Subgroups — #27300446

Order 13568.

298 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	2	yes	maximal
2	#20272799	C₂	1	2	yes	maximal
3	#20272795	C₂	1	2	yes	maximal
4	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	#16810444	—	2	4	no	—
7	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#376	C₄	1	1	yes	characteristic
14	#13133157	—	2	4	no	—
15	#13133157	—	2	4	no	—
16	C₂×#11719908	—	2	4	no	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	#97×C₅₃	—	2	2	no	—
25	#97×C₅₃	—	2	2	no	—
26	#97×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	C₈	1	4	yes	—
29	C₄×C₈×C₅₃	D₄	1	2	yes	—
30	C₄×C₈×C₅₃	Q₈	1	2	yes	—
31	#95×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	—	2	1	no	—
34	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₂×C₈×C₅₃	#41	1	2	yes	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₃	C₂×C₈	1	2	yes	—
55	#11719908	—	4	4	no	—
56	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
72	C₈×C₅₃	#124	1	2	yes	—
73	C₈×C₅₃	#125	1	2	yes	—
74	#12881	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	#423	1	1	yes	characteristic
77	C₂²×C₅₃	#431	1	1	yes	characteristic
78	C₂²×C₅₃	#432	1	1	yes	characteristic
79	C₄×C₅₃	—	4	2	no	—
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	#382	1	1	yes	characteristic
83	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₄×C₅₃	#431	1	1	yes	characteristic
85	C₄×C₅₃	#432	1	1	yes	characteristic
86	C₂×#364	—	53	2	no	—
87	#1726	D₅₃	1	1	yes	characteristic
88	#1377	—	53	2	no	—
89	#1375	—	53	2	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	#1831	1	1	yes	characteristic
93	C₂×C₅₃	#2166	1	2	yes	—
94	C₂×M₃₂	—	53	4	no	—
95	C₂×#97	Dic₅₃	1	1	yes	characteristic
96	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
97	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
98	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
99	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	#376	Dic₅₃	1	1	yes	characteristic
102	#364	—	106	4	no	—
103	C₅₃	#12881	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
111	M₃₂	—	106	4	no	—
112	M₃₂	—	106	4	no	—
113	C₂×C₁₆	—	106	4	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	#88322	1	4	yes	—
119	C₄×C₈	Dic₁₀₆	1	2	yes	—
120	C₄×C₈	D₂₁₂	1	2	yes	—
121	#95	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	#11719917	1	2	yes	—
125	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	C₂×#88322	1	2	yes	—
131	C₂×C₈	C₂×#88322	1	2	yes	—
132	C₂×C₈	#11719917	1	2	yes	—
133	C₂×C₈	#11719917	1	2	yes	—
134	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
135	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
136	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
137	C₂×C₈	#11719945	1	2	yes	—
138	C₂×C₈	#11719945	1	2	yes	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₂×#88322	1	2	yes	—
144	C₄²	C₄⋊Dic₅₃	1	2	yes	—
145	C₁₆	—	212	4	no	—
146	C₂³	C₂×#11719917	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
154	C₂×C₄	#13133176	1	1	yes	characteristic
155	C₂×C₄	#13133176	1	1	yes	characteristic
156	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
157	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	C₄⋊#88322	1	4	yes	—
162	C₈	#13133205	1	2	yes	—
163	C₈	#13133209	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	#16810647	1	1	yes	characteristic
166	C₂²	#16810676	1	1	yes	characteristic
167	C₂²	#16810680	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	#16810489	1	1	yes	characteristic
172	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
173	C₄	#16810676	1	1	yes	characteristic
174	C₄	#16810680	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#20274832	1	1	yes	characteristic
178	C₂	#20276531	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1726	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12881
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
103. C₅₃	#12881	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
#20272799	2
#20272795	2
C₅₃×#1726	2
#12881	53