Subgroups — #25250268

Order 11008.

570 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	Q₈×C₄₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×C₄₃⋊C₁₆	C₂	1	6	yes	maximal, complemented
3	#19136394	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	6	yes	—
7	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	3	yes	—
8	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	6	yes	—
9	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	6	yes	—
11	C₄×C₄₃⋊C₁₆	C₂²	1	6	yes	—
12	C₄×#41×C₄₃	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₄₃	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₄₃	C₄	1	3	yes	—
16	C₈×D₄×C₄₃	C₂²	1	3	yes	—
17	C₄₃×#431	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
19	C₂×C₄₃⋊C₁₆	—	4	6	no	—
20	C₂×C₄₃⋊C₁₆	C₂³	1	6	yes	—
21	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
22	C₂×#41×C₄₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₄₃	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
25	C₄×Q₈×C₄₃	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₄₃	C₈	1	3	yes	—
28	C₄×D₄×C₄₃	C₂×C₄	1	3	yes	—
29	#117×C₄₃	C₈	1	3	yes	—
30	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₄₃	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₄₃	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₄₃	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₄₃	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₄₃	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₄₃	C₂³	1	3	yes	—
37	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
38	#41×C₄₃	C₁₆	1	8	yes	complemented
39	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄×C₄₃	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₄₃	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₄₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₄₃	—	2	3	no	—
45	C₂×C₈×C₄₃	—	2	3	no	—
46	C₂×C₈×C₄₃	—	4	3	no	—
47	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₄₃	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₄₃	C₂×C₈	1	3	yes	—
52	C₄⋊C₄×C₄₃	C₂×C₈	1	3	yes	—
53	C₂²⋊C₄×C₄₃	C₂×C₈	1	6	yes	—
54	C₄²×C₄₃	C₂×C₈	1	3	yes	—
55	C₄²×C₄₃	C₂×C₈	1	3	yes	—
56	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₄₃	C₂²×C₄	1	3	yes	—
58	C₄₃⋊C₁₆	—	2	6	no	—
59	C₄₃⋊C₁₆	—	2	2	no	—
60	C₂³×C₄₃	C₂²×C₈	1	3	yes	—
61	Q₈×C₄₃	C₂×C₁₆	1	4	yes	complemented
62	D₄×C₄₃	C₂×C₁₆	1	12	yes	complemented
63	C₂×C₄×C₄₃	—	2	3	no	—
64	C₂×C₄×C₄₃	—	2	3	no	—
65	C₂×C₄×C₄₃	—	4	3	no	—
66	C₂×C₄×C₄₃	C₂×C₁₆	1	12	yes	complemented
67	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
73	C₈×C₄₃	—	2	3	no	—
74	C₈×C₄₃	—	2	1	no	—
75	C₈×C₄₃	—	2	3	no	—
76	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
77	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
78	#33502	—	43	1	no	maximal, Sylow
79	C₂²×C₄₃	—	2	3	no	—
80	C₂²×C₄₃	—	2	3	no	—
81	C₂²×C₄₃	C₂²×C₁₆	1	6	yes	—
82	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	—	2	3	no	—
85	C₄×C₄₃	—	2	3	no	—
86	C₄×C₄₃	C₂²×C₁₆	1	2	yes	—
87	C₄×C₄₃	C₂²×C₁₆	1	6	yes	—
88	C₄×C₄₃	#520	1	1	yes	characteristic
89	C₄×C₄₃	#568	1	1	yes	characteristic
90	C₈×#41	D₄₃	1	1	yes	characteristic
91	Q₈×C₁₆	—	43	2	no	—
92	D₄×C₁₆	—	43	6	no	—
93	#2162	—	43	3	no	—
94	C₂×C₄⋊C₁₆	—	43	3	no	—
95	C₂×C₄₃	—	2	6	no	—
96	C₂×C₄₃	#2988	1	1	yes	characteristic
97	C₂×C₄₃	C₂³×C₁₆	1	1	yes	characteristic
98	C₂×C₄₃	#3407	1	1	yes	characteristic
99	C₄×#41	Dic₄₃	1	1	yes	characteristic
100	C₂²×C₁₆	—	43	6	no	—
101	C₈×Q₈	Dic₄₃	1	1	yes	characteristic
102	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
103	C₈×D₄	Dic₄₃	1	3	yes	—
104	C₈×D₄	C₂×D₄₃	1	3	yes	—
105	#431	Dic₄₃	1	3	yes	—
106	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
107	C₄⋊C₁₆	—	43	3	no	—
108	C₄⋊C₁₆	—	43	6	no	—
109	C₄⋊C₁₆	—	43	1	no	—
110	C₂²⋊C₁₆	—	43	6	no	—
111	C₄×C₁₆	—	43	6	no	—
112	C₄₃	#33502	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	C₄₃⋊C₈	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₄₃	1	3	yes	—
115	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
116	C₄×Q₈	C₄₃⋊C₈	1	1	yes	characteristic
117	C₄×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
118	C₄×D₄	C₄₃⋊C₈	1	3	yes	—
119	C₄×D₄	C₂×Dic₄₃	1	3	yes	—
120	#117	C₄₃⋊C₈	1	3	yes	—
121	C₂×C₄²	C₂×Dic₄₃	1	3	yes	—
122	C₂×C₁₆	—	43	2	no	—
123	C₂×C₁₆	—	43	6	no	—
124	C₂×C₁₆	—	172	6	no	—
125	C₄⋊C₈	C₂×Dic₄₃	1	3	yes	—
126	C₄⋊C₈	C₂×Dic₄₃	1	3	yes	—
127	C₂²⋊C₈	C₂×Dic₄₃	1	6	yes	—
128	C₄×C₈	C₂×Dic₄₃	1	3	yes	—
129	C₄×C₈	C₂×Dic₄₃	1	3	yes	—
130	C₄×C₈	C₂²×D₄₃	1	3	yes	—
131	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
132	#41	C₄₃⋊C₁₆	1	8	yes	complemented
133	C₂×Q₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
134	C₂×D₄	C₂×C₄₃⋊C₈	1	3	yes	—
135	C₂²×C₄	C₂×C₄₃⋊C₈	1	3	yes	—
136	C₂²×C₄	C₂×C₄₃⋊C₈	1	3	yes	—
137	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
138	C₂×C₈	—	2	3	no	—
139	C₂×C₈	—	2	3	no	—
140	C₂×C₈	—	4	3	no	—
141	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₄₃	1	3	yes	—
143	C₂×C₈	C₂²×Dic₄₃	1	3	yes	—
144	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
145	C₄⋊C₄	C₂×C₄₃⋊C₈	1	3	yes	—
146	C₄⋊C₄	C₂×C₄₃⋊C₈	1	3	yes	—
147	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	6	yes	—
148	C₄²	C₂×C₄₃⋊C₈	1	3	yes	—
149	C₄²	C₂×C₄₃⋊C₈	1	3	yes	—
150	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
151	C₄²	C₂²×Dic₄₃	1	3	yes	—
152	C₁₆	—	86	6	no	—
153	C₁₆	—	86	2	no	—
154	C₂³	C₂²×C₄₃⋊C₈	1	3	yes	—
155	Q₈	C₂×C₄₃⋊C₁₆	1	4	yes	complemented
156	D₄	C₂×C₄₃⋊C₁₆	1	12	yes	complemented
157	C₂×C₄	—	2	3	no	—
158	C₂×C₄	—	2	3	no	—
159	C₂×C₄	—	4	3	no	—
160	C₂×C₄	C₂×C₄₃⋊C₁₆	1	12	yes	complemented
161	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₄₃⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₄₃⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₄₃⋊C₈	1	3	yes	—
166	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
167	C₈	—	2	3	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	3	no	—
170	C₈	#13063371	1	1	yes	characteristic
171	C₈	#13063372	1	1	yes	characteristic
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	C₂²×C₄₃⋊C₁₆	1	6	yes	—
175	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
176	C₄	—	2	3	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂²×C₄₃⋊C₁₆	1	2	yes	—
180	C₄	C₂²×C₄₃⋊C₁₆	1	6	yes	—
181	C₄	#14583382	1	1	yes	characteristic
182	C₄	#14583666	1	1	yes	characteristic
183	C₂	—	2	6	no	—
184	C₂	#19143706	1	1	yes	characteristic
185	C₂	C₂³×C₄₃⋊C₁₆	1	1	yes	characteristic
186	C₂	#19147698	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator	C₂×C₄₃	C₂³×C₁₆
Fitting	C₈×#41×C₄₃	C₂
socle	C₂²×C₄₃	C₂³×C₈
prime radical	C₂×#41×C₄₃	C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₈×#41	D₄₃
hypocenter	C₄₃	#33502
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×C₄₃⋊C₁₆	C₂	semidirect
2. D₄×C₄₃⋊C₁₆	C₂	semidirect
3. #19136394	C₂	semidirect
4. C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
38. #41×C₄₃	C₁₆	semidirect
61. Q₈×C₄₃	C₂×C₁₆	semidirect
62. D₄×C₄₃	C₂×C₁₆	semidirect
66. C₂×C₄×C₄₃	C₂×C₁₆	semidirect
112. C₄₃	#33502	semidirect
132. #41	C₄₃⋊C₁₆	semidirect
155. Q₈	C₂×C₄₃⋊C₁₆	semidirect
156. D₄	C₂×C₄₃⋊C₁₆	semidirect
160. C₂×C₄	C₂×C₄₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄₃⋊C₁₆	2
D₄×C₄₃⋊C₁₆	2
#19136394	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₈×#41×C₄₃	2
#33502	43