Subgroups — #28293168

Order 13568.

6578 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	#20288525	C₂	1	16	yes	maximal, complemented
2	#20288521	C₂	1	32	yes	maximal, complemented
3	#20288120	C₂	1	12	yes	maximal
4	C₂×#16811101	C₂	1	2	yes	maximal
5	C₂×C₅₃×#519	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₆	C₂²	1	64	yes	complemented
7	#16811352	C₂²	1	64	yes	complemented
8	#16811349	C₂²	1	128	yes	complemented
9	#16811254	C₂²	1	48	yes	—
10	#16811251	C₂²	1	96	yes	—
11	#16811249	C₂²	1	16	yes	—
12	#16811180	C₂²	1	48	yes	—
13	#16811176	C₂²	1	96	yes	—
14	#16811101	C₂²	1	16	yes	—
15	#16811101	C₂²	1	16	yes	—
16	C₂×Q₈×Dic₅₃	C₂²	1	16	yes	—
17	C₂×#13133176	C₂²	1	12	yes	—
18	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#519	C₂²	1	16	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	8	yes	—
21	C₂×#117×C₅₃	C₂²	1	6	yes	—
22	#13133217	C₂³	1	96	yes	—
23	#13133216	C₂³	1	192	yes	—
24	#13133209	C₂³	1	96	yes	—
25	#13133208	C₂³	1	192	yes	—
26	#13133196	C₂³	1	64	yes	—
27	#13133194	C₂³	1	128	yes	—
28	Q₈×Dic₅₃	C₂³	1	128	yes	—
29	C₄×Dic₁₀₆	C₂³	1	192	yes	—
30	#13133176	C₂³	1	48	yes	—
31	#13133176	C₂³	1	48	yes	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
33	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
35	C₂²×Q₈×C₅₃	Q₈	1	16	yes	complemented
36	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
37	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
39	C₄×Q₈×C₅₃	C₂³	1	64	yes	—
40	#117×C₅₃	C₂³	1	48	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
42	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
44	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	C₂×Q₈	1	32	yes	—
46	C₂×Q₈×C₅₃	C₂×Q₈	1	64	yes	—
47	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
52	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
53	C₂×Dic₁₀₆	C₂⁴	1	64	yes	—
54	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
55	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
56	C₄²×C₅₃	C₂⁴	1	48	yes	—
57	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
58	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
59	C₄⋊Dic₅₃	C₂⁴	1	96	yes	—
60	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
61	#11719919	C₂⁴	1	128	yes	—
62	C₄×Dic₅₃	C₂⁴	1	96	yes	—
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
65	C₂³×C₅₃	C₂⁵	1	2	yes	—
66	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
67	Q₈×C₅₃	C₂²×Q₈	1	64	yes	—
68	C₂×C₄×C₅₃	—	4	8	no	—
69	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	—
70	C₂×C₄×C₅₃	C₂²×Q₈	1	48	yes	—
71	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
72	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
73	C₂×Dic₅₃	—	4	16	no	—
74	C₂×Dic₅₃	C₂⁵	1	32	yes	—
75	Dic₁₀₆	—	4	64	no	—
76	#62849	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	4	no	—
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	4	4	no	—
80	C₂²×C₅₃	C₂³×Q₈	1	4	yes	—
81	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
82	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
83	C₄×C₅₃	—	2	16	no	—
84	C₄×C₅₃	C₂³×Q₈	1	24	yes	—
85	Dic₅₃	—	2	32	no	—
86	#3502	—	53	32	no	—
87	#3502	—	53	16	no	—
88	#3458	—	53	12	no	—
89	C₂×#519	—	53	2	no	—
90	C₂×#519	D₅₃	1	1	yes	characteristic
91	C₂×C₅₃	—	2	4	no	—
92	C₂×C₅₃	—	2	2	no	—
93	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
94	C₂×C₅₃	2₋¹⁺⁶	1	2	yes	—
95	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
96	Q₈²	—	53	64	no	—
97	#557	—	53	128	no	—
98	#557	—	53	64	no	—
99	#544	—	53	96	no	—
100	#544	—	53	48	no	—
101	#543	—	53	16	no	—
102	#533	—	53	96	no	—
103	#533	—	53	48	no	—
104	#519	—	53	16	no	—
105	#519	—	53	16	no	—
106	#519	C₂×D₅₃	1	16	yes	—
107	C₂×C₄×Q₈	—	53	16	no	—
108	C₂×C₄×Q₈	C₂×D₅₃	1	8	yes	—
109	C₂×#117	—	53	12	no	—
110	C₂×#117	C₂×D₅₃	1	6	yes	—
111	C₅₃	#62849	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	Dic₁₀₆	1	16	yes	complemented
113	C₂²×Q₈	C₂²×D₅₃	1	8	yes	—
114	C₂²×Q₈	C₂²×D₅₃	1	4	yes	—
115	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
116	C₄⋊Q₈	—	53	192	no	—
117	C₄⋊Q₈	—	53	96	no	—
118	#125	—	53	192	no	—
119	#125	—	53	96	no	—
120	C₂²⋊Q₈	—	53	64	no	—
121	C₂²⋊Q₈	—	53	128	no	—
122	C₄×Q₈	—	53	128	no	—
123	C₄×Q₈	—	53	192	no	—
124	C₄×Q₈	C₂²×D₅₃	1	64	yes	—
125	#117	—	53	48	no	—
126	#117	—	53	48	no	—
127	#117	C₂²×D₅₃	1	48	yes	—
128	C₂×C₄⋊C₄	—	53	24	no	—
129	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
130	C₂×C₂²⋊C₄	—	53	8	no	—
131	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
132	C₂×C₄²	—	53	24	no	—
133	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
134	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
135	C₂×Q₈	—	53	64	no	—
136	C₂×Q₈	C₂×Dic₁₀₆	1	64	yes	—
137	C₂×Q₈	C₂×Dic₁₀₆	1	32	yes	—
138	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
139	C₂²×C₄	—	53	16	no	—
140	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
141	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
142	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
143	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
144	C₄⋊C₄	—	53	96	no	—
145	C₄⋊C₄	—	53	128	no	—
146	C₄⋊C₄	—	53	64	no	—
147	C₄⋊C₄	C₂³×D₅₃	1	48	yes	—
148	C₂²⋊C₄	—	53	16	no	—
149	C₂²⋊C₄	—	53	16	no	—
150	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
151	C₄²	—	53	96	no	—
152	C₄²	C₂³×D₅₃	1	48	yes	—
153	C₂³	—	4	2	no	—
154	C₂³	C₂²×Dic₁₀₆	1	4	yes	—
155	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
156	C₂³	C₂⁴×D₅₃	1	2	yes	—
157	Q₈	—	212	64	no	—
158	Q₈	C₂²×Dic₁₀₆	1	64	yes	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	—	53	32	no	—
161	C₂×C₄	—	212	16	no	—
162	C₂×C₄	C₂²×Dic₁₀₆	1	48	yes	—
163	C₂×C₄	C₂²×Dic₁₀₆	1	24	yes	—
164	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
165	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	4	4	no	—
169	C₂²	C₂³×Dic₁₀₆	1	2	yes	—
170	C₂²	C₂³×Dic₁₀₆	1	4	yes	—
171	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
172	C₄	—	2	16	no	—
173	C₄	—	106	32	no	—
174	C₄	C₂³×Dic₁₀₆	1	24	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	4	no	—
177	C₂	C₂⁴×Dic₁₀₆	1	1	yes	characteristic
178	C₂	#20289527	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₂×C₅₃×#519	C₂
prime radical	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#519	D₅₃
hypocenter	C₅₃	#62849
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
1. #20288525	C₂	semidirect
2. #20288521	C₂	semidirect
6. Q₈×Dic₁₀₆	C₂²	semidirect
7. #16811352	C₂²	semidirect
8. #16811349	C₂²	semidirect
35. C₂²×Q₈×C₅₃	Q₈	semidirect
111. C₅₃	#62849	semidirect
112. C₂²×Q₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20288525	2
#20288521	2
#20288120	2
C₂×#16811101	2
C₂×C₅₃×#519	2
#62849	53