Subgroups — #19135314

Order 5504.

260 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	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14582654	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
10	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
11	C₄×C₄₃⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	#95×C₄₃	C₄	1	2	yes	—
17	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₄₃	—	2	2	no	—
23	C₄⋊C₄×C₄₃	C₈	1	4	yes	complemented
24	C₄²×C₄₃	—	2	2	no	—
25	C₄²×C₄₃	—	2	2	no	—
26	C₄²×C₄₃	D₄	1	2	yes	—
27	C₄²×C₄₃	Q₈	1	2	yes	—
28	C₂×C₄₃⋊C₈	—	2	1	no	—
29	C₂×C₄₃⋊C₈	—	2	1	no	—
30	C₂×C₄₃⋊C₈	—	2	1	no	—
31	C₂×C₄₃⋊C₈	—	2	1	no	—
32	C₂×C₄₃⋊C₈	—	2	2	no	—
33	C₂×C₄₃⋊C₈	—	2	2	no	—
34	C₂×C₄₃⋊C₈	—	2	2	no	—
35	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
36	C₂×C₄₃⋊C₈	Q₈	1	2	yes	—
37	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₃	—	2	2	no	—
39	C₂×C₄×C₄₃	—	2	1	no	—
40	C₂×C₄×C₄₃	—	2	1	no	—
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₃	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
56	C₄₃⋊C₈	—	2	2	no	—
57	C₄₃⋊C₈	—	2	2	no	—
58	C₄₃⋊C₈	—	4	2	no	—
59	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₄₃	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₄₃	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₄₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₄₃	#131	1	1	yes	characteristic
65	C₂²×C₄₃	#131	1	1	yes	characteristic
66	C₄×C₄₃	—	2	2	no	—
67	C₄×C₄₃	—	2	4	no	—
68	C₄×C₄₃	—	4	2	no	—
69	C₄×C₄₃	C₄⋊C₈	1	4	yes	complemented
70	C₄×C₄₃	#121	1	1	yes	characteristic
71	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄×C₄₃	#125	1	1	yes	characteristic
73	C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
74	#1939	—	43	1	no	maximal, Sylow
75	C₂×C₄₃	#389	1	1	yes	characteristic
76	C₂×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄₃	#424	1	1	yes	characteristic
78	C₂×C₄₃	C₈×D₄	1	1	yes	characteristic
79	C₂×C₄₃	#436	1	1	yes	characteristic
80	C₂×C₄₃	C₈×Q₈	1	1	yes	characteristic
81	C₂×C₄₃	#446	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	43	2	no	—
83	C₂×C₄⋊C₈	—	43	1	no	—
84	C₂×C₄×C₈	—	43	1	no	—
85	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
86	#336	—	43	2	no	—
87	C₄₃	#1939	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	43	1	no	—
89	C₂²×C₈	—	43	1	no	—
90	C₂²×C₈	—	43	2	no	—
91	C₂×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
92	C₂×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₄₃	1	2	yes	—
94	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
95	C₄⋊C₈	—	86	4	no	—
96	C₄⋊C₈	—	86	2	no	—
97	C₄×C₈	—	86	2	no	—
98	#95	Dic₄₃	1	2	yes	—
99	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
100	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
101	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
102	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
104	C₂×C₈	—	43	2	no	—
105	C₂×C₈	—	43	2	no	—
106	C₂×C₈	—	86	1	no	—
107	C₂×C₈	—	86	1	no	—
108	C₂×C₈	—	86	1	no	—
109	C₂×C₈	—	86	1	no	—
110	C₂×C₈	—	86	2	no	—
111	C₂×C₈	—	86	2	no	—
112	C₂×C₈	—	86	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₄₃⋊C₈	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	Dic₈₆	1	2	yes	—
118	C₄²	D₁₇₂	1	2	yes	—
119	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₂×C₄₃⋊C₈	1	4	yes	—
128	C₂×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
129	C₂×C₄	#10621877	1	2	yes	—
130	C₂×C₄	C₄⋊Dic₄₃	1	2	yes	—
131	C₂×C₄	C₄⋊Dic₄₃	1	2	yes	—
132	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
134	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
135	C₂×C₄	#10621900	1	1	yes	characteristic
136	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
137	C₂×C₄	#10621902	1	1	yes	characteristic
138	C₈	—	86	2	no	—
139	C₈	—	86	2	no	—
140	C₈	—	172	2	no	—
141	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
142	C₂²	D₄×Dic₄₃	1	1	yes	characteristic
143	C₂²	Q₈×Dic₄₃	1	1	yes	characteristic
144	C₂²	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
145	C₂²	C₂×#10621877	1	1	yes	characteristic
146	C₂²	#13063332	1	1	yes	characteristic
147	C₂²	#13063332	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊(C₄₃⋊C₈)	1	4	yes	complemented
152	C₄	#13063300	1	1	yes	characteristic
153	C₄	#13063305	1	1	yes	characteristic
154	C₄	#13063316	1	1	yes	characteristic
155	C₄	#13063324	1	1	yes	characteristic
156	C₂	#14582798	1	1	yes	characteristic
157	C₂	C₂×C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
158	C₂	#14582926	1	1	yes	characteristic
159	C₂	D₄×C₄₃⋊C₈	1	1	yes	characteristic
160	C₂	#14582977	1	1	yes	characteristic
161	C₂	Q₈×C₄₃⋊C₈	1	1	yes	characteristic
162	C₂	#14583030	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×C₄
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#1939
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. C₄⋊C₄×C₄₃	C₈	semidirect
69. C₄×C₄₃	C₄⋊C₈	semidirect
87. C₄₃	#1939	semidirect
114. C₄⋊C₄	C₄₃⋊C₈	semidirect
151. C₄	C₄⋊(C₄₃⋊C₈)	semidirect