Subgroups — #25014708

Order 11008.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136384	C₂	1	2	yes	maximal
2	C₂×#14582720	C₂	1	1	yes	maximal, characteristic
3	#19134759	C₂	1	1	yes	maximal, characteristic
4	#19132705	C₂	1	2	yes	maximal
5	C₄₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	C₂²	1	2	yes	—
7	C₂×#13063259	C₂²	1	2	yes	—
8	C₂×#13063257	—	2	1	no	—
9	#14582765	—	2	1	no	—
10	#14582726	—	2	2	no	—
11	#14582724	—	2	2	no	—
12	#14582720	C₄	1	4	yes	complemented
13	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#427	C₂²	1	1	yes	characteristic
15	C₄₃×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#339	—	2	1	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	#13063265	—	2	2	no	—
20	#13063265	C₂×C₄	1	4	yes	—
21	C₂×C₄₃⋊C₁₆	—	2	2	no	—
22	#13063259	—	2	2	no	—
23	#13063259	—	2	2	no	—
24	#13063257	—	4	1	no	—
25	#13063257	—	4	1	no	—
26	C₄₃⋊#95	—	4	2	no	—
27	C₂×M₁₆×C₄₃	—	2	1	no	—
28	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	#117×C₄₃	—	2	1	no	—
31	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
32	#108×C₄₃	—	2	1	no	—
33	#108×C₄₃	Q₈	1	2	yes	—
34	#107×C₄₃	D₄	1	2	yes	—
35	#106×C₄₃	—	2	1	no	—
36	#97×C₄₃	—	4	1	no	—
37	C₄×C₈×C₄₃	—	4	1	no	—
38	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₃	Q₈	1	2	yes	—
40	C₂²×C₄×C₄₃	—	2	1	no	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₄₃	—	4	2	no	—
43	M₁₆×C₄₃	—	2	2	no	—
44	M₁₆×C₄₃	—	4	1	no	—
45	C₂×C₈×C₄₃	—	4	1	no	—
46	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
48	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
49	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
50	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₄₃	—	2	2	no	—
57	C₂²⋊C₄×C₄₃	—	4	1	no	—
58	C₄²×C₄₃	—	2	1	no	—
59	C₄²×C₄₃	—	4	1	no	—
60	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
61	C₄⋊Dic₄₃	—	4	2	no	—
62	C₄⋊Dic₄₃	—	4	2	no	—
63	C₄₃⋊C₁₆	—	2	4	no	—
64	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	4	2	no	—
67	C₂×C₄×C₄₃	—	4	2	no	—
68	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
74	C₂×Dic₄₃	—	8	4	no	—
75	C₂×Dic₄₃	—	8	2	no	—
76	C₈×C₄₃	—	4	2	no	—
77	C₈×C₄₃	#107	1	2	yes	—
78	C₈×C₄₃	#107	1	2	yes	—
79	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
80	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₄₃	#125	1	1	yes	characteristic
82	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
83	#13180	—	43	1	no	maximal, Sylow
84	C₂²×C₄₃	—	2	2	no	—
85	C₂²×C₄₃	#428	1	1	yes	characteristic
86	C₂²×C₄₃	#459	1	1	yes	characteristic
87	C₂²×C₄₃	#477	1	1	yes	characteristic
88	C₄×C₄₃	—	4	2	no	—
89	C₄×C₄₃	—	8	2	no	—
90	C₄×C₄₃	#389	1	1	yes	characteristic
91	C₄×C₄₃	C₂×#107	1	1	yes	characteristic
92	C₄×C₄₃	#462	1	1	yes	characteristic
93	C₄×C₄₃	#474	1	1	yes	characteristic
94	Dic₄₃	—	16	2	no	—
95	#2159	—	43	2	no	—
96	C₂×#364	—	43	1	no	—
97	#1846	—	43	1	no	—
98	#1772	D₄₃	1	1	yes	characteristic
99	#1387	—	43	2	no	—
100	C₂×C₄₃	—	2	1	no	—
101	C₂×C₄₃	—	2	1	no	—
102	C₂×C₄₃	#1947	1	1	yes	characteristic
103	C₂×C₄₃	#2212	1	2	yes	—
104	C₂×M₃₂	—	43	2	no	—
105	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
106	#427	C₂×D₄₃	1	1	yes	characteristic
107	C₂×#107	—	43	2	no	—
108	C₂×#106	—	86	1	no	—
109	#405	—	2	1	no	—
110	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
111	#382	—	86	1	no	—
112	#367	—	86	2	no	—
113	#366	—	86	2	no	—
114	#364	—	43	4	no	—
115	#339	—	2	1	no	—
116	C₄₃	#13180	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	2	1	no	—
118	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
120	#117	—	2	1	no	—
121	C₂×C₄⋊C₄	—	86	2	no	—
122	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
123	M₃₂	—	43	4	no	—
124	M₃₂	—	86	2	no	—
125	C₂×C₁₆	—	86	2	no	—
126	#108	—	2	1	no	—
127	#108	Dic₈₆	1	2	yes	—
128	#107	—	86	2	no	—
129	#107	—	86	2	no	—
130	#107	D₁₇₂	1	2	yes	—
131	#106	—	2	1	no	—
132	#106	—	172	1	no	—
133	#106	—	172	1	no	—
134	#97	—	4	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	Dic₈₆	1	2	yes	—
137	C₄×C₈	C₄×D₄₃	1	2	yes	—
138	#95	—	172	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	172	2	no	—
141	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
146	C₂×C₈	#10621882	1	1	yes	characteristic
147	C₂×C₈	#10621882	1	1	yes	characteristic
148	C₂×C₈	#10621885	1	2	yes	—
149	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
151	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
152	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
153	C₂×C₈	#10621898	1	1	yes	characteristic
154	C₂×C₈	#10621898	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	172	2	no	—
157	C₄⋊C₄	—	172	2	no	—
158	C₂²⋊C₄	—	4	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	Dic₄₃⋊C₄	1	2	yes	—
162	C₁₆	—	86	4	no	—
163	C₂³	#13063350	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	344	4	no	—
168	C₂×C₄	—	344	2	no	—
169	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
170	C₂×C₄	#13063299	1	1	yes	characteristic
171	C₂×C₄	#13063304	1	1	yes	characteristic
172	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
173	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
174	C₂×C₄	#13063355	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	#13063258	1	2	yes	—
177	C₈	#13063258	1	2	yes	—
178	C₈	C₄×D₁₇₂	1	1	yes	characteristic
179	C₈	C₄×Dic₈₆	1	1	yes	characteristic
180	C₈	#13063317	1	1	yes	characteristic
181	C₈	#13063325	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#14582942	1	1	yes	characteristic
184	C₂²	#14583094	1	1	yes	characteristic
185	C₂²	#14583194	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	688	2	no	—
189	C₄	#14582797	1	1	yes	characteristic
190	C₄	C₂×#13063258	1	1	yes	characteristic
191	C₄	#14583109	1	1	yes	characteristic
192	C₄	#14583179	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#19135353	1	1	yes	characteristic
196	C₂	#19136664	1	2	yes	—

	subgroup	G/H
center	C₂²	#14583094
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1772	C₂
socle	C₂²×C₄₃	#459
prime radical	C₂³×C₄₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1772	D₄₃
hypocenter	C₄₃	#13180
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
12. #14582720	C₄	semidirect
116. C₄₃	#13180	semidirect

Maximal subgroups

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

subgroup	index
#19136384	2
C₂×#14582720	2
#19134759	2
#19132705	2
C₄₃×#1772	2
#13180	43