Subgroups — #25008142

Order 11008.

270 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136351	C₂	1	2	yes	maximal
2	#19132659	C₂	1	4	yes	maximal, complemented
3	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063265	—	2	2	no	—
5	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
6	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₄₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₈²×C₄₃	C₄	1	2	yes	—
11	C₈²×C₄₃	C₂²	1	2	yes	—
12	#13063265	—	4	4	no	—
13	C₂×C₄₃⋊C₁₆	—	2	4	no	—
14	C₂²×C₈×C₄₃	—	2	1	no	—
15	C₂²×C₈×C₄₃	C₈	1	2	yes	—
16	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₄₃	—	2	1	no	—
20	C₄×C₈×C₄₃	C₈	1	1	yes	characteristic
21	C₄×C₈×C₄₃	C₈	1	1	yes	characteristic
22	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₄₃	—	2	4	no	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₂×C₈×C₄₃	—	2	4	no	—
35	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
40	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₄₃	M₁₆	1	1	yes	characteristic
45	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₄₃⋊C₁₆	—	4	4	no	—
47	C₂³×C₄₃	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₃	—	2	2	no	—
49	C₂×C₄×C₄₃	—	2	2	no	—
50	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₄₃	#106	1	2	yes	—
54	C₂×C₄×C₄₃	#108	1	2	yes	—
55	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₄₃	—	2	4	no	—
59	C₈×C₄₃	—	2	1	no	—
60	C₈×C₄₃	—	2	1	no	—
61	C₈×C₄₃	—	2	2	no	—
62	C₈×C₄₃	#106	1	2	yes	—
63	C₈×C₄₃	M₃₂	1	4	yes	complemented
64	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
65	#11996	—	43	1	no	maximal, Sylow
66	C₂²×C₄₃	—	2	1	no	—
67	C₂²×C₄₃	—	2	1	no	—
68	C₂²×C₄₃	#334	1	2	yes	—
69	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	#334	1	2	yes	—
73	C₄×C₄₃	#364	1	2	yes	—
74	C₄×C₄₃	C₂×#106	1	1	yes	characteristic
75	C₄×C₄₃	C₂×#108	1	1	yes	characteristic
76	C₄×C₄₃	C₂×M₃₂	1	2	yes	—
77	#2150	—	43	2	no	—
78	C₂×C₈²	D₄₃	1	1	yes	characteristic
79	#1367	—	43	4	no	—
80	C₂×C₄₃	—	2	2	no	—
81	C₂×C₄₃	C₂×#334	1	1	yes	characteristic
82	C₂×C₄₃	#2150	1	1	yes	characteristic
83	C₂×C₄₃	C₂×#364	1	1	yes	characteristic
84	C₂×M₃₂	—	86	2	no	—
85	C₂×C₄×C₈	Dic₄₃	1	1	yes	characteristic
86	C₂×C₄×C₈	Dic₄₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
88	C₄⋊C₁₆	—	43	4	no	—
89	C₄⋊C₁₆	—	86	2	no	—
90	C₈²	Dic₄₃	1	2	yes	—
91	C₈²	C₂×D₄₃	1	2	yes	—
92	C₄₃	#11996	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	C₄₃⋊C₈	1	2	yes	—
95	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
96	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
98	M₃₂	—	172	4	no	—
99	C₂×C₁₆	—	86	4	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₄₃⋊C₈	1	1	yes	characteristic
102	C₄×C₈	C₄₃⋊C₈	1	1	yes	characteristic
103	C₄×C₈	Dic₈₆	1	1	yes	characteristic
104	C₄×C₈	D₁₇₂	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
106	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
107	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
110	C₂²×C₄	#10621877	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	#10621873	1	4	yes	—
117	C₂×C₈	C₂×C₄₃⋊C₈	1	2	yes	—
118	C₂×C₈	C₂×C₄₃⋊C₈	1	2	yes	—
119	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
122	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
123	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
124	C₄²	#10621877	1	1	yes	characteristic
125	C₄²	C₄⋊Dic₄₃	1	1	yes	characteristic
126	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
127	C₁₆	—	172	4	no	—
128	C₂³	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
132	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
133	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
134	C₂×C₄	#13063257	1	2	yes	—
135	C₂×C₄	#13063261	1	2	yes	—
136	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
137	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
138	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	2	2	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	#13063257	1	2	yes	—
144	C₈	#13063265	1	4	yes	complemented
145	C₈	C₂×#10621873	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	#14582650	1	2	yes	—
149	C₂²	C₂×C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#14582650	1	2	yes	—
153	C₄	#14582720	1	2	yes	—
154	C₄	C₂×#13063257	1	1	yes	characteristic
155	C₄	C₂×#13063261	1	1	yes	characteristic
156	C₄	C₂×#13063265	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	C₂×#14582650	1	1	yes	characteristic
159	C₂	#19136351	1	1	yes	characteristic
160	C₂	C₂×#14582720	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₂×C₈²×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₈
prime radical	C₂³×C₄₃	C₄⋊C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#11996
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #19132659	C₂	semidirect
63. C₈×C₄₃	M₃₂	semidirect
92. C₄₃	#11996	semidirect
144. C₈	#13063265	semidirect

Maximal subgroups

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

subgroup	index
#19136351	2
#19132659	2
C₂×C₈²×C₄₃	2
#11996	43