Subgroups — #25107002

Order 11008.

786 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136993	C₂	1	2	yes	maximal
2	#19135365	C₂	1	8	yes	maximal, complemented
3	#19134628	C₂	1	4	yes	maximal
4	C₂×C₄₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063259	C₂²	1	4	yes	—
6	C₂×#13063257	C₂²	1	4	yes	—
7	#14582767	—	2	2	no	—
8	#14582765	C₂²	1	4	yes	—
9	#14582762	—	2	2	no	—
10	#14582759	C₂²	1	4	yes	—
11	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
12	C₄×C₂²⋊Dic₄₃	—	2	2	no	—
13	#14582745	—	2	2	no	—
14	#14582662	C₂²	1	8	yes	—
15	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#423	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
20	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
21	C₂×C₄×Dic₄₃	—	2	4	no	—
22	#13063259	—	4	4	no	—
23	#13063257	—	4	4	no	—
24	C₄₃⋊#95	—	2	4	no	—
25	C₄₃⋊#95	—	2	4	no	—
26	C₄₃⋊#95	—	2	4	no	—
27	C₄₃⋊#95	—	2	4	no	—
28	C₂³×C₄×C₄₃	D₄	1	2	yes	—
29	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₄₃	—	4	2	no	—
31	C₂×M₁₆×C₄₃	—	4	2	no	—
32	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
38	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
39	C₄⋊C₈×C₄₃	C₂×C₄	1	16	yes	—
40	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	—	2	2	no	—
42	C₂²×C₄×C₄₃	—	2	2	no	—
43	C₂²×C₄×C₄₃	—	4	4	no	—
44	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
45	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₄₃	—	2	8	no	—
52	M₁₆×C₄₃	—	4	8	no	—
53	C₂×C₈×C₄₃	—	2	4	no	—
54	C₂×C₈×C₄₃	—	2	4	no	—
55	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
56	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
57	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
58	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
59	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
60	C₂²⋊Dic₄₃	—	4	8	no	—
61	C₄⋊Dic₄₃	—	2	8	no	—
62	C₄⋊Dic₄₃	—	2	8	no	—
63	C₄×Dic₄₃	—	4	8	no	—
64	C₂³×C₄₃	—	2	2	no	—
65	C₂³×C₄₃	—	2	4	no	—
66	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	4	no	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	2	2	no	—
73	C₂×C₄×C₄₃	—	2	4	no	—
74	C₂×C₄×C₄₃	—	4	8	no	—
75	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
81	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
82	C₂×Dic₄₃	—	4	8	no	—
83	C₂×Dic₄₃	—	4	8	no	—
84	C₂×Dic₄₃	—	4	8	no	—
85	C₈×C₄₃	—	2	8	no	—
86	#21965	—	43	1	no	maximal, Sylow
87	C₂²×C₄₃	—	2	2	no	—
88	C₂²×C₄₃	—	2	4	no	—
89	C₂²×C₄₃	—	4	4	no	—
90	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄₃	#537	1	2	yes	—
92	C₂²×C₄₃	#575	1	4	yes	—
93	C₄×C₄₃	—	2	2	no	—
94	C₄×C₄₃	—	2	4	no	—
95	C₄×C₄₃	—	2	4	no	—
96	C₄×C₄₃	#518	1	1	yes	characteristic
97	C₄×C₄₃	#519	1	1	yes	characteristic
98	C₄×C₄₃	#544	1	2	yes	—
99	Dic₄₃	—	8	8	no	—
100	C₂×#423	D₄₃	1	1	yes	characteristic
101	#2283	—	43	2	no	—
102	#1949	—	43	8	no	—
103	#1809	—	43	4	no	—
104	C₂×C₄₃	—	4	2	no	—
105	C₂×C₄₃	#2316	1	1	yes	characteristic
106	C₂×C₄₃	#2911	1	2	yes	—
107	C₂×C₄₃	#3256	1	4	yes	—
108	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
109	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
110	C₂×#107	—	43	4	no	—
111	C₂×#106	—	43	4	no	—
112	#423	Dic₄₃	1	8	yes	complemented
113	C₂×C₄⋊C₈	C₂×D₄₃	1	4	yes	—
114	#383	—	86	2	no	—
115	#382	—	43	4	no	—
116	#381	—	86	2	no	—
117	#380	—	43	4	no	—
118	C₄×C₄⋊C₄	—	43	4	no	—
119	C₄×C₂²⋊C₄	—	86	2	no	—
120	#376	—	86	2	no	—
121	#340	—	43	8	no	—
122	C₄₃	#21965	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	D₁₇₂	1	2	yes	—
124	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
125	C₂×M₁₆	—	4	2	no	—
126	C₂×M₁₆	—	4	2	no	—
127	C₂×M₁₆	C₂×Dic₄₃	1	8	yes	—
128	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
129	C₂×C₄⋊C₄	—	43	8	no	—
130	C₂×C₂²⋊C₄	—	86	4	no	—
131	C₂×C₄²	—	86	4	no	—
132	C₂×C₄²	Dic₈₆	1	4	yes	—
133	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
134	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
135	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
136	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
137	#107	—	172	4	no	—
138	#106	—	172	4	no	—
139	C₄⋊C₈	C₂×Dic₄₃	1	16	yes	—
140	#95	—	86	4	no	—
141	#95	—	86	4	no	—
142	#95	—	86	4	no	—
143	#95	—	86	4	no	—
144	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	—	86	8	no	—
149	C₂²×C₄	C₄⋊Dic₄₃	1	8	yes	—
150	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
151	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
152	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
155	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
156	M₁₆	—	4	8	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
160	C₄⋊C₄	—	86	8	no	—
161	C₄⋊C₄	—	86	8	no	—
162	C₂²⋊C₄	—	172	8	no	—
163	C₄²	—	172	8	no	—
164	C₄²	C₄⋊Dic₄₃	1	4	yes	—
165	C₄²	C₄⋊Dic₄₃	1	4	yes	—
166	C₄²	C₂×Dic₈₆	1	4	yes	—
167	C₄²	C₂²×Dic₄₃	1	4	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	2	no	—
170	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
171	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	172	8	no	—
180	C₂×C₄	—	172	8	no	—
181	C₂×C₄	—	172	8	no	—
182	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
185	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
186	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
187	C₂×C₄	#13063370	1	2	yes	—
188	C₂×C₄	#13063373	1	2	yes	—
189	C₈	—	2	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
194	C₂²	#14583480	1	2	yes	—
195	C₂²	#14583699	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	344	8	no	—
200	C₄	#14583374	1	1	yes	characteristic
201	C₄	#14583377	1	1	yes	characteristic
202	C₄	#14583530	1	2	yes	—
203	C₂	—	4	2	no	—
204	C₂	#19137213	1	1	yes	characteristic
205	C₂	#19143185	1	2	yes	—
206	C₂	#19145996	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#423	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#423	D₄₃
hypocenter	C₄₃	#21965
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
2. #19135365	C₂	semidirect
17. C₄₃×#423	C₄	semidirect
112. #423	Dic₄₃	semidirect
122. C₄₃	#21965	semidirect

Maximal subgroups

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

subgroup	index
#19136993	2
#19135365	2
#19134628	2
C₂×C₄₃×#423	2
#21965	43