Subgroups — #28248485

Order 13568.

1810 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283656	C₂	1	12	yes	maximal, complemented
2	#20283650	C₂	1	12	yes	maximal, complemented
3	#20283279	C₂	1	3	yes	maximal
4	C₂×#16810579	C₂	1	3	yes	maximal
5	C₅₃×#3458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	3	yes	—
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	#16810754	C₂²	1	48	yes	complemented
9	Q₈×#88322	C₂²	1	16	yes	complemented
10	#16810650	C₂²	1	24	yes	—
11	#16810647	C₂²	1	12	yes	—
12	#16810579	C₂²	1	12	yes	—
13	#16810579	C₂²	1	12	yes	—
14	C₂×#13133128	C₂²	1	6	yes	—
15	C₂×C₄×#88322	C₂²	1	6	yes	—
16	C₅₃×#544	C₄	1	12	yes	—
17	C₅₃×#543	C₄	1	4	yes	—
18	C₅₃×#533	C₂²	1	12	yes	—
19	C₂×#117×C₅₃	C₂²	1	3	yes	—
20	C₂×#11719917	—	2	12	no	—
21	C₂×#11719917	C₂³	1	12	yes	—
22	C₂²×#88322	Q₈	1	8	yes	complemented
23	C₂²×#88322	C₂³	1	6	yes	—
24	C₂²×#88322	C₂³	1	3	yes	—
25	C₂²×#88322	C₂³	1	3	yes	—
26	C₄⋊#88322	C₂³	1	48	yes	—
27	#13133128	C₂³	1	24	yes	—
28	C₄×#88322	C₂³	1	12	yes	—
29	C₄×#88322	C₂³	1	12	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₃	C₂×C₄	1	24	yes	—
32	#125×C₅₃	C₂×C₄	1	24	yes	—
33	C₂²⋊Q₈×C₅₃	C₂×C₄	1	48	yes	—
34	C₄×Q₈×C₅₃	C₂³	1	16	yes	—
35	#117×C₅₃	C₂×C₄	1	12	yes	—
36	#117×C₅₃	C₂³	1	12	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
38	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
40	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₅₃	#41	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
47	C₄⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
48	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
49	C₂²⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
51	C₄²×C₅₃	C₂⁴	1	12	yes	—
52	#11719917	—	2	24	no	—
53	C₂×#88322	—	2	12	no	—
54	C₂×#88322	C₂×Q₈	1	12	yes	—
55	C₂×#88322	C₂×Q₈	1	12	yes	—
56	C₂×#88322	C₂⁴	1	12	yes	—
57	C₂×#88322	C₂⁴	1	1	yes	characteristic
58	C₂×#88322	C₂⁴	1	3	yes	—
59	C₂³×C₅₃	—	4	2	no	—
60	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
61	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
62	Q₈×C₅₃	—	4	16	no	—
63	C₂×C₄×C₅₃	—	4	12	no	—
64	C₂×C₄×C₅₃	C₄×Q₈	1	12	yes	—
65	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
67	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
68	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
69	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
70	#88322	—	2	12	no	—
71	#88322	C₂²×Q₈	1	2	yes	—
72	#88322	C₂²×Q₈	1	6	yes	—
73	#60191	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	6	no	—
75	C₂²×C₅₃	—	4	4	no	—
76	C₂²×C₅₃	C₂×C₄×Q₈	1	6	yes	—
77	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
78	C₄×C₅₃	—	2	24	no	—
79	C₄×C₅₃	C₂×C₄×Q₈	1	6	yes	—
80	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
81	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
82	#3458	D₅₃	1	1	yes	characteristic
83	#2961	—	53	12	no	—
84	#2960	—	53	12	no	—
85	#2905	—	53	3	no	—
86	C₂×#405	—	53	3	no	—
87	C₂×C₅₃	—	2	6	no	—
88	C₂×C₅₃	C₂²×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₅₃	#3573	1	2	yes	—
90	C₂²×M₁₆	—	53	3	no	—
91	C₂³×C₈	—	53	1	no	—
92	#544	Dic₅₃	1	12	yes	—
93	#543	Dic₅₃	1	4	yes	—
94	#533	C₂×D₅₃	1	12	yes	—
95	C₂×#117	C₂×D₅₃	1	3	yes	—
96	#446	—	53	48	no	—
97	C₈×Q₈	—	53	16	no	—
98	#424	—	53	24	no	—
99	#423	—	53	12	no	—
100	#405	—	53	12	no	—
101	#405	—	53	12	no	—
102	C₂×#97	—	53	6	no	—
103	C₂×C₄×C₈	—	53	6	no	—
104	C₅₃	#60191	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×M₁₆	—	53	12	no	—
107	C₂×M₁₆	—	106	12	no	—
108	C₂²×C₈	—	53	8	no	—
109	C₂²×C₈	—	53	3	no	—
110	C₂²×C₈	—	53	3	no	—
111	C₂²×C₈	—	53	6	no	—
112	C₄⋊Q₈	C₂×Dic₅₃	1	24	yes	—
113	#125	C₂×Dic₅₃	1	24	yes	—
114	C₂²⋊Q₈	C₂×Dic₅₃	1	48	yes	—
115	C₄×Q₈	C₂²×D₅₃	1	16	yes	—
116	#117	C₂×Dic₅₃	1	12	yes	—
117	#117	C₂²×D₅₃	1	12	yes	—
118	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
119	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	6	yes	—
120	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
121	C₄⋊C₈	—	53	48	no	—
122	#97	—	53	24	no	—
123	C₄×C₈	—	53	12	no	—
124	C₄×C₈	—	53	12	no	—
125	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
126	C₂×Q₈	C₂²×Dic₅₃	1	16	yes	—
127	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
128	C₂²×C₄	#11719949	1	4	yes	—
129	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
130	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
131	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
132	M₁₆	—	106	24	no	—
133	C₂×C₈	—	53	12	no	—
134	C₂×C₈	—	53	12	no	—
135	C₂×C₈	—	53	1	no	—
136	C₂×C₈	—	53	3	no	—
137	C₂×C₈	—	53	12	no	—
138	C₂×C₈	—	106	12	no	—
139	C₄⋊C₄	C₂²×Dic₅₃	1	24	yes	—
140	C₄⋊C₄	C₂²×Dic₅₃	1	48	yes	—
141	C₂²⋊C₄	C₂²×Dic₅₃	1	24	yes	—
142	C₄²	C₂²×Dic₅₃	1	12	yes	—
143	C₄²	C₂³×D₅₃	1	12	yes	—
144	C₂³	—	4	2	no	—
145	C₂³	Q₈×Dic₅₃	1	4	yes	—
146	C₂³	C₂³×Dic₅₃	1	3	yes	—
147	Q₈	—	4	16	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	Q₈×Dic₅₃	1	12	yes	—
150	C₂×C₄	C₂³×Dic₅₃	1	24	yes	—
151	C₂×C₄	C₂³×Dic₅₃	1	3	yes	—
152	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
153	C₂×C₄	C₂×#11719949	1	6	yes	—
154	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
155	C₈	—	53	2	no	—
156	C₈	—	53	6	no	—
157	C₈	—	106	12	no	—
158	C₂²	—	2	6	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×Q₈×Dic₅₃	1	6	yes	—
161	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
162	C₄	—	2	24	no	—
163	C₄	C₂×Q₈×Dic₅₃	1	6	yes	—
164	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
165	C₄	C₂²×#11719949	1	1	yes	characteristic
166	C₂	—	2	6	no	—
167	C₂	C₂²×Q₈×Dic₅₃	1	1	yes	characteristic
168	C₂	#20289427	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#3458	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	#3458	D₅₃
hypocenter	C₅₃	#60191
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #20283656	C₂	semidirect
2. #20283650	C₂	semidirect
8. #16810754	C₂²	semidirect
9. Q₈×#88322	C₂²	semidirect
22. C₂²×#88322	Q₈	semidirect
104. C₅₃	#60191	semidirect

Maximal subgroups

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

subgroup	index
#20283656	2
#20283650	2
#20283279	2
C₂×#16810579	2
C₅₃×#3458	2
#60191	53