Subgroups — #25110266

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
2	#19137343	C₂	1	2	yes	maximal
3	#19135568	C₂	1	2	yes	maximal
4	#19135188	C₂	1	4	yes	maximal
5	#19135167	C₂	1	4	yes	maximal
6	#19134726	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063325	C₂²	1	2	yes	—
9	#14583304	C₄	1	8	yes	complemented
10	C₂×#13063259	C₂²	1	2	yes	—
11	C₂×#13063259	C₂²	1	2	yes	—
12	#14582803	—	2	4	no	—
13	#14582797	C₂²	1	4	yes	—
14	#14582782	—	2	4	no	—
15	#14582765	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
17	#14582662	C₂²	1	2	yes	—
18	#14582662	C₂²	1	2	yes	—
19	#14582662	C₂²	1	8	yes	—
20	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#376	C₂²	1	1	yes	characteristic
23	C₄₃×#336	C₂²	1	2	yes	—
24	C₄₃×#336	C₂²	1	2	yes	—
25	C₂²×Dic₈₆	—	2	2	no	—
26	#13063325	—	2	4	no	—
27	#13063325	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊Dic₄₃	—	2	2	no	—
29	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
30	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
31	C₂×Dic₄₃⋊C₄	—	2	4	no	—
32	C₂×C₄×Dic₄₃	—	2	4	no	—
33	#13063259	C₂×C₄	1	8	yes	—
34	#13063259	C₂×C₄	1	8	yes	—
35	C₄₃⋊#95	—	2	4	no	—
36	C₄₃⋊#95	—	2	2	no	—
37	C₄₃⋊#95	—	2	4	no	—
38	C₄₃⋊#95	—	2	2	no	—
39	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	#97×C₄₃	—	2	2	no	—
45	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
46	#95×C₄₃	D₄	1	2	yes	—
47	#95×C₄₃	D₄	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₄₃	—	2	4	no	—
53	C₂²×Dic₄₃	—	2	4	no	—
54	C₂×Dic₈₆	—	2	8	no	—
55	C₂×Dic₈₆	—	8	4	no	—
56	C₂×C₈×C₄₃	—	2	2	no	—
57	C₂×C₈×C₄₃	—	2	2	no	—
58	C₂×C₈×C₄₃	—	2	2	no	—
59	C₂×C₈×C₄₃	—	2	2	no	—
60	C₂×C₈×C₄₃	—	2	2	no	—
61	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	—	2	2	no	—
65	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
66	C₄²×C₄₃	#41	1	2	yes	—
67	C₄⋊Dic₄₃	—	2	8	no	—
68	C₄⋊Dic₄₃	—	2	4	no	—
69	C₄⋊Dic₄₃	—	2	4	no	—
70	C₄⋊Dic₄₃	—	2	4	no	—
71	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
72	Dic₄₃⋊C₄	—	8	4	no	—
73	C₄×Dic₄₃	—	8	4	no	—
74	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	4	no	—
78	C₂×C₄×C₄₃	—	2	4	no	—
79	C₂×C₄×C₄₃	—	2	4	no	—
80	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
84	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
85	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
86	C₂×Dic₄₃	—	2	8	no	—
87	C₂×Dic₄₃	—	4	4	no	—
88	C₂×Dic₄₃	—	4	4	no	—
89	C₂×Dic₄₃	—	4	4	no	—
90	C₂×Dic₄₃	—	4	4	no	—
91	C₂×Dic₄₃	—	4	4	no	—
92	Dic₈₆	—	8	8	no	—
93	C₈×C₄₃	—	2	2	no	—
94	C₈×C₄₃	—	2	2	no	—
95	C₈×C₄₃	—	4	2	no	—
96	#22254	—	43	1	no	maximal, Sylow
97	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂²×C₄₃	#536	1	2	yes	—
99	C₂²×C₄₃	#576	1	2	yes	—
100	C₂²×C₄₃	#578	1	2	yes	—
101	C₄×C₄₃	—	2	2	no	—
102	C₄×C₄₃	—	2	2	no	—
103	C₄×C₄₃	—	4	4	no	—
104	C₄×C₄₃	#519	1	1	yes	characteristic
105	C₄×C₄₃	#519	1	1	yes	characteristic
106	C₄×C₄₃	#541	1	2	yes	—
107	Dic₄₃	—	4	8	no	—
108	Dic₄₃	—	8	4	no	—
109	C₂×#500	—	43	1	no	—
110	#2331	—	43	2	no	—
111	#1993	—	43	2	no	—
112	#1921	—	43	4	no	—
113	#1918	—	43	4	no	—
114	#1835	—	43	1	no	—
115	#1746	D₄₃	1	1	yes	characteristic
116	C₂×C₄₃	#2327	1	1	yes	characteristic
117	C₂×C₄₃	#2948	1	1	yes	characteristic
118	C₂×C₄₃	#2950	1	1	yes	characteristic
119	C₂×C₄₃	#3243	1	4	yes	—
120	C₂×C₄⋊Q₈	—	43	2	no	—
121	#500	—	43	8	no	—
122	C₂×#107	—	43	2	no	—
123	C₂×#107	—	43	2	no	—
124	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
126	#391	—	86	4	no	—
127	#389	—	43	4	no	—
128	#387	—	86	4	no	—
129	#382	—	43	2	no	—
130	C₄×C₄⋊C₄	—	43	4	no	—
131	#376	C₂×D₄₃	1	1	yes	characteristic
132	#340	—	43	2	no	—
133	#340	—	43	2	no	—
134	#340	—	43	8	no	—
135	#336	C₂×D₄₃	1	2	yes	—
136	#336	C₂×D₄₃	1	2	yes	—
137	C₄₃	#22254	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	86	2	no	—
139	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
140	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
142	C₄⋊Q₈	—	43	8	no	—
143	C₄⋊Q₈	—	86	4	no	—
144	C₂×C₄⋊C₄	—	43	4	no	—
145	C₂×C₄⋊C₄	—	43	4	no	—
146	C₂×C₄⋊C₄	—	86	2	no	—
147	C₂×C₄⋊C₄	—	86	4	no	—
148	C₂×C₄²	—	86	4	no	—
149	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
150	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
151	#107	—	43	8	no	—
152	#107	—	43	8	no	—
153	#97	—	2	2	no	—
154	C₄×C₈	C₄×D₄₃	1	4	yes	—
155	#95	—	86	4	no	—
156	#95	—	86	2	no	—
157	#95	—	86	2	no	—
158	#95	—	86	4	no	—
159	#95	D₁₇₂	1	2	yes	—
160	#95	D₁₇₂	1	2	yes	—
161	C₂×Q₈	—	86	8	no	—
162	C₂×Q₈	—	344	4	no	—
163	C₂²×C₄	—	86	4	no	—
164	C₂²×C₄	—	86	4	no	—
165	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
166	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
168	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
176	C₄⋊C₄	—	43	8	no	—
177	C₄⋊C₄	—	86	8	no	—
178	C₄⋊C₄	—	86	4	no	—
179	C₄⋊C₄	—	86	4	no	—
180	C₄⋊C₄	—	86	4	no	—
181	C₄⋊C₄	—	344	4	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	344	4	no	—
185	C₄²	C₂×C₄×D₄₃	1	2	yes	—
186	C₄²	#10621898	1	2	yes	—
187	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
188	Q₈	—	344	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	86	8	no	—
195	C₂×C₄	—	172	4	no	—
196	C₂×C₄	—	172	4	no	—
197	C₂×C₄	—	172	4	no	—
198	C₂×C₄	—	172	4	no	—
199	C₂×C₄	—	172	4	no	—
200	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
201	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
202	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
203	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
204	C₂×C₄	#13063372	1	2	yes	—
205	C₂×C₄	#13063372	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	4	2	no	—
209	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
210	C₂²	#14583475	1	2	yes	—
211	C₂²	#14583704	1	2	yes	—
212	C₂²	#14583716	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	4	no	—
216	C₄	—	344	4	no	—
217	C₄	—	172	8	no	—
218	C₄	#14583376	1	1	yes	characteristic
219	C₄	#14583376	1	1	yes	characteristic
220	C₄	#14583511	1	2	yes	—
221	C₂	#19137314	1	1	yes	characteristic
222	C₂	#19143448	1	1	yes	characteristic
223	C₂	#19143464	1	1	yes	characteristic
224	C₂	#19145880	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1746	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1746	D₄₃
hypocenter	C₄₃	#22254
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1746

Complements

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

N	H ≅ G/N	type
9. #14583304	C₄	semidirect
137. C₄₃	#22254	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583304	2
#19137343	2
#19135568	2
#19135188	2
#19135167	2
#19134726	2
C₄₃×#1746	2
#22254	43