Subgroups — #19132825

Order 5504.

656 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	C₂²×C₄⋊Dic₄₃	C₂	1	2	yes	maximal
2	C₂×C₄₃⋊#95	C₂	1	4	yes	maximal
3	C₂²×C₄²×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₄₃	C₂²	1	4	yes	—
5	C₂×C₄⋊Dic₄₃	—	2	4	no	—
6	C₂×C₄⋊Dic₄₃	—	2	4	no	—
7	C₂×C₄⋊Dic₄₃	C₄	1	8	yes	complemented
8	C₄₃⋊#95	—	2	8	no	—
9	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
10	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₄₃	—	2	4	no	—
12	C₂×C₄²×C₄₃	C₄	1	4	yes	complemented
13	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₄₃	—	2	4	no	—
15	C₂²×C₄×C₄₃	—	2	4	no	—
16	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₃	D₄	1	4	yes	—
20	C₂²×C₄×C₄₃	D₄	1	2	yes	—
21	C₂²×C₄×C₄₃	D₄	1	4	yes	—
22	C₂²×C₄×C₄₃	Q₈	1	4	yes	—
23	C₂²×C₄×C₄₃	Q₈	1	2	yes	—
24	C₂²×Dic₄₃	—	2	8	no	—
25	C₂²×Dic₄₃	—	2	4	no	—
26	C₂²×Dic₄₃	—	2	4	no	—
27	C₂²×Dic₄₃	—	2	8	no	—
28	C₂²×Dic₄₃	C₂×C₄	1	8	yes	—
29	C₄²×C₄₃	—	2	8	no	—
30	C₄⋊Dic₄₃	—	2	16	no	—
31	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
32	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
33	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂³×C₄₃	C₂×D₄	1	2	yes	—
35	C₂³×C₄₃	C₂×D₄	1	2	yes	—
36	C₂³×C₄₃	C₂×Q₈	1	1	yes	characteristic
37	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
38	C₂³×C₄₃	#41	1	4	yes	—
39	C₂×C₄×C₄₃	—	2	4	no	—
40	C₂×C₄×C₄₃	—	2	4	no	—
41	C₂×C₄×C₄₃	—	2	8	no	—
42	C₂×C₄×C₄₃	—	2	8	no	—
43	C₂×C₄×C₄₃	C₄²	1	4	yes	—
44	C₂×C₄×C₄₃	C₂²⋊C₄	1	8	yes	—
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
47	C₂×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₄₃	C₄⋊C₄	1	8	yes	complemented
49	C₂×Dic₄₃	—	2	8	no	—
50	C₂×Dic₄₃	—	2	16	no	—
51	C₂×Dic₄₃	—	4	8	no	—
52	C₂×Dic₄₃	—	4	4	no	—
53	C₂×Dic₄₃	—	4	4	no	—
54	C₂²×C₄₃	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
60	C₂²×C₄₃	#117	1	2	yes	—
61	C₂²×C₄₃	C₄×D₄	1	4	yes	—
62	C₂²×C₄₃	C₄×Q₈	1	4	yes	—
63	C₂²×C₄₃	#121	1	4	yes	—
64	C₂²×C₄₃	C₂²⋊Q₈	1	4	yes	—
65	C₂²×C₄₃	#124	1	2	yes	—
66	C₂²×C₄₃	#125	1	2	yes	—
67	C₂²×C₄₃	#127	1	1	yes	characteristic
68	C₂²×C₄₃	C₄⋊Q₈	1	2	yes	—
69	C₂²×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₄×C₄₃	—	2	8	no	—
71	C₄×C₄₃	#95	1	8	yes	complemented
72	Dic₄₃	—	4	8	no	—
73	C₄⋊#95	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	C₂×#95	1	1	yes	characteristic
75	C₂×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
76	C₂×C₄₃	#380	1	2	yes	—
77	C₂×C₄₃	#382	1	1	yes	characteristic
78	C₂×C₄₃	#384	1	1	yes	characteristic
79	C₂×C₄₃	#389	1	4	yes	—
80	C₂×C₄₃	#390	1	2	yes	—
81	C₂×C₄₃	#391	1	2	yes	—
82	C₂²×C₄⋊C₄	—	43	2	no	—
83	C₂²×C₄²	D₄₃	1	1	yes	characteristic
84	C₂×#95	—	43	4	no	—
85	C₄₃	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
86	C₂³×C₄	—	43	4	no	—
87	C₂³×C₄	C₂×D₄₃	1	2	yes	—
88	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	43	8	no	—
90	C₂×C₄⋊C₄	—	86	4	no	—
91	C₂×C₄⋊C₄	—	86	4	no	—
92	C₂×C₄²	—	2	4	no	—
93	C₂×C₄²	Dic₄₃	1	4	yes	complemented
94	#95	—	86	8	no	—
95	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
96	C₂²×C₄	—	2	4	no	—
97	C₂²×C₄	—	2	4	no	—
98	C₂²×C₄	—	43	8	no	—
99	C₂²×C₄	—	86	8	no	—
100	C₂²×C₄	—	86	4	no	—
101	C₂²×C₄	—	86	8	no	—
102	C₂²×C₄	—	86	4	no	—
103	C₂²×C₄	Dic₈₆	1	4	yes	—
104	C₂²×C₄	Dic₈₆	1	2	yes	—
105	C₂²×C₄	C₄×D₄₃	1	4	yes	—
106	C₂²×C₄	D₁₇₂	1	4	yes	—
107	C₂²×C₄	D₁₇₂	1	2	yes	—
108	C₂²×C₄	C₂×Dic₄₃	1	4	yes	—
109	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
110	C₂²×C₄	C₂²⋊D₄₃	1	4	yes	—
111	C₄⋊C₄	—	86	16	no	—
112	C₄²	—	2	8	no	—
113	C₂³	C₂×Dic₈₆	1	1	yes	characteristic
114	C₂³	C₂×Dic₈₆	1	2	yes	—
115	C₂³	C₂×C₄×D₄₃	1	2	yes	—
116	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
117	C₂³	C₂×D₁₇₂	1	2	yes	—
118	C₂³	#10621898	1	4	yes	—
119	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
120	C₂³	C₂×C₂²⋊D₄₃	1	2	yes	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	8	no	—
125	C₂×C₄	—	86	8	no	—
126	C₂×C₄	—	86	16	no	—
127	C₂×C₄	—	172	8	no	—
128	C₂×C₄	—	172	4	no	—
129	C₂×C₄	—	172	4	no	—
130	C₂×C₄	C₄×Dic₄₃	1	4	yes	—
131	C₂×C₄	Dic₄₃⋊C₄	1	8	yes	—
132	C₂×C₄	C₄⋊Dic₄₃	1	4	yes	—
133	C₂×C₄	C₄⋊Dic₄₃	1	8	yes	complemented
134	C₂×C₄	(C₂×D₄₃)⋊C₄	1	8	yes	—
135	C₂×C₄	C₂²⋊Dic₄₃	1	4	yes	—
136	C₂²	C₂×C₄×Dic₄₃	1	1	yes	characteristic
137	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
138	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
139	C₂²	C₂×Dic₄₃⋊C₄	1	2	yes	—
140	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
141	C₂²	C₂×C₄⋊Dic₄₃	1	2	yes	—
142	C₂²	#13063284	1	2	yes	—
143	C₂²	C₄×D₁₇₂	1	4	yes	—
144	C₂²	C₄×Dic₈₆	1	4	yes	—
145	C₂²	#13063299	1	4	yes	—
146	C₂²	#13063304	1	4	yes	—
147	C₂²	#13063313	1	2	yes	—
148	C₂²	#13063317	1	2	yes	—
149	C₂²	#13063322	1	1	yes	characteristic
150	C₂²	#13063325	1	2	yes	—
151	C₂²	#13063325	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	172	8	no	—
154	C₄	C₄₃⋊#95	1	8	yes	complemented
155	C₂	C₂×C₄₃⋊#95	1	1	yes	characteristic
156	C₂	C₄×C₄⋊Dic₄₃	1	2	yes	—
157	C₂	#14582759	1	2	yes	—
158	C₂	#14582765	1	1	yes	characteristic
159	C₂	#14582769	1	1	yes	characteristic
160	C₂	#14582797	1	4	yes	—
161	C₂	#14582801	1	2	yes	—
162	C₂	#14582805	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄²
Fitting	C₂²×C₄²×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂³
hypercenter	C₂²×C₄²	D₄₃
hypocenter	C₄₃	C₄⋊#95
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊Dic₄₃	C₄	semidirect
12. C₂×C₄²×C₄₃	C₄	semidirect
48. C₂×C₄×C₄₃	C₄⋊C₄	semidirect
71. C₄×C₄₃	#95	semidirect
85. C₄₃	C₄⋊#95	semidirect
93. C₂×C₄²	Dic₄₃	semidirect
133. C₂×C₄	C₄⋊Dic₄₃	semidirect
154. C₄	C₄₃⋊#95	semidirect