Subgroups — #27349683

Order 13568.

890 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#88322	C₂	1	2	yes	maximal
2	#20272998	C₂	1	3	yes	maximal, complemented
3	M₁₆×#88322	C₂	1	6	yes	maximal, complemented
4	C₂×#16810344	C₂	1	3	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	C₄	1	4	yes	—
7	D₄×#88322	C₄	1	12	yes	—
8	#16810676	C₂²	1	6	yes	—
9	C₂×C₄×#88322	C₂²	1	6	yes	—
10	#16810344	C₂²	1	1	yes	characteristic
11	#16810344	C₂²	1	6	yes	—
12	#16810344	C₂²	1	3	yes	—
13	C₈×#88322	C₂²	1	6	yes	—
14	C₂×#131×C₅₃	C₄	1	2	yes	—
15	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#405	C₄	1	6	yes	—
17	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
19	C₂²×#88322	C₂×C₄	1	12	yes	—
20	C₄⋊#88322	C₂×C₄	1	12	yes	—
21	C₂²⋊#88322	C₂×C₄	1	12	yes	—
22	C₄×#88322	C₂×C₄	1	12	yes	—
23	C₄×#88322	C₂³	1	2	yes	—
24	C₄×#88322	C₂³	1	6	yes	—
25	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
26	#131×C₅₃	C₈	1	16	yes	complemented
27	C₂×M₁₆×C₅₃	C₂×C₄	1	6	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
29	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×D₄×C₅₃	C₂×C₄	1	6	yes	—
31	#117×C₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
33	#97×C₅₃	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
35	C₄×C₈×C₅₃	C₂³	1	3	yes	—
36	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	#41×C₅₃	C₂×C₈	1	8	yes	—
38	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×D₄×C₅₃	C₄²	1	3	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
42	M₁₆×C₅₃	C₂×C₈	1	24	yes	complemented
43	C₂×C₈×C₅₃	C₂×C₈	1	24	yes	complemented
44	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
46	C₄⋊C₄×C₅₃	C₄²	1	6	yes	—
47	C₂²⋊C₄×C₅₃	C₄²	1	6	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
49	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
50	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×#88322	—	4	12	no	—
52	C₂×#88322	C₂²×C₄	1	4	yes	—
53	C₂×#88322	C₂²×C₄	1	12	yes	—
54	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
55	Q₈×C₅₃	C₄×C₈	1	4	yes	—
56	D₄×C₅₃	C₄×C₈	1	12	yes	—
57	C₂×C₄×C₅₃	—	2	12	no	—
58	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
59	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
61	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₅₃	C₂²×C₈	1	4	yes	—
64	C₈×C₅₃	C₂²×C₈	1	12	yes	—
65	#88322	—	2	4	no	—
66	#88322	—	2	12	no	—
67	#17316	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	6	no	—
69	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
70	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
71	C₄×C₅₃	—	2	6	no	—
72	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
73	C₄×C₅₃	C₂³×C₈	1	2	yes	—
74	C₄×C₅₃	#568	1	4	yes	—
75	C₈×#41	—	53	2	no	—
76	C₄×#131	D₅₃	1	1	yes	characteristic
77	#1450	—	53	3	no	—
78	C₈×M₁₆	—	53	6	no	—
79	C₂×#322	—	53	3	no	—
80	C₂×C₅₃	—	2	6	no	—
81	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
82	C₂×C₅₃	#2875	1	2	yes	—
83	C₂×#131	Dic₅₃	1	2	yes	—
84	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
85	C₈×Q₈	—	53	4	no	—
86	C₈×D₄	—	53	12	no	—
87	#431	—	53	6	no	—
88	#405	Dic₅₃	1	6	yes	—
89	C₄×M₁₆	C₂×D₅₃	1	3	yes	—
90	C₂×C₄×C₈	—	53	6	no	—
91	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
92	#322	—	53	1	no	—
93	#322	—	53	6	no	—
94	#322	—	53	3	no	—
95	C₈²	—	53	6	no	—
96	C₅₃	#17316	1	1	yes	characteristic, complemented, Sylow
97	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
98	#131	#88322	1	16	yes	complemented
99	C₂×M₁₆	C₂×Dic₅₃	1	6	yes	—
100	C₂²×C₈	—	53	12	no	—
101	C₂²×C₈	C₂×Dic₅₃	1	6	yes	—
102	C₄×Q₈	C₄×D₅₃	1	2	yes	—
103	C₄×D₄	C₄×D₅₃	1	6	yes	—
104	#117	C₂×Dic₅₃	1	3	yes	—
105	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
106	C₄⋊C₈	—	53	12	no	—
107	C₂²⋊C₈	—	53	12	no	—
108	#97	C₂×Dic₅₃	1	6	yes	—
109	C₄×C₈	—	53	12	no	—
110	C₄×C₈	—	53	6	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
113	C₄×C₈	C₂²×D₅₃	1	3	yes	—
114	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	#41	C₂×#88322	1	8	yes	—
116	C₂×Q₈	C₄×Dic₅₃	1	1	yes	characteristic
117	C₂×D₄	C₄×Dic₅₃	1	3	yes	—
118	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
119	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
120	M₁₆	C₂×#88322	1	24	yes	complemented
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	53	12	no	—
123	C₂×C₈	—	212	12	no	—
124	C₂×C₈	C₂×#88322	1	24	yes	complemented
125	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
126	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
127	C₄⋊C₄	C₄×Dic₅₃	1	6	yes	—
128	C₂²⋊C₄	C₄×Dic₅₃	1	6	yes	—
129	C₄²	C₂×C₄×D₅₃	1	6	yes	—
130	C₄²	C₂²×Dic₅₃	1	3	yes	—
131	C₄²	C₂³×D₅₃	1	1	yes	characteristic
132	C₂³	C₂×C₄×Dic₅₃	1	3	yes	—
133	Q₈	C₄×#88322	1	4	yes	—
134	D₄	C₄×#88322	1	12	yes	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
137	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
138	C₂×C₄	C₂²×#88322	1	12	yes	—
139	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
140	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
141	C₈	—	106	12	no	—
142	C₈	—	106	4	no	—
143	C₈	C₂²×#88322	1	4	yes	—
144	C₈	C₂²×#88322	1	12	yes	—
145	C₂²	—	2	6	no	—
146	C₂²	C₂×C₄×#88322	1	6	yes	—
147	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	C₂×C₄×#88322	1	6	yes	—
150	C₄	C₂³×#88322	1	2	yes	—
151	C₄	#16811388	1	4	yes	—
152	C₂	—	2	6	no	—
153	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
154	C₂	#20283141	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#17316
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20272998	C₂	semidirect
3. M₁₆×#88322	C₂	semidirect
4. C₂×#16810344	C₂	semidirect
26. #131×C₅₃	C₈	semidirect
42. M₁₆×C₅₃	C₂×C₈	semidirect
43. C₂×C₈×C₅₃	C₂×C₈	semidirect
96. C₅₃	#17316	semidirect
98. #131	#88322	semidirect
120. M₁₆	C₂×#88322	semidirect
124. C₂×C₈	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
#41×#88322	2
#20272998	2
M₁₆×#88322	2
C₂×#16810344	2
C₄×#131×C₅₃	2
#17316	53