Subgroups — #24987841

Order 11008.

430 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063259	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₄₃	C₂	1	2	yes	maximal
3	C₂×#14582650	C₂	1	1	yes	maximal, characteristic
4	#19132435	C₂	1	2	yes	maximal
5	C₂×C₄₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063259	C₄	1	2	yes	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₈×Dic₄₃	—	2	2	no	—
9	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	#14582662	C₄	1	2	yes	—
11	#14582653	—	2	2	no	—
12	#14582650	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#322	C₄	1	4	yes	complemented
17	C₂²×C₄₃⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₄₃	—	2	2	no	—
20	#13063259	C₈	1	8	yes	complemented
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
23	C₈×Dic₄₃	—	4	4	no	—
24	C₄₃⋊#95	—	2	2	no	—
25	C₂²×C₈×C₄₃	—	2	1	no	—
26	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₃	D₄	1	2	yes	—
29	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₄₃	—	2	1	no	—
31	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
39	C₂²×Dic₄₃	—	2	4	no	—
40	C₂×C₈×C₄₃	—	2	2	no	—
41	C₂×C₈×C₄₃	—	2	2	no	—
42	C₂×C₈×C₄₃	—	2	2	no	—
43	C₂×C₈×C₄₃	—	2	2	no	—
44	C₂×C₈×C₄₃	—	4	2	no	—
45	C₂×C₈×C₄₃	C₄²	1	2	yes	—
46	C₂×C₈×C₄₃	C₄²	1	2	yes	—
47	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
49	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₄₃	#41	1	1	yes	characteristic
53	C₄⋊Dic₄₃	—	2	4	no	—
54	C₄⋊Dic₄₃	C₂×C₈	1	8	yes	—
55	C₄×Dic₄₃	—	4	2	no	—
56	C₄×Dic₄₃	—	4	2	no	—
57	C₂×C₄₃⋊C₈	—	2	4	no	—
58	C₂×C₄₃⋊C₈	—	4	2	no	—
59	C₂×C₄₃⋊C₈	—	4	2	no	—
60	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₄₃	#131	1	2	yes	—
70	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₄₃	#137	1	2	yes	—
72	C₂×Dic₄₃	—	2	8	no	—
73	C₂×Dic₄₃	—	4	4	no	—
74	C₂×Dic₄₃	—	4	4	no	—
75	C₈×C₄₃	—	2	2	no	—
76	C₈×C₄₃	—	2	4	no	—
77	C₈×C₄₃	—	4	2	no	—
78	C₈×C₄₃	C₄×C₈	1	4	yes	complemented
79	C₄₃⋊C₈	—	4	4	no	—
80	#8284	—	43	1	no	maximal, Sylow
81	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄₃	C₈×D₄	1	2	yes	—
83	C₂²×C₄₃	#441	1	1	yes	characteristic
84	C₂²×C₄₃	#442	1	1	yes	characteristic
85	C₂²×C₄₃	#444	1	2	yes	—
86	C₄×C₄₃	—	2	1	no	—
87	C₄×C₄₃	—	2	1	no	—
88	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₄₃	#405	1	1	yes	characteristic
90	C₄×C₄₃	#428	1	2	yes	—
91	C₄×C₄₃	#440	1	2	yes	—
92	C₄×C₄₃	C₈×Q₈	1	2	yes	—
93	Dic₄₃	—	4	8	no	—
94	C₄×#107	—	43	1	no	—
95	C₈×C₄⋊C₄	—	43	2	no	—
96	C₂×#334	—	43	1	no	—
97	C₂×#322	D₄₃	1	1	yes	characteristic
98	#1279	—	43	2	no	—
99	C₂×C₄₃	#1579	1	2	yes	—
100	C₂×C₄₃	#1580	1	2	yes	—
101	C₂×C₄₃	C₈×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄₃	#1773	1	1	yes	characteristic
103	C₂×C₄₃	#1776	1	1	yes	characteristic
104	C₂×#107	—	43	2	no	—
105	C₂×C₄⋊C₈	—	43	2	no	—
106	C₂×C₄×C₈	—	86	2	no	—
107	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	43	2	no	—
111	#340	—	43	2	no	—
112	#336	—	86	2	no	—
113	#334	—	43	4	no	—
114	#322	Dic₄₃	1	4	yes	complemented
115	C₄₃	#8284	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	86	2	no	—
118	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
119	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
120	C₂²×C₈	D₁₇₂	1	2	yes	—
121	C₂×C₄⋊C₄	—	43	4	no	—
122	C₂×C₄²	—	86	2	no	—
123	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
124	#107	—	43	8	no	—
125	C₄⋊C₈	—	43	4	no	—
126	C₄⋊C₈	—	86	2	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	172	4	no	—
129	C₄×C₈	Dic₈₆	1	2	yes	—
130	C₄×C₈	C₄×D₄₃	1	2	yes	—
131	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
132	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
133	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
134	#95	—	86	2	no	—
135	C₂²×C₄	—	86	4	no	—
136	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
138	C₂²×C₄	#10621898	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	86	4	no	—
145	C₂×C₈	—	172	2	no	—
146	C₂×C₈	—	172	2	no	—
147	C₂×C₈	C₈×D₄₃	1	4	yes	—
148	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
149	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
150	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	—
151	C₄⋊C₄	—	43	8	no	—
152	C₄⋊C₄	—	86	4	no	—
153	C₄²	—	172	2	no	—
154	C₄²	—	172	2	no	—
155	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
157	C₄²	#10621898	1	1	yes	characteristic
158	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
159	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	86	8	no	—
162	C₂×C₄	—	172	4	no	—
163	C₂×C₄	—	172	4	no	—
164	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
166	C₂×C₄	#13063284	1	1	yes	characteristic
167	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₈₆	1	1	yes	characteristic
169	C₂×C₄	C₄×Dic₈₆	1	1	yes	characteristic
170	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
171	C₂×C₄	#13063331	1	2	yes	—
172	C₂×C₄	#13063353	1	2	yes	—
173	C₂×C₄	#13063358	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	—	2	4	no	—
176	C₈	—	4	2	no	—
177	C₈	—	172	4	no	—
178	C₈	C₈×Dic₄₃	1	4	yes	complemented
179	C₂²	C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
180	C₂²	C₈×D₁₇₂	1	2	yes	—
181	C₂²	#14583007	1	1	yes	characteristic
182	C₂²	#14583012	1	1	yes	characteristic
183	C₂²	#14583022	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	172	8	no	—
187	C₄	C₂×C₈×Dic₄₃	1	1	yes	characteristic
188	C₄	#14582854	1	1	yes	characteristic
189	C₄	#14582941	1	2	yes	—
190	C₄	#14583001	1	2	yes	—
191	C₄	C₈×Dic₈₆	1	2	yes	—
192	C₂	#19133415	1	2	yes	—
193	C₂	#19133420	1	2	yes	—
194	C₂	C₈×C₄⋊Dic₄₃	1	1	yes	characteristic
195	C₂	#19134479	1	1	yes	characteristic
196	C₂	#19134492	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#322	C₂
socle, prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#322	D₄₃
hypocenter	C₄₃	#8284
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
12. #14582650	C₄	semidirect
16. C₄₃×#322	C₄	semidirect
20. #13063259	C₈	semidirect
78. C₈×C₄₃	C₄×C₈	semidirect
114. #322	Dic₄₃	semidirect
115. C₄₃	#8284	semidirect
178. C₈	C₈×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063259	2
C₈×C₄⋊Dic₄₃	2
C₂×#14582650	2
#19132435	2
C₂×C₄₃×#322	2
#8284	43