Subgroups — C₂³×#13133152 #28247780

Order 13568.

2846 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	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133152	C₂	1	28	yes	maximal, direct product factor
4	C₂³×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	7	no	—
6	C₂²×#11719917	C₂²	1	14	yes	—
7	C₂³×#88322	—	2	1	no	—
8	C₂³×#88322	C₄	1	14	yes	—
9	C₂³×#88322	C₄	1	2	yes	—
10	C₂³×#88322	C₂²	1	14	yes	—
11	C₂×#13133152	C₂²	1	112	yes	direct product factor
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	—	2	7	no	—
14	C₂²×M₁₆×C₅₃	C₂²	1	14	yes	—
15	C₂³×C₈×C₅₃	—	2	1	no	—
16	C₂×#11719917	—	2	42	no	—
17	C₂×#11719917	C₂³	1	28	yes	—
18	C₂²×#88322	—	2	14	no	—
19	C₂²×#88322	C₂×C₄	1	84	yes	—
20	C₂²×#88322	C₂×C₄	1	28	yes	—
21	C₂²×#88322	C₂³	1	28	yes	—
22	#13133152	C₂³	1	64	yes	direct product factor
23	C₂⁵×C₅₃	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	D₄	1	7	yes	—
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	Q₈	1	7	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
30	C₂×M₁₆×C₅₃	—	2	42	no	—
31	C₂×M₁₆×C₅₃	C₂³	1	28	yes	—
32	C₂²×C₈×C₅₃	—	2	14	no	—
33	C₂⁴×C₅₃	—	2	8	no	—
34	C₂⁴×C₅₃	C₄⋊C₄	1	7	yes	—
35	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂⁴×C₅₃	C₂×Q₈	1	7	yes	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	42	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	7	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	7	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	7	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	7	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	21	yes	—
45	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
46	M₁₆×C₅₃	—	2	28	no	—
47	M₁₆×C₅₃	C₂⁴	1	8	yes	—
48	C₂×C₈×C₅₃	—	2	28	no	—
49	#11719917	—	2	28	no	—
50	#11719917	C₂⁴	1	8	yes	—
51	C₂×#88322	—	2	28	no	—
52	C₂×#88322	C₂²×C₄	1	56	yes	—
53	C₂×#88322	C₂²×C₄	1	56	yes	—
54	C₂×#88322	C₂⁴	1	8	yes	—
55	C₂³×C₅₃	—	2	28	no	—
56	C₂³×C₅₃	—	2	28	no	—
57	C₂³×C₅₃	#108	1	8	yes	complemented
58	C₂³×C₅₃	C₂×C₄⋊C₄	1	21	yes	—
59	C₂³×C₅₃	C₂×C₄⋊C₄	1	7	yes	—
60	C₂³×C₅₃	C₂²×Q₈	1	7	yes	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	42	yes	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	21	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	21	yes	—
64	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
65	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
66	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
67	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
68	C₂×C₄×C₅₃	C₂²×Q₈	1	7	yes	—
69	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
70	C₈×C₅₃	—	2	8	no	—
71	#88322	—	2	8	no	—
72	#88322	C₂³×C₄	1	16	yes	—
73	C₂³×#108	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	42	no	—
75	C₂²×C₅₃	—	2	14	no	—
76	C₂²×C₅₃	C₂×#108	1	28	yes	complemented
77	C₂²×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
78	C₂²×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
79	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
80	C₄×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
81	C₄×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
82	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
83	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
84	C₂³×M₁₆	—	53	1	no	—
85	C₂³×M₁₆	D₅₃	1	1	yes	characteristic
86	C₂⁴×C₈	—	53	1	no	—
87	C₂²×#108	—	53	28	no	—
88	C₂×C₅₃	—	2	7	no	—
89	C₂×C₅₃	—	2	1	no	—
90	C₂×C₅₃	C₂²×#108	1	14	yes	complemented
91	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
92	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×M₁₆	—	2	7	no	—
94	C₂²×M₁₆	—	53	14	no	—
95	C₂²×M₁₆	—	106	7	no	—
96	C₂²×M₁₆	C₂×D₅₃	1	14	yes	—
97	C₂³×C₈	—	2	1	no	—
98	C₂³×C₈	—	53	2	no	—
99	C₂³×C₈	—	53	14	no	—
100	C₂³×C₈	—	53	14	no	—
101	C₂³×C₈	—	106	1	no	—
102	C₂×#108	—	53	112	no	—
103	C₅₃	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
104	C₂⁵	Dic₁₀₆	1	1	yes	characteristic
105	C₂³×C₄	Dic₁₀₆	1	7	yes	—
106	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
107	C₂³×C₄	C₄×D₅₃	1	7	yes	—
108	C₂³×C₄	C₂²⋊D₅₃	1	7	yes	—
109	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
111	C₂×M₁₆	—	2	42	no	—
112	C₂×M₁₆	—	53	28	no	—
113	C₂×M₁₆	—	106	42	no	—
114	C₂×M₁₆	C₂²×D₅₃	1	28	yes	—
115	C₂²×C₈	—	2	14	no	—
116	C₂²×C₈	—	53	84	no	—
117	C₂²×C₈	—	53	28	no	—
118	C₂²×C₈	—	53	28	no	—
119	C₂²×C₈	—	106	14	no	—
120	#108	—	53	64	no	—
121	C₂⁴	—	2	8	no	—
122	C₂⁴	#11719919	1	7	yes	—
123	C₂⁴	#11719919	1	1	yes	characteristic
124	C₂⁴	C₂×Dic₁₀₆	1	7	yes	—
125	C₂²×C₄	#11719919	1	42	yes	—
126	C₂²×C₄	#11719919	1	7	yes	—
127	C₂²×C₄	#11719919	1	7	yes	—
128	C₂²×C₄	C₂×Dic₁₀₆	1	21	yes	—
129	C₂²×C₄	C₂×C₄×D₅₃	1	21	yes	—
130	C₂²×C₄	C₂×C₄×D₅₃	1	7	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₅₃	1	7	yes	—
132	C₂²×C₄	C₂×C₂²⋊D₅₃	1	21	yes	—
133	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
134	M₁₆	—	2	28	no	—
135	M₁₆	—	53	8	no	—
136	M₁₆	—	106	28	no	—
137	M₁₆	C₂³×D₅₃	1	8	yes	—
138	C₂×C₈	—	2	28	no	—
139	C₂×C₈	—	53	56	no	—
140	C₂×C₈	—	53	56	no	—
141	C₂×C₈	—	53	8	no	—
142	C₂×C₈	—	106	28	no	—
143	C₂³	—	2	28	no	—
144	C₂³	—	2	28	no	—
145	C₂³	#13133152	1	8	yes	direct product factor
146	C₂³	C₂×#11719919	1	21	yes	—
147	C₂³	C₂×#11719919	1	7	yes	—
148	C₂³	C₂²×Dic₁₀₆	1	7	yes	—
149	C₂×C₄	C₂×#11719919	1	42	yes	—
150	C₂×C₄	C₂×#11719919	1	21	yes	—
151	C₂×C₄	C₂×#11719919	1	21	yes	—
152	C₂×C₄	C₂²×C₄×D₅₃	1	7	yes	—
153	C₂×C₄	C₂²×C₄×D₅₃	1	7	yes	—
154	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
156	C₂×C₄	C₂²×Dic₁₀₆	1	7	yes	—
157	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
158	C₈	—	2	8	no	—
159	C₈	—	53	16	no	—
160	C₈	—	106	8	no	—
161	C₂²	—	2	42	no	—
162	C₂²	—	2	14	no	—
163	C₂²	C₂×#13133152	1	28	yes	direct product factor
164	C₂²	C₂²×#11719919	1	7	yes	—
165	C₂²	C₂²×#11719919	1	7	yes	—
166	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
167	C₄	C₂²×#11719919	1	7	yes	—
168	C₄	C₂²×#11719919	1	7	yes	—
169	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
170	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂	—	2	7	no	—
172	C₂	—	2	1	no	—
173	C₂	C₂²×#13133152	1	14	yes	direct product factor
174	C₂	C₂³×#11719919	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×M₁₆×C₅₃	C₂
socle	C₂⁴×C₅₃	C₄⋊C₄
prime radical	C₂⁵×C₅₃	Q₈
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×M₁₆	D₅₃
hypocenter	C₅₃	C₂³×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×#13133152	C₂	direct (N × H)
11. C₂×#13133152	C₂²	direct (N × H)
22. #13133152	C₂³	direct (N × H)
57. C₂³×C₅₃	#108	semidirect
76. C₂²×C₅₃	C₂×#108	semidirect
90. C₂×C₅₃	C₂²×#108	semidirect
103. C₅₃	C₂³×#108	semidirect
145. C₂³	#13133152	direct (N × H)
163. C₂²	C₂×#13133152	direct (N × H)
173. C₂	C₂²×#13133152	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719917	2
C₂⁴×#88322	2
C₂²×#13133152	2
C₂³×M₁₆×C₅₃	2
C₂³×#108	53

Subgroups — C23×#13133152 #28247780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133152 #28247780