Subgroups — #19143542

Order 5504.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583456	C₂	1	1	yes	maximal, characteristic
2	#14583029	C₂	1	2	yes	maximal, complemented
3	#14583027	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₈₆	C₂	1	2	yes	maximal, complemented
5	#14582927	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#14582857	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
10	#13063325	C₄	1	1	yes	characteristic
11	#13063325	C₄	1	1	yes	characteristic
12	#13063317	C₄	1	2	yes	—
13	#13063304	C₄	1	2	yes	—
14	#13063304	C₄	1	2	yes	—
15	C₄×Dic₈₆	C₂²	1	2	yes	—
16	C₄×Dic₈₆	C₂²	1	2	yes	—
17	#13063284	C₂²	1	2	yes	—
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
21	Dic₄₃⋊C₈	C₂²	1	4	yes	—
22	Dic₄₃⋊C₈	C₂²	1	4	yes	—
23	#13063235	C₂²	1	4	yes	—
24	C₈×Dic₄₃	C₂²	1	4	yes	—
25	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
28	#97×C₄₃	C₂²	1	2	yes	—
29	C₄×C₈×C₄₃	C₂²	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
35	M₁₆×C₄₃	—	2	2	no	—
36	M₁₆×C₄₃	Q₈	1	4	yes	complemented
37	C₂×C₈×C₄₃	C₂³	1	2	yes	—
38	C₂×C₈×C₄₃	C₂³	1	2	yes	—
39	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
48	Dic₄₃⋊C₄	C₂×C₄	1	4	yes	—
49	Dic₄₃⋊C₄	C₂×C₄	1	4	yes	—
50	C₄×Dic₄₃	C₂³	1	4	yes	—
51	C₂×C₄₃⋊C₈	—	2	4	no	—
52	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
53	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
54	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	4	1	no	—
57	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₄₃	#41	1	2	yes	—
64	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
66	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
67	Dic₈₆	—	2	4	no	—
68	Dic₈₆	—	4	2	no	—
69	C₈×C₄₃	—	2	2	no	—
70	C₈×C₄₃	C₂×Q₈	1	4	yes	—
71	C₄₃⋊C₈	—	2	2	no	—
72	C₄₃⋊C₈	—	2	2	no	—
73	C₂²×C₄₃	—	2	1	no	—
74	C₂²×C₄₃	—	2	1	no	—
75	C₂²×C₄₃	C₄×Q₈	1	2	yes	—
76	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₄₃	—	2	1	no	—
78	C₄×C₄₃	—	2	1	no	—
79	C₄×C₄₃	—	2	1	no	—
80	C₄×C₄₃	C₄×Q₈	1	2	yes	—
81	C₄×C₄₃	#131	1	2	yes	—
82	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
84	Dic₄₃	—	2	4	no	—
85	Dic₄₃	—	2	4	no	—
86	#2962	—	43	1	no	maximal, Sylow
87	C₂×C₄₃	—	2	2	no	—
88	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
90	C₂×C₄₃	#568	1	1	yes	characteristic
91	#533	—	43	1	no	—
92	#446	—	43	4	no	—
93	#446	—	43	2	no	—
94	C₈×Q₈	—	43	2	no	—
95	#424	—	43	2	no	—
96	C₂×C₄⋊C₈	—	43	1	no	—
97	#405	—	43	2	no	—
98	C₄×M₁₆	D₄₃	1	1	yes	characteristic
99	C₄₃	#2962	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
101	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
102	C₂²×C₈	—	43	2	no	—
103	C₄⋊Q₈	—	43	1	no	—
104	C₄⋊Q₈	—	43	1	no	—
105	#125	—	43	2	no	—
106	C₂²⋊Q₈	—	43	2	no	—
107	C₂²⋊Q₈	—	43	2	no	—
108	C₄×Q₈	—	43	2	no	—
109	C₄×Q₈	—	43	2	no	—
110	#117	—	43	2	no	—
111	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
112	C₄⋊C₈	—	43	4	no	—
113	C₄⋊C₈	—	43	4	no	—
114	C₄⋊C₈	—	43	1	no	—
115	C₄⋊C₈	—	43	1	no	—
116	C₄⋊C₈	—	43	2	no	—
117	#97	—	43	4	no	—
118	#97	C₂×D₄₃	1	2	yes	—
119	C₄×C₈	—	43	4	no	—
120	C₄×C₈	C₂×D₄₃	1	2	yes	—
121	C₂×Q₈	—	43	2	no	—
122	C₂×Q₈	—	43	2	no	—
123	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₈₆	1	4	yes	complemented
128	C₂×C₈	—	43	2	no	—
129	C₂×C₈	—	43	2	no	—
130	C₂×C₈	—	86	4	no	—
131	C₂×C₈	C₂²×D₄₃	1	2	yes	—
132	C₂×C₈	C₂²×D₄₃	1	2	yes	—
133	C₄⋊C₄	—	43	2	no	—
134	C₄⋊C₄	—	43	2	no	—
135	C₄⋊C₄	—	43	4	no	—
136	C₄⋊C₄	—	43	4	no	—
137	C₄⋊C₄	—	43	2	no	—
138	C₄⋊C₄	—	43	2	no	—
139	C₂²⋊C₄	—	43	4	no	—
140	C₄²	—	43	4	no	—
141	C₄²	C₄×D₄₃	1	1	yes	characteristic
142	C₄²	C₄×D₄₃	1	1	yes	characteristic
143	C₄²	C₂²×D₄₃	1	1	yes	characteristic
144	C₄²	C₂²×D₄₃	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
146	Q₈	—	86	4	no	—
147	Q₈	—	172	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	43	4	no	—
151	C₂×C₄	—	43	4	no	—
152	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
153	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
158	C₂×C₄	#10621898	1	2	yes	—
159	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	86	2	no	—
162	C₈	—	86	2	no	—
163	C₈	C₂×Dic₈₆	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₈₆	1	2	yes	—
167	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	86	4	no	—
172	C₄	—	86	4	no	—
173	C₄	C₄×Dic₈₆	1	2	yes	—
174	C₄	#13063333	1	2	yes	—
175	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
176	C₄	C₂×#10621898	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₈₆	1	1	yes	characteristic
179	C₂	C₂×#13063333	1	1	yes	characteristic
180	C₂	#14583665	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₄×M₁₆×C₄₃	C₂
hypercenter	C₄×M₁₆	D₄₃
hypocenter	C₄₃	#2962
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #14583029	C₂	semidirect
3. #14583027	C₂	semidirect
4. C₈×Dic₈₆	C₂	semidirect
36. M₁₆×C₄₃	Q₈	semidirect
99. C₄₃	#2962	semidirect
127. M₁₆	Dic₈₆	semidirect