	86	4	no	—
148	C₂×M₁₆	C₄×D₄₃	1	4	yes	—
149	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	43	4	no	—
152	C₂²×C₈	—	43	1	no	—
153	C₂²×C₈	—	43	1	no	—
154	C₂²×C₈	—	86	4	no	—
155	C₂²×C₈	—	86	2	no	—
156	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
157	#117	—	43	4	no	—
158	#117	—	43	4	no	—
159	C₂×C₄⋊C₄	—	43	2	no	—
160	C₂×C₂²⋊C₄	—	43	2	no	—
161	C₂×C₄²	—	43	2	no	—
162	C₄⋊C₈	—	86	16	no	—
163	#97	—	43	8	no	—
164	C₄×C₈	—	43	8	no	—
165	C₂⁴	Q₈×D₄₃	1	1	yes	characteristic
166	C₂²×C₄	—	43	4	no	—
167	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
168	C₂²×C₄	#10621898	1	4	yes	—
169	C₂²×C₄	D₄×D₄₃	1	2	yes	—
170	C₂²×C₄	#10621900	1	4	yes	—
171	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	M₁₆	—	2	4	no	—
175	M₁₆	—	2	4	no	—
176	M₁₆	—	4	4	no	—
177	M₁₆	—	86	4	no	—
178	M₁₆	—	86	4	no	—
179	M₁₆	—	172	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	43	2	no	—
184	C₂×C₈	—	43	2	no	—
185	C₂×C₈	—	86	8	no	—
186	C₂×C₈	—	86	2	no	—
187	C₂×C₈	—	86	2	no	—
188	C₂×C₈	—	86	4	no	—
189	C₂×C₈	—	172	4	no	—
190	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
191	C₄⋊C₄	—	43	8	no	—
192	C₂²⋊C₄	—	43	8	no	—
193	C₄²	—	43	4	no	—
194	C₄²	—	43	4	no	—
195	C₂³	—	4	2	no	—
196	C₂³	C₄⋊C₄×D₄₃	1	2	yes	—
197	C₂³	#13063315	1	4	yes	—
198	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
199	C₂×C₄	—	43	8	no	—
200	C₂×C₄	—	172	4	no	—
201	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
202	C₂×C₄	#13063281	1	4	yes	—
203	C₂×C₄	#13063282	1	4	yes	—
204	C₂×C₄	#13063311	1	4	yes	—
205	C₂×C₄	#13063311	1	4	yes	—
206	C₂×C₄	#13063315	1	4	yes	—
207	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
209	C₂×C₄	C₂×#10621900	1	2	yes	—
210	C₂×C₄	C₂×#10621898	1	2	yes	—
211	C₈	—	2	4	no	—
212	C₈	—	4	4	no	—
213	C₈	—	86	2	no	—
214	C₈	—	86	2	no	—
215	C₈	—	172	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	4	4	no	—
219	C₂²	#14582763	1	4	yes	—
220	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
221	C₂²	C₂×#13063315	1	2	yes	—
222	C₄	—	86	8	no	—
223	C₄	#14582763	1	4	yes	—
224	C₄	C₂×#13063281	1	1	yes	characteristic
225	C₄	C₂×#13063282	1	1	yes	characteristic
226	C₄	C₂×#13063311	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	2	4	no	—
229	C₂	C₂×#14582763	1	1	yes	characteristic
230	C₂	#19143205	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×#1738	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1738	D₄₃
hypocenter	C₄₃	#20768
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1738

Complements

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

N	H ≅ G/N	type
3. #19134734	C₂	semidirect
4. #19134731	C₂	semidirect
138. C₄₃	#20768	semidirect

Maximal subgroups

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

subgroup	index
#19143158	2
C₂×#14582857	2
#19134734	2
#19134731	2
#19134322	2
C₂×#14582663	2
C₄₃×#1738	2
#20768	43