Subgroups — #25250288

Order 11008.

532 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#10621902	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136428	C₂	1	3	yes	maximal, complemented
4	#19136426	C₂	1	3	yes	maximal, complemented
5	#19136392	C₂	1	3	yes	maximal, complemented
6	#19136349	C₂	1	3	yes	maximal, complemented
7	Q₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10621902	C₄	1	1	yes	characteristic
9	C₂×#13063264	C₂²	1	6	yes	—
10	Q₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
11	C₈×D₁₇₂	C₂²	1	3	yes	—
12	#14582964	C₄	1	3	yes	—
13	#14582951	C₄	1	3	yes	—
14	C₄×C₈×D₄₃	C₂²	1	3	yes	—
15	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	3	yes	—
16	Dic₄₃⋊C₁₆	C₂²	1	3	yes	—
17	Dic₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	(C₂×D₄₃)⋊C₁₆	C₂²	1	3	yes	—
19	(C₂×D₄₃)⋊C₁₆	C₂²	1	3	yes	—
20	C₄×C₄₃⋊C₁₆	C₂²	1	3	yes	—
21	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×C₄⋊C₁₆	C₂²	1	3	yes	—
23	C₄×C₁₆×C₄₃	C₂²	1	3	yes	—
24	C₂×#10621902	C₈	1	1	yes	characteristic
25	C₂×C₈×D₄₃	C₂×C₄	1	3	yes	—
26	C₂×C₈×D₄₃	C₂³	1	3	yes	—
27	Q₈×Dic₄₃	C₈	1	1	yes	characteristic
28	C₄×D₁₇₂	C₂×C₄	1	3	yes	—
29	#13063287	C₈	1	3	yes	—
30	#13063283	C₈	1	3	yes	—
31	C₄²×D₄₃	C₂×C₄	1	3	yes	—
32	#13063264	—	4	6	no	—
33	C₂×C₄₃⋊C₁₆	C₂³	1	3	yes	—
34	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
35	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	3	yes	—
36	(C₂×D₄₃)⋊C₈	C₂×C₄	1	6	yes	—
37	C₄×C₄₃⋊C₈	C₂×C₄	1	3	yes	—
38	C₈×Dic₄₃	C₂×C₄	1	3	yes	—
39	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₄₃	C₂³	1	3	yes	—
42	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₃	C₂×C₄	1	3	yes	—
44	C₄×C₈×C₄₃	C₂³	1	3	yes	—
45	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
46	#10621902	—	2	4	no	—
47	C₂×D₁₇₂	C₂×C₈	1	3	yes	—
48	C₂×C₄×D₄₃	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₄₃	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₄₃	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₄₃	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₄⋊C₄×C₄₃	C₂×C₈	1	3	yes	—
54	C₄²×C₄₃	C₂²×C₄	1	3	yes	—
55	(C₂×D₄₃)⋊C₄	C₂×C₈	1	6	yes	—
56	C₄⋊Dic₄₃	C₂×C₈	1	3	yes	—
57	C₄×Dic₄₃	C₂×C₈	1	3	yes	—
58	C₄×Dic₄₃	C₂×C₈	1	3	yes	—
59	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄₃⋊C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
62	C₈×D₄₃	—	2	3	no	—
63	C₈×D₄₃	—	2	3	no	—
64	C₈×D₄₃	—	4	3	no	—
65	C₁₆×C₄₃	—	2	3	no	—
66	C₁₆×C₄₃	—	2	1	no	—
67	C₄₃⋊C₁₆	—	2	3	no	—
68	C₄₃⋊C₁₆	—	2	1	no	—
69	C₂²×D₄₃	C₂²×C₈	1	3	yes	—
70	Q₈×C₄₃	M₃₂	1	4	yes	complemented
71	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×Dic₄₃	C₂²×C₈	1	3	yes	—
76	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
77	D₁₇₂	—	2	6	no	—
78	C₄×D₄₃	—	2	3	no	—
79	C₄×D₄₃	—	2	3	no	—
80	C₄×D₄₃	—	2	6	no	—
81	C₄×D₄₃	—	2	6	no	—
82	C₈×C₄₃	—	2	3	no	—
83	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
84	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₄₃⋊C₈	—	2	3	no	—
86	C₄₃⋊C₈	—	2	1	no	—
87	#33505	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
89	C₂×D₄₃	—	2	3	no	—
90	C₂×D₄₃	—	2	3	no	—
91	C₂×D₄₃	—	2	3	no	—
92	C₄×C₄₃	—	2	3	no	—
93	C₄×C₄₃	C₂×M₃₂	1	6	yes	—
94	C₄×C₄₃	#520	1	1	yes	characteristic
95	C₄×C₄₃	#568	1	1	yes	characteristic
96	Dic₄₃	—	2	1	no	—
97	Dic₄₃	—	2	3	no	—
98	Dic₄₃	#504	1	2	yes	—
99	C₈×#41	—	43	1	no	—
100	Q₈×C₁₆	—	43	1	no	—
101	Q₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
102	#2169	—	43	3	no	—
103	#2169	—	43	3	no	—
104	#2162	—	43	3	no	—
105	#2150	—	43	3	no	—
106	C₂×C₄₃	#2988	1	1	yes	characteristic
107	C₂×C₄₃	C₂²×M₃₂	1	1	yes	characteristic
108	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
109	D₄₃	—	4	3	no	—
110	C₄×#41	—	43	1	no	—
111	C₂×M₃₂	—	43	6	no	—
112	C₈×Q₈	—	43	1	no	—
113	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
114	C₈×D₄	—	43	3	no	—
115	C₈×D₄	—	43	3	no	—
116	#431	—	43	3	no	—
117	C₂×C₄×C₈	—	43	3	no	—
118	C₄⋊C₁₆	—	43	3	no	—
119	C₄⋊C₁₆	—	43	1	no	—
120	C₄⋊C₁₆	—	43	3	no	—
121	C₄⋊C₁₆	C₂×D₄₃	1	3	yes	—
122	C₂²⋊C₁₆	—	43	3	no	—
123	C₂²⋊C₁₆	—	43	3	no	—
124	C₄×C₁₆	—	43	3	no	—
125	C₄×C₁₆	C₂×D₄₃	1	3	yes	—
126	C₄₃	#33505	1	1	yes	characteristic, complemented, Sylow
127	C₂×#41	—	43	1	no	—
128	C₂²×C₈	—	43	3	no	—
129	C₂²×C₈	—	43	3	no	—
130	C₄×Q₈	—	43	1	no	—
131	C₄×Q₈	C₄×D₄₃	1	1	yes	characteristic
132	C₄×D₄	—	43	3	no	—
133	C₄×D₄	—	43	3	no	—
134	#117	—	43	3	no	—
135	C₂×C₄²	—	43	3	no	—
136	M₃₂	—	172	6	no	—
137	C₂×C₁₆	—	43	1	no	—
138	C₂×C₁₆	—	43	3	no	—
139	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
140	C₂×C₁₆	C₂²×D₄₃	1	3	yes	—
141	C₄⋊C₈	—	43	3	no	—
142	C₄⋊C₈	C₄×D₄₃	1	3	yes	—
143	C₂²⋊C₈	—	43	6	no	—
144	C₄×C₈	—	43	3	no	—
145	C₄×C₈	—	43	1	no	—
146	C₄×C₈	—	43	3	no	—
147	C₄×C₈	C₂²×D₄₃	1	3	yes	—
148	#41	—	86	4	no	—
149	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
150	C₂×D₄	—	43	3	no	—
151	C₂²×C₄	—	43	3	no	—
152	C₂²×C₄	—	43	3	no	—
153	C₂²×C₄	—	43	3	no	—
154	C₂×C₈	—	43	1	no	—
155	C₂×C₈	—	43	3	no	—
156	C₂×C₈	—	86	3	no	—
157	C₂×C₈	—	86	3	no	—
158	C₂×C₈	—	172	3	no	—
159	C₂×C₈	C₂×C₄×D₄₃	1	3	yes	—
160	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
161	C₄⋊C₄	—	43	3	no	—
162	C₄⋊C₄	C₈×D₄₃	1	3	yes	—
163	C₂²⋊C₄	—	43	6	no	—
164	C₄²	—	43	3	no	—
165	C₄²	—	43	3	no	—
166	C₄²	—	43	1	no	—
167	C₄²	C₂×C₄×D₄₃	1	3	yes	—
168	C₁₆	—	2	3	no	—
169	C₁₆	—	2	1	no	—
170	C₁₆	—	86	3	no	—
171	C₁₆	—	86	1	no	—
172	C₂³	—	43	3	no	—
173	Q₈	#13063264	1	4	yes	complemented
174	D₄	—	86	6	no	—
175	C₂×C₄	—	43	1	no	—
176	C₂×C₄	—	43	1	no	—
177	C₂×C₄	—	43	3	no	—
178	C₂×C₄	—	86	6	no	—
179	C₂×C₄	—	86	6	no	—
180	C₂×C₄	—	86	3	no	—
181	C₂×C₄	—	86	3	no	—
182	C₂×C₄	C₂×C₈×D₄₃	1	3	yes	—
183	C₂×C₄	C₂×C₈×D₄₃	1	3	yes	—
184	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
185	C₈	—	2	3	no	—
186	C₈	—	86	3	no	—
187	C₈	—	86	1	no	—
188	C₈	#13063371	1	1	yes	characteristic
189	C₈	#13063373	1	1	yes	characteristic
190	C₂²	—	86	3	no	—
191	C₂²	—	86	3	no	—
192	C₂²	—	86	3	no	—
193	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	43	2	no	—
196	C₄	—	86	3	no	—
197	C₄	—	86	1	no	—
198	C₄	C₂×#13063264	1	6	yes	—
199	C₄	#14583381	1	1	yes	characteristic
200	C₄	#14583664	1	1	yes	characteristic
201	C₂	—	172	3	no	—
202	C₂	#19143705	1	1	yes	characteristic
203	C₂	C₂²×#13063264	1	1	yes	characteristic
204	C₂	#504×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	Q₈×C₁₆×C₄₃	C₂
prime radical	C₂×#10621902	C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	Q₈×C₁₆	D₄₃
hypocenter	C₄₃	#33505
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄₃⋊C₁₆	C₂	semidirect
3. #19136428	C₂	semidirect
4. #19136426	C₂	semidirect
5. #19136392	C₂	semidirect
6. #19136349	C₂	semidirect
7. Q₈×C₁₆×C₄₃	C₂	semidirect
70. Q₈×C₄₃	M₃₂	semidirect
101. Q₈×C₁₆	D₄₃	semidirect
126. C₄₃	#33505	semidirect
173. Q₈	#13063264	semidirect

Maximal subgroups

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

subgroup	index
C₈×#10621902	2
Q₈×C₄₃⋊C₁₆	2
#19136428	2
#19136426	2
#19136392	2
#19136349	2
Q₈×C₁₆×C₄₃	2
#33505	43