Subgroups — #20283810

Order 6784.

322 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	#16810754	C₂	1	4	yes	maximal, complemented
2	#16810754	C₂	1	2	yes	maximal, complemented
3	Q₈×#88322	C₂	1	2	yes	maximal, complemented
4	#16810684	C₂	1	4	yes	maximal
5	#16810680	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₄⋊#88322	C₂²	1	4	yes	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	C₄⋊#88322	C₂²	1	4	yes	—
12	C₂²⋊#88322	C₂²	1	2	yes	—
13	C₂²⋊#88322	C₂²	1	2	yes	—
14	#13133128	C₂²	1	2	yes	—
15	#13133128	C₂²	1	4	yes	—
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄×#88322	C₂²	1	4	yes	—
18	C₄×#88322	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
21	#125×C₅₃	C₄	1	2	yes	—
22	C₂²⋊Q₈×C₅₃	C₄	1	2	yes	—
23	C₂²⋊Q₈×C₅₃	C₄	1	2	yes	—
24	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
26	#117×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
36	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
40	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄²×C₅₃	C₂³	1	4	yes	—
45	C₂×#88322	C₂³	1	2	yes	—
46	C₂×#88322	C₂³	1	2	yes	—
47	C₂×#88322	C₂³	1	4	yes	—
48	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₅₃	—	2	4	no	—
50	Q₈×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	#88322	—	2	2	no	—
63	#88322	—	2	2	no	—
64	#88322	—	2	4	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	4	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	4	no	—
75	C₄×C₅₃	#131	1	2	yes	—
76	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
77	#2985	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	—	4	1	no	—
79	C₂×C₅₃	#519	1	1	yes	characteristic
80	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
81	C₂×C₅₃	#568	1	1	yes	characteristic
82	#533	D₅₃	1	1	yes	characteristic
83	#446	—	53	2	no	—
84	#446	—	53	4	no	—
85	C₈×Q₈	—	53	2	no	—
86	#433	—	53	4	no	—
87	#432	—	53	1	no	—
88	#431	—	53	1	no	—
89	C₅₃	#2985	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
91	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
92	#125	Dic₅₃	1	2	yes	—
93	C₂²⋊Q₈	Dic₅₃	1	2	yes	—
94	C₂²⋊Q₈	Dic₅₃	1	2	yes	—
95	C₄×Q₈	C₂×D₅₃	1	2	yes	—
96	C₄×Q₈	C₂×D₅₃	1	2	yes	—
97	#117	C₂×D₅₃	1	2	yes	—
98	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	53	2	no	—
100	C₄⋊C₈	—	53	2	no	—
101	C₄⋊C₈	—	53	4	no	—
102	C₄⋊C₈	—	53	4	no	—
103	C₂²⋊C₈	—	53	2	no	—
104	C₂²⋊C₈	—	53	2	no	—
105	#97	—	53	2	no	—
106	#97	—	53	4	no	—
107	C₄×C₈	—	53	1	no	—
108	C₄×C₈	—	53	1	no	—
109	C₄×C₈	—	53	4	no	—
110	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
111	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
112	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
115	C₂×C₈	—	53	2	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	53	4	no	—
118	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
119	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
120	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
121	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
122	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
123	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
124	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
125	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
126	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
127	C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₄²	C₂²×D₅₃	1	4	yes	—
129	C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
137	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
141	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
142	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
144	C₈	—	106	2	no	—
145	C₈	—	106	2	no	—
146	C₈	—	106	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	4	no	—
157	C₄	#13133224	1	2	yes	—
158	C₄	#13133265	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#16811101	1	1	yes	characteristic
161	C₂	C₂×#13133224	1	1	yes	characteristic
162	C₂	#16811390	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#2985
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
2. #16810754	C₂	semidirect
3. Q₈×#88322	C₂	semidirect
89. C₅₃	#2985	semidirect