Subgroups — #24983787

Order 11008.

246 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	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#19134759	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₃⋊C₁₆	—	2	1	no	—
5	C₂×#13063259	C₄	1	2	yes	—
6	C₂×#13063257	—	2	1	no	—
7	#14582765	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₃⋊C₁₆)	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#334	—	2	1	no	—
13	C₄₃×#334	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₄₃	—	2	2	no	—
15	C₂×C₄₃⋊C₁₆	—	2	2	no	—
16	C₂×C₄₃⋊C₁₆	—	4	1	no	—
17	C₂×C₄₃⋊C₁₆	—	4	1	no	—
18	#13063259	—	4	2	no	—
19	#13063257	—	4	1	no	—
20	#13063257	—	4	1	no	—
21	C₄₃⋊#95	—	4	1	no	—
22	C₄₃⋊#95	—	4	1	no	—
23	C₂²×C₈×C₄₃	—	4	1	no	—
24	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₄₃	—	2	2	no	—
28	C₄⋊C₈×C₄₃	—	4	1	no	—
29	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₄₃	—	4	2	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	—	2	1	no	—
39	C₂×C₈×C₄₃	—	4	2	no	—
40	C₂×C₈×C₄₃	—	8	2	no	—
41	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
46	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₄₃	Q₁₆	1	1	yes	characteristic
49	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
52	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
53	C₄⋊Dic₄₃	—	4	2	no	—
54	C₄⋊Dic₄₃	—	4	2	no	—
55	C₄₃⋊C₁₆	—	4	2	no	—
56	C₂³×C₄₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	—	2	2	no	—
59	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#100	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#101	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
67	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
69	C₂×Dic₄₃	—	8	2	no	—
70	C₂×Dic₄₃	—	8	2	no	—
71	C₂×Dic₄₃	—	8	2	no	—
72	C₈×C₄₃	—	2	1	no	—
73	C₈×C₄₃	—	2	1	no	—
74	C₈×C₄₃	—	8	2	no	—
75	C₈×C₄₃	#102	1	1	yes	characteristic
76	C₈×C₄₃	#103	1	1	yes	characteristic
77	C₈×C₄₃	#106	1	1	yes	characteristic
78	C₈×C₄₃	#107	1	1	yes	characteristic
79	#7275	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	#331	1	1	yes	characteristic
81	C₂²×C₄₃	#332	1	1	yes	characteristic
82	C₂²×C₄₃	#340	1	1	yes	characteristic
83	C₂²×C₄₃	#359	1	2	yes	—
84	C₂²×C₄₃	#360	1	2	yes	—
85	C₄×C₄₃	—	2	1	no	—
86	C₄×C₄₃	—	2	1	no	—
87	C₄×C₄₃	—	2	1	no	—
88	C₄×C₄₃	—	2	1	no	—
89	C₄×C₄₃	#332	1	1	yes	characteristic
90	C₄×C₄₃	#333	1	1	yes	characteristic
91	C₄×C₄₃	#340	1	1	yes	characteristic
92	C₄×C₄₃	#343	1	1	yes	characteristic
93	Dic₄₃	—	16	2	no	—
94	C₂×C₄⋊C₁₆	—	43	1	no	—
95	#1846	—	43	1	no	—
96	C₂×#334	D₄₃	1	1	yes	characteristic
97	C₂×C₄₃	#1296	1	1	yes	characteristic
98	C₂×C₄₃	#1360	1	2	yes	—
99	C₂×C₄₃	#1363	1	2	yes	—
100	C₂×C₄₃	#1382	1	1	yes	characteristic
101	C₂×C₄₃	#1386	1	1	yes	characteristic
102	C₂²×C₁₆	—	86	1	no	—
103	C₂×#107	—	43	2	no	—
104	C₂×#106	—	86	1	no	—
105	C₂×C₄⋊C₈	—	2	1	no	—
106	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
107	#382	—	86	1	no	—
108	C₄⋊C₁₆	—	43	2	no	—
109	C₄⋊C₁₆	—	86	1	no	—
110	#334	—	2	1	no	—
111	#334	Dic₄₃	1	2	yes	complemented
112	C₄₃	#7275	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
115	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	86	2	no	—
117	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
118	C₂×C₁₆	—	86	2	no	—
119	C₂×C₁₆	—	172	1	no	—
120	C₂×C₁₆	—	172	1	no	—
121	#107	—	172	2	no	—
122	#106	—	172	1	no	—
123	#106	—	172	1	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄×C₈	Dic₈₆	1	1	yes	characteristic
127	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
130	#95	—	172	1	no	—
131	#95	—	172	1	no	—
132	C₂²×C₄	—	172	2	no	—
133	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	#10621882	1	1	yes	characteristic
135	C₂²×C₄	#10621884	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	#10621873	1	1	yes	characteristic
141	C₂×C₈	#10621873	1	1	yes	characteristic
142	C₂×C₈	D₃₄₄	1	1	yes	characteristic
143	C₂×C₈	Dic₁₇₂	1	1	yes	characteristic
144	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
145	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
146	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
148	C₄⋊C₄	—	172	2	no	—
149	C₄⋊C₄	—	172	2	no	—
150	C₄²	Dic₄₃⋊C₄	1	1	yes	characteristic
151	C₄²	#10621883	1	1	yes	characteristic
152	C₄²	#10621885	1	1	yes	characteristic
153	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
154	C₁₆	—	172	2	no	—
155	C₂³	#13063245	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	344	2	no	—
159	C₂×C₄	—	344	2	no	—
160	C₂×C₄	—	344	2	no	—
161	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
162	C₂×C₄	#13063241	1	1	yes	characteristic
163	C₂×C₄	#13063243	1	1	yes	characteristic
164	C₂×C₄	#13063246	1	1	yes	characteristic
165	C₂×C₄	#13063247	1	1	yes	characteristic
166	C₂×C₄	#13063248	1	1	yes	characteristic
167	C₂×C₄	#13063249	1	1	yes	characteristic
168	C₂×C₄	#13063250	1	1	yes	characteristic
169	C₂×C₄	#13063256	1	1	yes	characteristic
170	C₂×C₄	#13063258	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	8	2	no	—
174	C₈	#13063247	1	1	yes	characteristic
175	C₈	#13063250	1	1	yes	characteristic
176	C₈	#13063257	1	1	yes	characteristic
177	C₈	#13063259	1	1	yes	characteristic
178	C₂²	#14582642	1	1	yes	characteristic
179	C₂²	#14582646	1	1	yes	characteristic
180	C₂²	#14582661	1	1	yes	characteristic
181	C₂²	#14582705	1	2	yes	—
182	C₂²	#14582708	1	2	yes	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	688	2	no	—
188	C₄	#14582644	1	1	yes	characteristic
189	C₄	#14582647	1	1	yes	characteristic
190	C₄	#14582662	1	1	yes	characteristic
191	C₄	#14582667	1	1	yes	characteristic
192	C₂	#19132479	1	1	yes	characteristic
193	C₂	#19132637	1	2	yes	—
194	C₂	#19132648	1	2	yes	—
195	C₂	#19132692	1	1	yes	characteristic
196	C₂	#19132702	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063245
commutator	C₂×C₈×C₄₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₂×C₄₃×#334	C₂
socle, prime radical	C₂³×C₄₃	#102
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	#7275
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₄₃⋊C₁₆)	C₄	semidirect
13. C₄₃×#334	C₄	semidirect
111. #334	Dic₄₃	semidirect
112. C₄₃	#7275	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₃⋊C₁₆)	2
#19134759	2
C₂×C₄₃×#334	2
#7275	43