Finite Group Databasebeta

Subgroups — #19136554

Order 5504.

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

normal subgroups only

This group has 220 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	#14583704	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583337	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583169	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582713	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582707	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#13063371	—	2	1	no	—
9	#13063353	—	2	1	no	—
10	#13063353	—	2	2	no	—
11	#13063350	—	2	1	no	—
12	#13063347	—	2	1	no	—
13	#13063347	C₂²	1	1	yes	characteristic
14	C₂×D₃₄₄	—	2	1	no	—
15	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
16	C₂×D₃₄₄	C₂²	1	1	yes	characteristic, complemented
17	#13063270	—	2	1	no	—
18	D₆₈₈	—	2	2	no	—
19	D₆₈₈	—	2	1	no	—
20	#13063261	C₂²	1	1	yes	characteristic, complemented
21	#13063241	—	2	1	no	—
22	#131×C₄₃	C₂²	1	1	yes	characteristic
23	M₃₂×C₄₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
25	#41×C₄₃	D₄	1	1	yes	characteristic
26	#10621902	—	4	1	no	—
27	D₄×D₄₃	—	4	2	no	—
28	D₄×D₄₃	—	4	1	no	—
29	#10621898	—	2	2	no	—
30	#10621898	—	2	1	no	—
31	C₂×D₁₇₂	—	2	2	no	—
32	C₂×D₁₇₂	—	2	1	no	—
33	C₂×D₁₇₂	—	2	1	no	—
34	M₁₆×C₄₃	—	2	1	no	—
35	M₁₆×C₄₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
38	#10621884	—	4	1	no	—
39	#10621882	—	4	1	no	—
40	#10621877	—	2	1	no	—
41	Dic₁₇₂	D₄	1	1	yes	characteristic
42	D₃₄₄	—	2	2	no	—
43	D₃₄₄	—	2	1	no	—
44	D₃₄₄	—	2	1	no	—
45	D₃₄₄	—	2	1	no	—
46	D₃₄₄	D₄	1	1	yes	characteristic
47	D₃₄₄	D₄	1	2	yes	—
48	#10621873	—	2	2	no	—
49	#10621873	—	2	1	no	—
50	C₁₆×C₄₃	—	2	1	no	—
51	C₁₆×C₄₃	—	2	1	no	—
52	C₂²×D₄₃	—	4	2	no	—
53	C₂²×D₄₃	—	4	1	no	—
54	C₂²×D₄₃	—	4	1	no	—
55	Q₈×C₄₃	D₈	1	1	yes	characteristic
56	D₄×C₄₃	—	2	1	no	—
57	D₄×C₄₃	D₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	—	2	1	no	—
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₄₃	—	4	2	no	—
61	C₂²⋊D₄₃	—	4	1	no	—
62	D₁₇₂	—	2	2	no	—
63	D₁₇₂	—	2	1	no	—
64	D₁₇₂	—	4	2	no	—
65	D₁₇₂	—	4	1	no	—
66	D₁₇₂	—	4	1	no	—
67	D₁₇₂	—	4	1	no	—
68	C₄×D₄₃	—	4	2	no	—
69	C₄×D₄₃	—	4	1	no	—
70	Dic₈₆	—	2	1	no	—
71	C₈×C₄₃	—	2	1	no	—
72	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₄₃⋊C₈	—	4	1	no	—
75	C₂²×C₄₃	—	2	1	no	—
76	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₄₃	—	4	1	no	—
78	C₂×D₄₃	—	4	2	no	—
79	C₂×D₄₃	—	8	1	no	—
80	C₂×D₄₃	—	16	1	no	—
81	C₂×D₄₃	—	16	2	no	—
82	C₂×D₄₃	—	16	1	no	—
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	#120	1	1	yes	characteristic
85	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
86	Dic₄₃	—	4	1	no	—
87	#2192	—	43	1	no	maximal, Sylow
88	C₂×C₄₃	—	2	1	no	—
89	C₂×C₄₃	—	4	1	no	—
90	C₂×C₄₃	#447	1	1	yes	characteristic
91	D₄₃	—	8	2	no	—
92	D₄₃	—	8	1	no	—
93	D₄₃	—	16	1	no	—
94	#576	—	43	1	no	—
95	#509	—	43	1	no	—
96	C₂×D₁₆	—	43	1	no	—
97	#472	—	43	1	no	—
98	#361	—	43	1	no	—
99	#359	—	43	1	no	—
100	#350	D₄₃	1	1	yes	characteristic, complemented
101	C₄₃	#2192	1	1	yes	characteristic, complemented, Sylow
102	H(5,𝔽₂)	—	86	1	no	—
103	AGL(1,ℤ/8)	—	86	2	no	—
104	AGL(1,ℤ/8)	—	86	1	no	—
105	AGL(1,ℤ/8)	—	86	1	no	—
106	#135	—	43	1	no	—
107	#135	—	86	1	no	—
108	C₂×D₈	—	43	1	no	—
109	C₂×D₈	—	43	1	no	—
110	C₂×D₈	—	86	1	no	—
111	#131	C₂×D₄₃	1	1	yes	characteristic
112	SD₃₂	—	86	1	no	—
113	D₁₆	—	86	2	no	—
114	D₁₆	—	86	1	no	—
115	M₃₂	C₂×D₄₃	1	1	yes	characteristic, complemented
116	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
117	#108	—	43	1	no	—
118	#100	—	86	1	no	—
119	#41	—	86	2	no	—
120	#41	—	86	1	no	—
121	#41	—	172	1	no	—
122	#41	D₁₇₂	1	1	yes	characteristic
123	C₂×D₄	—	86	2	no	—
124	C₂×D₄	—	86	1	no	—
125	C₂×D₄	—	86	1	no	—
126	C₂×D₄	—	172	2	no	—
127	C₂×D₄	—	172	1	no	—
128	Q₁₆	—	43	1	no	—
129	SD₁₆	—	86	2	no	—
130	SD₁₆	—	86	1	no	—
131	SD₁₆	—	172	1	no	—
132	D₈	—	43	2	no	—
133	D₈	—	43	1	no	—
134	D₈	—	86	2	no	—
135	D₈	—	86	1	no	—
136	D₈	—	86	1	no	—
137	D₈	—	86	1	no	—
138	D₈	—	172	1	no	—
139	M₁₆	—	2	1	no	—
140	M₁₆	—	86	1	no	—
141	M₁₆	D₁₇₂	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
144	C₁₆	—	2	1	no	—
145	C₁₆	—	2	1	no	—
146	C₂³	—	172	2	no	—
147	C₂³	—	172	1	no	—
148	C₂³	—	172	1	no	—
149	Q₈	—	86	1	no	—
150	Q₈	D₃₄₄	1	1	yes	characteristic
151	D₄	—	2	1	no	—
152	D₄	—	86	2	no	—
153	D₄	—	86	1	no	—
154	D₄	—	172	2	no	—
155	D₄	—	172	2	no	—
156	D₄	—	172	1	no	—
157	D₄	—	172	1	no	—
158	D₄	—	172	1	no	—
159	D₄	—	172	1	no	—
160	D₄	D₃₄₄	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	172	2	no	—
163	C₂×C₄	—	172	1	no	—
164	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	172	1	no	—
167	C₈	D₄×D₄₃	1	1	yes	characteristic
168	C₈	D₄×D₄₃	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	344	1	no	—
171	C₂²	—	172	2	no	—
172	C₂²	—	172	1	no	—
173	C₂²	—	688	1	no	—
174	C₂²	—	688	1	no	—
175	C₂²	—	688	2	no	—
176	C₂²	#13063353	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	172	1	no	—
179	C₄	#13063295	1	1	yes	characteristic
180	C₄	C₂×D₃₄₄	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	344	2	no	—
184	C₂	—	344	1	no	—
185	C₂	—	688	1	no	—
186	C₂	#14583035	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14583035
commutator	C₂×C₈×C₄₃	C₂³
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₄₃×#350	C₂
socle	C₂×C₄₃	#447
hypercenter	#350	D₄₃
hypocenter	C₄₃	#2192
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
1. #14583704	C₂	semidirect
2. #14583337	C₂	semidirect
3. C₂×D₆₈₈	C₂	semidirect
4. #14583169	C₂	semidirect
5. #14582713	C₂	semidirect
6. #14582707	C₂	semidirect
7. C₄₃×#350	C₂	semidirect
16. C₂×D₃₄₄	C₂²	semidirect
20. #13063261	C₂²	semidirect
23. M₃₂×C₄₃	C₂²	semidirect
24. C₂×C₁₆×C₄₃	C₂²	semidirect
100. #350	D₄₃	semidirect
101. C₄₃	#2192	semidirect
115. M₃₂	C₂×D₄₃	semidirect
116. C₂×C₁₆	C₂×D₄₃	semidirect