Subgroups — C₂³×#13063316 #26003264

Order 11008.

8414 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063316	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₄₃	C₂	1	2	yes	maximal
3	C₂³×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₄₃⋊C₄	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₄₃	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₄₃	C₂²	1	2	yes	—
10	C₂×#13063316	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊Dic₄₃	C₂²	1	56	yes	—
12	C₂²×C₄⋊Dic₄₃	C₂²	1	28	yes	—
13	C₂²×Dic₄₃⋊C₄	C₂²	1	56	yes	—
14	C₂²×C₄×Dic₄₃	C₂²	1	28	yes	—
15	C₂⁴×C₄×C₄₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₃	C₂²	1	28	yes	—
18	C₂³×Dic₄₃	—	2	8	no	—
19	C₂³×Dic₄₃	—	2	8	no	—
20	C₂³×Dic₄₃	—	2	16	no	—
21	C₂³×Dic₄₃	C₂³	1	14	yes	—
22	C₂³×Dic₄₃	C₂³	1	14	yes	—
23	C₂³×Dic₄₃	C₂³	1	28	yes	—
24	#13063316	C₂³	1	512	yes	direct product factor
25	C₂×C₄⋊Dic₄₃	C₂³	1	224	yes	—
26	C₂×C₄⋊Dic₄₃	C₂³	1	112	yes	—
27	C₂×Dic₄₃⋊C₄	C₂³	1	224	yes	—
28	C₂×C₄×Dic₄₃	C₂³	1	112	yes	—
29	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	—	2	16	no	—
31	C₂³×C₄×C₄₃	Q₈	1	16	yes	—
32	C₂³×C₄×C₄₃	C₂³	1	28	yes	—
33	C₂³×C₄×C₄₃	C₂³	1	14	yes	—
34	C₂×C₄⋊C₄×C₄₃	C₂³	1	112	yes	—
35	C₂⁴×C₄₃	C₂×Q₈	1	8	yes	—
36	C₂⁴×C₄₃	#41	1	8	yes	—
37	C₂⁴×C₄₃	#41	1	8	yes	—
38	C₂⁴×C₄₃	C₂⁴	1	7	yes	—
39	C₂²×C₄×C₄₃	—	2	112	no	—
40	C₂²×C₄×C₄₃	C₂×Q₈	1	112	yes	—
41	C₂²×C₄×C₄₃	C₂⁴	1	56	yes	—
42	C₂²×C₄×C₄₃	C₂⁴	1	28	yes	—
43	C₂²×Dic₄₃	—	2	56	no	—
44	C₂²×Dic₄₃	—	2	56	no	—
45	C₂²×Dic₄₃	—	2	112	no	—
46	C₂²×Dic₄₃	C₂⁴	1	28	yes	—
47	C₂²×Dic₄₃	C₂⁴	1	28	yes	—
48	C₂²×Dic₄₃	C₂⁴	1	56	yes	—
49	C₄⋊C₄×C₄₃	C₂⁴	1	64	yes	—
50	C₄⋊Dic₄₃	C₂⁴	1	128	yes	—
51	C₄⋊Dic₄₃	C₂⁴	1	64	yes	—
52	Dic₄₃⋊C₄	C₂⁴	1	128	yes	—
53	C₄×Dic₄₃	C₂⁴	1	64	yes	—
54	C₂³×C₄₃	#125	1	64	yes	complemented
55	C₂³×C₄₃	C₂²×Q₈	1	28	yes	—
56	C₂³×C₄₃	C₂×#41	1	28	yes	—
57	C₂³×C₄₃	C₂×#41	1	28	yes	—
58	C₂³×C₄₃	C₂⁵	1	7	yes	—
59	C₂×C₄×C₄₃	—	2	112	no	—
60	C₂×C₄×C₄₃	C₂²×Q₈	1	112	yes	—
61	C₂×C₄×C₄₃	C₂⁵	1	16	yes	—
62	C₂×C₄×C₄₃	C₂⁵	1	8	yes	—
63	C₂×Dic₄₃	—	2	56	no	—
64	C₂×Dic₄₃	—	2	56	no	—
65	C₂×Dic₄₃	—	2	112	no	—
66	C₂×Dic₄₃	C₂⁵	1	8	yes	—
67	C₂×Dic₄₃	C₂⁵	1	8	yes	—
68	C₂×Dic₄₃	C₂⁵	1	16	yes	—
69	C₂³×#125	—	43	1	no	maximal, Sylow
70	C₂²×C₄₃	C₂×#125	1	112	yes	complemented
71	C₂²×C₄₃	C₂³×Q₈	1	14	yes	—
72	C₂²×C₄₃	C₂²×#41	1	14	yes	—
73	C₂²×C₄₃	C₂²×#41	1	14	yes	—
74	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
75	C₄×C₄₃	—	2	16	no	—
76	C₄×C₄₃	C₂³×Q₈	1	16	yes	—
77	Dic₄₃	—	2	8	no	—
78	Dic₄₃	—	2	8	no	—
79	Dic₄₃	—	2	16	no	—
80	C₂²×#125	—	43	56	no	—
81	C₂³×C₄⋊C₄	—	43	2	no	—
82	C₂³×C₄⋊C₄	—	43	2	no	—
83	C₂³×C₄⋊C₄	—	43	1	no	—
84	C₂³×C₄⋊C₄	D₄₃	1	1	yes	characteristic
85	C₂³×C₄²	—	43	1	no	—
86	C₂×C₄₃	C₂²×#125	1	28	yes	complemented
87	C₂×C₄₃	C₂⁴×Q₈	1	1	yes	characteristic
88	C₂×C₄₃	C₂³×#41	1	1	yes	characteristic
89	C₂×C₄₃	C₂³×#41	1	1	yes	characteristic
90	C₂⁴×C₄	—	43	1	no	—
91	C₂⁴×C₄	—	43	1	no	—
92	C₂⁴×C₄	—	43	2	no	—
93	C₂⁴×C₄	C₂×D₄₃	1	2	yes	—
94	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
95	C₂×#125	—	43	448	no	—
96	C₂²×C₄⋊C₄	—	43	56	no	—
97	C₂²×C₄⋊C₄	—	43	28	no	—
98	C₂²×C₄⋊C₄	—	43	56	no	—
99	C₂²×C₄⋊C₄	C₂×D₄₃	1	28	yes	—
100	C₂²×C₄²	—	43	28	no	—
101	C₄₃	C₂³×#125	1	1	yes	characteristic, complemented, Sylow
102	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
103	C₂³×C₄	—	2	16	no	—
104	C₂³×C₄	—	43	28	no	—
105	C₂³×C₄	—	43	14	no	—
106	C₂³×C₄	—	43	14	no	—
107	C₂³×C₄	—	86	8	no	—
108	C₂³×C₄	—	86	8	no	—
109	C₂³×C₄	—	86	16	no	—
110	C₂³×C₄	Dic₈₆	1	16	yes	—
111	C₂³×C₄	C₂²×D₄₃	1	28	yes	—
112	C₂³×C₄	C₂²×D₄₃	1	14	yes	—
113	#125	—	43	512	no	—
114	C₂×C₄⋊C₄	—	43	224	no	—
115	C₂×C₄⋊C₄	—	43	224	no	—
116	C₂×C₄⋊C₄	—	43	112	no	—
117	C₂×C₄⋊C₄	C₂²×D₄₃	1	112	yes	—
118	C₂×C₄²	—	43	112	no	—
119	C₂⁴	C₂×Dic₈₆	1	8	yes	—
120	C₂⁴	#10621900	1	8	yes	—
121	C₂⁴	#10621902	1	8	yes	—
122	C₂⁴	C₂³×D₄₃	1	7	yes	—
123	C₂²×C₄	—	2	112	no	—
124	C₂²×C₄	—	43	28	no	—
125	C₂²×C₄	—	43	28	no	—
126	C₂²×C₄	—	43	56	no	—
127	C₂²×C₄	—	86	56	no	—
128	C₂²×C₄	—	86	112	no	—
129	C₂²×C₄	—	86	56	no	—
130	C₂²×C₄	C₂×Dic₈₆	1	112	yes	—
131	C₂²×C₄	C₂³×D₄₃	1	56	yes	—
132	C₂²×C₄	C₂³×D₄₃	1	28	yes	—
133	C₄⋊C₄	—	43	128	no	—
134	C₄⋊C₄	—	43	128	no	—
135	C₄⋊C₄	—	43	64	no	—
136	C₄⋊C₄	C₂³×D₄₃	1	64	yes	—
137	C₄²	—	43	64	no	—
138	C₂³	#13063316	1	64	yes	direct product factor
139	C₂³	C₂²×Dic₈₆	1	28	yes	—
140	C₂³	C₂×#10621900	1	28	yes	—
141	C₂³	C₂×#10621902	1	28	yes	—
142	C₂³	C₂⁴×D₄₃	1	7	yes	—
143	C₂×C₄	—	2	112	no	—
144	C₂×C₄	—	43	8	no	—
145	C₂×C₄	—	43	8	no	—
146	C₂×C₄	—	43	16	no	—
147	C₂×C₄	—	86	56	no	—
148	C₂×C₄	—	86	56	no	—
149	C₂×C₄	—	86	112	no	—
150	C₂×C₄	C₂²×Dic₈₆	1	112	yes	—
151	C₂×C₄	C₂⁴×D₄₃	1	16	yes	—
152	C₂×C₄	C₂⁴×D₄₃	1	8	yes	—
153	C₂²	C₂×#13063316	1	112	yes	direct product factor
154	C₂²	C₂³×Dic₈₆	1	14	yes	—
155	C₂²	C₂²×#10621900	1	14	yes	—
156	C₂²	C₂²×#10621902	1	14	yes	—
157	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
158	C₄	—	2	16	no	—
159	C₄	—	86	8	no	—
160	C₄	—	86	8	no	—
161	C₄	—	86	16	no	—
162	C₄	C₂³×Dic₈₆	1	16	yes	—
163	C₂	C₂²×#13063316	1	28	yes	direct product factor
164	C₂	C₂⁴×Dic₈₆	1	1	yes	characteristic
165	C₂	C₂³×#10621900	1	1	yes	characteristic
166	C₂	C₂³×#10621902	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₄₃
commutator, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂³×C₄⋊C₄×C₄₃	C₂
socle, prime radical	C₂⁵×C₄₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₄₃
hypocenter	C₄₃	C₂³×#125
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063316	C₂	direct (N × H)
10. C₂×#13063316	C₂²	direct (N × H)
24. #13063316	C₂³	direct (N × H)
54. C₂³×C₄₃	#125	semidirect
70. C₂²×C₄₃	C₂×#125	semidirect
86. C₂×C₄₃	C₂²×#125	semidirect
101. C₄₃	C₂³×#125	semidirect
138. C₂³	#13063316	direct (N × H)
153. C₂²	C₂×#13063316	direct (N × H)
163. C₂	C₂²×#13063316	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13063316	2
C₂³×C₄⋊Dic₄₃	2
C₂³×Dic₄₃⋊C₄	2
C₂³×C₄×Dic₄₃	2
C₂³×C₄⋊C₄×C₄₃	2
C₂³×#125	43

Subgroups — C23×#13063316 #26003264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13063316 #26003264