Finite Group Databasebeta

Subgroups — #19136595

Order 5504.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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	#14583641	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583086	C₂	1	2	yes	maximal, complemented
3	#14582694	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582687	C₂	1	2	yes	maximal, complemented
5	C₄₃×#455	C₂	1	1	yes	maximal, characteristic, complemented
6	#13063314	—	2	1	no	—
7	#13063313	—	2	1	no	—
8	#13063311	—	2	2	no	—
9	#13063295	—	2	2	no	—
10	#13063293	—	2	2	no	—
11	#13063293	C₂²	1	2	yes	—
12	#13063240	—	2	2	no	—
13	#13063240	C₂²	1	2	yes	—
14	H(5,𝔽₂)×C₄₃	C₂²	1	1	yes	characteristic
15	#137×C₄₃	—	2	1	no	—
16	#124×C₄₃	C₂²	1	1	yes	characteristic
17	#104×C₄₃	—	2	1	no	—
18	#100×C₄₃	C₂²	1	1	yes	characteristic
19	C₂³×D₄₃	D₄	1	2	yes	complemented
20	#41×C₄₃	—	2	1	no	—
21	#41×C₄₃	—	4	1	no	—
22	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic, complemented
23	C₂×D₄×C₄₃	—	2	2	no	—
24	C₂×D₄×C₄₃	—	4	1	no	—
25	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₄₃	—	2	2	no	—
27	C₂×C₂²⋊D₄₃	—	4	2	no	—
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×Dic₈₆	—	2	1	no	—
31	Q₁₆×C₄₃	—	4	1	no	—
32	SD₁₆×C₄₃	—	4	1	no	—
33	M₁₆×C₄₃	—	2	1	no	—
34	C₂²⋊C₄×C₄₃	—	2	2	no	—
35	C₄²×C₄₃	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊Dic₄₃	—	2	2	no	—
37	C₂²⋊Dic₄₃	D₄	1	2	yes	complemented
38	(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	(C₂×D₄₃)⋊C₄	—	2	2	no	—
40	(C₂×D₄₃)⋊C₄	—	2	2	no	—
41	(C₂×D₄₃)⋊C₄	—	4	2	no	—
42	C₄×Dic₄₃	—	2	1	no	—
43	C₂³×C₄₃	—	2	2	no	—
44	C₂³×C₄₃	C₂×D₄	1	2	yes	—
45	C₂²×D₄₃	—	2	2	no	—
46	C₂²×D₄₃	—	4	2	no	—
47	C₂²×D₄₃	—	8	2	no	—
48	Q₈×C₄₃	—	2	1	no	—
49	Q₈×C₄₃	—	4	1	no	—
50	D₄×C₄₃	—	2	1	no	—
51	D₄×C₄₃	—	4	1	no	—
52	D₄×C₄₃	—	4	2	no	—
53	D₄×C₄₃	—	4	1	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	1	no	—
57	C₂×C₄×C₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₄₃	—	8	2	no	—
60	C₂²⋊D₄₃	—	8	4	no	—
61	C₂×Dic₄₃	—	2	2	no	—
62	C₂×Dic₄₃	—	4	2	no	—
63	D₁₇₂	—	8	1	no	—
64	D₁₇₂	—	8	1	no	—
65	Dic₈₆	—	8	1	no	—
66	C₈×C₄₃	—	4	1	no	—
67	C₂²×C₄₃	—	2	2	no	—
68	C₂²×C₄₃	—	4	2	no	—
69	C₂²×C₄₃	—	4	1	no	—
70	C₂²×C₄₃	—	8	2	no	—
71	C₂²×C₄₃	#120	1	1	yes	characteristic
72	C₂×D₄₃	—	4	2	no	—
73	C₂×D₄₃	—	4	2	no	—
74	C₂×D₄₃	—	4	4	no	—
75	C₂×D₄₃	—	4	2	no	—
76	C₂×D₄₃	—	8	2	no	—
77	C₄×C₄₃	—	2	1	no	—
78	C₄×C₄₃	—	4	1	no	—
79	C₄×C₄₃	—	4	1	no	—
80	C₄×C₄₃	—	4	1	no	—
81	Dic₄₃	—	4	2	no	—
82	Dic₄₃	—	8	2	no	—
83	#2199	—	43	1	no	maximal, Sylow
84	C₂×C₄₃	—	2	1	no	—
85	C₂×C₄₃	—	4	2	no	—
86	C₂×C₄₃	—	8	1	no	—
87	C₂×C₄₃	B(4,𝔽₂)	1	1	yes	characteristic
88	D₄₃	—	8	2	no	—
89	H(3,𝔽₄)	—	43	1	no	—
90	B(4,𝔽₂)	—	43	2	no	—
91	#455	D₄₃	1	1	yes	characteristic, complemented
92	#354	—	43	1	no	—
93	AΣL(1,𝔽₁₆)	—	43	2	no	—
94	C₄₃	#2199	1	1	yes	characteristic, complemented, Sylow
95	H(5,𝔽₂)	C₂×D₄₃	1	1	yes	characteristic
96	#137	—	2	1	no	—
97	#124	—	86	2	no	—
98	#124	—	86	1	no	—
99	#124	—	86	1	no	—
100	#124	C₂×D₄₃	1	1	yes	characteristic
101	#120	—	43	2	no	—
102	#120	—	86	2	no	—
103	#120	—	86	2	no	—
104	#104	—	2	1	no	—
105	#100	C₂×D₄₃	1	1	yes	characteristic
106	#99	—	43	2	no	—
107	#99	—	86	2	no	—
108	C₂⁴	—	43	2	no	—
109	#41	—	2	1	no	—
110	#41	—	4	1	no	—
111	C₂×Q₈	—	86	1	no	—
112	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic, complemented
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	4	1	no	—
115	C₂×D₄	—	86	1	no	—
116	C₂×D₄	—	86	1	no	—
117	C₂×D₄	—	86	2	no	—
118	C₂×D₄	—	172	2	no	—
119	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
120	Q₁₆	—	4	1	no	—
121	SD₁₆	—	4	1	no	—
122	M₁₆	—	2	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	43	2	no	—
125	C₂²⋊C₄	—	86	2	no	—
126	C₂²⋊C₄	—	86	2	no	—
127	C₂²⋊C₄	—	86	2	no	—
128	C₂²⋊C₄	—	86	2	no	—
129	C₂²⋊C₄	—	172	2	no	—
130	C₄²	—	86	1	no	—
131	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic, complemented
132	C₂³	—	2	2	no	—
133	C₂³	—	86	2	no	—
134	C₂³	—	344	2	no	—
135	C₂³	—	172	2	no	—
136	C₂³	D₄×D₄₃	1	2	yes	—
137	Q₈	—	2	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	344	1	no	—
140	D₄	—	2	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	344	1	no	—
145	D₄	—	344	1	no	—
146	D₄	—	344	4	no	—
147	D₄	—	344	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	86	2	no	—
153	C₂×C₄	—	172	2	no	—
154	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	344	2	no	—
161	C₂²	—	172	2	no	—
162	C₂²	—	172	2	no	—
163	C₂²	—	172	4	no	—
164	C₂²	—	172	2	no	—
165	C₂²	#13063293	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	344	2	no	—
171	C₄	—	172	2	no	—
172	C₂	—	2	1	no	—
173	C₂	—	4	2	no	—
174	C₂	—	8	1	no	—
175	C₂	—	344	2	no	—
176	C₂	#14583085	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14583085
commutator	C₂×D₄×C₄₃	C₂³
Frattini	C₂×D₄	C₂²×D₄₃
Fitting	C₄₃×#455	C₂
socle	C₂×C₄₃	B(4,𝔽₂)
hypercenter	#455	D₄₃
hypocenter	C₄₃	#2199
LCS	G ▷ C₂×D₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #455

Complements

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

N	H ≅ G/N	type
1. #14583641	C₂	semidirect
2. #14583086	C₂	semidirect
3. #14582694	C₂	semidirect
4. #14582687	C₂	semidirect
5. C₄₃×#455	C₂	semidirect
19. C₂³×D₄₃	D₄	semidirect
22. C₂×Q₈×C₄₃	D₄	semidirect
35. C₄²×C₄₃	D₄	semidirect
37. C₂²⋊Dic₄₃	D₄	semidirect
91. #455	D₄₃	semidirect
94. C₄₃	#2199	semidirect
112. C₂×Q₈	C₂²⋊D₄₃	semidirect
131. C₄²	C₂²⋊D₄₃	semidirect