Subgroups — #24983466

Order 11008.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₂×#14582652	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₃⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
6	C₄×C₄₃⋊C₁₆	—	2	1	no	—
7	C₄×C₄₃⋊C₁₆	—	2	1	no	—
8	#14582652	—	2	1	no	—
9	#14582652	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#335	—	2	1	no	—
13	C₄₃×#335	C₄	1	2	yes	—
14	C₂²×C₄₃⋊C₈	—	4	1	no	—
15	C₂×C₄₃⋊C₁₆	—	2	2	no	—
16	C₂×C₄₃⋊C₁₆	—	2	2	no	—
17	C₂×C₄₃⋊C₁₆	C₈	1	4	yes	complemented
18	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
19	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
20	C₂²×C₈×C₄₃	—	4	1	no	—
21	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₄₃	—	2	2	no	—
25	C₄⋊C₈×C₄₃	—	4	1	no	—
26	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₂×C₈×C₄₃	—	4	2	no	—
35	C₂×C₈×C₄₃	—	8	1	no	—
36	C₂×C₈×C₄₃	—	8	1	no	—
37	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
43	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
46	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₄₃⋊C₈	—	4	2	no	—
48	C₂×C₄₃⋊C₈	—	8	1	no	—
49	C₂×C₄₃⋊C₈	—	8	1	no	—
50	C₄₃⋊C₁₆	—	2	4	no	—
51	C₂³×C₄₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#108	1	2	yes	—
61	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₃	SD₃₂	1	1	yes	characteristic
63	C₂×C₄×C₄₃	SD₃₂	1	1	yes	characteristic
64	C₂×C₄×C₄₃	Q₃₂	1	1	yes	characteristic
65	C₈×C₄₃	—	2	1	no	—
66	C₈×C₄₃	—	2	1	no	—
67	C₈×C₄₃	—	8	2	no	—
68	C₈×C₄₃	C₄×C₈	1	1	yes	characteristic
69	C₈×C₄₃	#97	1	1	yes	characteristic
70	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
71	C₄₃⋊C₈	—	8	2	no	—
72	#7154	—	43	1	no	maximal, Sylow
73	C₂²×C₄₃	#335	1	2	yes	—
74	C₂²×C₄₃	#340	1	1	yes	characteristic
75	C₂²×C₄₃	#366	1	1	yes	characteristic
76	C₂²×C₄₃	#367	1	1	yes	characteristic
77	C₂²×C₄₃	#368	1	2	yes	—
78	C₄×C₄₃	—	2	1	no	—
79	C₄×C₄₃	—	2	1	no	—
80	C₄×C₄₃	#334	1	2	yes	—
81	C₄×C₄₃	#336	1	1	yes	characteristic
82	C₄×C₄₃	#341	1	1	yes	characteristic
83	C₄×C₄₃	#357	1	1	yes	characteristic
84	C₄×C₄₃	#357	1	1	yes	characteristic
85	C₄×C₄₃	#358	1	1	yes	characteristic
86	C₄×C₄₃	#358	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	43	1	no	—
88	C₂×#335	—	43	1	no	—
89	C₂×#335	D₄₃	1	1	yes	characteristic
90	C₂×C₄₃	#1277	1	1	yes	characteristic
91	C₂×C₄₃	#1371	1	2	yes	—
92	C₂×C₄₃	#1372	1	2	yes	—
93	C₂×C₄₃	#1380	1	1	yes	characteristic
94	C₂×C₄₃	#1381	1	1	yes	characteristic
95	C₂²×C₁₆	—	43	2	no	—
96	C₂×C₄⋊C₈	—	2	1	no	—
97	C₂×C₄⋊C₈	—	86	1	no	—
98	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
99	C₄×C₁₆	—	86	1	no	—
100	C₄×C₁₆	—	86	1	no	—
101	#335	—	2	1	no	—
102	#335	—	43	2	no	—
103	#335	—	86	1	no	—
104	#335	Dic₄₃	1	2	yes	—
105	C₄₃	#7154	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	—	172	1	no	—
108	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
109	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
110	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
111	C₂×C₁₆	—	43	4	no	—
112	C₂×C₁₆	—	86	2	no	—
113	C₂×C₁₆	—	86	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	86	2	no	—
117	C₄⋊C₈	—	172	1	no	—
118	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
119	C₄×C₈	D₁₇₂	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
122	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
123	C₂²×C₄	#10621882	1	1	yes	characteristic
124	C₂²×C₄	#10621885	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	—	344	1	no	—
130	C₂×C₈	—	344	1	no	—
131	C₂×C₈	—	172	2	no	—
132	C₂×C₈	C₈×D₄₃	1	2	yes	—
133	C₂×C₈	#10621872	1	2	yes	—
134	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
135	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
136	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
137	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
138	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
139	C₄²	#10621883	1	1	yes	characteristic
140	C₄²	#10621884	1	1	yes	characteristic
141	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
142	C₁₆	—	86	4	no	—
143	C₂³	#13063258	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
146	C₂×C₄	#13063245	1	1	yes	characteristic
147	C₂×C₄	#13063246	1	1	yes	characteristic
148	C₂×C₄	#13063248	1	1	yes	characteristic
149	C₂×C₄	#13063249	1	1	yes	characteristic
150	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
151	C₂×C₄	#13063256	1	1	yes	characteristic
152	C₂×C₄	#13063260	1	2	yes	—
153	C₂×C₄	#13063266	1	1	yes	characteristic
154	C₂×C₄	#13063268	1	1	yes	characteristic
155	C₂×C₄	#13063269	1	1	yes	characteristic
156	C₂×C₄	#13063271	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	8	2	no	—
160	C₈	—	344	2	no	—
161	C₈	C₈×Dic₄₃	1	1	yes	characteristic
162	C₈	#13063235	1	1	yes	characteristic
163	C₈	(C₂×D₄₃)⋊C₈	1	2	yes	—
164	C₂²	#14582651	1	2	yes	—
165	C₂²	#14582661	1	1	yes	characteristic
166	C₂²	#14582724	1	1	yes	characteristic
167	C₂²	#14582726	1	1	yes	characteristic
168	C₂²	#14582728	1	2	yes	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	#14582649	1	2	yes	—
172	C₄	#14582653	1	1	yes	characteristic
173	C₄	#14582663	1	1	yes	characteristic
174	C₄	#14582699	1	1	yes	characteristic
175	C₄	#14582700	1	1	yes	characteristic
176	C₄	#14582702	1	1	yes	characteristic
177	C₄	#14582703	1	1	yes	characteristic
178	C₂	#19132429	1	1	yes	characteristic
179	C₂	#19132666	1	2	yes	—
180	C₂	#19132668	1	2	yes	—
181	C₂	#19132686	1	1	yes	characteristic
182	C₂	#19132688	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10621882
commutator	C₈×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₂×C₄₃×#335	C₂
socle, prime radical	C₂³×C₄₃	#107
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₂×#335	D₄₃
hypocenter	C₄₃	#7154
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₂×C₄₃⋊C₁₆	C₈	semidirect
105. C₄₃	#7154	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
C₂×#14582652	2
C₂×C₄₃×#335	2
#7154	43