Finite Group Databasebeta

Subgroups — #19136046

Order 5504.

190 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14582917	C₂	1	1	yes	maximal, characteristic
2	#14582896	C₂	1	1	yes	maximal, characteristic
3	#14582881	C₂	1	1	yes	maximal, characteristic
4	#14582671	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582664	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582655	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621898	C₂²	1	1	yes	characteristic
9	C₂×#10621877	C₂²	1	1	yes	characteristic
10	C₂×#10621877	C₂²	1	1	yes	characteristic
11	#13063284	C₂²	1	1	yes	characteristic
12	#13063253	—	2	2	no	—
13	#13063250	—	2	1	no	—
14	#13063247	—	2	1	no	—
15	#13063243	—	2	1	no	—
16	#13063241	—	2	1	no	—
17	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	#117×C₄₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₄₃	—	2	2	no	—
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₄₃	—	2	1	no	—
23	#10621898	—	2	1	no	—
24	#10621898	—	2	1	no	—
25	#10621898	—	2	1	no	—
26	#10621898	—	2	1	no	—
27	C₂×D₁₇₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₄₃	—	2	1	no	—
29	C₂×Dic₈₆	D₄	1	1	yes	characteristic
30	M₁₆×C₄₃	—	2	1	no	—
31	M₁₆×C₄₃	—	2	1	no	—
32	C₂×C₈×C₄₃	—	2	1	no	—
33	C₂×C₈×C₄₃	—	2	1	no	—
34	C₂×C₈×C₄₃	—	2	1	no	—
35	C₂×C₈×C₄₃	—	2	1	no	—
36	C₄⋊C₄×C₄₃	—	2	1	no	—
37	C₂²⋊C₄×C₄₃	—	2	1	no	—
38	C₄²×C₄₃	D₄	1	2	yes	—
39	C₂²⋊Dic₄₃	—	2	1	no	—
40	C₄⋊Dic₄₃	—	2	1	no	—
41	C₄×Dic₄₃	—	2	1	no	—
42	#10621877	—	2	1	no	—
43	#10621877	—	2	1	no	—
44	#10621877	—	2	1	no	—
45	#10621877	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₂³×C₄₃	#41	1	1	yes	characteristic
49	C₂²×D₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
57	C₂²⋊D₄₃	—	4	1	no	—
58	C₂²⋊D₄₃	—	4	1	no	—
59	C₂×Dic₄₃	—	2	1	no	—
60	C₂×Dic₄₃	—	4	1	no	—
61	D₁₇₂	—	2	1	no	—
62	D₁₇₂	—	2	1	no	—
63	C₄×D₄₃	—	4	1	no	—
64	C₄×D₄₃	—	4	1	no	—
65	Dic₈₆	—	2	1	no	—
66	Dic₈₆	—	2	1	no	—
67	C₈×C₄₃	—	4	1	no	—
68	C₈×C₄₃	—	4	1	no	—
69	C₄₃⋊C₈	—	4	1	no	—
70	C₄₃⋊C₈	—	4	1	no	—
71	C₂²×C₄₃	—	4	1	no	—
72	C₂²×C₄₃	#123	1	1	yes	characteristic
73	C₂²×C₄₃	#123	1	1	yes	characteristic
74	C₂²×C₄₃	#126	1	1	yes	characteristic
75	C₂×D₄₃	—	4	1	no	—
76	C₂×D₄₃	—	8	1	no	—
77	C₄×C₄₃	—	4	2	no	—
78	C₄×C₄₃	#121	1	1	yes	characteristic
79	C₄×C₄₃	#123	1	1	yes	characteristic
80	C₄×C₄₃	#124	1	1	yes	characteristic
81	C₄×C₄₃	#126	1	1	yes	characteristic
82	Dic₄₃	—	4	1	no	—
83	Dic₄₃	—	8	1	no	—
84	#2080	—	43	1	no	maximal, Sylow
85	C₂×C₄₃	—	2	1	no	—
86	C₂×C₄₃	—	2	1	no	—
87	C₂×C₄₃	—	2	1	no	—
88	C₂×C₄₃	#397	1	1	yes	characteristic
89	D₄₃	—	8	1	no	—
90	#424	D₄₃	1	1	yes	characteristic, complemented
91	#421	—	43	1	no	—
92	#416	—	43	1	no	—
93	#413	—	43	1	no	—
94	#344	—	43	1	no	—
95	#341	—	43	1	no	—
96	H(3,ℤ/4)	—	43	1	no	—
97	C₄₃	#2080	1	1	yes	characteristic, complemented, Sylow
98	C₂×#41	—	43	1	no	—
99	C₂×M₁₆	—	43	1	no	—
100	C₂×M₁₆	—	43	1	no	—
101	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
103	#117	—	43	1	no	—
104	#117	C₂×D₄₃	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	#104	—	86	2	no	—
107	#103	—	86	1	no	—
108	#102	—	86	1	no	—
109	#101	—	86	1	no	—
110	#100	—	86	1	no	—
111	#41	—	86	1	no	—
112	#41	—	86	1	no	—
113	#41	—	86	1	no	—
114	#41	—	86	1	no	—
115	C₂×Q₈	—	43	1	no	—
116	C₂×D₄	—	43	1	no	—
117	C₂×D₄	—	86	1	no	—
118	C₂²×C₄	—	86	1	no	—
119	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
120	M₁₆	—	2	1	no	—
121	M₁₆	—	2	1	no	—
122	M₁₆	—	86	1	no	—
123	M₁₆	—	86	1	no	—
124	M₁₆	—	86	1	no	—
125	M₁₆	—	86	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	86	1	no	—
131	C₂×C₈	—	86	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	86	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	86	1	no	—
136	C₄²	—	86	1	no	—
137	C₄²	C₂²⋊D₄₃	1	2	yes	—
138	C₂³	—	86	1	no	—
139	C₂³	#10621898	1	1	yes	characteristic
140	Q₈	—	86	1	no	—
141	Q₈	—	86	1	no	—
142	D₄	—	86	1	no	—
143	D₄	—	86	1	no	—
144	D₄	—	172	1	no	—
145	D₄	—	172	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	86	1	no	—
148	C₂×C₄	—	172	1	no	—
149	C₂×C₄	—	172	1	no	—
150	C₂×C₄	—	172	1	no	—
151	C₂×C₄	#10621898	1	1	yes	characteristic
152	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
153	C₂×C₄	#10621900	1	1	yes	characteristic
154	C₂×C₄	#10621902	1	1	yes	characteristic
155	C₂×C₄	#10621902	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
157	C₈	—	4	1	no	—
158	C₈	—	4	1	no	—
159	C₈	—	172	1	no	—
160	C₈	—	172	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	344	1	no	—
163	C₂²	—	172	1	no	—
164	C₂²	#13063308	1	1	yes	characteristic
165	C₂²	#13063310	1	1	yes	characteristic
166	C₂²	#13063320	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	344	1	no	—
169	C₄	—	172	1	no	—
170	C₄	#13063296	1	1	yes	characteristic
171	C₄	#13063310	1	1	yes	characteristic
172	C₄	#13063314	1	1	yes	characteristic
173	C₄	#13063320	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	344	1	no	—
178	C₂	#14582825	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13063314
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#424	C₂
socle	C₂×C₄₃	#397
hypercenter	#424	D₄₃
hypocenter	C₄₃	#2080
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #14582671	C₂	semidirect
5. #14582664	C₂	semidirect
6. #14582655	C₂	semidirect
7. C₄₃×#424	C₂	semidirect
90. #424	D₄₃	semidirect
97. C₄₃	#2080	semidirect