Subgroups — #25094504

Order 11008.

1150 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582923	C₂	1	2	yes	maximal
2	C₂×C₄×#10621877	C₂	1	2	yes	maximal
3	#19134452	C₂	1	8	yes	maximal, complemented
4	#19134289	C₂	1	2	yes	maximal
5	C₂×C₄₃×#376	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	4	yes	—
7	#14582923	C₄	1	16	yes	complemented
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
9	C₄×#10621877	—	2	8	no	—
10	C₂×#13063236	C₂²	1	4	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
12	#14582654	C₂²	1	8	yes	—
13	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#376	C₄	1	4	yes	—
17	C₄₃×#376	C₂²	1	4	yes	—
18	C₂×#95×C₄₃	C₄	1	4	yes	—
19	C₂×#10621877	—	2	8	no	—
20	C₂×#10621877	—	2	8	no	—
21	C₂×#10621877	C₂×C₄	1	16	yes	—
22	C₂²×C₄₃⋊C₈	C₂³	1	8	yes	—
23	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	32	yes	complemented
24	#13063236	—	2	8	no	—
25	C₄×C₄₃⋊C₈	—	2	8	no	—
26	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₄₃	—	2	4	no	—
31	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₃	D₄	1	4	yes	—
36	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
37	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
40	#95×C₄₃	C₂×C₄	1	16	yes	—
41	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	—	2	4	no	—
43	C₂²×C₄×C₄₃	—	4	4	no	—
44	C₂²×C₄×C₄₃	—	4	4	no	—
45	C₂²×C₄×C₄₃	—	4	4	no	—
46	C₂²×C₄×C₄₃	C₄²	1	8	yes	—
47	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₄₃	#41	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₄²×C₄₃	—	2	4	no	—
58	C₄²×C₄₃	—	2	4	no	—
59	C₄²×C₄₃	C₄²	1	8	yes	—
60	C₄²×C₄₃	C₄⋊C₄	1	8	yes	—
61	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
62	C₄²×C₄₃	C₂×D₄	1	4	yes	—
63	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
64	#10621877	—	2	32	no	—
65	C₂×C₄₃⋊C₈	—	2	8	no	—
66	C₂×C₄₃⋊C₈	—	2	8	no	—
67	C₂×C₄₃⋊C₈	C₂²×C₄	1	16	yes	—
68	C₂³×C₄₃	—	2	4	no	—
69	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
70	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₄₃	#117	1	2	yes	—
72	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	—	2	8	no	—
74	C₂×C₄×C₄₃	—	2	8	no	—
75	C₂×C₄×C₄₃	—	2	4	no	—
76	C₂×C₄×C₄₃	—	2	8	no	—
77	C₂×C₄×C₄₃	—	4	8	no	—
78	C₂×C₄×C₄₃	—	4	8	no	—
79	C₂×C₄×C₄₃	—	4	8	no	—
80	C₂×C₄×C₄₃	C₂×C₄²	1	8	yes	—
81	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
82	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	8	yes	—
83	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₄₃	#117	1	2	yes	—
85	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
86	C₂×C₄×C₄₃	C₄×Q₈	1	8	yes	—
87	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
91	C₄₃⋊C₈	—	2	16	no	—
92	#20429	—	43	1	no	maximal, Sylow
93	C₂²×C₄₃	—	2	4	no	—
94	C₂²×C₄₃	—	2	4	no	—
95	C₂²×C₄₃	—	2	4	no	—
96	C₂²×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
97	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
98	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
100	C₂²×C₄₃	#568	1	4	yes	—
101	C₄×C₄₃	—	2	8	no	—
102	C₄×C₄₃	—	2	8	no	—
103	C₄×C₄₃	—	4	4	no	—
104	C₄×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
105	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
106	C₄×C₄₃	C₂×C₄×Q₈	1	2	yes	—
107	C₂×#423	—	43	2	no	—
108	C₂×C₄×M₁₆	—	43	2	no	—
109	C₂×#376	D₄₃	1	1	yes	characteristic
110	#1768	—	43	8	no	—
111	#1728	—	43	2	no	—
112	C₂×C₄₃	—	2	4	no	—
113	C₂×C₄₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄₃	#2873	1	2	yes	—
115	C₂×C₄₃	#2905	1	2	yes	—
116	C₂×C₄₃	#2921	1	2	yes	—
117	C₂²×M₁₆	—	43	4	no	—
118	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
119	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
120	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
121	#423	—	43	16	no	—
122	C₂×C₄⋊C₈	—	43	8	no	—
123	C₄×M₁₆	—	86	8	no	—
124	C₂×#97	—	43	4	no	—
125	C₂×C₄×C₈	—	43	4	no	—
126	#376	Dic₄₃	1	4	yes	—
127	#376	C₂×D₄₃	1	4	yes	—
128	C₂×#95	Dic₄₃	1	4	yes	—
129	#336	—	43	8	no	—
130	C₄₃	#20429	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
132	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
133	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
134	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
135	C₂×M₁₆	—	43	16	no	—
136	C₂×M₁₆	—	86	8	no	—
137	C₂×M₁₆	—	86	8	no	—
138	C₂²×C₈	—	43	8	no	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	Dic₈₆	1	4	yes	—
141	C₂×C₄²	C₄×D₄₃	1	8	yes	—
142	C₂×C₄²	D₁₇₂	1	4	yes	—
143	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
144	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
145	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
146	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
147	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
148	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
149	C₄⋊C₈	—	43	32	no	—
150	#97	—	86	8	no	—
151	C₄×C₈	—	86	8	no	—
152	#95	C₂×Dic₄₃	1	16	yes	—
153	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	C₄×Dic₄₃	1	8	yes	—
159	C₂²×C₄	C₄⋊Dic₄₃	1	8	yes	—
160	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
161	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
162	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
163	C₂²×C₄	#10621898	1	4	yes	—
164	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
165	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
166	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
167	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
169	M₁₆	—	86	32	no	—
170	C₂×C₈	—	43	16	no	—
171	C₂×C₈	—	86	8	no	—
172	C₂×C₈	—	86	8	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₄×Dic₄₃	1	8	yes	—
176	C₄²	C₄⋊Dic₄₃	1	8	yes	—
177	C₄²	C₂×Dic₈₆	1	4	yes	—
178	C₄²	C₂×C₄×D₄₃	1	8	yes	—
179	C₄²	C₂×D₁₇₂	1	4	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
182	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
183	C₂³	#13063284	1	2	yes	—
184	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	C₂×C₄×Dic₄₃	1	8	yes	—
193	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
194	C₂×C₄	C₂×C₄⋊Dic₄₃	1	8	yes	—
195	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
196	C₂×C₄	#13063284	1	2	yes	—
197	C₂×C₄	C₄×D₁₇₂	1	8	yes	—
198	C₂×C₄	C₄×Dic₈₆	1	8	yes	—
199	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
200	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
201	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
202	C₂×C₄	C₂×#10621898	1	2	yes	—
203	C₈	—	86	16	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	4	no	—
207	C₂²	C₄×C₄⋊Dic₄₃	1	4	yes	—
208	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
209	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
210	C₂²	C₂×#13063284	1	1	yes	characteristic
211	C₂²	#14583666	1	4	yes	—
212	C₄	—	2	8	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	4	no	—
215	C₄	C₄×C₄⋊Dic₄₃	1	4	yes	—
216	C₄	C₂×C₄×D₁₇₂	1	2	yes	—
217	C₄	C₂×C₄×Dic₈₆	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	C₂×C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
220	C₂	#19143014	1	2	yes	—
221	C₂	#19143159	1	2	yes	—
222	C₂	#19143233	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₂×C₄₃×#376	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#376	D₄₃
hypocenter	C₄₃	#20429
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
3. #19134452	C₂	semidirect
7. #14582923	C₄	semidirect
23. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
130. C₄₃	#20429	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582923	2
C₂×C₄×#10621877	2
#19134452	2
#19134289	2
C₂×C₄₃×#376	2
#20429	43