Finite Group Databasebeta

Subgroups — #19146388

Order 5504.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14583705	C₂	1	4	yes	maximal, complemented
2	#14583159	C₂	1	4	yes	maximal, complemented
3	#14583081	C₂	1	2	yes	maximal, complemented
4	#14583066	C₂	1	2	yes	maximal, complemented
5	#14583017	C₂	1	2	yes	maximal, complemented
6	C₄₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
7	#13063373	C₂²	1	2	yes	—
8	#13063370	C₂²	1	2	yes	complemented
9	#13063356	C₂²	1	4	yes	complemented
10	#13063354	—	2	4	no	—
11	#13063352	—	2	4	no	—
12	#13063351	C₂²	1	4	yes	—
13	#13063347	C₂²	1	2	yes	—
14	#13063346	—	2	4	no	—
15	SD₁₆×D₄₃	—	2	4	no	—
16	#13063331	C₂²	1	4	yes	complemented
17	#13063261	C₂²	1	2	yes	complemented
18	#13063253	C₂²	1	4	yes	complemented
19	#13063244	C₂²	1	2	yes	complemented
20	#13063242	C₂²	1	2	yes	complemented
21	C₂×SD₁₆×C₄₃	C₂²	1	4	yes	complemented
22	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
23	#127×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
25	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
26	C₂×D₄×C₄₃	—	4	1	no	—
27	C₂×D₄×C₄₃	C₂³	1	2	yes	—
28	C₂×C₂²⋊D₄₃	—	4	2	no	—
29	#10621902	—	2	4	no	—
30	Q₈×D₄₃	—	2	4	no	—
31	#10621900	—	2	4	no	—
32	D₄×D₄₃	—	2	4	no	—
33	#10621898	—	4	2	no	—
34	#10621898	C₂³	1	4	yes	—
35	SD₁₆×C₄₃	—	2	4	no	—
36	SD₁₆×C₄₃	—	2	4	no	—
37	C₂×C₈×C₄₃	C₂³	1	2	yes	—
38	C₄⋊C₄×C₄₃	—	4	1	no	—
39	C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	#10621885	—	2	4	no	—
41	#10621884	—	2	4	no	—
42	#10621883	—	2	4	no	—
43	#10621882	—	2	4	no	—
44	#10621877	C₂³	1	4	yes	—
45	Dic₁₇₂	D₄	1	2	yes	complemented
46	D₃₄₄	D₄	1	2	yes	complemented
47	#10621873	D₄	1	4	yes	complemented
48	#10621872	—	2	4	no	—
49	C₈×D₄₃	—	2	4	no	—
50	C₂³×C₄₃	—	2	2	no	—
51	C₂²×D₄₃	—	4	2	no	—
52	Q₈×C₄₃	—	2	4	no	—
53	D₄×C₄₃	—	2	4	no	—
54	D₄×C₄₃	—	8	2	no	—
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
58	C₂²⋊D₄₃	—	2	4	no	—
59	C₂²⋊D₄₃	—	4	4	no	—
60	C₂×Dic₄₃	—	4	2	no	—
61	D₁₇₂	—	4	2	no	—
62	D₁₇₂	C₂×D₄	1	4	yes	—
63	C₄×D₄₃	—	2	4	no	—
64	C₄×D₄₃	—	4	4	no	—
65	Dic₈₆	—	4	2	no	—
66	Dic₈₆	C₂×D₄	1	4	yes	—
67	C₈×C₄₃	C₂×D₄	1	2	yes	—
68	C₈×C₄₃	C₂×D₄	1	2	yes	—
69	C₄₃⋊C₈	—	2	4	no	—
70	C₂²×C₄₃	—	4	2	no	—
71	C₂²×C₄₃	—	8	2	no	—
72	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×D₄₃	—	2	4	no	—
74	C₂×D₄₃	—	16	2	no	—
75	C₄×C₄₃	—	2	2	no	—
76	C₄×C₄₃	—	4	2	no	—
77	C₄×C₄₃	C₂²×D₄	1	2	yes	—
78	Dic₄₃	—	2	4	no	—
79	#3291	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	2	1	no	—
81	C₂×C₄₃	—	8	2	no	—
82	C₂×C₄₃	D₄²	1	1	yes	characteristic
83	D₄₃	—	4	4	no	—
84	#577	—	43	4	no	—
85	#492	D₄₃	1	1	yes	characteristic, complemented
86	#471	—	43	4	no	—
87	#456	—	43	2	no	—
88	#453	—	43	2	no	—
89	#443	—	43	2	no	—
90	C₄₃	#3291	1	1	yes	characteristic, complemented, Sylow
91	2₋¹⁺⁴	—	43	2	no	—
92	H(5,𝔽₂)	—	43	2	no	—
93	#137	—	43	4	no	—
94	#137	—	86	4	no	—
95	AGL(1,ℤ/8)	—	43	4	no	—
96	AGL(1,ℤ/8)	—	86	4	no	—
97	#135	—	43	2	no	—
98	#135	—	86	4	no	—
99	C₂×SD₁₆	—	86	4	no	—
100	C₂×SD₁₆	C₂×D₄₃	1	4	yes	complemented
101	#131	—	43	4	no	—
102	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
103	#127	C₂×D₄₃	1	1	yes	characteristic
104	#108	—	43	2	no	—
105	#104	—	43	4	no	—
106	#101	—	43	2	no	—
107	#100	—	43	2	no	—
108	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
109	#41	—	43	4	no	—
110	#41	—	86	4	no	—
111	#41	—	86	4	no	—
112	#41	—	172	2	no	—
113	C₂×Q₈	—	86	4	no	—
114	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	86	4	no	—
117	C₂×D₄	—	172	2	no	—
118	C₂×D₄	C₂²×D₄₃	1	2	yes	—
119	Q₁₆	—	43	2	no	—
120	Q₁₆	—	86	4	no	—
121	SD₁₆	—	2	4	no	—
122	SD₁₆	—	2	4	no	—
123	SD₁₆	—	43	4	no	—
124	SD₁₆	—	86	4	no	—
125	SD₁₆	—	86	4	no	—
126	D₈	—	43	2	no	—
127	D₈	—	86	4	no	—
128	M₁₆	—	43	4	no	—
129	M₁₆	—	86	4	no	—
130	C₂×C₈	—	86	4	no	—
131	C₂×C₈	C₂²×D₄₃	1	2	yes	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄²	C₂²×D₄₃	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	—	172	2	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	43	4	no	—
138	Q₈	—	172	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	8	2	no	—
141	D₄	—	43	4	no	—
142	D₄	—	86	4	no	—
143	D₄	—	172	2	no	—
144	D₄	—	172	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	86	4	no	—
148	C₂×C₄	—	172	4	no	—
149	C₂×C₄	—	172	2	no	—
150	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
151	C₈	—	86	4	no	—
152	C₈	D₄×D₄₃	1	2	yes	—
153	C₈	D₄×D₄₃	1	2	yes	—
154	C₂²	—	4	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	86	4	no	—
157	C₂²	—	688	2	no	—
158	C₂²	#13063370	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	86	4	no	—
162	C₄	C₂×D₄×D₄₃	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	172	4	no	—
166	C₂	#14583534	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14583534
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#492	C₂
socle	C₂×C₄₃	D₄²
hypercenter	#492	D₄₃
hypocenter	C₄₃	#3291
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #14583705	C₂	semidirect
2. #14583159	C₂	semidirect
3. #14583081	C₂	semidirect
4. #14583066	C₂	semidirect
5. #14583017	C₂	semidirect
6. C₄₃×#492	C₂	semidirect
8. #13063370	C₂²	semidirect
9. #13063356	C₂²	semidirect
16. #13063331	C₂²	semidirect
17. #13063261	C₂²	semidirect
18. #13063253	C₂²	semidirect
19. #13063244	C₂²	semidirect
20. #13063242	C₂²	semidirect
21. C₂×SD₁₆×C₄₃	C₂²	semidirect
22. C₄⋊Q₈×C₄₃	C₂²	semidirect
24. C₄×C₈×C₄₃	C₂²	semidirect
45. Dic₁₇₂	D₄	semidirect
46. D₃₄₄	D₄	semidirect
47. #10621873	D₄	semidirect
85. #492	D₄₃	semidirect
90. C₄₃	#3291	semidirect
100. C₂×SD₁₆	C₂×D₄₃	semidirect
102. C₄⋊Q₈	C₂×D₄₃	semidirect
108. C₄×C₈	C₂×D₄₃	semidirect