Subgroups — #27277833

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	4	no	—
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	#16810376	—	2	8	no	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×#88322	—	2	4	no	—
13	C₄⋊#88322	—	2	8	no	—
14	C₄⋊#88322	—	4	8	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	D₄	1	2	yes	—
20	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₅₃	—	2	4	no	—
23	C₄×C₈×C₅₃	—	2	2	no	—
24	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
25	C₄×C₈×C₅₃	D₄	1	4	yes	—
26	C₄×C₈×C₅₃	Q₈	1	4	yes	—
27	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
38	C₄²×C₅₃	—	2	4	no	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	C₂×C₈	1	2	yes	—
41	C₄²×C₅₃	M₁₆	1	2	yes	—
42	C₄²×C₅₃	D₈	1	4	yes	—
43	C₄²×C₅₃	Q₁₆	1	4	yes	—
44	C₄²×C₅₃	C₂×D₄	1	2	yes	—
45	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
46	C₂×#88322	—	4	4	no	—
47	C₂×#88322	—	4	8	no	—
48	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
53	C₂×C₄×C₅₃	#107	1	4	yes	—
54	C₂×C₄×C₅₃	#107	1	4	yes	—
55	C₂×C₄×C₅₃	#108	1	4	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
65	C₈×C₅₃	—	2	4	no	—
66	C₈×C₅₃	C₄⋊C₈	1	8	yes	complemented
67	#88322	—	8	4	no	—
68	#8806	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	#384	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂²×C₅₃	#423	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×#107	1	2	yes	—
73	C₂²×C₅₃	C₂×#108	1	2	yes	—
74	C₄×C₅₃	—	2	4	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	#335	1	8	yes	complemented
77	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
78	C₄×C₅₃	#423	1	2	yes	—
79	C₄×C₅₃	#493	1	1	yes	characteristic
80	C₄×C₅₃	#494	1	1	yes	characteristic
81	C₄×C₅₃	#500	1	2	yes	—
82	#1842	—	53	2	no	—
83	C₄²×C₈	D₅₃	1	1	yes	characteristic
84	C₂×#335	—	53	4	no	—
85	C₂×C₅₃	C₂×#335	1	2	yes	—
86	C₂×C₅₃	#1567	1	2	yes	—
87	C₂×C₅₃	#1842	1	1	yes	characteristic
88	C₂×C₅₃	#1845	1	1	yes	characteristic
89	C₂×C₅₃	#1848	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	53	4	no	—
91	C₂×C₄⋊C₈	—	106	4	no	—
92	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
93	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
94	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
96	C₄³	C₂×D₅₃	1	1	yes	characteristic
97	#335	—	106	8	no	—
98	C₅₃	#8806	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	106	4	no	—
100	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
101	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
102	C₂×C₄²	Dic₁₀₆	1	2	yes	—
103	C₂×C₄²	D₂₁₂	1	2	yes	—
104	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
107	C₄⋊C₈	—	106	8	no	—
108	C₄⋊C₈	—	212	8	no	—
109	C₄×C₈	—	2	4	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	#88322	1	4	yes	complemented
112	C₄×C₈	Dic₁₀₆	1	4	yes	—
113	C₄×C₈	D₂₁₂	1	4	yes	—
114	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
115	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
116	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
118	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	212	4	no	—
122	C₂×C₈	—	212	8	no	—
123	C₂×C₈	C₂×#88322	1	4	yes	—
124	C₂×C₈	#11719917	1	4	yes	—
125	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
126	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
127	C₄²	—	2	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	D₄₂₄	1	4	yes	—
130	C₄²	Dic₂₁₂	1	4	yes	—
131	C₄²	C₂×#88322	1	2	yes	—
132	C₄²	#11719917	1	2	yes	—
133	C₄²	C₂×Dic₁₀₆	1	2	yes	—
134	C₄²	C₂×D₂₁₂	1	2	yes	—
135	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₄⋊#88322	1	4	yes	—
140	C₂×C₄	#13133151	1	4	yes	—
141	C₂×C₄	#13133151	1	4	yes	—
142	C₂×C₄	#13133153	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
144	C₂×C₄	#13133214	1	1	yes	characteristic
145	C₂×C₄	#13133217	1	2	yes	—
146	C₂×C₄	#13133217	1	1	yes	characteristic
147	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
148	C₂×C₄	C₂×#11719917	1	2	yes	—
149	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
150	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
151	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	—	424	4	no	—
154	C₈	C₄⋊#88322	1	8	yes	complemented
155	C₂²	#16810493	1	1	yes	characteristic
156	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
157	C₂²	#16810647	1	1	yes	characteristic
158	C₂²	C₂×#13133151	1	2	yes	—
159	C₂²	C₂×#13133153	1	2	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	#16810376	1	8	yes	complemented
163	C₄	C₂×C₄⋊#88322	1	2	yes	—
164	C₄	#16810647	1	2	yes	—
165	C₄	#16811000	1	1	yes	characteristic
166	C₄	#16811003	1	1	yes	characteristic
167	C₄	#16811028	1	2	yes	—
168	C₂	C₂×#16810376	1	2	yes	—
169	C₂	#20273474	1	2	yes	—
170	C₂	#20274867	1	1	yes	characteristic
171	C₂	#20274875	1	1	yes	characteristic
172	C₂	#20274887	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8806
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₅₃	C₈	semidirect
66. C₈×C₅₃	C₄⋊C₈	semidirect
76. C₄×C₅₃	#335	semidirect
98. C₅₃	#8806	semidirect
111. C₄×C₈	#88322	semidirect
154. C₈	C₄⋊#88322	semidirect
162. C₄	#16810376	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₂×#16810376	2
C₄²×C₈×C₅₃	2
#8806	53