Subgroups — #25094082

Order 11008.

1078 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19134347	C₂	1	4	yes	maximal, complemented
4	#19134341	C₂	1	4	yes	maximal, complemented
5	#19134292	C₂	1	4	yes	maximal
6	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10621877	C₂²	1	2	yes	—
8	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
9	C₄×#10621877	C₄	1	8	yes	complemented
10	C₂×#13063236	C₂²	1	2	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
15	#14582654	C₂²	1	16	yes	—
16	C₂×#117×C₄₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₄₃	C₄	1	4	yes	—
19	C₄×C₂²⋊C₄×C₄₃	C₄	1	4	yes	—
20	C₄₃×#376	C₂²	1	2	yes	—
21	C₄³×C₄₃	C₂²	1	2	yes	—
22	C₂×#10621877	—	2	8	no	—
23	C₂×#10621877	C₂×C₄	1	8	yes	—
24	C₂²×C₄₃⋊C₈	—	2	8	no	—
25	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
27	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
28	#13063236	C₂×C₄	1	8	yes	—
29	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
30	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
31	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	#117×C₄₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	#95×C₄₃	C₂×C₄	1	8	yes	—
45	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	—	2	8	no	—
47	C₂²×C₄×C₄₃	C₄²	1	8	yes	—
48	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	#41	1	8	yes	—
55	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₄₃	—	2	8	no	—
57	C₂²⋊C₄×C₄₃	—	2	8	no	—
58	C₄²×C₄₃	—	2	16	no	—
59	C₄²×C₄₃	—	2	8	no	—
60	C₄²×C₄₃	C₄²	1	8	yes	—
61	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
64	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
65	#10621877	—	2	16	no	—
66	C₂×C₄₃⋊C₈	—	2	16	no	—
67	C₂×C₄₃⋊C₈	—	2	8	no	—
68	C₂×C₄₃⋊C₈	—	2	8	no	—
69	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
70	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
71	C₂³×C₄₃	—	2	4	no	—
72	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
73	C₂³×C₄₃	#117	1	4	yes	—
74	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	—	2	4	no	—
76	C₂×C₄×C₄₃	—	2	16	no	—
77	C₂×C₄×C₄₃	—	2	8	no	—
78	C₂×C₄×C₄₃	—	2	16	no	—
79	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
80	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
82	C₂×C₄×C₄₃	#117	1	8	yes	—
83	C₂×C₄×C₄₃	#117	1	8	yes	—
84	C₂×C₄×C₄₃	#117	1	4	yes	—
85	C₂×C₄×C₄₃	#131	1	8	yes	—
86	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
89	C₄₃⋊C₈	—	2	8	no	—
90	C₄₃⋊C₈	—	2	8	no	—
91	#20366	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	—	2	4	no	—
93	C₂²×C₄₃	—	2	8	no	—
94	C₂²×C₄₃	#376	1	4	yes	—
95	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₄₃	C₂×#117	1	2	yes	—
97	C₂²×C₄₃	C₂×#131	1	4	yes	—
98	C₄×C₄₃	—	2	4	no	—
99	C₄×C₄₃	—	4	8	no	—
100	C₄×C₄₃	#376	1	4	yes	—
101	C₄×C₄₃	#405	1	8	yes	—
102	C₄×C₄₃	C₂×#117	1	2	yes	—
103	C₄×C₄₃	C₂×#117	1	2	yes	—
104	C₂×C₄×M₁₆	—	43	1	no	—
105	C₂²×C₄×C₈	—	43	1	no	—
106	C₄×#117	D₄₃	1	1	yes	characteristic
107	#1746	—	43	4	no	—
108	#1744	—	43	4	no	—
109	#1729	—	43	4	no	—
110	C₂×C₄₃	—	2	4	no	—
111	C₂×C₄₃	C₂×#376	1	1	yes	characteristic
112	C₂×C₄₃	C₂×#405	1	2	yes	—
113	C₂×C₄₃	#2922	1	4	yes	—
114	C₂²×M₁₆	—	43	2	no	—
115	C₂³×C₈	—	43	2	no	—
116	C₂×#117	C₂×D₄₃	1	2	yes	—
117	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
118	C₄×M₁₆	—	43	8	no	—
119	C₂×#97	—	43	2	no	—
120	C₂×C₄×C₈	—	43	2	no	—
121	C₂×C₄×C₈	—	43	2	no	—
122	C₂×C₄×C₈	—	43	2	no	—
123	C₂×C₄×C₈	—	43	8	no	—
124	C₄×C₄⋊C₄	Dic₄₃	1	4	yes	—
125	C₄×C₂²⋊C₄	Dic₄₃	1	4	yes	—
126	#376	C₂×D₄₃	1	2	yes	—
127	C₄³	C₂×D₄₃	1	2	yes	—
128	#336	—	43	16	no	—
129	C₄₃	#20366	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
131	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
132	C₂×M₁₆	—	43	8	no	—
133	C₂×M₁₆	—	86	8	no	—
134	C₂²×C₈	—	43	8	no	—
135	C₂²×C₈	—	43	4	no	—
136	C₂²×C₈	—	43	4	no	—
137	C₂²×C₈	—	86	8	no	—
138	#117	—	2	8	no	—
139	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
140	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
141	C₂×C₄²	C₄×D₄₃	1	4	yes	—
142	C₂×C₄²	C₄×D₄₃	1	4	yes	—
143	C₂×C₄²	C₂×Dic₄₃	1	8	yes	—
144	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
145	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
146	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
148	#97	—	43	8	no	—
149	C₄×C₈	—	43	8	no	—
150	C₄×C₈	—	43	8	no	—
151	C₄×C₈	—	43	8	no	—
152	#95	C₂×Dic₄₃	1	8	yes	—
153	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	C₄×Dic₄₃	1	8	yes	—
156	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
157	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
158	C₂²×C₄	#10621898	1	8	yes	—
159	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
160	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
163	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
164	M₁₆	—	86	16	no	—
165	C₂×C₈	—	43	8	no	—
166	C₂×C₈	—	43	8	no	—
167	C₂×C₈	—	86	16	no	—
168	C₂×C₈	—	86	8	no	—
169	C₂×C₈	—	86	8	no	—
170	C₄⋊C₄	—	2	8	no	—
171	C₂²⋊C₄	—	2	8	no	—
172	C₄²	—	2	16	no	—
173	C₄²	—	2	8	no	—
174	C₄²	C₄×Dic₄₃	1	8	yes	—
175	C₄²	C₂×C₄×D₄₃	1	2	yes	—
176	C₄²	C₂×C₄×D₄₃	1	2	yes	—
177	C₄²	C₂×C₄×D₄₃	1	2	yes	—
178	C₄²	C₂×C₄×D₄₃	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
181	C₂³	#13063284	1	4	yes	—
182	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
188	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
189	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
190	C₂×C₄	#13063282	1	8	yes	—
191	C₂×C₄	#13063282	1	8	yes	—
192	C₂×C₄	#13063284	1	4	yes	—
193	C₂×C₄	#13063332	1	8	yes	—
194	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
196	C₂×C₄	C₂×#10621898	1	4	yes	—
197	C₈	—	86	8	no	—
198	C₈	—	86	8	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	4	no	—
201	C₂²	#14582745	1	4	yes	—
202	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
203	C₂²	C₂×#13063284	1	2	yes	—
204	C₂²	C₂×#13063332	1	4	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	4	8	no	—
207	C₄	#14582745	1	4	yes	—
208	C₄	#14582855	1	8	yes	—
209	C₄	C₂×#13063282	1	2	yes	—
210	C₄	C₂×#13063282	1	2	yes	—
211	C₂	—	2	4	no	—
212	C₂	C₂×#14582745	1	1	yes	characteristic
213	C₂	C₂×#14582855	1	2	yes	—
214	C₂	#19143239	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#117	D₄₃
hypocenter	C₄₃	#20366
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #19134347	C₂	semidirect
4. #19134341	C₂	semidirect
9. C₄×#10621877	C₄	semidirect
11. C₂×C₄×C₄₃⋊C₈	C₄	semidirect
129. C₄₃	#20366	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621877	2
C₂²×C₄×C₄₃⋊C₈	2
#19134347	2
#19134341	2
#19134292	2
C₄×#117×C₄₃	2
#20366	43