Subgroups — #25014627

Order 11008.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582725	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582725	C₂	1	1	yes	maximal, characteristic
3	#19135328	C₂	1	2	yes	maximal
4	#19132687	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063259	C₂²	1	2	yes	—
7	C₂×#13063259	C₂²	1	2	yes	—
8	#14582797	—	2	2	no	—
9	#14582725	—	2	2	no	—
10	#14582725	C₄	1	4	yes	complemented
11	#14582662	—	2	2	no	—
12	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₄₃	—	2	2	no	—
16	C₂×C₄⋊Dic₄₃	—	2	2	no	—
17	C₂×C₄⋊Dic₄₃	—	2	2	no	—
18	C₂×Dic₄₃⋊C₄	—	4	2	no	—
19	#13063259	—	2	2	no	—
20	#13063259	—	2	2	no	—
21	#13063259	—	2	2	no	—
22	#13063259	C₂×C₄	1	4	yes	—
23	C₄₃⋊#95	—	4	2	no	—
24	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₃	—	2	1	no	—
28	C₂×C₁₆×C₄₃	—	2	1	no	—
29	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
31	C₂×C₁₆×C₄₃	Q₈	1	2	yes	—
32	C₂×C₁₆×C₄₃	Q₈	1	2	yes	—
33	C₄×C₈×C₄₃	—	2	1	no	—
34	C₄×C₈×C₄₃	—	2	1	no	—
35	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₃	—	4	2	no	—
39	C₂²×Dic₄₃	—	4	2	no	—
40	C₂×C₈×C₄₃	—	2	1	no	—
41	C₂×C₈×C₄₃	—	2	1	no	—
42	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
49	C₄²×C₄₃	—	2	1	no	—
50	C₄²×C₄₃	—	2	1	no	—
51	C₄⋊Dic₄₃	—	2	4	no	—
52	C₄⋊Dic₄₃	—	4	2	no	—
53	C₄⋊Dic₄₃	—	4	2	no	—
54	C₄⋊Dic₄₃	—	4	2	no	—
55	Dic₄₃⋊C₄	—	8	4	no	—
56	C₁₆×C₄₃	—	2	2	no	—
57	C₁₆×C₄₃	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	2	1	no	—
60	C₂×C₄×C₄₃	—	2	1	no	—
61	C₂×C₄×C₄₃	—	2	1	no	—
62	C₂×C₄×C₄₃	—	2	1	no	—
63	C₂×C₄×C₄₃	D₁₆	1	2	yes	—
64	C₂×C₄×C₄₃	Q₃₂	1	2	yes	—
65	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
71	C₂×Dic₄₃	—	4	4	no	—
72	C₂×Dic₄₃	—	8	2	no	—
73	C₂×Dic₄₃	—	8	2	no	—
74	C₂×Dic₄₃	—	8	2	no	—
75	C₂×Dic₄₃	—	8	2	no	—
76	C₂×Dic₄₃	—	8	2	no	—
77	C₈×C₄₃	—	2	2	no	—
78	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₈×C₄₃	#125	1	1	yes	characteristic
82	#13165	—	43	1	no	maximal, Sylow
83	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
84	C₂²×C₄₃	#466	1	1	yes	characteristic
85	C₂²×C₄₃	#500	1	1	yes	characteristic
86	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
87	C₂²×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
88	C₂²×C₄₃	#508	1	1	yes	characteristic
89	C₂²×C₄₃	#508	1	1	yes	characteristic
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	—	2	2	no	—
92	C₄×C₄₃	#389	1	1	yes	characteristic
93	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
94	C₄×C₄₃	#467	1	1	yes	characteristic
95	C₄×C₄₃	#499	1	1	yes	characteristic
96	Dic₄₃	—	8	4	no	—
97	Dic₄₃	—	16	2	no	—
98	C₂×#366	—	43	1	no	—
99	C₂×#366	—	43	1	no	—
100	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic
101	#1942	—	43	2	no	—
102	#1380	—	43	2	no	—
103	C₂×C₄₃	#1942	1	1	yes	characteristic
104	C₂×C₄₃	C₄×D₁₆	1	1	yes	characteristic
105	C₂×C₄₃	C₄×Q₃₂	1	1	yes	characteristic
106	C₂×C₄₃	#2215	1	1	yes	characteristic
107	C₂×C₄₃	#2216	1	1	yes	characteristic
108	C₂×C₄₃	#2252	1	1	yes	characteristic
109	C₂×C₄₃	#2253	1	1	yes	characteristic
110	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
111	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
112	C₂×#107	—	43	2	no	—
113	C₂×#107	—	43	2	no	—
114	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
115	#389	—	86	2	no	—
116	#366	—	43	4	no	—
117	#366	—	86	2	no	—
118	C₄×C₁₆	—	2	2	no	—
119	#340	—	86	2	no	—
120	C₄₃	#13165	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	86	2	no	—
124	C₂×C₄⋊C₄	—	86	2	no	—
125	C₂×C₄⋊C₄	—	172	2	no	—
126	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	Dic₈₆	1	2	yes	—
130	C₂×C₁₆	Dic₈₆	1	2	yes	—
131	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
133	#107	—	43	4	no	—
134	#107	—	86	2	no	—
135	#107	—	86	2	no	—
136	#107	—	86	2	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	#95	—	172	2	no	—
140	C₂²×C₄	—	172	2	no	—
141	C₂²×C₄	—	172	2	no	—
142	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
143	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	Dic₁₇₂	1	2	yes	—
148	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
149	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
151	C₂×C₈	#10621898	1	1	yes	characteristic
152	C₂×C₈	#10621898	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
154	C₄⋊C₄	—	86	4	no	—
155	C₄⋊C₄	—	344	4	no	—
156	C₄⋊C₄	—	172	2	no	—
157	C₄⋊C₄	—	172	2	no	—
158	C₄⋊C₄	—	172	2	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	Dic₄₃⋊C₄	1	4	yes	complemented
163	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	344	2	no	—
169	C₂×C₄	—	344	2	no	—
170	C₂×C₄	—	344	2	no	—
171	C₂×C₄	—	344	2	no	—
172	C₂×C₄	—	344	2	no	—
173	C₂×C₄	—	172	4	no	—
174	C₂×C₄	D₆₈₈	1	2	yes	—
175	C₂×C₄	Dic₃₄₄	1	2	yes	—
176	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
177	C₂×C₄	#13063299	1	1	yes	characteristic
178	C₂×C₄	#13063325	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
180	C₂×C₄	#13063347	1	1	yes	characteristic
181	C₂×C₄	#13063347	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
184	C₈	C₄×Dic₈₆	1	1	yes	characteristic
185	C₈	#13063304	1	1	yes	characteristic
186	C₈	#13063317	1	1	yes	characteristic
187	C₂²	C₄×D₃₄₄	1	1	yes	characteristic
188	C₂²	#14583135	1	1	yes	characteristic
189	C₂²	#14583304	1	1	yes	characteristic
190	C₂²	C₂×D₆₈₈	1	1	yes	characteristic
191	C₂²	C₂×Dic₃₄₄	1	1	yes	characteristic
192	C₂²	#14583331	1	1	yes	characteristic
193	C₂²	#14583331	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	344	4	no	—
197	C₄	—	688	2	no	—
198	C₄	#14582797	1	1	yes	characteristic
199	C₄	C₄×Dic₁₇₂	1	1	yes	characteristic
200	C₄	#14583140	1	1	yes	characteristic
201	C₄	#14583301	1	1	yes	characteristic
202	C₂	#19135328	1	1	yes	characteristic
203	C₂	C₄×D₆₈₈	1	1	yes	characteristic
204	C₂	C₄×Dic₃₄₄	1	1	yes	characteristic
205	C₂	#19136681	1	1	yes	characteristic
206	C₂	#19136686	1	1	yes	characteristic
207	C₂	#19136870	1	1	yes	characteristic
208	C₂	#19136874	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	#13165
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #14582725	C₄	semidirect
57. C₁₆×C₄₃	C₄⋊C₄	semidirect
120. C₄₃	#13165	semidirect
162. C₁₆	Dic₄₃⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582725	2
#19135328	2
#19132687	2
C₂×C₄×C₁₆×C₄₃	2
#13165	43