Subgroups — #24983513

Order 11008.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₄×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₃⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
6	C₄×C₄₃⋊C₁₆	—	2	1	no	—
7	C₄×C₄₃⋊C₁₆	C₄	1	2	yes	complemented
8	#14582652	—	2	1	no	—
9	#14582652	C₄	1	2	yes	complemented
10	C₂×#107×C₄₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
13	C₄₃×#340	—	2	1	no	—
14	C₂²×C₄₃⋊C₈	—	4	1	no	—
15	C₂×C₄₃⋊C₁₆	—	2	4	no	—
16	C₂×C₄₃⋊C₁₆	—	2	2	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
19	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
22	C₂×C₄²×C₄₃	—	4	1	no	—
23	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
24	#107×C₄₃	—	2	2	no	—
25	#107×C₄₃	C₈	1	4	yes	complemented
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	Q₈	1	1	yes	characteristic
30	#95×C₄₃	—	4	1	no	—
31	C₂²×C₄×C₄₃	—	4	2	no	—
32	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₄₃	—	2	2	no	—
36	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
42	C₄⋊C₄×C₄₃	—	2	4	no	—
43	C₄⋊C₄×C₄₃	—	4	2	no	—
44	C₄²×C₄₃	—	8	1	no	—
45	C₄²×C₄₃	—	8	1	no	—
46	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
50	C₂×C₄₃⋊C₈	—	4	2	no	—
51	C₂×C₄₃⋊C₈	—	8	1	no	—
52	C₂×C₄₃⋊C₈	—	8	1	no	—
53	C₄₃⋊C₁₆	—	2	4	no	—
54	C₂³×C₄₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	4	4	no	—
57	C₂×C₄×C₄₃	—	8	2	no	—
58	C₂×C₄×C₄₃	—	8	2	no	—
59	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#104	1	2	yes	—
65	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
67	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₃	SD₃₂	1	1	yes	characteristic
69	C₂×C₄×C₄₃	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₄₃	Q₃₂	1	1	yes	characteristic
71	C₈×C₄₃	—	2	1	no	—
72	C₈×C₄₃	—	2	1	no	—
73	C₈×C₄₃	C₄×C₈	1	1	yes	characteristic
74	C₈×C₄₃	#97	1	1	yes	characteristic
75	C₈×C₄₃	C₄⋊C₈	1	2	yes	—
76	C₄₃⋊C₈	—	8	2	no	—
77	#7171	—	43	1	no	maximal, Sylow
78	C₂²×C₄₃	D₄⋊C₈	1	2	yes	—
79	C₂²×C₄₃	#340	1	1	yes	characteristic
80	C₂²×C₄₃	#357	1	1	yes	characteristic
81	C₂²×C₄₃	#358	1	1	yes	characteristic
82	C₂²×C₄₃	#359	1	2	yes	—
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	—	2	1	no	—
85	C₄×C₄₃	—	8	4	no	—
86	C₄×C₄₃	Q₈⋊C₈	1	2	yes	—
87	C₄×C₄₃	#336	1	1	yes	characteristic
88	C₄×C₄₃	#339	1	1	yes	characteristic
89	C₄×C₄₃	#357	1	1	yes	characteristic
90	C₄×C₄₃	#358	1	1	yes	characteristic
91	C₄×C₄₃	#366	1	1	yes	characteristic
92	C₄×C₄₃	#367	1	1	yes	characteristic
93	C₂×C₄×C₁₆	—	43	1	no	—
94	C₄×#107	D₄₃	1	1	yes	characteristic
95	C₂×#335	—	43	1	no	—
96	C₂×C₄₃	#1279	1	1	yes	characteristic
97	C₂×C₄₃	#1331	1	2	yes	—
98	C₂×C₄₃	#1332	1	2	yes	—
99	C₂×C₄₃	#1380	1	1	yes	characteristic
100	C₂×C₄₃	#1382	1	1	yes	characteristic
101	C₂²×C₁₆	—	86	1	no	—
102	C₂×#107	Dic₄₃	1	2	yes	—
103	C₂×C₄⋊C₈	—	86	1	no	—
104	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄×C₁₆	—	43	2	no	—
107	C₄×C₁₆	—	86	1	no	—
108	#340	—	2	1	no	—
109	#335	—	43	2	no	—
110	#335	—	86	1	no	—
111	C₄₃	#7171	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	172	1	no	—
113	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
118	C₂×C₁₆	—	86	4	no	—
119	C₂×C₁₆	—	86	2	no	—
120	#107	—	2	2	no	—
121	#107	C₄₃⋊C₈	1	4	yes	complemented
122	C₄⋊C₈	—	86	2	no	—
123	C₄⋊C₈	—	172	1	no	—
124	C₄×C₈	Dic₈₆	1	1	yes	characteristic
125	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
126	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
127	C₄×C₈	D₁₇₂	1	1	yes	characteristic
128	#95	—	4	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	#10621882	1	1	yes	characteristic
131	C₂²×C₄	#10621883	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	344	1	no	—
135	C₂×C₈	—	344	1	no	—
136	C₂×C₈	—	172	2	no	—
137	C₂×C₈	C₂×C₄₃⋊C₈	1	2	yes	—
138	C₂×C₈	#10621877	1	2	yes	—
139	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
140	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
142	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	Dic₄₃⋊C₄	1	1	yes	characteristic
148	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
149	C₄²	#10621884	1	1	yes	characteristic
150	C₄²	#10621885	1	1	yes	characteristic
151	C₁₆	—	86	4	no	—
152	C₂³	#13063246	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
159	C₂×C₄	#13063245	1	1	yes	characteristic
160	C₂×C₄	#13063248	1	1	yes	characteristic
161	C₂×C₄	#13063249	1	1	yes	characteristic
162	C₂×C₄	#13063252	1	2	yes	—
163	C₂×C₄	#13063256	1	1	yes	characteristic
164	C₂×C₄	#13063258	1	1	yes	characteristic
165	C₂×C₄	#13063266	1	1	yes	characteristic
166	C₂×C₄	#13063268	1	1	yes	characteristic
167	C₂×C₄	#13063269	1	1	yes	characteristic
168	C₂×C₄	#13063271	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	344	2	no	—
172	C₈	C₄×C₄₃⋊C₈	1	1	yes	characteristic
173	C₈	#13063236	1	1	yes	characteristic
174	C₈	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
175	C₂²	D₄⋊(C₄₃⋊C₈)	1	2	yes	—
176	C₂²	#14582661	1	1	yes	characteristic
177	C₂²	#14582700	1	1	yes	characteristic
178	C₂²	#14582703	1	1	yes	characteristic
179	C₂²	#14582706	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	Q₈⋊(C₄₃⋊C₈)	1	2	yes	—
184	C₄	#14582654	1	1	yes	characteristic
185	C₄	#14582658	1	1	yes	characteristic
186	C₄	#14582699	1	1	yes	characteristic
187	C₄	#14582702	1	1	yes	characteristic
188	C₄	#14582724	1	1	yes	characteristic
189	C₄	#14582726	1	1	yes	characteristic
190	C₂	#19132436	1	1	yes	characteristic
191	C₂	#19132556	1	2	yes	—
192	C₂	#19132559	1	2	yes	—
193	C₂	#19132686	1	1	yes	characteristic
194	C₂	#19132690	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10621882
commutator	C₈×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	#102
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	#7171
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. C₄×C₄₃⋊C₁₆	C₄	semidirect
9. #14582652	C₄	semidirect
25. #107×C₄₃	C₈	semidirect
111. C₄₃	#7171	semidirect
121. #107	C₄₃⋊C₈	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
C₄×#107×C₄₃	2
#7171	43