Subgroups — C₂³×C₂²⋊#88322 #28247519

Order 13568.

7646 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×#88322	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊#88322	C₂	1	28	yes	maximal, direct product factor
3	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	—	2	16	no	—
5	C₂³×#88322	C₂²	1	28	yes	complemented
6	C₂×C₂²⋊#88322	C₂²	1	112	yes	direct product factor
7	C₂⁶×C₅₃	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₅₃	—	2	16	no	—
9	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
10	C₂⁴×C₄×C₅₃	C₄	1	14	yes	—
11	C₂⁴×C₄×C₅₃	C₂²	1	14	yes	—
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×#88322	—	2	112	no	—
14	C₂²×#88322	C₂³	1	56	yes	complemented
15	C₂²⋊#88322	C₂³	1	64	yes	direct product factor
16	C₂⁵×C₅₃	—	2	8	no	—
17	C₂⁵×C₅₃	—	2	8	no	—
18	C₂⁵×C₅₃	C₈	1	16	yes	complemented
19	C₂⁵×C₅₃	C₂×C₄	1	14	yes	—
20	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	—	2	8	no	—
22	C₂³×C₄×C₅₃	—	2	112	no	—
23	C₂³×C₄×C₅₃	—	2	112	no	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	14	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	84	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
27	C₂³×C₄×C₅₃	D₄	1	16	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	28	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
30	C₂⁴×C₅₃	—	2	128	no	—
31	C₂⁴×C₅₃	—	2	56	no	—
32	C₂⁴×C₅₃	—	2	56	no	—
33	C₂⁴×C₅₃	C₂²⋊C₄	1	8	yes	—
34	C₂⁴×C₅₃	C₂×C₈	1	8	yes	—
35	C₂⁴×C₅₃	C₂×C₈	1	112	yes	complemented
36	C₂⁴×C₅₃	M₁₆	1	8	yes	—
37	C₂⁴×C₅₃	C₂²×C₄	1	28	yes	—
38	C₂⁴×C₅₃	C₂²×C₄	1	7	yes	—
39	C₂²×C₄×C₅₃	—	2	56	no	—
40	C₂²×C₄×C₅₃	—	2	336	no	—
41	C₂²×C₄×C₅₃	—	2	112	no	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	56	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	56	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	56	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
49	C₂×#88322	—	2	112	no	—
50	C₂×#88322	C₂⁴	1	16	yes	complemented
51	C₂³×C₅₃	—	2	56	no	—
52	C₂³×C₅₃	—	2	56	no	—
53	C₂³×C₅₃	—	2	448	no	—
54	C₂³×C₅₃	C₂²⋊C₈	1	64	yes	complemented
55	C₂³×C₅₃	C₂×C₂²⋊C₄	1	28	yes	—
56	C₂³×C₅₃	C₂²×C₈	1	28	yes	—
57	C₂³×C₅₃	C₂²×C₈	1	112	yes	complemented
58	C₂³×C₅₃	C₂×M₁₆	1	28	yes	—
59	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
60	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
61	C₂×C₄×C₅₃	—	2	56	no	—
62	C₂×C₄×C₅₃	—	2	112	no	—
63	C₂×C₄×C₅₃	—	2	16	no	—
64	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	84	yes	—
65	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
67	C₂×C₄×C₅₃	C₂²×D₄	1	28	yes	—
68	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
69	#88322	—	2	16	no	—
70	C₂³×C₂²⋊C₈	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	224	no	—
72	C₂²×C₅₃	—	2	8	no	—
73	C₂²×C₅₃	—	2	8	no	—
74	C₂²×C₅₃	C₂×C₂²⋊C₈	1	112	yes	complemented
75	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
76	C₂²×C₅₃	C₂³×C₈	1	14	yes	—
77	C₂²×C₅₃	C₂³×C₈	1	16	yes	complemented
78	C₂²×C₅₃	C₂²×M₁₆	1	14	yes	—
79	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
80	C₄×C₅₃	—	2	8	no	—
81	C₄×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
82	C₄×C₅₃	C₂³×D₄	1	2	yes	—
83	C₂⁵×C₄	D₅₃	1	1	yes	characteristic
84	C₂⁴×C₈	—	53	2	no	—
85	C₂²×C₂²⋊C₈	—	53	28	no	—
86	C₂×C₅₃	—	2	16	no	—
87	C₂×C₅₃	C₂²×C₂²⋊C₈	1	28	yes	complemented
88	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
90	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
91	C₂⁶	Dic₅₃	1	1	yes	characteristic
92	C₂⁴×C₄	—	2	16	no	—
93	C₂⁴×C₄	Dic₅₃	1	1	yes	characteristic
94	C₂⁴×C₄	Dic₅₃	1	14	yes	—
95	C₂⁴×C₄	C₂×D₅₃	1	14	yes	—
96	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
97	C₂³×C₈	—	53	28	no	—
98	C₂³×C₈	—	106	16	no	—
99	C₂×C₂²⋊C₈	—	53	112	no	—
100	C₅₃	C₂³×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
101	C₂⁵	—	2	8	no	—
102	C₂⁵	—	2	8	no	—
103	C₂⁵	#88322	1	16	yes	complemented
104	C₂⁵	C₂×Dic₅₃	1	14	yes	—
105	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂³×C₄	—	2	8	no	—
107	C₂³×C₄	—	2	112	no	—
108	C₂³×C₄	—	2	112	no	—
109	C₂³×C₄	C₂×Dic₅₃	1	14	yes	—
110	C₂³×C₄	C₂×Dic₅₃	1	84	yes	—
111	C₂³×C₄	C₂×Dic₅₃	1	7	yes	—
112	C₂³×C₄	C₂²⋊D₅₃	1	16	yes	—
113	C₂³×C₄	C₂²×D₅₃	1	28	yes	—
114	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
115	C₂²×C₈	—	53	56	no	—
116	C₂²×C₈	—	106	112	no	—
117	C₂²⋊C₈	—	53	64	no	—
118	C₂⁴	—	2	128	no	—
119	C₂⁴	—	2	56	no	—
120	C₂⁴	—	2	56	no	—
121	C₂⁴	C₂×#88322	1	8	yes	—
122	C₂⁴	C₂×#88322	1	112	yes	complemented
123	C₂⁴	#11719917	1	8	yes	—
124	C₂⁴	C₂²⋊Dic₅₃	1	8	yes	—
125	C₂⁴	C₂²×Dic₅₃	1	28	yes	—
126	C₂⁴	C₂²×Dic₅₃	1	7	yes	—
127	C₂²×C₄	—	2	56	no	—
128	C₂²×C₄	—	2	336	no	—
129	C₂²×C₄	—	2	112	no	—
130	C₂²×C₄	C₂²⋊Dic₅₃	1	56	yes	—
131	C₂²×C₄	C₂²×Dic₅₃	1	28	yes	—
132	C₂²×C₄	C₂²×Dic₅₃	1	56	yes	—
133	C₂²×C₄	C₂²×Dic₅₃	1	21	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₅₃	1	56	yes	—
135	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
136	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
137	C₂×C₈	—	53	16	no	—
138	C₂×C₈	—	106	112	no	—
139	C₂³	—	2	56	no	—
140	C₂³	—	2	448	no	—
141	C₂³	—	2	56	no	—
142	C₂³	C₂²⋊#88322	1	64	yes	direct product factor
143	C₂³	C₂×C₂²⋊Dic₅₃	1	28	yes	—
144	C₂³	C₂²×#88322	1	28	yes	—
145	C₂³	C₂²×#88322	1	112	yes	complemented
146	C₂³	C₂×#11719917	1	28	yes	—
147	C₂³	C₂³×Dic₅₃	1	8	yes	—
148	C₂³	C₂³×Dic₅₃	1	7	yes	—
149	C₂×C₄	—	2	56	no	—
150	C₂×C₄	—	2	112	no	—
151	C₂×C₄	—	2	16	no	—
152	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	84	yes	—
153	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
154	C₂×C₄	C₂³×Dic₅₃	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₅₃	1	28	yes	—
156	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
157	C₈	—	106	16	no	—
158	C₂²	—	2	224	no	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	8	no	—
161	C₂²	C₂×C₂²⋊#88322	1	112	yes	direct product factor
162	C₂²	C₂²×C₂²⋊Dic₅₃	1	14	yes	—
163	C₂²	C₂³×#88322	1	14	yes	—
164	C₂²	C₂³×#88322	1	16	yes	complemented
165	C₂²	C₂²×#11719917	1	14	yes	—
166	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	C₂²×C₂²⋊Dic₅₃	1	14	yes	—
169	C₄	C₂³×C₂²⋊D₅₃	1	2	yes	—
170	C₂	—	2	16	no	—
171	C₂	C₂²×C₂²⋊#88322	1	28	yes	direct product factor
172	C₂	C₂³×C₂²⋊Dic₅₃	1	1	yes	characteristic
173	C₂	C₂⁴×#88322	1	1	yes	characteristic
174	C₂	C₂³×#11719917	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂×C₄
prime radical	C₂⁶×C₅₃	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂³×C₂²⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#88322	C₂	semidirect
2. C₂²×C₂²⋊#88322	C₂	direct (N × H)
5. C₂³×#88322	C₂²	semidirect
6. C₂×C₂²⋊#88322	C₂²	direct (N × H)
14. C₂²×#88322	C₂³	semidirect
15. C₂²⋊#88322	C₂³	direct (N × H)
18. C₂⁵×C₅₃	C₈	semidirect
35. C₂⁴×C₅₃	C₂×C₈	semidirect
50. C₂×#88322	C₂⁴	semidirect
54. C₂³×C₅₃	C₂²⋊C₈	semidirect
57. C₂³×C₅₃	C₂²×C₈	semidirect
74. C₂²×C₅₃	C₂×C₂²⋊C₈	semidirect
77. C₂²×C₅₃	C₂³×C₈	semidirect
87. C₂×C₅₃	C₂²×C₂²⋊C₈	semidirect
100. C₅₃	C₂³×C₂²⋊C₈	semidirect
103. C₂⁵	#88322	semidirect
122. C₂⁴	C₂×#88322	semidirect
142. C₂³	C₂²⋊#88322	direct (N × H)
145. C₂³	C₂²×#88322	semidirect
161. C₂²	C₂×C₂²⋊#88322	direct (N × H)
164. C₂²	C₂³×#88322	semidirect
171. C₂	C₂²×C₂²⋊#88322	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
C₂²×C₂²⋊#88322	2
C₂⁵×C₄×C₅₃	2
C₂³×C₂²⋊C₈	53

Subgroups — C23×C22⋊#88322 #28247519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊#88322 #28247519