Finite Group Databasebeta

Subgroups — #19134127

Order 5504.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583245	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583190	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583091	C₂	1	1	yes	maximal, characteristic
4	#14582976	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582651	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582647	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621882	—	2	1	no	—
9	C₂×#10621872	—	2	1	no	—
10	#13063325	C₂²	1	1	yes	characteristic
11	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	#13063257	—	2	1	no	—
13	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#13063247	—	2	1	no	—
15	#13063246	—	2	1	no	—
16	#13063246	—	2	1	no	—
17	#13063246	—	2	1	no	—
18	(C₂×D₄₃)⋊C₈	—	2	1	no	—
19	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₄₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₄₃	—	2	1	no	—
24	C₂×D₄×C₄₃	—	2	1	no	—
25	C₂×D₄×C₄₃	—	2	1	no	—
26	C₂×D₁₇₂	D₄	1	1	yes	characteristic
27	C₂×C₄×D₄₃	—	2	1	no	—
28	C₂×Dic₈₆	—	2	1	no	—
29	C₂×C₈×C₄₃	—	2	1	no	—
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₄²×C₄₃	C₂³	1	1	yes	characteristic
32	#10621882	—	4	2	no	—
33	(C₂×D₄₃)⋊C₄	—	2	1	no	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄₃⋊C₈	—	2	1	no	—
38	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
40	#10621872	—	4	2	no	—
41	C₂³×C₄₃	—	4	1	no	—
42	C₂²×D₄₃	—	2	1	no	—
43	D₄×C₄₃	—	4	1	no	—
44	D₄×C₄₃	—	4	1	no	—
45	D₄×C₄₃	—	4	1	no	—
46	D₄×C₄₃	—	4	1	no	—
47	D₄×C₄₃	—	8	1	no	—
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
51	C₂×Dic₄₃	—	2	1	no	—
52	C₂×Dic₄₃	—	4	1	no	—
53	D₁₇₂	—	2	1	no	—
54	D₁₇₂	—	2	1	no	—
55	C₄×D₄₃	—	4	1	no	—
56	C₄×D₄₃	—	4	1	no	—
57	Dic₈₆	—	8	1	no	—
58	C₈×C₄₃	—	4	1	no	—
59	C₈×C₄₃	—	4	1	no	—
60	C₄₃⋊C₈	—	2	1	no	—
61	C₄₃⋊C₈	—	2	1	no	—
62	C₄₃⋊C₈	—	4	1	no	—
63	C₂²×C₄₃	—	8	1	no	—
64	C₂²×C₄₃	—	8	1	no	—
65	C₂²×C₄₃	—	8	1	no	—
66	C₂²×C₄₃	#121	1	1	yes	characteristic
67	C₂×D₄₃	—	4	1	no	—
68	C₂×D₄₃	—	4	1	no	—
69	C₂×D₄₃	—	4	1	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	#135	1	1	yes	characteristic
72	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₄₃	#137	1	1	yes	characteristic
75	Dic₄₃	—	4	1	no	—
76	Dic₄₃	—	8	1	no	—
77	#1697	—	43	1	no	maximal, Sylow
78	C₂×C₄₃	—	16	1	no	—
79	C₂×C₄₃	#463	1	1	yes	characteristic
80	C₂×C₄₃	#468	1	1	yes	characteristic
81	C₂×C₄₃	#472	1	1	yes	characteristic
82	D₄₃	—	8	1	no	—
83	#486	—	43	1	no	—
84	#476	—	43	1	no	—
85	#459	—	43	1	no	—
86	#436	—	43	1	no	—
87	#335	—	43	1	no	—
88	#333	—	43	1	no	—
89	#331	D₄₃	1	1	yes	characteristic, complemented
90	C₄₃	#1697	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	86	1	no	—
92	C₂×M₁₆	—	86	1	no	—
93	C₄⋊Q₈	—	43	1	no	—
94	#127	C₂×D₄₃	1	1	yes	characteristic
95	C₄×D₄	—	43	1	no	—
96	#106	—	86	1	no	—
97	C₄⋊C₈	—	43	1	no	—
98	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
100	#102	—	86	1	no	—
101	#102	—	86	1	no	—
102	#102	—	86	1	no	—
103	#102	—	86	1	no	—
104	C₂²⋊C₈	—	86	1	no	—
105	C₄×C₈	—	43	1	no	—
106	C₂×Q₈	—	86	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	43	1	no	—
111	C₂²×C₄	—	86	1	no	—
112	D₈	—	172	2	no	—
113	M₁₆	—	172	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	43	1	no	—
117	C₂×C₈	—	43	1	no	—
118	C₂×C₈	—	86	1	no	—
119	C₄⋊C₄	—	43	1	no	—
120	C₄⋊C₄	—	86	1	no	—
121	C₄⋊C₄	—	86	1	no	—
122	C₂²⋊C₄	—	86	1	no	—
123	C₄²	C₂²×D₄₃	1	1	yes	characteristic
124	C₂³	—	4	1	no	—
125	C₂³	—	86	1	no	—
126	Q₈	—	344	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	86	1	no	—
133	D₄	—	86	1	no	—
134	C₂×C₄	—	86	1	no	—
135	C₂×C₄	—	172	1	no	—
136	C₂×C₄	—	172	1	no	—
137	C₂×C₄	—	172	1	no	—
138	C₂×C₄	#10621898	1	1	yes	characteristic
139	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
140	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	4	1	no	—
143	C₈	—	86	1	no	—
144	C₈	—	86	1	no	—
145	C₈	—	172	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	172	1	no	—
150	C₂²	—	172	1	no	—
151	C₂²	—	172	1	no	—
152	C₂²	#13063298	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	344	1	no	—
155	C₄	—	172	1	no	—
156	C₄	#13063346	1	1	yes	characteristic
157	C₄	#13063349	1	1	yes	characteristic
158	C₄	#13063349	1	1	yes	characteristic
159	C₄	#13063354	1	1	yes	characteristic
160	C₂	—	16	1	no	—
161	C₂	—	344	1	no	—
162	C₂	#14583113	1	1	yes	characteristic
163	C₂	#14583144	1	1	yes	characteristic
164	C₂	#14583168	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13063298
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#331	C₂
socle	C₂²×C₄₃	#121
hypercenter	#331	D₄₃
hypocenter	C₄₃	#1697
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

Complements

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

N	H ≅ G/N	type
1. #14583245	C₂	semidirect
2. #14583190	C₂	semidirect
4. #14582976	C₂	semidirect
5. #14582651	C₂	semidirect
6. #14582647	C₂	semidirect
7. C₄₃×#331	C₂	semidirect
89. #331	D₄₃	semidirect
90. C₄₃	#1697	semidirect