Subgroups — C₂³×#13133141 #28247604

Order 13568.

5502 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133141	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₃	—	2	1	no	—
6	C₂³×#88322	—	2	8	no	—
7	C₂³×#88322	C₂²	1	14	yes	—
8	C₂²×C₄⋊Dic₅₃	—	2	7	no	—
9	C₂²×C₄⋊Dic₅₃	C₂²	1	14	yes	—
10	C₂×#13133141	C₂²	1	112	yes	direct product factor
11	C₂³×Q₈×C₅₃	—	2	8	no	—
12	C₂³×Q₈×C₅₃	—	2	7	no	—
13	C₂³×Q₈×C₅₃	C₄	1	16	yes	complemented
14	C₂³×Q₈×C₅₃	C₂²	1	14	yes	—
15	C₂⁴×C₄×C₅₃	—	2	1	no	—
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×Dic₅₃	—	2	14	no	—
18	C₂³×Dic₅₃	—	4	8	no	—
19	C₂²×#88322	—	2	56	no	—
20	C₂²×#88322	C₂³	1	28	yes	—
21	C₂×C₄⋊Dic₅₃	—	2	42	no	—
22	C₂×C₄⋊Dic₅₃	C₂³	1	28	yes	—
23	#13133141	C₂³	1	64	yes	direct product factor
24	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×Q₈×C₅₃	—	2	112	no	—
26	C₂²×Q₈×C₅₃	—	2	56	no	—
27	C₂²×Q₈×C₅₃	—	2	42	no	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	112	yes	complemented
29	C₂²×Q₈×C₅₃	C₂³	1	28	yes	—
30	C₂³×C₄×C₅₃	—	2	16	no	—
31	C₂³×C₄×C₅₃	—	2	14	no	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₅₃	D₄	1	7	yes	—
34	C₂³×C₄×C₅₃	D₄	1	8	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
36	C₂⁴×C₅₃	C₂²⋊C₄	1	8	yes	—
37	C₂⁴×C₅₃	SD₁₆	1	8	yes	—
38	C₂⁴×C₅₃	Q₁₆	1	8	yes	—
39	C₂⁴×C₅₃	C₂×D₄	1	7	yes	—
40	C₂×Q₈×C₅₃	—	2	336	no	—
41	C₂×Q₈×C₅₃	—	2	56	no	—
42	C₂×Q₈×C₅₃	—	2	28	no	—
43	C₂×Q₈×C₅₃	C₂²×C₄	1	112	yes	complemented
44	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
45	C₂²×C₄×C₅₃	—	2	112	no	—
46	C₂²×C₄×C₅₃	—	2	28	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	56	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	28	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
52	C₂²×Dic₅₃	—	2	28	no	—
53	C₂²×Dic₅₃	—	4	56	no	—
54	C₄⋊Dic₅₃	—	2	28	no	—
55	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
56	C₂×#88322	—	2	56	no	—
57	C₂×#88322	C₂⁴	1	8	yes	—
58	C₂³×C₅₃	#103	1	64	yes	complemented
59	C₂³×C₅₃	C₂×C₂²⋊C₄	1	28	yes	—
60	C₂³×C₅₃	C₂×SD₁₆	1	28	yes	—
61	C₂³×C₅₃	C₂×Q₁₆	1	28	yes	—
62	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
63	Q₈×C₅₃	—	2	112	no	—
64	Q₈×C₅₃	—	2	8	no	—
65	Q₈×C₅₃	C₂³×C₄	1	16	yes	complemented
66	C₂×C₄×C₅₃	—	2	112	no	—
67	C₂×C₄×C₅₃	—	2	8	no	—
68	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	84	yes	—
69	C₂×C₄×C₅₃	C₂³×C₄	1	14	yes	—
70	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
71	C₂×C₄×C₅₃	C₂²×D₄	1	14	yes	—
72	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
73	C₂×Dic₅₃	—	2	8	no	—
74	C₂×Dic₅₃	—	4	56	no	—
75	#88322	—	2	8	no	—
76	C₂³×#103	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₂×#103	1	112	yes	complemented
78	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
79	C₂²×C₅₃	C₂²×SD₁₆	1	14	yes	—
80	C₂²×C₅₃	C₂²×Q₁₆	1	14	yes	—
81	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
82	C₄×C₅₃	—	2	16	no	—
83	C₄×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
84	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
86	Dic₅₃	—	4	8	no	—
87	C₂⁴×Q₈	D₅₃	1	1	yes	characteristic
88	C₂⁴×C₈	—	53	1	no	—
89	C₂³×C₄⋊C₄	—	53	1	no	—
90	C₂²×#103	—	53	28	no	—
91	C₂×C₅₃	C₂²×#103	1	28	yes	complemented
92	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₅₃	C₂³×SD₁₆	1	1	yes	characteristic
94	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
95	C₂³×Q₈	—	2	7	no	—
96	C₂³×Q₈	—	2	8	no	—
97	C₂³×Q₈	Dic₅₃	1	16	yes	complemented
98	C₂³×Q₈	C₂×D₅₃	1	14	yes	—
99	C₂⁴×C₄	—	2	1	no	—
100	C₂⁴×C₄	—	106	1	no	—
101	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
102	C₂³×C₈	—	53	14	no	—
103	C₂³×C₈	—	106	8	no	—
104	C₂²×C₄⋊C₄	—	53	14	no	—
105	C₂²×C₄⋊C₄	—	106	7	no	—
106	C₂×#103	—	53	112	no	—
107	C₅₃	C₂³×#103	1	1	yes	characteristic, complemented, Sylow
108	C₂⁵	C₂²⋊D₅₃	1	1	yes	characteristic
109	C₂²×Q₈	—	2	56	no	—
110	C₂²×Q₈	—	2	42	no	—
111	C₂²×Q₈	—	2	112	no	—
112	C₂²×Q₈	C₂×Dic₅₃	1	112	yes	complemented
113	C₂²×Q₈	C₂²×D₅₃	1	28	yes	—
114	C₂³×C₄	—	2	14	no	—
115	C₂³×C₄	—	2	16	no	—
116	C₂³×C₄	—	106	14	no	—
117	C₂³×C₄	—	212	8	no	—
118	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
119	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
120	C₂³×C₄	C₂²⋊D₅₃	1	7	yes	—
121	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
122	C₂²×C₈	—	53	28	no	—
123	C₂²×C₈	—	106	56	no	—
124	C₂×C₄⋊C₄	—	53	28	no	—
125	C₂×C₄⋊C₄	—	106	42	no	—
126	#103	—	53	64	no	—
127	C₂⁴	#11719924	1	8	yes	—
128	C₂⁴	#11719925	1	8	yes	—
129	C₂⁴	C₂²⋊Dic₅₃	1	8	yes	—
130	C₂⁴	C₂×C₂²⋊D₅₃	1	7	yes	—
131	C₂×Q₈	—	2	336	no	—
132	C₂×Q₈	—	2	56	no	—
133	C₂×Q₈	—	2	28	no	—
134	C₂×Q₈	C₂²×Dic₅₃	1	112	yes	complemented
135	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
136	C₂²×C₄	—	2	112	no	—
137	C₂²×C₄	—	2	28	no	—
138	C₂²×C₄	—	106	28	no	—
139	C₂²×C₄	—	212	56	no	—
140	C₂²×C₄	C₂²⋊Dic₅₃	1	56	yes	—
141	C₂²×C₄	C₂²×Dic₅₃	1	28	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₅₃	1	21	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₃	1	28	yes	—
144	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
145	C₂×C₈	—	53	8	no	—
146	C₂×C₈	—	106	56	no	—
147	C₄⋊C₄	—	53	8	no	—
148	C₄⋊C₄	—	106	28	no	—
149	C₂³	#13133141	1	64	yes	direct product factor
150	C₂³	C₂×C₂²⋊Dic₅₃	1	28	yes	—
151	C₂³	C₂×#11719924	1	28	yes	—
152	C₂³	C₂×#11719925	1	28	yes	—
153	C₂³	C₂²×C₂²⋊D₅₃	1	7	yes	—
154	Q₈	—	2	112	no	—
155	Q₈	—	2	8	no	—
156	Q₈	C₂³×Dic₅₃	1	16	yes	complemented
157	C₂×C₄	—	2	112	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	106	8	no	—
160	C₂×C₄	—	212	56	no	—
161	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	84	yes	—
162	C₂×C₄	C₂³×Dic₅₃	1	14	yes	—
163	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₅₃	1	14	yes	—
165	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
166	C₈	—	106	8	no	—
167	C₂²	C₂×#13133141	1	112	yes	direct product factor
168	C₂²	C₂²×C₂²⋊Dic₅₃	1	14	yes	—
169	C₂²	C₂²×#11719924	1	14	yes	—
170	C₂²	C₂²×#11719925	1	14	yes	—
171	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄	—	2	16	no	—
173	C₄	—	212	8	no	—
174	C₄	C₂²×C₂²⋊Dic₅₃	1	14	yes	—
175	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
176	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂	C₂²×#13133141	1	28	yes	direct product factor
178	C₂	C₂³×C₂²⋊Dic₅₃	1	1	yes	characteristic
179	C₂	C₂³×#11719924	1	1	yes	characteristic
180	C₂	C₂³×#11719925	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂⁴×Q₈×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	D₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂⁴×Q₈	D₅₃
hypocenter	C₅₃	C₂³×#103
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#13133141	C₂	direct (N × H)
10. C₂×#13133141	C₂²	direct (N × H)
13. C₂³×Q₈×C₅₃	C₄	semidirect
23. #13133141	C₂³	direct (N × H)
28. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
43. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
58. C₂³×C₅₃	#103	semidirect
65. Q₈×C₅₃	C₂³×C₄	semidirect
77. C₂²×C₅₃	C₂×#103	semidirect
91. C₂×C₅₃	C₂²×#103	semidirect
97. C₂³×Q₈	Dic₅₃	semidirect
107. C₅₃	C₂³×#103	semidirect
112. C₂²×Q₈	C₂×Dic₅₃	semidirect
134. C₂×Q₈	C₂²×Dic₅₃	semidirect
149. C₂³	#13133141	direct (N × H)
156. Q₈	C₂³×Dic₅₃	semidirect
167. C₂²	C₂×#13133141	direct (N × H)
177. C₂	C₂²×#13133141	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×#88322	2
C₂³×C₄⋊Dic₅₃	2
C₂²×#13133141	2
C₂⁴×Q₈×C₅₃	2
C₂³×#103	53

Subgroups — C23×#13133141 #28247604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133141 #28247604