Finite Group Databasebeta

Subgroups — #14583708

Order 2752.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13063373	C₂	1	1	yes	maximal, characteristic, complemented
2	#13063371	C₂	1	1	yes	maximal, characteristic, complemented
3	#13063357	C₂	1	2	yes	maximal, complemented
4	#13063353	C₂	1	1	yes	maximal, characteristic, complemented
5	#13063352	C₂	1	2	yes	maximal, complemented
6	#13063348	C₂	1	2	yes	maximal, complemented
7	#13063345	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621884	C₂	1	1	yes	maximal, characteristic, complemented
9	SD₁₆×D₄₃	C₂	1	2	yes	maximal, complemented
10	#13063333	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
14	#10621902	—	2	1	no	—
15	#10621902	C₂²	1	2	yes	—
16	#10621902	C₂²	1	2	yes	complemented
17	Q₈×D₄₃	—	2	1	no	—
18	Q₈×D₄₃	C₂²	1	2	yes	complemented
19	D₄×D₄₃	—	2	2	no	—
20	D₄×D₄₃	C₂²	1	2	yes	—
21	#10621898	—	2	2	no	—
22	#10621898	—	2	1	no	—
23	#10621898	C₂²	1	1	yes	characteristic
24	#10621898	C₂²	1	1	yes	characteristic, complemented
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	C₂²	1	1	yes	characteristic
27	Q₁₆×C₄₃	C₂²	1	2	yes	complemented
28	SD₁₆×C₄₃	C₂²	1	2	yes	complemented
29	M₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
30	#10621885	C₂²	1	1	yes	characteristic, complemented
31	#10621884	C₂²	1	2	yes	complemented
32	#10621884	C₂²	1	2	yes	complemented
33	#10621884	C₂²	1	1	yes	characteristic, complemented
34	#10621883	C₂²	1	1	yes	characteristic, complemented
35	#10621882	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	D₃₄₄	C₂²	1	2	yes	complemented
38	#10621873	C₂²	1	2	yes	complemented
39	#10621872	C₂²	1	2	yes	complemented
40	C₈×D₄₃	C₂²	1	2	yes	complemented
41	C₂²×D₄₃	—	2	1	no	—
42	C₂²×D₄₃	—	2	2	no	—
43	Q₈×C₄₃	—	2	1	no	—
44	Q₈×C₄₃	C₂³	1	2	yes	—
45	Q₈×C₄₃	C₂³	1	1	yes	characteristic
46	D₄×C₄₃	—	2	1	no	—
47	D₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₄₃	—	2	1	no	—
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²⋊D₄₃	—	2	1	no	—
52	C₂²⋊D₄₃	—	2	2	no	—
53	C₂²⋊D₄₃	D₄	1	2	yes	—
54	D₁₇₂	—	2	2	no	—
55	D₁₇₂	—	2	2	no	—
56	D₁₇₂	—	2	1	no	—
57	D₁₇₂	D₄	1	1	yes	characteristic
58	D₁₇₂	C₂³	1	2	yes	—
59	D₁₇₂	C₂³	1	1	yes	characteristic
60	C₄×D₄₃	—	2	2	no	—
61	C₄×D₄₃	—	2	1	no	—
62	C₄×D₄₃	—	2	2	no	—
63	C₄×D₄₃	—	2	2	no	—
64	C₄×D₄₃	C₂³	1	2	yes	—
65	Dic₈₆	—	2	2	no	—
66	Dic₈₆	D₄	1	1	yes	characteristic
67	Dic₈₆	C₂³	1	1	yes	characteristic
68	C₈×C₄₃	C₂³	1	2	yes	—
69	C₄₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₄₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₄₃	—	2	1	no	—
72	C₂²×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×D₄₃	—	2	2	no	—
74	C₂×D₄₃	—	2	1	no	—
75	C₂×D₄₃	—	8	2	no	—
76	C₂×D₄₃	—	8	1	no	—
77	C₂×D₄₃	C₂×D₄	1	2	yes	—
78	C₄×C₄₃	—	2	2	no	—
79	C₄×C₄₃	—	2	1	no	—
80	C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	Dic₄₃	—	2	1	no	—
83	Dic₄₃	C₂×D₄	1	2	yes	—
84	C₂×C₄₃	—	2	1	no	—
85	C₂×C₄₃	—	4	1	no	—
86	C₂×C₄₃	C₂²×D₄	1	1	yes	characteristic
87	D₄₃	—	2	2	no	—
88	D₄₃	—	4	1	no	—
89	D₄₃	—	4	2	no	—
90	#577	—	43	1	no	maximal, Sylow
91	C₄₃	#577	1	1	yes	characteristic, complemented, Sylow
92	2₋¹⁺⁴	—	43	1	no	—
93	H(5,𝔽₂)	—	43	1	no	—
94	#137	—	43	2	no	—
95	#137	D₄₃	1	1	yes	characteristic, complemented
96	AGL(1,ℤ/8)	—	43	2	no	—
97	AGL(1,ℤ/8)	—	43	1	no	—
98	#135	—	43	1	no	—
99	#135	—	43	2	no	—
100	C₂×SD₁₆	—	43	1	no	—
101	C₂×SD₁₆	—	43	2	no	—
102	#131	—	43	1	no	—
103	#41	—	43	1	no	—
104	#41	—	43	2	no	—
105	#41	—	43	2	no	—
106	#41	—	43	1	no	—
107	#41	—	86	2	no	—
108	#41	—	86	1	no	—
109	#41	—	86	1	no	—
110	#41	C₂×D₄₃	1	1	yes	characteristic
111	C₂×Q₈	—	43	2	no	—
112	C₂×Q₈	—	86	1	no	—
113	C₂×Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
114	C₂×D₄	—	43	2	no	—
115	C₂×D₄	—	43	1	no	—
116	C₂×D₄	—	86	1	no	—
117	C₂×D₄	—	86	2	no	—
118	Q₁₆	—	43	1	no	—
119	Q₁₆	C₂×D₄₃	1	2	yes	complemented
120	SD₁₆	—	43	2	no	—
121	SD₁₆	—	43	2	no	—
122	SD₁₆	—	43	1	no	—
123	SD₁₆	—	43	1	no	—
124	SD₁₆	—	43	2	no	—
125	SD₁₆	C₂×D₄₃	1	2	yes	complemented
126	D₈	—	43	1	no	—
127	D₈	—	43	2	no	—
128	M₁₆	—	43	2	no	—
129	M₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
130	C₂×C₈	—	43	1	no	—
131	C₂×C₈	—	43	2	no	—
132	C₂³	—	86	2	no	—
133	C₂³	—	86	1	no	—
134	Q₈	—	2	1	no	—
135	Q₈	—	43	1	no	—
136	Q₈	—	43	1	no	—
137	Q₈	—	86	2	no	—
138	Q₈	C₂²×D₄₃	1	2	yes	—
139	Q₈	C₂²×D₄₃	1	1	yes	characteristic
140	D₄	—	2	1	no	—
141	D₄	—	43	2	no	—
142	D₄	—	43	1	no	—
143	D₄	—	43	1	no	—
144	D₄	—	43	2	no	—
145	D₄	—	86	2	no	—
146	D₄	—	86	2	no	—
147	D₄	—	86	2	no	—
148	D₄	—	86	1	no	—
149	D₄	—	86	1	no	—
150	D₄	C₂²×D₄₃	1	1	yes	characteristic
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	43	2	no	—
154	C₂×C₄	—	86	2	no	—
155	C₂×C₄	—	86	2	no	—
156	C₂×C₄	—	86	2	no	—
157	C₂×C₄	—	86	1	no	—
158	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
159	C₈	—	43	1	no	—
160	C₈	—	43	1	no	—
161	C₈	C₂²×D₄₃	1	2	yes	—
162	C₂²	—	2	1	no	—
163	C₂²	—	43	2	no	—
164	C₂²	—	86	1	no	—
165	C₂²	—	86	2	no	—
166	C₂²	—	344	1	no	—
167	C₂²	—	344	2	no	—
168	C₂²	D₄×D₄₃	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	—	43	2	no	—
172	C₄	—	86	1	no	—
173	C₄	D₄×D₄₃	1	1	yes	characteristic
174	C₄	C₂³×D₄₃	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	86	2	no	—
178	C₂	—	172	2	no	—
179	C₂	—	172	1	no	—
180	C₂	C₂×D₄×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₄₃
commutator, cosocle	C₄×C₄₃	C₂⁴
Frattini	C₄	C₂³×D₄₃
Fitting	#137×C₄₃	C₂
socle	C₂×C₄₃	C₂²×D₄
hypercenter	#137	D₄₃
hypocenter	C₄₃	#577
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #13063373	C₂	semidirect
2. #13063371	C₂	semidirect
3. #13063357	C₂	semidirect
4. #13063353	C₂	semidirect
5. #13063352	C₂	semidirect
6. #13063348	C₂	semidirect
7. #13063345	C₂	semidirect
8. C₂×#10621884	C₂	semidirect
9. SD₁₆×D₄₃	C₂	semidirect
10. #13063333	C₂	semidirect
11. #137×C₄₃	C₂	semidirect
13. C₂×Q₈×C₄₃	C₂²	semidirect
16. #10621902	C₂²	semidirect
18. Q₈×D₄₃	C₂²	semidirect
24. #10621898	C₂²	semidirect
27. Q₁₆×C₄₃	C₂²	semidirect
28. SD₁₆×C₄₃	C₂²	semidirect
29. M₁₆×C₄₃	C₂²	semidirect
30. #10621885	C₂²	semidirect
31. #10621884	C₂²	semidirect
32. #10621884	C₂²	semidirect
33. #10621884	C₂²	semidirect
34. #10621883	C₂²	semidirect
35. #10621882	C₂²	semidirect
36. C₂×C₄₃⋊C₈	C₂²	semidirect
37. D₃₄₄	C₂²	semidirect
38. #10621873	C₂²	semidirect
39. #10621872	C₂²	semidirect
40. C₈×D₄₃	C₂²	semidirect
91. C₄₃	#577	semidirect
95. #137	D₄₃	semidirect
113. C₂×Q₈	C₂×D₄₃	semidirect
119. Q₁₆	C₂×D₄₃	semidirect
125. SD₁₆	C₂×D₄₃	semidirect
129. M₁₆	C₂×D₄₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13063373	2
#13063371	2
#13063357	2
#13063353	2
#13063352	2
#13063348	2
#13063345	2
C₂×#10621884	2
SD₁₆×D₄₃	2
#13063333	2
#137×C₄₃	2
#577	43