Subgroups — #20275434

Order 6784.

260 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#88322	C₂	1	2	yes	maximal
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	#16810378	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	—	2	2	no	—
10	C₄⋊#88322	—	2	4	no	—
11	C₄×#88322	—	2	2	no	—
12	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#95×C₅₃	C₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	C₈	1	4	yes	complemented
24	C₄²×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	D₄	1	2	yes	—
27	C₄²×C₅₃	Q₈	1	2	yes	—
28	C₂×#88322	—	2	1	no	—
29	C₂×#88322	—	2	1	no	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	D₄	1	2	yes	—
36	C₂×#88322	Q₈	1	2	yes	—
37	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	#88322	—	2	2	no	—
57	#88322	—	2	2	no	—
58	#88322	—	4	2	no	—
59	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₅₃	#131	1	1	yes	characteristic
65	C₂²×C₅₃	#131	1	1	yes	characteristic
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	4	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	C₄⋊C₈	1	4	yes	complemented
70	C₄×C₅₃	#121	1	1	yes	characteristic
71	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄×C₅₃	#125	1	1	yes	characteristic
73	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
74	#1939	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	#389	1	1	yes	characteristic
76	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₅₃	#424	1	1	yes	characteristic
78	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
79	C₂×C₅₃	#436	1	1	yes	characteristic
80	C₂×C₅₃	C₈×Q₈	1	1	yes	characteristic
81	C₂×C₅₃	#446	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	53	2	no	—
83	C₂×C₄⋊C₈	—	53	1	no	—
84	C₂×C₄×C₈	—	53	1	no	—
85	C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
86	#336	—	53	2	no	—
87	C₅₃	#1939	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	1	no	—
89	C₂²×C₈	—	53	1	no	—
90	C₂²×C₈	—	53	2	no	—
91	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
92	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₅₃	1	2	yes	—
94	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
95	C₄⋊C₈	—	106	4	no	—
96	C₄⋊C₈	—	106	2	no	—
97	C₄×C₈	—	106	2	no	—
98	#95	Dic₅₃	1	2	yes	—
99	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
100	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
101	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
102	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
104	C₂×C₈	—	53	2	no	—
105	C₂×C₈	—	53	2	no	—
106	C₂×C₈	—	106	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	—	106	2	no	—
112	C₂×C₈	—	106	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	#88322	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	Dic₁₀₆	1	2	yes	—
118	C₄²	D₂₁₂	1	2	yes	—
119	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₂×#88322	1	4	yes	—
128	C₂×C₄	C₂×#88322	1	2	yes	—
129	C₂×C₄	#11719917	1	2	yes	—
130	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
131	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
132	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
134	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄	#11719947	1	1	yes	characteristic
136	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
137	C₂×C₄	#11719949	1	1	yes	characteristic
138	C₈	—	106	2	no	—
139	C₈	—	106	2	no	—
140	C₈	—	212	2	no	—
141	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
142	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
143	C₂²	Q₈×Dic₅₃	1	1	yes	characteristic
144	C₂²	C₂²×#88322	1	1	yes	characteristic
145	C₂²	C₂×#11719917	1	1	yes	characteristic
146	C₂²	#13133224	1	1	yes	characteristic
147	C₂²	#13133224	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊#88322	1	4	yes	complemented
152	C₄	#13133192	1	1	yes	characteristic
153	C₄	#13133197	1	1	yes	characteristic
154	C₄	#13133208	1	1	yes	characteristic
155	C₄	#13133216	1	1	yes	characteristic
156	C₂	#16810522	1	1	yes	characteristic
157	C₂	C₂×C₄⋊#88322	1	1	yes	characteristic
158	C₂	#16810650	1	1	yes	characteristic
159	C₂	D₄×#88322	1	1	yes	characteristic
160	C₂	#16810701	1	1	yes	characteristic
161	C₂	Q₈×#88322	1	1	yes	characteristic
162	C₂	#16810754	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#1939
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. C₄⋊C₄×C₅₃	C₈	semidirect
69. C₄×C₅₃	C₄⋊C₈	semidirect
87. C₅₃	#1939	semidirect
114. C₄⋊C₄	#88322	semidirect
151. C₄	C₄⋊#88322	semidirect