Subgroups — #25250228

Order 11008.

556 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136421	C₂	1	8	yes	maximal, complemented
2	#19136398	C₂	1	2	yes	maximal, complemented
3	#19136209	C₂	1	4	yes	maximal, complemented
4	C₄₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₂²	1	4	yes	—
6	C₂²×C₄₃⋊C₁₆	C₂²	1	4	yes	—
7	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
8	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
9	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	8	yes	—
10	#14582675	C₂²	1	4	yes	—
11	C₂×#131×C₄₃	C₂²	1	2	yes	—
12	C₄₃×#532	C₄	1	1	yes	characteristic
13	C₄₃×#436	C₄	1	4	yes	—
14	C₈×D₄×C₄₃	C₂²	1	4	yes	—
15	C₄₃×#423	C₄	1	1	yes	characteristic
16	C₄₃×#408	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	#13063265	—	4	4	no	—
19	C₂×C₄₃⋊C₁₆	—	4	4	no	—
20	C₂×C₄₃⋊C₁₆	C₂³	1	8	yes	—
21	C₂×#41×C₄₃	C₂×C₄	1	2	yes	—
22	#131×C₄₃	—	4	2	no	—
23	#131×C₄₃	—	4	2	no	—
24	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₄₃	C₈	1	1	yes	characteristic
30	#127×C₄₃	C₈	1	1	yes	characteristic
31	#124×C₄₃	C₈	1	2	yes	—
32	#121×C₄₃	C₈	1	4	yes	—
33	C₄×D₄×C₄₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
36	C₂²⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₃	C₂³	1	2	yes	—
39	#41×C₄₃	—	4	2	no	—
40	#41×C₄₃	—	4	2	no	—
41	C₂×Q₈×C₄₃	C₂×C₈	1	2	yes	—
42	C₂×D₄×C₄₃	C₂×C₈	1	4	yes	—
43	C₂×D₄×C₄₃	C₂×C₈	1	4	yes	—
44	C₂×D₄×C₄₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₄₃	—	4	2	no	—
49	M₁₆×C₄₃	—	4	4	no	—
50	C₂×C₈×C₄₃	—	2	2	no	—
51	C₂×C₈×C₄₃	—	2	4	no	—
52	C₂×C₈×C₄₃	—	2	4	no	—
53	C₂×C₈×C₄₃	—	4	1	no	—
54	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₄₃	C₂×C₈	1	4	yes	—
59	C₂²⋊C₄×C₄₃	C₂×C₈	1	8	yes	—
60	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
63	C₄₃⋊C₁₆	—	2	8	no	—
64	C₂³×C₄₃	C₂²×C₈	1	4	yes	—
65	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₄₃	—	4	2	no	—
67	D₄×C₄₃	—	4	4	no	—
68	D₄×C₄₃	—	4	2	no	—
69	D₄×C₄₃	—	4	4	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	—	2	1	no	—
73	C₂×C₄×C₄₃	—	2	4	no	—
74	C₂×C₄×C₄₃	—	4	2	no	—
75	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₄₃	—	2	4	no	—
81	C₈×C₄₃	—	2	1	no	—
82	C₈×C₄₃	—	2	2	no	—
83	C₈×C₄₃	H(5,𝔽₂)	1	2	yes	—
84	#33494	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	—	2	2	no	—
86	C₂²×C₄₃	—	2	8	no	—
87	C₂²×C₄₃	—	2	4	no	—
88	C₂²×C₄₃	—	2	1	no	—
89	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	—	2	2	no	—
92	C₄×C₄₃	—	2	1	no	—
93	C₄×C₄₃	—	2	4	no	—
94	C₄×C₄₃	#518	1	1	yes	characteristic
95	C₄×C₄₃	#568	1	1	yes	characteristic
96	#2931	D₄₃	1	1	yes	characteristic
97	#2168	—	43	8	no	—
98	#2163	—	43	2	no	—
99	#2113	—	43	4	no	—
100	C₂×C₄₃	—	4	1	no	—
101	C₂×C₄₃	—	4	4	no	—
102	C₂×C₄₃	#2972	1	1	yes	characteristic
103	C₂×C₄₃	#3407	1	2	yes	—
104	C₂×#131	C₂×D₄₃	1	2	yes	—
105	#532	Dic₄₃	1	1	yes	characteristic
106	C₂×M₃₂	—	43	4	no	—
107	C₂²×C₁₆	—	43	4	no	—
108	#436	Dic₄₃	1	4	yes	—
109	C₈×D₄	C₂×D₄₃	1	4	yes	—
110	#423	Dic₄₃	1	1	yes	characteristic
111	#408	Dic₄₃	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
113	C₄⋊C₁₆	—	43	4	no	—
114	C₂²⋊C₁₆	—	43	4	no	—
115	C₂²⋊C₁₆	—	43	8	no	—
116	#346	—	43	4	no	—
117	C₄₃	#33494	1	1	yes	characteristic, complemented, Sylow
118	C₂×#41	C₂×Dic₄₃	1	2	yes	—
119	#131	—	4	2	no	—
120	#131	—	4	2	no	—
121	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
122	C₂×M₁₆	C₂×Dic₄₃	1	4	yes	—
123	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
126	C₄⋊Q₈	C₄₃⋊C₈	1	1	yes	characteristic
127	#127	C₄₃⋊C₈	1	1	yes	characteristic
128	#124	C₄₃⋊C₈	1	2	yes	—
129	#121	C₄₃⋊C₈	1	4	yes	—
130	C₄×D₄	C₂×Dic₄₃	1	4	yes	—
131	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
132	M₃₂	—	172	4	no	—
133	C₂×C₁₆	—	43	8	no	—
134	C₂×C₁₆	—	172	4	no	—
135	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
136	C₂²⋊C₈	C₂×Dic₄₃	1	8	yes	—
137	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
138	C₄×C₈	C₂²×D₄₃	1	2	yes	—
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₂×C₄₃⋊C₈	1	2	yes	—
142	C₂×D₄	C₂×C₄₃⋊C₈	1	4	yes	—
143	C₂×D₄	C₂×C₄₃⋊C₈	1	4	yes	—
144	C₂×D₄	C₂×C₄₃⋊C₈	1	2	yes	—
145	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
147	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
155	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
156	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
157	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂×C₄₃⋊C₈	1	4	yes	—
159	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	8	yes	—
160	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
161	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
162	C₄²	C₂²×Dic₄₃	1	2	yes	—
163	C₁₆	—	86	8	no	—
164	C₂³	C₂²×C₄₃⋊C₈	1	4	yes	—
165	C₂³	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
177	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
178	C₂×C₄	C₂²×C₄₃⋊C₈	1	4	yes	—
179	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	—	2	4	no	—
182	C₈	—	2	2	no	—
183	C₈	#13063370	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	8	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	4	no	—
188	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	4	no	—
193	C₄	#14583374	1	1	yes	characteristic
194	C₄	#14583666	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	—	4	4	no	—
197	C₂	#19143619	1	1	yes	characteristic
198	C₂	#19147698	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₄₃×#2931	C₂
prime radical	C₄₃×#532	C₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2931	D₄₃
hypocenter	C₄₃	#33494
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #19136421	C₂	semidirect
2. #19136398	C₂	semidirect
3. #19136209	C₂	semidirect
117. C₄₃	#33494	semidirect

Maximal subgroups

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

subgroup	index
#19136421	2
#19136398	2
#19136209	2
C₄₃×#2931	2
#33494	43