Finite Group Databasebeta

Subgroups — #19134129

Order 5504.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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	#14583246	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583190	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583095	C₂	1	1	yes	maximal, characteristic
4	#14582978	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582647	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₃₄₄	—	2	1	no	—
9	C₂×#10621872	—	2	1	no	—
10	#13063325	C₂²	1	1	yes	characteristic
11	#13063322	C₂²	1	1	yes	characteristic
12	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
13	#13063257	—	2	1	no	—
14	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
15	#13063247	—	2	1	no	—
16	#13063247	—	2	1	no	—
17	#13063247	—	2	1	no	—
18	#13063247	—	2	1	no	—
19	(C₂×D₄₃)⋊C₈	—	2	1	no	—
20	C₄⋊C₈×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×D₁₇₂	—	2	1	no	—
24	C₂×D₁₇₂	—	2	1	no	—
25	C₂×D₁₇₂	—	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	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	(C₂×D₄₃)⋊C₄	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	C₄⋊Dic₄₃	—	2	1	no	—
37	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₈	—	2	1	no	—
39	D₃₄₄	—	4	2	no	—
40	#10621872	—	4	2	no	—
41	C₂²×D₄₃	—	2	1	no	—
42	C₂²×D₄₃	—	4	1	no	—
43	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
46	C₂×Dic₄₃	—	2	1	no	—
47	C₂×Dic₄₃	—	4	1	no	—
48	D₁₇₂	—	2	1	no	—
49	D₁₇₂	—	2	1	no	—
50	D₁₇₂	—	4	1	no	—
51	D₁₇₂	—	4	1	no	—
52	D₁₇₂	—	4	1	no	—
53	D₁₇₂	—	4	1	no	—
54	D₁₇₂	—	8	1	no	—
55	C₄×D₄₃	—	4	1	no	—
56	C₄×D₄₃	—	4	1	no	—
57	Dic₈₆	—	8	1	no	—
58	C₈×C₄₃	—	2	1	no	—
59	C₈×C₄₃	—	2	1	no	—
60	C₈×C₄₃	—	4	1	no	—
61	C₈×C₄₃	—	4	1	no	—
62	C₄₃⋊C₈	—	4	1	no	—
63	C₂²×C₄₃	#121	1	1	yes	characteristic
64	C₂×D₄₃	—	4	1	no	—
65	C₂×D₄₃	—	4	1	no	—
66	C₂×D₄₃	—	4	1	no	—
67	C₂×D₄₃	—	8	1	no	—
68	C₂×D₄₃	—	8	1	no	—
69	C₂×D₄₃	—	8	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	#463	1	1	yes	characteristic
79	C₂×C₄₃	#468	1	1	yes	characteristic
80	C₂×C₄₃	#472	1	1	yes	characteristic
81	D₄₃	—	8	1	no	—
82	D₄₃	—	16	1	no	—
83	#486	—	43	1	no	—
84	#476	—	43	1	no	—
85	#459	—	43	1	no	—
86	#436	—	43	1	no	—
87	#335	D₄₃	1	1	yes	characteristic, complemented
88	#333	—	43	1	no	—
89	#331	—	43	1	no	—
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	—	43	1	no	—
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	C₂×D₄₃	1	1	yes	characteristic
106	C₂×Q₈	—	86	1	no	—
107	C₂×D₄	—	43	1	no	—
108	C₂×D₄	—	86	1	no	—
109	C₂×D₄	—	86	1	no	—
110	C₂×D₄	—	86	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	—	86	1	no	—
117	C₂×C₈	D₁₇₂	1	1	yes	characteristic
118	C₂×C₈	D₁₇₂	1	1	yes	characteristic
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	—	86	1	no	—
125	C₂³	—	172	1	no	—
126	Q₈	—	344	1	no	—
127	D₄	—	86	1	no	—
128	D₄	—	86	1	no	—
129	D₄	—	344	1	no	—
130	D₄	—	172	1	no	—
131	D₄	—	172	1	no	—
132	D₄	—	172	1	no	—
133	D₄	—	172	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	C₂×D₁₇₂	1	1	yes	characteristic
139	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
140	C₂×C₄	#10621902	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	172	1	no	—
146	C₂²	—	344	1	no	—
147	C₂²	—	344	1	no	—
148	C₂²	—	344	1	no	—
149	C₂²	—	172	1	no	—
150	C₂²	—	172	1	no	—
151	C₂²	—	172	1	no	—
152	C₂²	#13063300	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	344	1	no	—
155	C₄	—	172	1	no	—
156	C₄	#13063347	1	1	yes	characteristic
157	C₄	#13063349	1	1	yes	characteristic
158	C₄	#13063353	1	1	yes	characteristic
159	C₄	#13063357	1	1	yes	characteristic
160	C₂	—	344	1	no	—
161	C₂	—	688	1	no	—
162	C₂	#14583115	1	1	yes	characteristic
163	C₂	#14583146	1	1	yes	characteristic
164	C₂	#14583170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13063300
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#335	C₂
socle	C₂²×C₄₃	#121
hypercenter	#335	D₄₃
hypocenter	C₄₃	#1697
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
1. #14583246	C₂	semidirect
2. #14583190	C₂	semidirect
4. #14582978	C₂	semidirect
5. #14582647	C₂	semidirect
6. #14582642	C₂	semidirect
7. C₄₃×#335	C₂	semidirect
87. #335	D₄₃	semidirect
90. C₄₃	#1697	semidirect