Subgroups — #27289282

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719917	C₂	1	2	yes	maximal
2	C₂×#16810388	C₂	1	4	yes	maximal
3	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	4	yes	—
5	C₄×#11719917	—	2	4	no	—
6	C₄×#11719917	C₄	1	8	yes	—
7	C₂×#13133128	—	2	2	no	—
8	C₂×C₄×#88322	—	2	2	no	—
9	#16810388	C₄	1	16	yes	complemented
10	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#97×C₅₃	—	2	4	no	—
13	C₂×#97×C₅₃	C₄	1	4	yes	—
14	C₂×#11719917	—	2	4	no	—
15	C₂×#11719917	—	2	8	no	—
16	C₂×#11719917	C₂×C₄	1	8	yes	—
17	C₂×#11719917	C₂×C₄	1	16	yes	—
18	C₂²×#88322	—	2	4	no	—
19	#13133128	—	2	8	no	—
20	C₄×#88322	—	2	8	no	—
21	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	4	yes	—
31	#97×C₅₃	—	2	8	no	—
32	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	4	4	no	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₅₃	#41	1	4	yes	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	2	16	no	—
45	C₂×C₈×C₅₃	C₄²	1	8	yes	—
46	C₄²×C₅₃	C₄²	1	4	yes	—
47	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
48	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
49	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
50	#11719917	—	2	8	no	—
51	#11719917	—	2	16	no	—
52	#11719917	C₄²	1	16	yes	—
53	C₂×#88322	—	2	8	no	—
54	C₂×#88322	—	2	16	no	—
55	C₂³×C₅₃	—	2	4	no	—
56	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
59	C₂×C₄×C₅₃	—	2	8	no	—
60	C₂×C₄×C₅₃	—	4	8	no	—
61	C₂×C₄×C₅₃	#95	1	8	yes	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
63	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#117	1	4	yes	—
68	C₂×C₄×C₅₃	#117	1	2	yes	—
69	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
70	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
71	C₈×C₅₃	—	2	8	no	—
72	#88322	—	2	16	no	—
73	#10705	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	4	no	—
75	C₂²×C₅₃	—	2	8	no	—
76	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
77	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
78	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
79	C₂²×C₅₃	#430	1	4	yes	—
80	C₄×C₅₃	—	2	8	no	—
81	C₄×C₅₃	C₄³	1	1	yes	characteristic
82	C₄×C₅₃	#376	1	1	yes	characteristic
83	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
84	C₄×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
85	C₂×C₄×M₁₆	—	53	2	no	—
86	C₂²×#97	D₅₃	1	1	yes	characteristic
87	C₂×#341	—	53	4	no	—
88	C₂×C₅₃	—	2	4	no	—
89	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
90	C₂×C₅₃	#1738	1	2	yes	—
91	C₂×C₅₃	#1778	1	4	yes	—
92	C₂²×M₁₆	—	53	4	no	—
93	C₂³×C₈	C₂×D₅₃	1	2	yes	—
94	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
95	C₄×M₁₆	—	53	8	no	—
96	C₄×M₁₆	—	106	4	no	—
97	C₂×#97	—	2	4	no	—
98	C₂×#97	—	106	2	no	—
99	C₂×#97	Dic₅₃	1	4	yes	—
100	C₂×C₄×C₈	—	106	2	no	—
101	#341	—	53	16	no	—
102	C₅₃	#10705	1	1	yes	characteristic, complemented, Sylow
103	C₂³×C₄	Dic₁₀₆	1	2	yes	—
104	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	C₂×M₁₆	—	53	16	no	—
106	C₂×M₁₆	—	53	8	no	—
107	C₂×M₁₆	—	106	8	no	—
108	C₂×M₁₆	—	106	4	no	—
109	C₂²×C₈	—	2	4	no	—
110	C₂²×C₈	—	2	4	no	—
111	C₂²×C₈	—	106	4	no	—
112	C₂²×C₈	C₄×D₅₃	1	8	yes	—
113	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
114	C₂×C₄²	C₄×D₅₃	1	4	yes	—
115	C₂×C₄²	D₂₁₂	1	2	yes	—
116	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
117	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
118	#97	—	2	8	no	—
119	#97	—	106	8	no	—
120	C₄×C₈	—	106	8	no	—
121	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
122	C₂²×C₄	—	4	4	no	—
123	C₂²×C₄	#11719919	1	8	yes	—
124	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
125	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
126	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
127	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
128	C₂²×C₄	#11719945	1	4	yes	—
129	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
131	M₁₆	—	53	16	no	—
132	M₁₆	—	106	8	no	—
133	M₁₆	—	106	16	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	16	no	—
137	C₂×C₈	—	106	8	no	—
138	C₂×C₈	—	106	16	no	—
139	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
140	C₄²	C₄×Dic₅₃	1	4	yes	—
141	C₄²	#11719921	1	4	yes	—
142	C₄²	#11719921	1	4	yes	—
143	C₄²	C₂²⋊Dic₅₃	1	4	yes	—
144	C₂³	—	2	4	no	—
145	C₂³	C₂×#11719919	1	2	yes	—
146	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
147	C₂³	C₄×Dic₁₀₆	1	4	yes	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	4	8	no	—
150	C₂×C₄	#13133124	1	8	yes	—
151	C₂×C₄	C₄²×D₅₃	1	4	yes	—
152	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
154	C₂×C₄	C₂×#11719921	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
156	C₂×C₄	#13133174	1	4	yes	—
157	C₂×C₄	#13133176	1	2	yes	—
158	C₂×C₄	C₄×C₂²⋊D₅₃	1	8	yes	—
159	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	—	106	16	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×#13133124	1	1	yes	characteristic
165	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
166	C₂²	C₄×#11719919	1	4	yes	—
167	C₂²	#16810672	1	4	yes	—
168	C₄	—	2	8	no	—
169	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
170	C₄	#16810469	1	1	yes	characteristic
171	C₄	C₄×C₂²⋊Dic₅₃	1	2	yes	—
172	C₄	C₄×#11719921	1	4	yes	—
173	C₂	—	2	4	no	—
174	C₂	C₄×#13133124	1	1	yes	characteristic
175	C₂	#20274442	1	2	yes	—
176	C₂	#20274621	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	#10705
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
9. #16810388	C₄	semidirect
102. C₅₃	#10705	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719917	2
C₂×#16810388	2
C₂²×#97×C₅₃	2
#10705	53