Subgroups — #24990405

Order 11008.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134733	C₂	1	2	yes	maximal
2	#19133354	C₂	1	2	yes	maximal
3	#19133351	C₂	1	2	yes	maximal
4	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic
5	#14582923	C₂²	1	4	yes	—
6	#14582664	—	2	4	no	—
7	#14582652	—	2	4	no	—
8	#14582650	—	2	4	no	—
9	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₈²×C₄₃	C₄	1	1	yes	characteristic
13	C₈²×C₄₃	C₄	1	1	yes	characteristic
14	C₈²×C₄₃	C₄	1	1	yes	characteristic
15	C₈²×C₄₃	C₄	1	1	yes	characteristic
16	C₂×#10621877	—	2	4	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
18	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
19	C₂²×C₈×C₄₃	D₄	1	2	yes	—
20	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
21	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
22	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	D₄	1	2	yes	—
30	C₄×C₈×C₄₃	D₄	1	2	yes	—
31	C₄×C₈×C₄₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₄₃	—	2	4	no	—
36	C₂×C₈×C₄₃	—	2	4	no	—
37	C₂×C₈×C₄₃	—	2	4	no	—
38	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
48	#10621877	—	4	8	no	—
49	C₂×C₄₃⋊C₈	—	4	4	no	—
50	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#135	1	4	yes	—
60	C₈×C₄₃	—	2	2	no	—
61	C₈×C₄₃	—	2	2	no	—
62	C₈×C₄₃	—	2	2	no	—
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	#108	1	4	yes	—
65	C₈×C₄₃	#108	1	4	yes	—
66	C₄₃⋊C₈	—	8	4	no	—
67	#8824	—	43	1	no	maximal, Sylow
68	C₂²×C₄₃	—	2	1	no	—
69	C₂²×C₄₃	—	2	1	no	—
70	C₂²×C₄₃	#384	1	1	yes	characteristic
71	C₂²×C₄₃	#499	1	2	yes	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	#427	1	2	yes	—
75	C₄×C₄₃	C₂×#108	1	2	yes	—
76	C₄×C₄₃	C₂×#108	1	2	yes	—
77	C₄×C₄₃	#495	1	2	yes	—
78	#1837	—	43	2	no	—
79	#1567	—	43	2	no	—
80	#1566	—	43	2	no	—
81	C₂×C₈²	D₄₃	1	1	yes	characteristic
82	C₂×C₄₃	—	2	2	no	—
83	C₂×C₄₃	#1846	1	1	yes	characteristic
84	C₂×C₄₃	#1848	1	1	yes	characteristic
85	C₂×C₄₃	#1848	1	1	yes	characteristic
86	#423	—	43	4	no	—
87	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
90	#341	—	86	4	no	—
91	#335	—	86	4	no	—
92	#334	—	86	4	no	—
93	C₈²	Dic₄₃	1	1	yes	characteristic
94	C₈²	Dic₄₃	1	1	yes	characteristic
95	C₈²	Dic₄₃	1	1	yes	characteristic
96	C₈²	Dic₄₃	1	1	yes	characteristic
97	C₄₃	#8824	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	86	4	no	—
99	C₂²×C₈	Dic₈₆	1	2	yes	—
100	C₂²×C₈	Dic₈₆	1	2	yes	—
101	C₂²×C₈	D₁₇₂	1	2	yes	—
102	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
103	C₄⋊C₈	—	86	4	no	—
104	C₄⋊C₈	—	86	4	no	—
105	C₄×C₈	Dic₈₆	1	2	yes	—
106	C₄×C₈	D₁₇₂	1	2	yes	—
107	C₄×C₈	D₁₇₂	1	2	yes	—
108	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
116	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
117	M₁₆	—	172	8	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	172	4	no	—
122	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
123	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
124	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
125	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
126	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
128	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
129	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
130	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
131	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
132	C₂³	#13063325	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
138	C₂×C₄	#13063322	1	1	yes	characteristic
139	C₂×C₄	#13063325	1	1	yes	characteristic
140	C₂×C₄	#13063325	1	1	yes	characteristic
141	C₂×C₄	#13063347	1	4	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	344	4	no	—
147	C₈	#13063261	1	4	yes	—
148	C₈	#13063261	1	4	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#14582769	1	1	yes	characteristic
152	C₂²	#14583301	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#14582937	1	2	yes	—
156	C₄	C₂×#13063261	1	2	yes	—
157	C₄	C₂×#13063261	1	2	yes	—
158	C₄	#14583283	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#19134759	1	1	yes	characteristic
161	C₂	#19134767	1	1	yes	characteristic
162	C₂	#19134767	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063322
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₈²×C₄₃	C₂
socle	C₂²×C₄₃	#384
prime radical	C₂³×C₄₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#8824
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
97. C₄₃	#8824	semidirect

Maximal subgroups

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

subgroup	index
#19134733	2
#19133354	2
#19133351	2
C₂×C₈²×C₄₃	2
#8824	43