Subgroups — #25254899

Order 11008.

3158 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137002	C₂	1	6	yes	maximal, complemented
2	#19136993	C₂	1	12	yes	maximal, complemented
3	#19136978	C₂	1	6	yes	maximal
4	C₄×#13063284	C₂	1	6	yes	maximal, complemented
5	C₂³×C₄²×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063284	C₂²	1	12	yes	—
7	#14582769	C₂²	1	4	yes	complemented
8	#14582767	C₂²	1	8	yes	complemented
9	#14582765	C₂²	1	12	yes	complemented
10	#14582762	C₂²	1	12	yes	—
11	#14582759	C₂²	1	24	yes	—
12	#14582755	C₂²	1	4	yes	—
13	C₄×C₄⋊Dic₄₃	C₂²	1	24	yes	complemented
14	C₄×C₂²⋊Dic₄₃	C₂²	1	24	yes	—
15	#14582745	C₂²	1	12	yes	complemented
16	C₄²×Dic₄₃	C₂²	1	4	yes	complemented
17	C₂⁴×C₄×C₄₃	C₂²	1	3	yes	—
18	C₂²×C₄²×C₄₃	C₄	1	16	yes	complemented
19	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
20	C₂²×C₄²×C₄₃	C₂²	1	6	yes	—
21	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
22	#13063284	—	2	48	no	—
23	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
24	C₂×C₂²⋊Dic₄₃	C₂³	1	24	yes	—
25	C₂×C₄×Dic₄₃	C₂³	1	24	yes	—
26	C₄₃⋊#95	C₂³	1	48	yes	—
27	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₄₃	—	2	24	no	—
29	C₂³×C₄×C₄₃	C₂×C₄	1	24	yes	—
30	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
31	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
32	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
33	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	48	yes	—
35	C₂×C₄²×C₄₃	C₂×C₄	1	24	yes	—
36	C₂×C₄²×C₄₃	C₂×C₄	1	24	yes	—
37	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
38	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₂⁴×C₄₃	—	2	12	no	—
44	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
45	C₂⁴×C₄₃	C₂⁴	1	3	yes	—
46	C₂²×C₄×C₄₃	—	2	96	no	—
47	C₂²×C₄×C₄₃	—	2	24	no	—
48	C₂²×C₄×C₄₃	—	2	24	no	—
49	C₂²×C₄×C₄₃	—	2	12	no	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	24	yes	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	24	yes	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₄₃	#41	1	24	yes	—
55	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
56	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
57	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
58	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
59	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
60	C₄²×C₄₃	C₂²×C₄	1	12	yes	—
61	C₄²×C₄₃	C₂²×C₄	1	12	yes	—
62	C₄²×C₄₃	C₂²×C₄	1	24	yes	—
63	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
64	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
65	C₂²⋊Dic₄₃	—	2	48	no	—
66	C₄⋊Dic₄₃	—	2	48	no	—
67	C₄×Dic₄₃	—	2	48	no	—
68	C₂³×C₄₃	—	2	32	no	—
69	C₂³×C₄₃	—	2	24	no	—
70	C₂³×C₄₃	—	2	12	no	—
71	C₂³×C₄₃	C₂³×C₄	1	6	yes	—
72	C₂³×C₄₃	C₂×#41	1	12	yes	—
73	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
74	C₂×C₄×C₄₃	—	2	24	no	—
75	C₂×C₄×C₄₃	—	2	12	no	—
76	C₂×C₄×C₄₃	—	2	48	no	—
77	C₂×C₄×C₄₃	—	2	48	no	—
78	C₂×C₄×C₄₃	#117	1	48	yes	complemented
79	C₂×C₄×C₄₃	C₂³×C₄	1	12	yes	—
80	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
81	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
82	C₂×C₄×C₄₃	C₂×#41	1	12	yes	—
83	C₂×C₄×C₄₃	C₂×#41	1	12	yes	—
84	C₂×Dic₄₃	—	2	48	no	—
85	#34012	—	43	1	no	maximal, Sylow
86	C₂²×C₄₃	—	2	32	no	—
87	C₂²×C₄₃	—	2	12	no	—
88	C₂²×C₄₃	—	2	24	no	—
89	C₂²×C₄₃	C₂×#117	1	12	yes	—
90	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
91	C₂²×C₄₃	C₂²×#41	1	6	yes	—
92	C₄×C₄₃	—	2	12	no	—
93	C₄×C₄₃	C₂×#117	1	12	yes	—
94	C₄×C₄₃	C₂×#117	1	12	yes	—
95	Dic₄₃	—	4	16	no	—
96	C₂³×C₄²	D₄₃	1	1	yes	characteristic
97	#2285	—	43	6	no	—
98	#2283	—	43	12	no	—
99	#2280	—	43	6	no	—
100	C₄×#117	—	43	6	no	—
101	C₂×C₄₃	—	2	12	no	—
102	C₂×C₄₃	C₂²×#117	1	3	yes	—
103	C₂×C₄₃	#3444	1	4	yes	—
104	C₂⁴×C₄	C₂×D₄₃	1	3	yes	—
105	C₂×#117	—	43	12	no	—
106	C₂²×C₄²	Dic₄₃	1	16	yes	complemented
107	C₂²×C₄²	C₂×D₄₃	1	6	yes	—
108	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
109	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
110	#384	—	43	4	no	—
111	#383	—	43	8	no	—
112	#382	—	43	12	no	—
113	#381	—	43	12	no	—
114	#380	—	43	24	no	—
115	#379	—	43	4	no	—
116	C₄×C₄⋊C₄	—	43	24	no	—
117	C₄×C₂²⋊C₄	—	43	24	no	—
118	#376	—	43	12	no	—
119	C₄³	—	43	4	no	—
120	C₄₃	#34012	1	1	yes	characteristic, complemented, Sylow
121	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
122	C₂³×C₄	—	2	24	no	—
123	C₂³×C₄	C₂×Dic₄₃	1	24	yes	—
124	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
125	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
126	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
127	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
128	#117	—	86	48	no	—
129	C₂×C₄⋊C₄	—	43	24	no	—
130	C₂×C₂²⋊C₄	—	43	24	no	—
131	C₂×C₄²	—	43	24	no	—
132	C₂×C₄²	C₂×Dic₄₃	1	48	yes	—
133	C₂×C₄²	C₂×Dic₄₃	1	24	yes	—
134	C₂×C₄²	C₂×Dic₄₃	1	24	yes	—
135	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
136	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
137	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
138	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
139	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
141	#95	—	43	48	no	—
142	C₂⁴	—	2	12	no	—
143	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
144	C₂⁴	C₂³×D₄₃	1	3	yes	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	2	96	no	—
147	C₂²×C₄	—	2	24	no	—
148	C₂²×C₄	—	2	24	no	—
149	C₂²×C₄	—	43	16	no	—
150	C₂²×C₄	#10621898	1	24	yes	—
151	C₂²×C₄	C₂²×Dic₄₃	1	12	yes	—
152	C₂²×C₄	C₂²×Dic₄₃	1	24	yes	—
153	C₂²×C₄	C₂²×Dic₄₃	1	24	yes	—
154	C₂²×C₄	C₂²×Dic₄₃	1	12	yes	—
155	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
156	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
157	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
158	C₄⋊C₄	—	86	48	no	—
159	C₂²⋊C₄	—	86	48	no	—
160	C₄²	—	86	48	no	—
161	C₄²	C₂²×Dic₄₃	1	8	yes	—
162	C₄²	C₂²×Dic₄₃	1	12	yes	—
163	C₄²	C₂²×Dic₄₃	1	12	yes	—
164	C₄²	C₂²×Dic₄₃	1	24	yes	—
165	C₄²	C₂²×Dic₄₃	1	4	yes	—
166	C₄²	C₂²×Dic₄₃	1	4	yes	—
167	C₂³	—	2	32	no	—
168	C₂³	—	2	24	no	—
169	C₂³	—	2	12	no	—
170	C₂³	C₂³×Dic₄₃	1	6	yes	—
171	C₂³	C₂×#10621898	1	12	yes	—
172	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
173	C₂×C₄	—	2	24	no	—
174	C₂×C₄	—	2	12	no	—
175	C₂×C₄	—	2	48	no	—
176	C₂×C₄	—	2	48	no	—
177	C₂×C₄	—	86	48	no	—
178	C₂×C₄	#13063284	1	48	yes	complemented
179	C₂×C₄	C₂³×Dic₄₃	1	12	yes	—
180	C₂×C₄	C₂³×Dic₄₃	1	6	yes	—
181	C₂×C₄	C₂³×Dic₄₃	1	6	yes	—
182	C₂×C₄	C₂×#10621898	1	12	yes	—
183	C₂×C₄	C₂×#10621898	1	12	yes	—
184	C₂²	—	2	32	no	—
185	C₂²	—	2	24	no	—
186	C₂²	—	2	12	no	—
187	C₂²	C₂×#13063284	1	12	yes	—
188	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
189	C₂²	C₂²×#10621898	1	6	yes	—
190	C₄	—	2	12	no	—
191	C₄	—	172	16	no	—
192	C₄	C₂×#13063284	1	12	yes	—
193	C₄	C₂×#13063284	1	12	yes	—
194	C₂	—	2	12	no	—
195	C₂	C₂²×#13063284	1	3	yes	—
196	C₂	#19147891	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₃
Fitting	C₂³×C₄²×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
prime radical	C₂⁵×C₄₃	C₂³
hypercenter	C₂³×C₄²	D₄₃
hypocenter	C₄₃	#34012
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. #19137002	C₂	semidirect
2. #19136993	C₂	semidirect
4. C₄×#13063284	C₂	semidirect
7. #14582769	C₂²	semidirect
8. #14582767	C₂²	semidirect
9. #14582765	C₂²	semidirect
13. C₄×C₄⋊Dic₄₃	C₂²	semidirect
15. #14582745	C₂²	semidirect
16. C₄²×Dic₄₃	C₂²	semidirect
18. C₂²×C₄²×C₄₃	C₄	semidirect
78. C₂×C₄×C₄₃	#117	semidirect
106. C₂²×C₄²	Dic₄₃	semidirect
120. C₄₃	#34012	semidirect
178. C₂×C₄	#13063284	semidirect

Maximal subgroups

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

subgroup	index
#19137002	2
#19136993	2
#19136978	2
C₄×#13063284	2
C₂³×C₄²×C₄₃	2
#34012	43