Subgroups — #25014713

Order 11008.

310 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#19134755	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	—	2	1	no	—
8	C₂×#13063259	C₂²	1	2	yes	—
9	#14582769	—	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	C₂×C₄⋊Dic₄₃	—	4	2	no	—
20	#13063265	—	2	2	no	—
21	#13063265	C₂×C₄	1	4	yes	—
22	C₂×C₄₃⋊C₁₆	—	2	2	no	—
23	#13063259	—	2	2	no	—
24	#13063259	—	2	2	no	—
25	#13063259	—	4	1	no	—
26	#13063259	—	4	1	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	1	yes	characteristic
35	#107×C₄₃	D₄	1	1	yes	characteristic
36	#106×C₄₃	—	2	1	no	—
37	#97×C₄₃	—	4	1	no	—
38	C₄×C₈×C₄₃	—	4	1	no	—
39	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₄₃	Q₈	1	2	yes	—
41	C₂²×C₄×C₄₃	—	2	1	no	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₃	—	4	2	no	—
44	M₁₆×C₄₃	—	2	2	no	—
45	M₁₆×C₄₃	—	4	1	no	—
46	C₂×C₈×C₄₃	—	4	1	no	—
47	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
49	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
50	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
51	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₄₃	—	2	1	no	—
58	C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂²⋊C₄×C₄₃	—	4	1	no	—
60	C₄²×C₄₃	—	2	1	no	—
61	C₄²×C₄₃	—	4	1	no	—
62	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
63	C₄⋊Dic₄₃	—	4	2	no	—
64	C₄⋊Dic₄₃	—	4	2	no	—
65	C₄⋊Dic₄₃	—	8	4	no	—
66	C₄₃⋊C₁₆	—	2	4	no	—
67	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	4	1	no	—
72	C₂×C₄×C₄₃	—	4	1	no	—
73	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
79	C₂×Dic₄₃	—	8	4	no	—
80	C₂×Dic₄₃	—	8	2	no	—
81	C₈×C₄₃	—	4	2	no	—
82	C₈×C₄₃	#107	1	2	yes	—
83	C₈×C₄₃	#107	1	2	yes	—
84	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
85	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₄₃	#125	1	1	yes	characteristic
87	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
88	#13181	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	—	2	1	no	—
90	C₂²×C₄₃	—	2	1	no	—
91	C₂²×C₄₃	#428	1	1	yes	characteristic
92	C₂²×C₄₃	#459	1	1	yes	characteristic
93	C₂²×C₄₃	#477	1	1	yes	characteristic
94	C₄×C₄₃	—	2	4	no	—
95	C₄×C₄₃	—	8	1	no	—
96	C₄×C₄₃	—	8	1	no	—
97	C₄×C₄₃	#389	1	1	yes	characteristic
98	C₄×C₄₃	C₂×#107	1	1	yes	characteristic
99	C₄×C₄₃	#462	1	1	yes	characteristic
100	C₄×C₄₃	#474	1	1	yes	characteristic
101	Dic₄₃	—	16	2	no	—
102	#2159	—	43	2	no	—
103	C₂×#364	—	43	1	no	—
104	#1845	—	43	1	no	—
105	#1772	D₄₃	1	1	yes	characteristic
106	#1387	—	43	2	no	—
107	C₂×C₄₃	—	2	1	no	—
108	C₂×C₄₃	—	2	1	no	—
109	C₂×C₄₃	#1947	1	1	yes	characteristic
110	C₂×C₄₃	#2213	1	1	yes	characteristic
111	C₂×C₄₃	#2214	1	1	yes	characteristic
112	C₂×M₃₂	—	43	2	no	—
113	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
114	#427	C₂×D₄₃	1	1	yes	characteristic
115	C₂×#107	—	43	2	no	—
116	C₂×#107	—	86	1	no	—
117	#405	—	2	1	no	—
118	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
119	#384	—	86	1	no	—
120	#367	—	86	2	no	—
121	#366	—	86	2	no	—
122	#364	—	43	4	no	—
123	#339	—	2	1	no	—
124	C₄₃	#13181	1	1	yes	characteristic, complemented, Sylow
125	C₂×M₁₆	—	2	1	no	—
126	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄⋊C₄	—	86	2	no	—
130	C₂×C₄⋊C₄	—	172	2	no	—
131	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
132	M₃₂	—	43	4	no	—
133	M₃₂	—	86	2	no	—
134	C₂×C₁₆	—	86	2	no	—
135	#108	—	2	1	no	—
136	#108	Dic₈₆	1	2	yes	—
137	#107	—	86	2	no	—
138	#107	—	86	2	no	—
139	#107	—	172	1	no	—
140	#107	—	172	1	no	—
141	#107	D₁₇₂	1	1	yes	characteristic
142	#107	D₁₇₂	1	1	yes	characteristic
143	#106	—	2	1	no	—
144	#97	—	4	1	no	—
145	C₄×C₈	—	4	1	no	—
146	C₄×C₈	Dic₈₆	1	2	yes	—
147	C₄×C₈	C₄×D₄₃	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	172	2	no	—
150	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
155	C₂×C₈	#10621882	1	1	yes	characteristic
156	C₂×C₈	#10621882	1	1	yes	characteristic
157	C₂×C₈	#10621885	1	2	yes	—
158	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
161	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
162	C₂×C₈	#10621898	1	1	yes	characteristic
163	C₂×C₈	#10621898	1	1	yes	characteristic
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	344	4	no	—
167	C₄⋊C₄	—	172	2	no	—
168	C₄⋊C₄	—	172	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	Dic₄₃⋊C₄	1	2	yes	—
173	C₁₆	—	86	4	no	—
174	C₂³	#13063350	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	344	4	no	—
181	C₂×C₄	—	344	2	no	—
182	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
183	C₂×C₄	#13063299	1	1	yes	characteristic
184	C₂×C₄	#13063304	1	1	yes	characteristic
185	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
186	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
187	C₂×C₄	#13063355	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	#13063258	1	2	yes	—
190	C₈	#13063258	1	2	yes	—
191	C₈	C₄×D₁₇₂	1	1	yes	characteristic
192	C₈	C₄×Dic₈₆	1	1	yes	characteristic
193	C₈	#13063317	1	1	yes	characteristic
194	C₈	#13063325	1	1	yes	characteristic
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	#14582942	1	1	yes	characteristic
198	C₂²	#14583094	1	1	yes	characteristic
199	C₂²	#14583194	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	8	1	no	—
202	C₄	—	8	1	no	—
203	C₄	—	688	2	no	—
204	C₄	#14582797	1	1	yes	characteristic
205	C₄	C₂×#13063258	1	1	yes	characteristic
206	C₄	#14583109	1	1	yes	characteristic
207	C₄	#14583179	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#19135353	1	1	yes	characteristic
211	C₂	#19136671	1	1	yes	characteristic
212	C₂	#19136676	1	1	yes	characteristic

	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	#13181
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
124. C₄₃	#13181	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
#19134755	2
#19132705	2
C₄₃×#1772	2
#13181	43