Subgroups — #27295710

Order 13568.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272785	C₂	1	2	yes	maximal, complemented
4	#20272779	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	1	yes	characteristic
9	C₄⋊#11719908	C₂²	1	1	yes	characteristic
10	C₄⋊#11719908	C₂²	1	2	yes	—
11	C₂²⋊#11719908	—	2	1	no	—
12	#16810399	—	2	1	no	—
13	C₅₃×#431	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#322	C₄	1	2	yes	—
17	C₈²×C₅₃	C₂²	1	2	yes	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	1	no	—
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	4	2	no	—
22	C₂×M₁₆×C₅₃	—	2	1	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
27	C₂²⋊C₈×C₅₃	C₈	1	2	yes	—
28	#97×C₅₃	—	2	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
38	M₁₆×C₅₃	—	2	4	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	4	1	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	#11719908	—	4	1	no	—
53	#11719908	—	4	2	no	—
54	#11719908	—	4	1	no	—
55	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₅₃	—	2	4	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	2	4	no	—
71	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₈×C₅₃	#137	1	1	yes	characteristic
73	#12016	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	#364	1	2	yes	—
77	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	#428	1	1	yes	characteristic
83	C₄×C₅₃	#430	1	1	yes	characteristic
84	C₄×C₅₃	#504	1	2	yes	—
85	#2163	—	53	1	no	—
86	C₂×C₄⋊C₁₆	—	53	1	no	—
87	C₈×M₁₆	D₅₃	1	1	yes	characteristic
88	#1370	—	53	2	no	—
89	#1367	—	53	2	no	—
90	C₂×C₅₃	—	2	2	no	—
91	C₂×C₅₃	#1565	1	1	yes	characteristic
92	C₂×C₅₃	#2151	1	1	yes	characteristic
93	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
94	C₂²×C₁₆	—	106	1	no	—
95	#431	Dic₅₃	1	1	yes	characteristic
96	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₄⋊C₁₆	—	53	1	no	—
99	C₄⋊C₁₆	—	53	2	no	—
100	C₄⋊C₁₆	—	53	1	no	—
101	C₄⋊C₁₆	—	106	1	no	—
102	C₂²⋊C₁₆	—	106	1	no	—
103	#346	—	106	1	no	—
104	#322	Dic₅₃	1	2	yes	—
105	C₈²	C₂×D₅₃	1	2	yes	—
106	C₅₃	#12016	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
109	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
111	C₂×C₁₆	—	106	1	no	—
112	C₂×C₁₆	—	106	1	no	—
113	C₂×C₁₆	—	106	2	no	—
114	C₂×C₁₆	—	212	2	no	—
115	C₄⋊C₈	#88322	1	2	yes	—
116	C₂²⋊C₈	#88322	1	2	yes	—
117	#97	—	2	1	no	—
118	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
119	C₄×C₈	D₂₁₂	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
121	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
122	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
125	C₂²×C₄	#11719917	1	1	yes	characteristic
126	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₂×#88322	1	4	yes	—
133	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
136	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
137	C₄²	C₂×#88322	1	1	yes	characteristic
138	C₄²	#11719917	1	1	yes	characteristic
139	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₁₆	—	212	2	no	—
142	C₁₆	—	212	1	no	—
143	C₁₆	—	212	1	no	—
144	C₂³	C₄⋊#88322	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
150	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
151	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
154	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	4	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	#13133245	1	1	yes	characteristic
161	C₈	#13133250	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#16810444	1	2	yes	—
165	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#16810665	1	1	yes	characteristic
171	C₄	#16810672	1	1	yes	characteristic
172	C₄	#16811040	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	#20273467	1	1	yes	characteristic
175	C₂	#20276474	1	1	yes	characteristic
176	C₂	C₂×#16810444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#12016
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20272785	C₂	semidirect
4. #20272779	C₂	semidirect
106. C₅₃	#12016	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
C₂×C₄⋊#11719908	2
#20272785	2
#20272779	2
C₈×M₁₆×C₅₃	2
#12016	53