Subgroups — #25013054

Order 11008.

266 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136379	C₂	1	1	yes	maximal, characteristic
2	#19136369	C₂	1	2	yes	maximal
3	#19136191	C₂	1	1	yes	maximal, characteristic
4	#19134720	C₂	1	2	yes	maximal
5	C₂×C₄₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	C₂²	1	1	yes	characteristic
7	#14582937	C₂²	1	2	yes	—
8	#14582854	C₂²	1	1	yes	characteristic
9	#14582727	—	2	1	no	—
10	#14582725	—	2	1	no	—
11	#14582723	—	2	4	no	—
12	#14582674	C₄	1	2	yes	complemented
13	C₁₆×Dic₄₃	C₄	1	2	yes	complemented
14	#14582671	—	2	2	no	—
15	#14582656	—	2	2	no	—
16	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#108×C₄₃	C₂²	1	2	yes	—
18	C₄₃×#368	—	2	1	no	—
19	C₄₃×#368	—	2	1	no	—
20	C₂×#10621877	D₄	1	1	yes	characteristic
21	#13063284	—	2	2	no	—
22	#13063284	D₄	1	1	yes	characteristic
23	#13063265	D₄	1	2	yes	—
24	#13063265	Q₈	1	2	yes	—
25	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
26	#13063259	—	2	2	no	—
27	#13063257	—	2	2	no	—
28	#13063235	Q₈	1	2	yes	—
29	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₄₃	—	2	2	no	—
31	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₄₃	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₄₃	Q₈	1	1	yes	characteristic
38	#108×C₄₃	—	2	2	no	—
39	#108×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₄₃	—	4	2	no	—
42	M₁₆×C₄₃	—	4	2	no	—
43	C₂×C₈×C₄₃	—	4	2	no	—
44	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²⋊Dic₄₃	—	2	1	no	—
51	C₂²⋊Dic₄₃	—	4	2	no	—
52	C₄⋊Dic₄₃	—	2	1	no	—
53	C₄⋊Dic₄₃	—	4	2	no	—
54	C₄×Dic₄₃	—	4	2	no	—
55	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
56	#10621877	—	2	1	no	—
57	#10621877	—	2	1	no	—
58	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₄₃	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₄₃	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₄₃	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆×C₄₃	C₄⋊C₄	1	1	yes	characteristic
63	C₄₃⋊C₁₆	C₄⋊C₄	1	4	yes	—
64	C₂³×C₄₃	#135	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
71	C₂×Dic₄₃	—	2	2	no	—
72	C₂×Dic₄₃	—	8	2	no	—
73	C₈×C₄₃	—	8	2	no	—
74	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₄₃⋊C₈	—	2	2	no	—
79	#12911	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	—	4	1	no	—
81	C₂²×C₄₃	#427	1	1	yes	characteristic
82	C₂²×C₄₃	#495	1	1	yes	characteristic
83	C₂²×C₄₃	#499	1	1	yes	characteristic
84	C₄×C₄₃	#384	1	1	yes	characteristic
85	C₄×C₄₃	#427	1	1	yes	characteristic
86	C₄×C₄₃	#495	1	1	yes	characteristic
87	C₄×C₄₃	#499	1	1	yes	characteristic
88	Dic₄₃	—	4	2	no	—
89	Dic₄₃	—	16	2	no	—
90	C₂×#368	D₄₃	1	1	yes	characteristic
91	#2158	—	43	1	no	—
92	#2155	—	43	2	no	—
93	#2108	—	43	1	no	—
94	#1833	—	43	2	no	—
95	C₂×C₄₃	—	2	1	no	—
96	C₂×C₄₃	—	2	1	no	—
97	C₂×C₄₃	—	2	1	no	—
98	C₂×C₄₃	#1846	1	1	yes	characteristic
99	C₂×M₃₂	—	43	1	no	—
100	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
101	C₂×#108	C₂×D₄₃	1	2	yes	—
102	#427	—	43	2	no	—
103	#405	—	43	1	no	—
104	#368	—	2	1	no	—
105	#368	—	2	1	no	—
106	#367	—	86	1	no	—
107	#366	—	86	1	no	—
108	#365	—	86	4	no	—
109	#346	—	43	2	no	—
110	C₄×C₁₆	—	43	2	no	—
111	#344	—	86	2	no	—
112	H(3,ℤ/4)	—	86	2	no	—
113	C₄₃	#12911	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	43	1	no	—
116	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
117	#117	—	43	1	no	—
118	#117	—	86	2	no	—
119	M₃₂	—	43	2	no	—
120	M₃₂	—	43	2	no	—
121	C₂×C₁₆	—	43	2	no	—
122	C₂×C₁₆	Dic₈₆	1	1	yes	characteristic
123	C₂×C₁₆	Dic₈₆	1	1	yes	characteristic
124	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
125	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
126	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
127	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#107	—	86	2	no	—
131	#106	—	86	2	no	—
132	#97	—	43	2	no	—
133	C₄×C₈	—	43	2	no	—
134	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	M₁₆	—	86	1	no	—
138	M₁₆	—	86	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	43	2	no	—
141	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
142	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
143	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
147	C₄⋊C₄	—	86	1	no	—
148	C₄⋊C₄	—	172	2	no	—
149	C₂²⋊C₄	—	86	1	no	—
150	C₂²⋊C₄	—	172	2	no	—
151	C₄²	—	43	2	no	—
152	C₄²	—	172	2	no	—
153	C₁₆	—	43	4	no	—
154	C₁₆	Dic₄₃⋊C₄	1	1	yes	characteristic
155	C₁₆	Dic₄₃⋊C₄	1	1	yes	characteristic
156	C₁₆	Dic₄₃⋊C₄	1	1	yes	characteristic
157	C₁₆	Dic₄₃⋊C₄	1	1	yes	characteristic
158	C₂³	#13063348	1	1	yes	characteristic
159	C₂×C₄	—	86	2	no	—
160	C₂×C₄	—	344	2	no	—
161	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
162	C₂×C₄	#13063321	1	1	yes	characteristic
163	C₂×C₄	#13063324	1	1	yes	characteristic
164	C₂×C₄	#13063344	1	1	yes	characteristic
165	C₂×C₄	#13063344	1	1	yes	characteristic
166	C₂×C₄	#13063348	1	1	yes	characteristic
167	C₈	—	8	2	no	—
168	C₈	—	86	2	no	—
169	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
170	C₈	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
171	C₈	#13063323	1	1	yes	characteristic
172	C₈	#13063324	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	#14582936	1	1	yes	characteristic
175	C₂²	#14583284	1	1	yes	characteristic
176	C₂²	#14583300	1	1	yes	characteristic
177	C₄	—	172	2	no	—
178	C₄	—	688	2	no	—
179	C₄	#14582768	1	1	yes	characteristic
180	C₄	#14582936	1	1	yes	characteristic
181	C₄	#14583282	1	1	yes	characteristic
182	C₄	#14583300	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#19134758	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14582936
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#368	C₂
socle	C₂×C₄₃	#1846
prime radical	C₂³×C₄₃	#135
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#368	D₄₃
hypocenter	C₄₃	#12911
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
12. #14582674	C₄	semidirect
13. C₁₆×Dic₄₃	C₄	semidirect
113. C₄₃	#12911	semidirect

Maximal subgroups

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

subgroup	index
#19136379	2
#19136369	2
#19136191	2
#19134720	2
C₂×C₄₃×#368	2
#12911	43