Subgroups — #28249792

Order 13568.

1964 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	#20283847	C₂	1	6	yes	maximal, complemented
2	#20283782	C₂	1	9	yes	maximal, complemented
3	#41×#88322	C₂	1	6	yes	maximal, complemented
4	#20283427	C₂	1	9	yes	maximal, complemented
5	C₅₃×#3483	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133224	C₂²	1	6	yes	—
7	Q₈×#88322	C₂²	1	2	yes	—
8	Q₈×#88322	C₂²	1	6	yes	—
9	#16810701	C₂²	1	36	yes	—
10	D₄×#88322	C₂²	1	18	yes	—
11	D₄×#88322	C₂²	1	18	yes	—
12	#16810676	C₂²	1	18	yes	—
13	#16810647	C₂²	1	9	yes	—
14	#16810590	C₂²	1	18	yes	—
15	C₂×C₄×#88322	C₂²	1	9	yes	—
16	Q₈²×C₅₃	C₄	1	1	yes	characteristic
17	C₅₃×#555	C₄	1	9	yes	—
18	C₅₃×#550	C₄	1	6	yes	—
19	C₅₃×#532	C₂²	1	9	yes	—
20	C₄×#41×C₅₃	C₂²	1	6	yes	—
21	#13133224	—	4	12	no	—
22	C₂×#11719917	C₂³	1	18	yes	—
23	C₂²×#88322	C₂³	1	9	yes	—
24	C₂²×#88322	C₂³	1	9	yes	—
25	C₄⋊#88322	C₂³	1	18	yes	—
26	C₄⋊#88322	C₂³	1	6	yes	—
27	C₂²⋊#88322	C₂³	1	36	yes	—
28	C₄×#88322	C₂³	1	6	yes	—
29	C₄×#88322	C₂³	1	18	yes	—
30	C₂×#41×C₅₃	C₂×C₄	1	6	yes	—
31	C₂×#41×C₅₃	C₂³	1	6	yes	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	9	yes	—
33	#127×C₅₃	C₂×C₄	1	9	yes	—
34	#124×C₅₃	C₂×C₄	1	18	yes	—
35	#121×C₅₃	C₂×C₄	1	36	yes	—
36	C₄×Q₈×C₅₃	C₂×C₄	1	6	yes	—
37	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
38	C₄×D₄×C₅₃	C₂×C₄	1	18	yes	—
39	C₄×D₄×C₅₃	C₂³	1	18	yes	—
40	#117×C₅₃	C₂×C₄	1	18	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	9	yes	—
42	#41×C₅₃	—	2	24	no	—
43	#41×C₅₃	—	4	6	no	—
44	#41×C₅₃	—	4	6	no	—
45	C₂×Q₈×C₅₃	C₂²×C₄	1	6	yes	—
46	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×D₄×C₅₃	C₂²×C₄	1	18	yes	—
48	C₂×D₄×C₅₃	C₂²×C₄	1	18	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	18	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
51	C₄⋊C₄×C₅₃	C₂²×C₄	1	6	yes	—
52	C₄⋊C₄×C₅₃	C₂²×C₄	1	18	yes	—
53	C₂²⋊C₄×C₅₃	C₂²×C₄	1	36	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	9	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	9	yes	—
56	C₄²×C₅₃	C₂⁴	1	6	yes	—
57	#11719917	—	4	18	no	—
58	C₂×#88322	—	2	18	no	—
59	C₂×#88322	—	4	9	no	—
60	C₂×#88322	C₂⁴	1	6	yes	—
61	C₂×#88322	C₂⁴	1	9	yes	—
62	C₂×#88322	C₂⁴	1	1	yes	characteristic
63	C₂³×C₅₃	C₂³×C₄	1	9	yes	—
64	Q₈×C₅₃	—	4	6	no	—
65	Q₈×C₅₃	#131	1	8	yes	complemented
66	D₄×C₅₃	—	2	36	no	—
67	D₄×C₅₃	—	4	18	no	—
68	C₂×C₄×C₅₃	—	2	36	no	—
69	C₂×C₄×C₅₃	—	2	9	no	—
70	C₂×C₄×C₅₃	—	2	9	no	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
72	C₂×C₄×C₅₃	C₂³×C₄	1	9	yes	—
73	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
74	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
75	#88322	—	2	9	no	—
76	#88322	—	2	6	no	—
77	#88322	H(5,𝔽₂)	1	2	yes	—
78	#60308	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	9	no	—
80	C₂²×C₅₃	—	2	18	no	—
81	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
82	C₄×C₅₃	—	2	9	no	—
83	C₄×C₅₃	—	2	6	no	—
84	C₄×C₅₃	C₂×#131	1	12	yes	—
85	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
86	C₄×C₅₃	#585	1	1	yes	characteristic
87	#3483	D₅₃	1	1	yes	characteristic
88	#2991	—	53	6	no	—
89	#2979	—	53	9	no	—
90	C₈×#41	—	53	6	no	—
91	#2931	—	53	9	no	—
92	C₂×C₅₃	—	4	9	no	—
93	C₂×C₅₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
94	C₂×C₅₃	C₂²×#131	1	2	yes	—
95	C₂×#131	—	53	6	no	—
96	Q₈²	Dic₅₃	1	1	yes	characteristic
97	#555	Dic₅₃	1	9	yes	—
98	#550	Dic₅₃	1	6	yes	—
99	#532	C₂×D₅₃	1	9	yes	—
100	C₄×#41	C₂×D₅₃	1	6	yes	—
101	C₈×Q₈	—	53	2	no	—
102	C₈×Q₈	—	53	6	no	—
103	#436	—	53	36	no	—
104	C₈×D₄	—	53	18	no	—
105	C₈×D₄	—	53	18	no	—
106	#431	—	53	18	no	—
107	#423	—	53	9	no	—
108	#408	—	53	18	no	—
109	C₂×C₄×C₈	—	53	9	no	—
110	C₅₃	#60308	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	C₂×Dic₅₃	1	6	yes	—
112	C₂×#41	C₂²×D₅₃	1	6	yes	—
113	#131	—	212	12	no	—
114	C₂×M₁₆	—	53	18	no	—
115	C₂²×C₈	—	53	9	no	—
116	C₂²×C₈	—	53	9	no	—
117	C₄⋊Q₈	C₂×Dic₅₃	1	9	yes	—
118	#127	C₂×Dic₅₃	1	9	yes	—
119	#124	C₂×Dic₅₃	1	18	yes	—
120	#121	C₂×Dic₅₃	1	36	yes	—
121	C₄×Q₈	C₂×Dic₅₃	1	6	yes	—
122	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
123	C₄×D₄	C₂×Dic₅₃	1	18	yes	—
124	C₄×D₄	C₂²×D₅₃	1	18	yes	—
125	#117	C₂×Dic₅₃	1	18	yes	—
126	C₂×C₄²	C₂²×D₅₃	1	9	yes	—
127	C₄⋊C₈	—	53	18	no	—
128	C₄⋊C₈	—	53	6	no	—
129	C₂²⋊C₈	—	53	36	no	—
130	C₄×C₈	—	53	6	no	—
131	C₄×C₈	—	53	18	no	—
132	#41	—	2	24	no	—
133	#41	—	4	6	no	—
134	#41	—	4	6	no	—
135	C₂×Q₈	C₂²×Dic₅₃	1	6	yes	—
136	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
137	C₂×D₄	C₂²×Dic₅₃	1	18	yes	—
138	C₂×D₄	C₂²×Dic₅₃	1	18	yes	—
139	C₂²×C₄	C₂²×Dic₅₃	1	18	yes	—
140	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
141	M₁₆	—	212	18	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	6	no	—
144	C₂×C₈	—	53	9	no	—
145	C₂×C₈	—	106	18	no	—
146	C₂×C₈	—	212	9	no	—
147	C₄⋊C₄	C₂²×Dic₅₃	1	6	yes	—
148	C₄⋊C₄	C₂²×Dic₅₃	1	18	yes	—
149	C₂²⋊C₄	C₂²×Dic₅₃	1	36	yes	—
150	C₄²	C₂²×Dic₅₃	1	9	yes	—
151	C₄²	C₂²×Dic₅₃	1	9	yes	—
152	C₄²	C₂³×D₅₃	1	6	yes	—
153	C₂³	C₂³×Dic₅₃	1	9	yes	—
154	Q₈	—	4	6	no	—
155	Q₈	#13133224	1	8	yes	complemented
156	D₄	—	2	36	no	—
157	D₄	—	4	18	no	—
158	C₂×C₄	—	2	36	no	—
159	C₂×C₄	—	2	9	no	—
160	C₂×C₄	—	2	9	no	—
161	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
162	C₂×C₄	C₂³×Dic₅₃	1	9	yes	—
163	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
164	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
165	C₈	—	53	2	no	—
166	C₈	—	106	9	no	—
167	C₈	—	106	6	no	—
168	C₂²	—	2	18	no	—
169	C₂²	—	2	9	no	—
170	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
171	C₄	—	2	6	no	—
172	C₄	—	2	9	no	—
173	C₄	C₂×#13133224	1	12	yes	—
174	C₄	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
175	C₄	#16811458	1	1	yes	characteristic
176	C₂	—	4	9	no	—
177	C₂	H(5,𝔽₂)×Dic₅₃	1	1	yes	characteristic
178	C₂	C₂²×#13133224	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁴×C₄
Fitting, prime radical	C₅₃×#3483	C₂
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	#3483	D₅₃
hypocenter	C₅₃	#60308
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #3483

Complements

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

N	H ≅ G/N	type
1. #20283847	C₂	semidirect
2. #20283782	C₂	semidirect
3. #41×#88322	C₂	semidirect
4. #20283427	C₂	semidirect
65. Q₈×C₅₃	#131	semidirect
110. C₅₃	#60308	semidirect
155. Q₈	#13133224	semidirect

Maximal subgroups

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

subgroup	index
#20283847	2
#20283782	2
#41×#88322	2
#20283427	2
C₅₃×#3483	2
#60308	53