Subgroups — #25008238

Order 11008.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#19133354	C₂	1	1	yes	maximal, characteristic
3	#19132666	C₂	1	4	yes	maximal, complemented
4	C₄₃×#1567	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₁₆	C₄	1	2	yes	—
6	#14582923	—	2	1	no	—
7	C₄×C₄₃⋊C₁₆	C₄	1	1	yes	characteristic
8	C₄×C₄₃⋊C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
10	#14582652	—	2	1	no	—
11	#14582652	C₂²	1	2	yes	—
12	C₄₃×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#335	—	2	1	no	—
15	C₄₃×#335	C₂²	1	2	yes	—
16	C₂×#10621877	—	4	1	no	—
17	C₂×C₄₃⋊C₁₆	—	2	4	no	—
18	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
22	C₂×M₁₆×C₄₃	—	4	1	no	—
23	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₄₃	—	2	2	no	—
27	C₄⋊C₈×C₄₃	—	4	1	no	—
28	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
35	M₁₆×C₄₃	—	8	2	no	—
36	C₂×C₈×C₄₃	—	2	2	no	—
37	C₂×C₈×C₄₃	—	2	2	no	—
38	C₂×C₈×C₄₃	—	4	2	no	—
39	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₄₃	D₈	1	1	yes	characteristic
45	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
46	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
47	#10621877	—	8	2	no	—
48	C₂×C₄₃⋊C₈	—	4	2	no	—
49	C₄₃⋊C₁₆	—	2	2	no	—
50	C₄₃⋊C₁₆	M₁₆	1	4	yes	complemented
51	C₂³×C₄₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	2	1	no	—
55	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
57	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
58	C₂×C₄×C₄₃	D₁₆	1	2	yes	—
59	C₂×C₄×C₄₃	Q₃₂	1	2	yes	—
60	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₈×C₄₃	—	2	1	no	—
64	C₈×C₄₃	—	2	1	no	—
65	C₈×C₄₃	—	8	2	no	—
66	C₈×C₄₃	#108	1	2	yes	—
67	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
68	C₄₃⋊C₈	—	8	2	no	—
69	#12022	—	43	1	no	maximal, Sylow
70	C₂²×C₄₃	—	2	1	no	—
71	C₂²×C₄₃	—	2	1	no	—
72	C₂²×C₄₃	#366	1	2	yes	—
73	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	#366	1	2	yes	—
77	C₄×C₄₃	#368	1	2	yes	—
78	C₄×C₄₃	#423	1	1	yes	characteristic
79	C₄×C₄₃	C₂×#108	1	1	yes	characteristic
80	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
81	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
82	C₂×C₄×C₁₆	—	43	1	no	—
83	#1567	—	43	1	no	—
84	#1567	D₄₃	1	1	yes	characteristic
85	#1371	—	43	4	no	—
86	C₂×C₄₃	—	2	2	no	—
87	C₂×C₄₃	#1567	1	1	yes	characteristic
88	C₂×C₄₃	C₂×#366	1	1	yes	characteristic
89	C₂×C₄₃	C₂×#368	1	1	yes	characteristic
90	C₂²×C₁₆	—	43	2	no	—
91	#423	—	2	1	no	—
92	#423	—	86	1	no	—
93	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
94	C₄×C₁₆	—	43	2	no	—
95	C₄×C₁₆	—	43	1	no	—
96	C₄×C₁₆	—	43	1	no	—
97	#335	—	2	1	no	—
98	#335	—	43	2	no	—
99	#335	—	86	1	no	—
100	#335	C₂×D₄₃	1	2	yes	—
101	C₄₃	#12022	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	172	1	no	—
104	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
105	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
106	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
107	C₂×C₁₆	—	43	2	no	—
108	C₂×C₁₆	—	43	2	no	—
109	C₂×C₁₆	—	86	4	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	86	2	no	—
113	C₄⋊C₈	—	172	1	no	—
114	C₄×C₈	Dic₈₆	1	1	yes	characteristic
115	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
118	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
119	C₂²×C₄	#10621882	1	1	yes	characteristic
120	C₂²×C₄	#10621885	1	1	yes	characteristic
121	M₁₆	—	8	2	no	—
122	M₁₆	—	344	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	172	2	no	—
127	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
131	C₄²	Dic₄₃⋊C₄	1	1	yes	characteristic
132	C₄²	#10621882	1	1	yes	characteristic
133	C₄²	#10621885	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
135	C₁₆	—	43	4	no	—
136	C₁₆	—	86	2	no	—
137	C₂³	#13063258	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	#13063258	1	1	yes	characteristic
142	C₂×C₄	#13063258	1	1	yes	characteristic
143	C₂×C₄	#13063258	1	1	yes	characteristic
144	C₂×C₄	#13063266	1	2	yes	—
145	C₂×C₄	#13063271	1	2	yes	—
146	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
148	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	344	2	no	—
153	C₈	#13063260	1	2	yes	—
154	C₈	M₁₆×D₄₃	1	2	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#14582724	1	2	yes	—
158	C₂²	C₂×#13063258	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#14582724	1	2	yes	—
162	C₄	#14582728	1	2	yes	—
163	C₄	#14582922	1	1	yes	characteristic
164	C₄	C₂×#13063260	1	1	yes	characteristic
165	C₄	C₂×#13063266	1	1	yes	characteristic
166	C₄	C₂×#13063271	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	#19133353	1	1	yes	characteristic
169	C₂	C₂×#14582724	1	1	yes	characteristic
170	C₂	C₂×#14582728	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1567	C₂
socle	C₂²×C₄₃	C₂×#107
prime radical	C₂³×C₄₃	#107
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1567	D₄₃
hypocenter	C₄₃	#12022
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #19132666	C₂	semidirect
50. C₄₃⋊C₁₆	M₁₆	semidirect
101. C₄₃	#12022	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₃⋊C₁₆	2
#19133354	2
#19132666	2
C₄₃×#1567	2
#12022	43