Subgroups — #25252089

Order 11008.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145109	C₂	1	2	yes	maximal
2	#19136877	C₂	1	2	yes	maximal, complemented
3	#19136874	C₂	1	4	yes	maximal, complemented
4	#19136870	C₂	1	2	yes	maximal, complemented
5	#19136379	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
7	#14583456	—	2	2	no	—
8	#14583304	C₂²	1	4	yes	—
9	#14583301	—	2	2	no	—
10	#14583301	C₂²	1	4	yes	—
11	#14583297	—	2	2	no	—
12	#14582937	—	2	2	no	—
13	#14582937	C₂²	1	4	yes	—
14	#14582727	C₂²	1	8	yes	—
15	#14582725	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
20	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
21	#13063325	—	2	4	no	—
22	#13063317	—	2	4	no	—
23	#13063304	—	4	4	no	—
24	C₄×Dic₈₆	—	4	4	no	—
25	#13063284	—	2	4	no	—
26	#13063259	—	2	4	no	—
27	#13063259	C₂³	1	8	yes	—
28	#13063257	—	2	4	no	—
29	#13063257	—	2	4	no	—
30	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₄₃	Q₈	1	8	yes	complemented
34	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
35	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
36	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	D₈	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₈₆	—	4	4	no	—
43	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
44	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
49	C₄²×C₄₃	D₈	1	2	yes	—
50	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊Dic₄₃	—	4	4	no	—
53	C₄⋊Dic₄₃	—	2	8	no	—
54	C₄⋊Dic₄₃	—	4	4	no	—
55	Dic₄₃⋊C₄	—	4	8	no	—
56	C₄×Dic₄₃	—	4	4	no	—
57	C₁₆×C₄₃	C₂×Q₈	1	4	yes	—
58	C₁₆×C₄₃	C₂×Q₈	1	4	yes	—
59	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	—	2	4	no	—
61	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₄₃	—	4	8	no	—
68	Dic₈₆	—	8	8	no	—
69	C₈×C₄₃	C₄⋊Q₈	1	2	yes	—
70	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
71	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
72	C₈×C₄₃	C₂²×Q₈	1	2	yes	—
73	#33726	—	43	1	no	maximal, Sylow
74	C₂²×C₄₃	—	2	2	no	—
75	C₂²×C₄₃	#500	1	2	yes	—
76	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
77	C₄×C₄₃	—	2	2	no	—
78	C₄×C₄₃	#500	1	2	yes	—
79	C₄×C₄₃	#508	1	4	yes	—
80	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
82	Dic₄₃	—	8	8	no	—
83	#3160	—	43	2	no	—
84	#2254	—	43	2	no	—
85	#2253	—	43	4	no	—
86	#2252	—	43	2	no	—
87	#2158	—	43	4	no	—
88	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic
89	C₂×C₄₃	—	2	2	no	—
90	C₂×C₄₃	C₂×#500	1	1	yes	characteristic
91	C₂×C₄₃	C₂×#508	1	2	yes	—
92	#533	—	86	2	no	—
93	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
94	#500	—	43	4	no	—
95	#499	—	43	4	no	—
96	#499	—	86	2	no	—
97	#498	—	86	2	no	—
98	#427	—	43	4	no	—
99	#427	—	86	2	no	—
100	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
101	#367	—	43	8	no	—
102	#366	—	43	8	no	—
103	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
104	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
105	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
106	C₄₃	#33726	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
109	C₄⋊Q₈	—	86	4	no	—
110	#125	—	86	4	no	—
111	C₂²⋊Q₈	—	172	4	no	—
112	C₄×Q₈	—	172	4	no	—
113	#117	—	86	4	no	—
114	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
115	C₂×C₁₆	Dic₈₆	1	8	yes	complemented
116	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
117	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
118	#107	—	43	8	no	—
119	#107	—	86	4	no	—
120	#106	—	86	4	no	—
121	#106	—	86	4	no	—
122	C₄×C₈	D₁₇₂	1	1	yes	characteristic
123	C₄×C₈	D₁₇₂	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
126	C₂×Q₈	—	172	4	no	—
127	C₂²×C₄	D₃₄₄	1	2	yes	—
128	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
129	C₂×C₈	Dic₁₇₂	1	4	yes	—
130	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
131	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
132	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
133	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
134	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
135	C₄⋊C₄	—	86	8	no	—
136	C₄⋊C₄	—	172	8	no	—
137	C₄⋊C₄	—	172	4	no	—
138	C₂²⋊C₄	—	172	4	no	—
139	C₄²	—	172	4	no	—
140	C₄²	D₃₄₄	1	2	yes	—
141	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
142	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
143	C₁₆	C₂×Dic₈₆	1	4	yes	—
144	C₁₆	C₂×Dic₈₆	1	4	yes	—
145	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
146	Q₈	—	344	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	172	8	no	—
149	C₂×C₄	#13063325	1	2	yes	—
150	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
151	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
152	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
154	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
155	C₈	#13063325	1	2	yes	—
156	C₈	C₂×Dic₁₇₂	1	2	yes	—
157	C₈	C₂×Dic₁₇₂	1	2	yes	—
158	C₈	C₂²×Dic₈₆	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#14583304	1	2	yes	—
161	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	344	8	no	—
164	C₄	#14583304	1	2	yes	—
165	C₄	#14583331	1	4	yes	—
166	C₄	C₂×#13063325	1	1	yes	characteristic
167	C₄	C₂²×Dic₁₇₂	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#14583304	1	1	yes	characteristic
170	C₂	C₂×#14583331	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	#33726
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #19136877	C₂	semidirect
3. #19136874	C₂	semidirect
4. #19136870	C₂	semidirect
33. C₂×C₁₆×C₄₃	Q₈	semidirect
106. C₄₃	#33726	semidirect
115. C₂×C₁₆	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19136877	2
#19136874	2
#19136870	2
#19136379	2
C₂×C₄×C₁₆×C₄₃	2
#33726	43