Subgroups — C₂³×#13133216 #28290782

Order 13568.

12190 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂²×#13133216	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#11719919	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	—	2	32	no	—
8	C₂³×Dic₁₀₆	C₂²	1	56	yes	—
9	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
10	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133216	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊Dic₅₃	C₂²	1	28	yes	—
14	C₂²×#11719919	C₂²	1	56	yes	—
15	C₂²×C₄×Dic₅₃	C₂²	1	28	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	—
19	C₂²×Dic₁₀₆	—	2	448	no	—
20	C₂²×Dic₁₀₆	C₂³	1	224	yes	—
21	C₂³×Dic₅₃	—	2	16	no	—
22	C₂³×Dic₅₃	D₄	1	16	yes	—
23	C₂³×Dic₅₃	Q₈	1	16	yes	complemented
24	C₂³×Dic₅₃	C₂³	1	28	yes	—
25	C₂³×Dic₅₃	C₂³	1	14	yes	—
26	C₂³×Dic₅₃	C₂³	1	14	yes	—
27	#13133216	C₂³	1	512	yes	direct product factor
28	C₂×C₄⋊Dic₅₃	C₂³	1	112	yes	—
29	C₂×#11719919	C₂³	1	224	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	112	yes	—
31	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	—	2	16	no	—
33	C₂³×C₄×C₅₃	Q₈	1	16	yes	complemented
34	C₂³×C₄×C₅₃	C₂³	1	28	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	112	yes	—
37	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
38	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
39	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
40	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
41	C₂²×C₄×C₅₃	—	2	112	no	—
42	C₂²×C₄×C₅₃	C₂×Q₈	1	112	yes	complemented
43	C₂²×C₄×C₅₃	C₂⁴	1	56	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
45	C₂²×Dic₅₃	—	2	112	no	—
46	C₂²×Dic₅₃	C₂×D₄	1	112	yes	—
47	C₂²×Dic₅₃	C₂×Q₈	1	112	yes	complemented
48	C₂²×Dic₅₃	C₂⁴	1	56	yes	—
49	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
50	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
51	C₂×Dic₁₀₆	—	2	896	no	—
52	C₂×Dic₁₀₆	C₂⁴	1	128	yes	—
53	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
54	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
55	#11719919	C₂⁴	1	128	yes	—
56	C₄×Dic₅₃	C₂⁴	1	64	yes	—
57	C₂³×C₅₃	C₄⋊Q₈	1	64	yes	complemented
58	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
59	C₂³×C₅₃	C₂²×Q₈	1	28	yes	—
60	C₂³×C₅₃	C₂²×Q₈	1	28	yes	—
61	C₂³×C₅₃	C₂⁵	1	7	yes	—
62	C₂×C₄×C₅₃	—	2	112	no	—
63	C₂×C₄×C₅₃	C₂²×Q₈	1	112	yes	complemented
64	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
65	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
66	C₂×Dic₅₃	—	2	112	no	—
67	C₂×Dic₅₃	C₂²×D₄	1	112	yes	—
68	C₂×Dic₅₃	C₂²×Q₈	1	112	yes	complemented
69	C₂×Dic₅₃	C₂⁵	1	16	yes	—
70	C₂×Dic₅₃	C₂⁵	1	8	yes	—
71	C₂×Dic₅₃	C₂⁵	1	8	yes	—
72	Dic₁₀₆	—	2	256	no	—
73	C₂³×C₄⋊Q₈	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	C₂×C₄⋊Q₈	1	112	yes	complemented
75	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
76	C₂²×C₅₃	C₂³×Q₈	1	14	yes	—
77	C₂²×C₅₃	C₂³×Q₈	1	14	yes	—
78	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
79	C₄×C₅₃	—	2	16	no	—
80	C₄×C₅₃	C₂³×Q₈	1	16	yes	complemented
81	Dic₅₃	—	2	16	no	—
82	Dic₅₃	C₂³×D₄	1	16	yes	—
83	Dic₅₃	C₂³×Q₈	1	16	yes	complemented
84	C₂⁴×Q₈	—	53	2	no	—
85	C₂²×C₄⋊Q₈	—	53	56	no	—
86	C₂³×C₄⋊C₄	—	53	2	no	—
87	C₂³×C₄⋊C₄	—	53	1	no	—
88	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
89	C₂³×C₄²	—	53	1	no	—
90	C₂×C₅₃	C₂²×C₄⋊Q₈	1	28	yes	complemented
91	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
92	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
93	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
94	C₂³×Q₈	—	53	56	no	—
95	C₂³×Q₈	—	106	32	no	—
96	C₂⁴×C₄	—	53	1	no	—
97	C₂⁴×C₄	—	53	1	no	—
98	C₂⁴×C₄	—	53	2	no	—
99	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
100	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
101	C₂×C₄⋊Q₈	—	53	448	no	—
102	C₂²×C₄⋊C₄	—	53	56	no	—
103	C₂²×C₄⋊C₄	—	53	28	no	—
104	C₂²×C₄⋊C₄	C₂×D₅₃	1	28	yes	—
105	C₂²×C₄²	—	53	28	no	—
106	C₅₃	C₂³×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
107	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
108	C₂²×Q₈	—	53	224	no	—
109	C₂²×Q₈	—	106	448	no	—
110	C₂³×C₄	—	2	16	no	—
111	C₂³×C₄	—	53	16	no	—
112	C₂³×C₄	—	53	16	no	—
113	C₂³×C₄	—	53	14	no	—
114	C₂³×C₄	—	53	14	no	—
115	C₂³×C₄	—	53	28	no	—
116	C₂³×C₄	—	106	16	no	—
117	C₂³×C₄	Dic₁₀₆	1	16	yes	complemented
118	C₂³×C₄	C₂²×D₅₃	1	28	yes	—
119	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
120	C₄⋊Q₈	—	53	512	no	—
121	C₂×C₄⋊C₄	—	53	224	no	—
122	C₂×C₄⋊C₄	—	53	112	no	—
123	C₂×C₄⋊C₄	C₂²×D₅₃	1	112	yes	—
124	C₂×C₄²	—	53	112	no	—
125	C₂⁴	C₂×Dic₁₀₆	1	8	yes	—
126	C₂⁴	D₄×D₅₃	1	8	yes	—
127	C₂⁴	Q₈×D₅₃	1	8	yes	—
128	C₂⁴	C₂³×D₅₃	1	7	yes	—
129	C₂×Q₈	—	53	128	no	—
130	C₂×Q₈	—	106	896	no	—
131	C₂²×C₄	—	2	112	no	—
132	C₂²×C₄	—	53	112	no	—
133	C₂²×C₄	—	53	112	no	—
134	C₂²×C₄	—	53	28	no	—
135	C₂²×C₄	—	53	28	no	—
136	C₂²×C₄	—	53	56	no	—
137	C₂²×C₄	—	106	112	no	—
138	C₂²×C₄	C₂×Dic₁₀₆	1	112	yes	complemented
139	C₂²×C₄	C₂³×D₅₃	1	56	yes	—
140	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
141	C₄⋊C₄	—	53	128	no	—
142	C₄⋊C₄	—	53	64	no	—
143	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
144	C₄²	—	53	64	no	—
145	C₂³	#13133216	1	64	yes	direct product factor
146	C₂³	C₂×D₄×D₅₃	1	28	yes	—
147	C₂³	C₂×Q₈×D₅₃	1	28	yes	—
148	C₂³	C₂²×Dic₁₀₆	1	28	yes	—
149	C₂³	C₂⁴×D₅₃	1	7	yes	—
150	Q₈	—	106	256	no	—
151	C₂×C₄	—	2	112	no	—
152	C₂×C₄	—	53	112	no	—
153	C₂×C₄	—	53	112	no	—
154	C₂×C₄	—	53	8	no	—
155	C₂×C₄	—	53	8	no	—
156	C₂×C₄	—	53	16	no	—
157	C₂×C₄	—	106	112	no	—
158	C₂×C₄	C₂²×Dic₁₀₆	1	112	yes	complemented
159	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
160	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
161	C₂²	C₂×#13133216	1	112	yes	direct product factor
162	C₂²	C₂²×D₄×D₅₃	1	14	yes	—
163	C₂²	C₂²×Q₈×D₅₃	1	14	yes	—
164	C₂²	C₂³×Dic₁₀₆	1	14	yes	—
165	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
166	C₄	—	2	16	no	—
167	C₄	—	53	16	no	—
168	C₄	—	53	16	no	—
169	C₄	—	106	16	no	—
170	C₄	C₂³×Dic₁₀₆	1	16	yes	complemented
171	C₂	C₂²×#13133216	1	28	yes	direct product factor
172	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
173	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
174	C₂	C₂⁴×Dic₁₀₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×C₄⋊Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133216	C₂	direct (N × H)
12. C₂×#13133216	C₂²	direct (N × H)
23. C₂³×Dic₅₃	Q₈	semidirect
27. #13133216	C₂³	direct (N × H)
33. C₂³×C₄×C₅₃	Q₈	semidirect
42. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
47. C₂²×Dic₅₃	C₂×Q₈	semidirect
57. C₂³×C₅₃	C₄⋊Q₈	semidirect
63. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
68. C₂×Dic₅₃	C₂²×Q₈	semidirect
74. C₂²×C₅₃	C₂×C₄⋊Q₈	semidirect
80. C₄×C₅₃	C₂³×Q₈	semidirect
83. Dic₅₃	C₂³×Q₈	semidirect
90. C₂×C₅₃	C₂²×C₄⋊Q₈	semidirect
106. C₅₃	C₂³×C₄⋊Q₈	semidirect
117. C₂³×C₄	Dic₁₀₆	semidirect
138. C₂²×C₄	C₂×Dic₁₀₆	semidirect
145. C₂³	#13133216	direct (N × H)
158. C₂×C₄	C₂²×Dic₁₀₆	semidirect
161. C₂²	C₂×#13133216	direct (N × H)
170. C₄	C₂³×Dic₁₀₆	semidirect
171. C₂	C₂²×#13133216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₀₆	2
C₂²×#13133216	2
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719919	2
C₂³×C₄×Dic₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×C₄⋊Q₈	53

Subgroups — C23×#13133216 #28290782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133216 #28290782