Subgroups — #25008199

Order 11008.

250 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136398	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19132664	C₂	1	2	yes	maximal, complemented
4	#19132658	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
9	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
10	#14582675	—	2	1	no	—
11	C₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
12	C₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₄₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₄₃×#346	—	2	1	no	—
19	C₂²×C₄₃⋊C₈	—	2	1	no	—
20	C₂²×C₄₃⋊C₈	C₈	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₈	C₈	1	1	yes	characteristic
22	C₂×C₄₃⋊C₁₆	—	2	2	no	—
23	C₄×C₄₃⋊C₈	—	2	1	no	—
24	C₄×C₄₃⋊C₈	C₈	1	1	yes	characteristic
25	C₄×C₄₃⋊C₈	C₈	1	1	yes	characteristic
26	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₄₃	—	2	2	no	—
34	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₄₃	—	2	2	no	—
42	C₂×C₈×C₄₃	—	2	2	no	—
43	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₄₃	M₁₆	1	1	yes	characteristic
50	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄₃⋊C₈	—	2	4	no	—
52	C₂×C₄₃⋊C₈	—	2	1	no	—
53	C₂×C₄₃⋊C₈	—	2	1	no	—
54	C₂×C₄₃⋊C₈	—	2	2	no	—
55	C₂×C₄₃⋊C₈	—	2	2	no	—
56	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
60	C₁₆×C₄₃	—	4	2	no	—
61	C₄₃⋊C₁₆	—	4	2	no	—
62	C₂³×C₄₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	1	no	—
65	C₂×C₄×C₄₃	—	2	1	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₄₃	—	2	1	no	—
75	C₈×C₄₃	—	2	1	no	—
76	C₈×C₄₃	#108	1	2	yes	—
77	C₈×C₄₃	#135	1	2	yes	—
78	C₄₃⋊C₈	—	2	2	no	—
79	C₄₃⋊C₈	—	2	2	no	—
80	C₄₃⋊C₈	—	2	1	no	—
81	C₄₃⋊C₈	—	2	1	no	—
82	C₄₃⋊C₈	M₃₂	1	2	yes	—
83	C₄₃⋊C₈	M₃₂	1	2	yes	—
84	#12013	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	—	2	1	no	—
86	C₂²×C₄₃	—	2	1	no	—
87	C₂²×C₄₃	#364	1	2	yes	—
88	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₄₃	—	2	1	no	—
90	C₄×C₄₃	—	2	1	no	—
91	C₄×C₄₃	—	2	1	no	—
92	C₄×C₄₃	—	2	1	no	—
93	C₄×C₄₃	#427	1	1	yes	characteristic
94	C₄×C₄₃	C₂×#108	1	1	yes	characteristic
95	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
96	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
97	#2163	—	43	1	no	—
98	#2163	D₄₃	1	1	yes	characteristic
99	C₂×C₈²	—	43	1	no	—
100	#1370	—	43	2	no	—
101	#1367	—	43	2	no	—
102	C₂×C₄₃	—	2	2	no	—
103	C₂×C₄₃	#1564	1	1	yes	characteristic
104	C₂×C₄₃	#2150	1	1	yes	characteristic
105	C₂×C₄₃	C₂×#364	1	1	yes	characteristic
106	C₂×C₄×C₈	—	43	1	no	—
107	C₂×C₄×C₈	—	43	1	no	—
108	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	43	2	no	—
110	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
111	C₂²⋊C₁₆	—	2	1	no	—
112	C₂²⋊C₁₆	—	86	1	no	—
113	#346	—	2	1	no	—
114	#346	—	86	1	no	—
115	C₈²	—	43	1	no	—
116	C₈²	—	43	1	no	—
117	C₈²	—	43	1	no	—
118	C₈²	—	43	1	no	—
119	C₄₃	#12013	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	43	1	no	—
121	C₂²×C₈	—	43	1	no	—
122	C₂²×C₈	—	86	1	no	—
123	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	86	2	no	—
128	C₄×C₈	—	43	1	no	—
129	C₄×C₈	—	43	1	no	—
130	C₄×C₈	—	43	1	no	—
131	C₄×C₈	—	43	1	no	—
132	C₄×C₈	—	43	1	no	—
133	C₄×C₈	—	43	1	no	—
134	C₄×C₈	—	86	1	no	—
135	C₄×C₈	Dic₈₆	1	1	yes	characteristic
136	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
139	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
140	C₂²×C₄	#10621872	1	1	yes	characteristic
141	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	43	1	no	—
145	C₂×C₈	—	43	1	no	—
146	C₂×C₈	—	43	1	no	—
147	C₂×C₈	—	43	1	no	—
148	C₂×C₈	—	86	2	no	—
149	C₂×C₈	—	86	2	no	—
150	C₂×C₈	—	86	4	no	—
151	C₂×C₈	—	86	1	no	—
152	C₂×C₈	—	86	1	no	—
153	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
154	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
157	C₄²	C₈×D₄₃	1	1	yes	characteristic
158	C₄²	#10621872	1	1	yes	characteristic
159	C₄²	Dic₄₃⋊C₄	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	172	2	no	—
163	C₂³	Dic₄₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	Dic₄₃⋊C₈	1	1	yes	characteristic
170	C₂×C₄	Dic₄₃⋊C₈	1	1	yes	characteristic
171	C₂×C₄	Dic₄₃⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
174	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	43	2	no	—
178	C₈	—	43	2	no	—
179	C₈	—	86	2	no	—
180	C₈	—	86	2	no	—
181	C₈	—	86	1	no	—
182	C₈	—	86	1	no	—
183	C₈	#13063260	1	2	yes	—
184	C₈	#13063345	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#14582719	1	2	yes	—
188	C₂²	C₂×Dic₄₃⋊C₈	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	#14582935	1	1	yes	characteristic
194	C₄	C₂×#13063260	1	1	yes	characteristic
195	C₄	M₃₂×D₄₃	1	1	yes	characteristic
196	C₄	M₃₂×D₄₃	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#19133341	1	1	yes	characteristic
199	C₂	#19136350	1	1	yes	characteristic
200	C₂	C₂×#14582719	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#2163	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₈
prime radical	C₂³×C₄₃	C₄⋊C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#2163	D₄₃
hypocenter	C₄₃	#12013
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #19132664	C₂	semidirect
4. #19132658	C₂	semidirect
119. C₄₃	#12013	semidirect

Maximal subgroups

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

subgroup	index
#19136398	2
C₂×C₈×C₄₃⋊C₈	2
#19132664	2
#19132658	2
C₄₃×#2163	2
#12013	43