Subgroups — #25095357

Order 11008.

922 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582952	C₂	1	2	yes	maximal
2	#19134464	C₂	1	4	yes	maximal, complemented
3	C₄⋊C₄×C₄₃⋊C₈	C₂	1	4	yes	maximal, complemented
4	#19134361	C₂	1	4	yes	maximal
5	C₄₃×#2285	C₂	1	1	yes	maximal, characteristic
6	#14582952	C₄	1	16	yes	complemented
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
10	C₂×#13063236	C₂²	1	4	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
13	#14582654	C₂²	1	8	yes	—
14	C₂×#117×C₄₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#384	C₄	1	2	yes	—
17	C₄₃×#382	C₄	1	2	yes	—
18	C₄₃×#380	C₄	1	4	yes	—
19	C₄×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
20	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
21	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
22	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
23	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
24	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	16	yes	—
25	#13063236	—	4	4	no	—
26	C₄×C₄₃⋊C₈	—	4	4	no	—
27	C₄×C₄₃⋊C₈	C₂×C₄	1	16	yes	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	#117×C₄₃	—	4	2	no	—
31	#117×C₄₃	—	4	2	no	—
32	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
40	#95×C₄₃	C₂×C₄	1	8	yes	—
41	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	—	2	8	no	—
43	C₂²×C₄×C₄₃	—	2	1	no	—
44	C₂²×C₄×C₄₃	—	2	1	no	—
45	C₂²×C₄×C₄₃	—	2	2	no	—
46	C₂²×C₄×C₄₃	C₄²	1	8	yes	—
47	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₄⋊C₄×C₄₃	—	2	16	no	—
54	C₄⋊C₄×C₄₃	—	4	4	no	—
55	C₂²⋊C₄×C₄₃	—	4	4	no	—
56	C₄²×C₄₃	—	2	4	no	—
57	C₄²×C₄₃	—	2	4	no	—
58	C₄²×C₄₃	C₄²	1	4	yes	—
59	C₄²×C₄₃	C₄²	1	4	yes	—
60	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
61	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
62	C₂×C₄₃⋊C₈	—	2	4	no	—
63	C₂×C₄₃⋊C₈	—	2	4	no	—
64	C₂×C₄₃⋊C₈	—	2	4	no	—
65	C₂×C₄₃⋊C₈	—	2	4	no	—
66	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
67	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
68	C₂³×C₄₃	—	2	1	no	—
69	C₂³×C₄₃	—	2	1	no	—
70	C₂³×C₄₃	—	2	4	no	—
71	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
72	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	—	2	1	no	—
74	C₂×C₄×C₄₃	—	2	1	no	—
75	C₂×C₄×C₄₃	—	2	4	no	—
76	C₂×C₄×C₄₃	—	2	4	no	—
77	C₂×C₄×C₄₃	—	2	4	no	—
78	C₂×C₄×C₄₃	—	2	16	no	—
79	C₂×C₄×C₄₃	—	2	8	no	—
80	C₂×C₄×C₄₃	—	2	16	no	—
81	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
82	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₄₃	#131	1	8	yes	—
85	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
86	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
89	C₄₃⋊C₈	—	2	8	no	—
90	C₄₃⋊C₈	—	2	8	no	—
91	#20538	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	—	2	4	no	—
93	C₂²×C₄₃	—	2	2	no	—
94	C₂²×C₄₃	—	4	2	no	—
95	C₂²×C₄₃	—	4	2	no	—
96	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
97	C₂²×C₄₃	#518	1	1	yes	characteristic
98	C₂²×C₄₃	#519	1	1	yes	characteristic
99	C₂²×C₄₃	C₂×#131	1	4	yes	—
100	C₄×C₄₃	—	2	4	no	—
101	C₄×C₄₃	—	2	2	no	—
102	C₄×C₄₃	—	4	8	no	—
103	C₄×C₄₃	#405	1	8	yes	—
104	C₄×C₄₃	#520	1	2	yes	—
105	C₄×C₄₃	#520	1	2	yes	—
106	C₂×#431	—	43	2	no	—
107	#2285	D₄₃	1	1	yes	characteristic
108	#1770	—	43	4	no	—
109	C₈×C₄⋊C₄	—	43	4	no	—
110	#1750	—	43	4	no	—
111	C₂×C₄₃	—	4	2	no	—
112	C₂×C₄₃	#2274	1	1	yes	characteristic
113	C₂×C₄₃	C₂×#405	1	2	yes	—
114	C₂×C₄₃	#2979	1	2	yes	—
115	C₂×C₄₃	#2986	1	2	yes	—
116	C₂×#117	C₂×D₄₃	1	2	yes	—
117	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
118	#431	—	43	16	no	—
119	C₂×C₄⋊C₈	—	43	2	no	—
120	C₂×C₄⋊C₈	—	43	2	no	—
121	C₂×C₂²⋊C₈	—	43	4	no	—
122	C₂×#97	—	43	4	no	—
123	C₂×C₄×C₈	—	43	4	no	—
124	C₂×C₄×C₈	—	43	4	no	—
125	#384	Dic₄₃	1	2	yes	—
126	#382	Dic₄₃	1	2	yes	—
127	#380	Dic₄₃	1	4	yes	—
128	C₄×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
129	#336	—	43	8	no	—
130	C₄₃	#20538	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
132	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
133	C₂²×C₈	—	43	4	no	—
134	C₂²×C₈	—	43	4	no	—
135	#117	—	4	2	no	—
136	#117	—	4	2	no	—
137	C₂×C₄⋊C₄	C₂×Dic₄₃	1	8	yes	—
138	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
139	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
140	C₂×C₄²	C₄×D₄₃	1	8	yes	—
141	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
143	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
144	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
145	C₄⋊C₈	—	43	8	no	—
146	C₄⋊C₈	—	43	8	no	—
147	C₂²⋊C₈	—	43	16	no	—
148	#97	—	172	4	no	—
149	C₄×C₈	—	43	16	no	—
150	C₄×C₈	—	172	4	no	—
151	#95	C₂×Dic₄₃	1	8	yes	—
152	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₄×Dic₄₃	1	8	yes	—
158	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
159	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
160	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
163	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
164	C₂×C₈	—	43	8	no	—
165	C₂×C₈	—	43	8	no	—
166	C₂×C₈	—	86	4	no	—
167	C₂×C₈	—	86	4	no	—
168	C₂×C₈	—	86	4	no	—
169	C₂×C₈	—	86	4	no	—
170	C₄⋊C₄	—	2	16	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₂²⋊C₄	—	4	4	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₄×Dic₄₃	1	4	yes	—
176	C₄²	C₄×Dic₄₃	1	4	yes	—
177	C₄²	C₂×C₄×D₄₃	1	4	yes	—
178	C₄²	C₂×C₄×D₄₃	1	4	yes	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
183	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	16	no	—
192	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
193	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
194	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
195	C₂×C₄	#13063332	1	8	yes	—
196	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
197	C₂×C₄	#13063370	1	1	yes	characteristic
198	C₂×C₄	#13063370	1	1	yes	characteristic
199	C₂×C₄	#13063373	1	2	yes	—
200	C₈	—	86	8	no	—
201	C₈	—	86	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
207	C₂²	#14583374	1	1	yes	characteristic
208	C₂²	#14583377	1	1	yes	characteristic
209	C₂²	C₂×#13063332	1	4	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	8	no	—
213	C₄	#14582855	1	8	yes	—
214	C₄	#14583380	1	2	yes	—
215	C₄	#14583380	1	2	yes	—
216	C₂	—	4	2	no	—
217	C₂	#19136952	1	1	yes	characteristic
218	C₂	C₂×#14582855	1	2	yes	—
219	C₂	#19143662	1	2	yes	—
220	C₂	#19143697	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₄₃×#2285	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2285	D₄₃
hypocenter	C₄₃	#20538
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
2. #19134464	C₂	semidirect
3. C₄⋊C₄×C₄₃⋊C₈	C₂	semidirect
6. #14582952	C₄	semidirect
130. C₄₃	#20538	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582952	2
#19134464	2
C₄⋊C₄×C₄₃⋊C₈	2
#19134361	2
C₄₃×#2285	2
#20538	43