Subgroups — #25250176

Order 11008.

548 conjugacy classes of subgroups.

The subgroup lattice has 208 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	#19136397	C₂	1	3	yes	maximal, complemented
6	C₄×#13063264	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	3	yes	—
10	C₂×#13063264	C₂²	1	3	yes	—
11	Q₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
12	C₈×D₁₇₂	C₂²	1	3	yes	—
13	#14582964	C₄	1	3	yes	—
14	#14582951	C₄	1	3	yes	—
15	C₄×C₈×D₄₃	C₂²	1	3	yes	—
16	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	3	yes	—
17	(C₂×D₄₃)⋊C₁₆	C₂²	1	6	yes	—
18	#14582674	C₂²	1	3	yes	—
19	#14582674	C₂²	1	1	yes	characteristic
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	—	2	6	no	—
33	#13063264	—	2	6	no	—
34	C₂×C₄₃⋊C₁₆	C₂³	1	3	yes	—
35	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
36	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	3	yes	—
37	(C₂×D₄₃)⋊C₈	C₂×C₄	1	6	yes	—
38	C₄×C₄₃⋊C₈	C₂×C₄	1	3	yes	—
39	C₈×Dic₄₃	C₂×C₄	1	3	yes	—
40	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₄₃	C₂³	1	3	yes	—
43	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₄₃	C₂×C₄	1	3	yes	—
45	C₄×C₈×C₄₃	C₂³	1	3	yes	—
46	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
47	#10621902	—	2	4	no	—
48	C₂×D₁₇₂	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₄₃	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₄₃	C₂×C₈	1	3	yes	—
51	C₂×C₄×D₄₃	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₄₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₄₃	C₂×C₈	1	3	yes	—
55	C₄²×C₄₃	C₂²×C₄	1	3	yes	—
56	(C₂×D₄₃)⋊C₄	C₂×C₈	1	6	yes	—
57	C₄⋊Dic₄₃	C₂×C₈	1	3	yes	—
58	C₄×Dic₄₃	C₂×C₈	1	3	yes	—
59	C₄×Dic₄₃	C₂×C₈	1	3	yes	—
60	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄₃⋊C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₄₃	—	2	3	no	—
64	C₈×D₄₃	—	2	3	no	—
65	C₈×D₄₃	—	2	6	no	—
66	C₁₆×C₄₃	—	2	3	no	—
67	C₁₆×C₄₃	#41	1	2	yes	complemented
68	C₄₃⋊C₁₆	—	2	3	no	—
69	C₄₃⋊C₁₆	#41	1	2	yes	complemented
70	C₂²×D₄₃	C₂²×C₈	1	3	yes	—
71	Q₈×C₄₃	M₃₂	1	4	yes	complemented
72	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×Dic₄₃	C₂²×C₈	1	3	yes	—
78	D₁₇₂	—	2	6	no	—
79	C₄×D₄₃	—	2	3	no	—
80	C₄×D₄₃	—	2	3	no	—
81	C₄×D₄₃	—	2	6	no	—
82	C₄×D₄₃	—	4	3	no	—
83	C₈×C₄₃	—	2	3	no	—
84	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
85	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
86	C₄₃⋊C₈	—	2	3	no	—
87	C₄₃⋊C₈	#131	1	2	yes	—
88	#33485	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
90	C₂×D₄₃	—	2	3	no	—
91	C₂×D₄₃	—	2	3	no	—
92	C₂×D₄₃	—	2	3	no	—
93	C₄×C₄₃	—	2	3	no	—
94	C₄×C₄₃	C₂×M₃₂	1	6	yes	—
95	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
96	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
97	Dic₄₃	—	2	1	no	—
98	Dic₄₃	—	2	1	no	—
99	Dic₄₃	—	2	3	no	—
100	C₈×#41	—	43	1	no	—
101	Q₈×C₁₆	—	43	1	no	—
102	Q₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
103	#2169	—	43	3	no	—
104	#2169	—	43	3	no	—
105	#2163	—	43	3	no	—
106	C₄×M₃₂	—	43	3	no	—
107	C₂×C₄₃	C₈×#41	1	1	yes	characteristic
108	C₂×C₄₃	C₂²×M₃₂	1	1	yes	characteristic
109	C₂×C₄₃	#3407	1	1	yes	characteristic
110	D₄₃	—	4	3	no	—
111	C₄×#41	—	43	1	no	—
112	C₂×M₃₂	—	43	3	no	—
113	C₂×M₃₂	—	43	3	no	—
114	C₈×Q₈	—	43	1	no	—
115	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
116	C₈×D₄	—	43	3	no	—
117	C₈×D₄	—	43	3	no	—
118	#431	—	43	3	no	—
119	C₂×C₄×C₈	—	43	3	no	—
120	C₄⋊C₁₆	—	43	3	no	—
121	C₄⋊C₁₆	C₂×D₄₃	1	3	yes	—
122	C₂²⋊C₁₆	—	43	6	no	—
123	#346	—	43	1	no	—
124	#346	—	43	3	no	—
125	C₄×C₁₆	—	43	3	no	—
126	C₄×C₁₆	C₂×D₄₃	1	3	yes	—
127	C₄₃	#33485	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	—	43	1	no	—
129	C₂²×C₈	—	43	3	no	—
130	C₂²×C₈	—	43	3	no	—
131	C₄×Q₈	—	43	1	no	—
132	C₄×Q₈	C₄×D₄₃	1	1	yes	characteristic
133	C₄×D₄	—	43	3	no	—
134	C₄×D₄	—	43	3	no	—
135	#117	—	43	3	no	—
136	C₂×C₄²	—	43	3	no	—
137	M₃₂	—	86	6	no	—
138	M₃₂	—	86	6	no	—
139	C₂×C₁₆	—	43	1	no	—
140	C₂×C₁₆	—	43	3	no	—
141	C₂×C₁₆	C₂²×D₄₃	1	3	yes	—
142	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
143	C₄⋊C₈	—	43	3	no	—
144	C₄⋊C₈	C₄×D₄₃	1	3	yes	—
145	C₂²⋊C₈	—	43	6	no	—
146	C₄×C₈	—	43	3	no	—
147	C₄×C₈	—	43	1	no	—
148	C₄×C₈	—	43	3	no	—
149	C₄×C₈	C₂²×D₄₃	1	3	yes	—
150	#41	—	86	4	no	—
151	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
152	C₂×D₄	—	43	3	no	—
153	C₂²×C₄	—	43	3	no	—
154	C₂²×C₄	—	43	3	no	—
155	C₂²×C₄	—	43	3	no	—
156	C₂×C₈	—	43	1	no	—
157	C₂×C₈	—	43	3	no	—
158	C₂×C₈	—	86	6	no	—
159	C₂×C₈	—	86	3	no	—
160	C₂×C₈	—	86	3	no	—
161	C₂×C₈	C₂×C₄×D₄₃	1	3	yes	—
162	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
163	C₄⋊C₄	—	43	3	no	—
164	C₄⋊C₄	C₈×D₄₃	1	3	yes	—
165	C₂²⋊C₄	—	43	6	no	—
166	C₄²	—	43	3	no	—
167	C₄²	—	43	3	no	—
168	C₄²	—	43	1	no	—
169	C₄²	C₂×C₄×D₄₃	1	3	yes	—
170	C₁₆	—	2	3	no	—
171	C₁₆	—	43	2	no	—
172	C₁₆	—	86	3	no	—
173	C₁₆	#10621902	1	2	yes	complemented
174	C₂³	—	43	3	no	—
175	Q₈	#13063264	1	4	yes	complemented
176	D₄	—	86	6	no	—
177	C₂×C₄	—	43	1	no	—
178	C₂×C₄	—	43	1	no	—
179	C₂×C₄	—	43	3	no	—
180	C₂×C₄	—	86	6	no	—
181	C₂×C₄	—	86	3	no	—
182	C₂×C₄	—	86	3	no	—
183	C₂×C₄	—	172	3	no	—
184	C₂×C₄	C₂×C₈×D₄₃	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₄₃	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	43	2	no	—
189	C₈	—	86	3	no	—
190	C₈	#41×D₄₃	1	1	yes	characteristic
191	C₈	C₂×#10621902	1	1	yes	characteristic
192	C₂²	—	86	3	no	—
193	C₂²	—	86	3	no	—
194	C₂²	—	86	3	no	—
195	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	86	3	no	—
198	C₄	—	86	1	no	—
199	C₄	—	86	1	no	—
200	C₄	C₂×#13063264	1	6	yes	—
201	C₄	C₄×#10621902	1	1	yes	characteristic
202	C₄	#131×D₄₃	1	1	yes	characteristic
203	C₂	—	172	3	no	—
204	C₂	C₈×#10621902	1	1	yes	characteristic
205	C₂	C₂²×#13063264	1	1	yes	characteristic
206	C₂	#19147696	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	#33485
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. #19136397	C₂	semidirect
6. C₄×#13063264	C₂	semidirect
7. Q₈×C₁₆×C₄₃	C₂	semidirect
67. C₁₆×C₄₃	#41	semidirect
69. C₄₃⋊C₁₆	#41	semidirect
71. Q₈×C₄₃	M₃₂	semidirect
102. Q₈×C₁₆	D₄₃	semidirect
127. C₄₃	#33485	semidirect
173. C₁₆	#10621902	semidirect
175. 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
#19136397	2
C₄×#13063264	2
Q₈×C₁₆×C₄₃	2
#33485	43