Subgroups — #25106969

Order 11008.

906 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137002	C₂	1	2	yes	maximal
2	#19135359	C₂	1	8	yes	maximal, complemented
3	#19134620	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063284	—	2	2	no	—
6	C₂×#13063257	C₂²	1	8	yes	—
7	#14582769	C₂²	1	4	yes	—
8	#14582765	—	2	2	no	—
9	#14582759	—	2	2	no	—
10	#14582759	C₂²	1	4	yes	—
11	C₄×C₄⋊Dic₄₃	—	2	2	no	—
12	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
13	#14582662	C₂²	1	8	yes	—
14	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₄₃	C₄	1	8	yes	complemented
17	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
18	#13063284	—	4	8	no	—
19	C₂×C₄⋊Dic₄₃	—	2	4	no	—
20	C₂×C₄⋊Dic₄₃	—	2	4	no	—
21	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
22	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
23	C₂×C₄×Dic₄₃	—	2	4	no	—
24	#13063257	—	4	8	no	—
25	C₄₃⋊#95	—	2	4	no	—
26	C₄₃⋊#95	—	2	4	no	—
27	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	—	4	4	no	—
31	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₃	D₄	1	2	yes	—
35	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
37	C₄⋊C₈×C₄₃	C₂×C₄	1	16	yes	—
38	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	2	2	no	—
41	C₂²×C₄×C₄₃	—	4	2	no	—
42	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₄₃	SD₁₆	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	C₂×C₈×C₄₃	—	2	8	no	—
53	C₂×C₈×C₄₃	—	4	8	no	—
54	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
55	C₄²×C₄₃	C₄⋊C₄	1	8	yes	—
56	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
57	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
58	C₂²⋊Dic₄₃	—	4	8	no	—
59	C₄⋊Dic₄₃	—	2	8	no	—
60	C₄⋊Dic₄₃	—	2	8	no	—
61	C₄⋊Dic₄₃	—	4	8	no	—
62	C₄⋊Dic₄₃	—	4	8	no	—
63	C₄×Dic₄₃	—	4	8	no	—
64	C₂³×C₄₃	—	2	2	no	—
65	C₂³×C₄₃	—	2	2	no	—
66	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₄₃	C₂×SD₁₆	1	4	yes	—
68	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	—	2	2	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	4	no	—
75	C₂×C₄×C₄₃	#106	1	16	yes	complemented
76	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
80	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
81	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
84	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
85	C₂×Dic₄₃	—	4	8	no	—
86	C₂×Dic₄₃	—	4	8	no	—
87	C₂×Dic₄₃	—	4	8	no	—
88	C₈×C₄₃	—	2	8	no	—
89	#21959	—	43	1	no	maximal, Sylow
90	C₂²×C₄₃	—	2	2	no	—
91	C₂²×C₄₃	—	2	2	no	—
92	C₂²×C₄₃	—	2	8	no	—
93	C₂²×C₄₃	C₂×#106	1	4	yes	—
94	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₄₃	#537	1	2	yes	—
96	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
97	C₂²×C₄₃	#575	1	2	yes	—
98	C₄×C₄₃	—	2	2	no	—
99	C₄×C₄₃	—	2	4	no	—
100	C₄×C₄₃	C₂×#106	1	8	yes	—
101	C₄×C₄₃	#518	1	1	yes	characteristic
102	C₄×C₄₃	#519	1	1	yes	characteristic
103	C₄×C₄₃	#544	1	2	yes	—
104	Dic₄₃	—	8	8	no	—
105	C₂²×C₄⋊C₈	D₄₃	1	1	yes	characteristic
106	#2285	—	43	2	no	—
107	#1948	—	43	8	no	—
108	#1807	—	43	4	no	—
109	C₂×C₄₃	—	2	4	no	—
110	C₂×C₄₃	#2316	1	1	yes	characteristic
111	C₂×C₄₃	C₂²×#106	1	1	yes	characteristic
112	C₂×C₄₃	#2911	1	1	yes	characteristic
113	C₂×C₄₃	#3246	1	4	yes	—
114	C₂³×C₈	C₂×D₄₃	1	2	yes	—
115	C₂×#117	—	86	2	no	—
116	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
117	C₂×#106	—	43	8	no	—
118	C₂×C₄⋊C₈	Dic₄₃	1	8	yes	complemented
119	C₂×C₄⋊C₈	C₂×D₄₃	1	4	yes	—
120	#384	—	43	4	no	—
121	#382	—	86	2	no	—
122	#380	—	43	4	no	—
123	#380	—	86	2	no	—
124	C₄×C₄⋊C₄	—	43	4	no	—
125	C₄×C₄⋊C₄	—	86	2	no	—
126	#340	—	43	8	no	—
127	C₄₃	#21959	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
129	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
130	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
131	C₂²×C₈	—	4	4	no	—
132	C₂²×C₈	C₂×Dic₄₃	1	8	yes	—
133	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
134	#117	—	172	8	no	—
135	C₂×C₄⋊C₄	—	43	8	no	—
136	C₂×C₄⋊C₄	—	86	4	no	—
137	C₂×C₄⋊C₄	—	86	4	no	—
138	C₂×C₂²⋊C₄	—	86	4	no	—
139	C₂×C₄²	—	86	4	no	—
140	C₂×C₄²	Dic₈₆	1	4	yes	—
141	C₂×C₄²	D₁₇₂	1	2	yes	—
142	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
143	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
144	#106	—	172	8	no	—
145	C₄⋊C₈	C₂×Dic₄₃	1	16	yes	—
146	#95	—	86	4	no	—
147	#95	—	86	4	no	—
148	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	86	8	no	—
153	C₂²×C₄	#10621873	1	8	yes	—
154	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
156	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
157	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
159	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
160	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
161	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	—	4	8	no	—
164	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
165	C₄⋊C₄	—	86	8	no	—
166	C₄⋊C₄	—	86	8	no	—
167	C₄⋊C₄	—	172	8	no	—
168	C₄⋊C₄	—	172	8	no	—
169	C₂²⋊C₄	—	172	8	no	—
170	C₄²	—	172	8	no	—
171	C₄²	C₄⋊Dic₄₃	1	8	yes	—
172	C₄²	C₂×Dic₈₆	1	4	yes	—
173	C₄²	C₂²×Dic₄₃	1	4	yes	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
177	C₂³	C₂×#10621873	1	4	yes	—
178	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	172	8	no	—
186	C₂×C₄	—	172	8	no	—
187	C₂×C₄	—	172	8	no	—
188	C₂×C₄	#13063257	1	16	yes	complemented
189	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
190	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
191	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
192	C₂×C₄	C₂×#10621873	1	4	yes	—
193	C₂×C₄	C₂×#10621873	1	4	yes	—
194	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
196	C₂×C₄	#13063370	1	2	yes	—
197	C₂×C₄	#13063373	1	2	yes	—
198	C₈	—	2	8	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	2	2	no	—
202	C₂²	C₂×#13063257	1	4	yes	—
203	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
204	C₂²	#14583480	1	2	yes	—
205	C₂²	C₂²×#10621873	1	2	yes	—
206	C₂²	#14583699	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	2	no	—
209	C₄	—	344	8	no	—
210	C₄	C₂×#13063257	1	8	yes	—
211	C₄	#14583374	1	1	yes	characteristic
212	C₄	#14583377	1	1	yes	characteristic
213	C₄	#14583530	1	2	yes	—
214	C₂	—	2	4	no	—
215	C₂	#19137213	1	1	yes	characteristic
216	C₂	C₂²×#13063257	1	1	yes	characteristic
217	C₂	#19143185	1	1	yes	characteristic
218	C₂	#19145912	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#21959
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #19135359	C₂	semidirect
16. C₂×C₄⋊C₈×C₄₃	C₄	semidirect
75. C₂×C₄×C₄₃	#106	semidirect
118. C₂×C₄⋊C₈	Dic₄₃	semidirect
127. C₄₃	#21959	semidirect
188. C₂×C₄	#13063257	semidirect

Maximal subgroups

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

subgroup	index
#19137002	2
#19135359	2
#19134620	2
C₂²×C₄⋊C₈×C₄₃	2
#21959	43