Subgroups — #27270620

Order 13568.

414 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	C₄×C₄⋊#88322	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88322	C₄	1	4	yes	—
4	C₂×C₄×#88322	—	2	2	no	—
5	#16810378	—	2	2	no	—
6	C₂×C₄×C₈×C₅₃	—	2	2	no	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#88322	—	2	4	no	—
10	C₄⋊#88322	—	2	4	no	—
11	C₄⋊#88322	C₈	1	8	yes	complemented
12	C₄×#88322	—	4	2	no	—
13	C₄×#88322	—	4	2	no	—
14	C₂²×C₈×C₅₃	—	2	2	no	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	D₄	1	2	yes	—
19	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	8	no	—
21	C₄×C₈×C₅₃	—	2	2	no	—
22	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
23	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	—	2	8	no	—
29	C₂×C₈×C₅₃	—	2	4	no	—
30	C₄²×C₅₃	—	2	4	no	—
31	C₄²×C₅₃	—	2	2	no	—
32	C₄²×C₅₃	C₂×C₈	1	4	yes	—
33	C₄²×C₅₃	C₂×C₈	1	2	yes	—
34	C₄²×C₅₃	M₁₆	1	4	yes	—
35	C₄²×C₅₃	M₁₆	1	2	yes	—
36	C₄²×C₅₃	D₈	1	1	yes	characteristic
37	C₄²×C₅₃	SD₁₆	1	2	yes	—
38	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
39	C₂×#88322	—	2	8	no	—
40	C₂×#88322	—	4	4	no	—
41	C₂×#88322	—	4	4	no	—
42	C₂³×C₅₃	#95	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#97	1	2	yes	—
49	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
51	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
52	C₂×C₄×C₅₃	#102	1	2	yes	—
53	C₂×C₄×C₅₃	#103	1	2	yes	—
54	C₂×C₄×C₅₃	#104	1	4	yes	—
55	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
56	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#108	1	2	yes	—
60	C₈×C₅₃	—	2	8	no	—
61	#88322	—	4	8	no	—
62	#7015	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	#336	1	2	yes	—
64	C₂²×C₅₃	#336	1	1	yes	characteristic
65	C₂²×C₅₃	#339	1	2	yes	—
66	C₂²×C₅₃	#340	1	1	yes	characteristic
67	C₂²×C₅₃	#341	1	1	yes	characteristic
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	2	4	no	—
70	C₄×C₅₃	C₈²	1	1	yes	characteristic
71	C₄×C₅₃	#322	1	1	yes	characteristic
72	C₄×C₅₃	#322	1	2	yes	—
73	C₄×C₅₃	D₄⋊C₈	1	4	yes	—
74	C₄×C₅₃	Q₈⋊C₈	1	4	yes	—
75	C₄×C₅₃	#334	1	2	yes	—
76	C₄×C₅₃	#335	1	2	yes	—
77	C₄×C₄⋊C₈	—	53	2	no	—
78	C₄²×C₈	D₅₃	1	1	yes	characteristic
79	C₂×C₅₃	#1273	1	1	yes	characteristic
80	C₂×C₅₃	#1276	1	1	yes	characteristic
81	C₂×C₅₃	#1277	1	1	yes	characteristic
82	C₂×C₅₃	#1278	1	2	yes	—
83	C₂×C₅₃	#1279	1	2	yes	—
84	C₂×C₄⋊C₈	—	53	4	no	—
85	C₂×C₄×C₈	—	2	2	no	—
86	C₂×C₄×C₈	—	106	2	no	—
87	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
88	C₄³	C₂×D₅₃	1	1	yes	characteristic
89	#336	—	106	2	no	—
90	C₅₃	#7015	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	106	4	no	—
93	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₅₃	1	2	yes	—
95	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
98	C₄⋊C₈	—	53	8	no	—
99	C₄⋊C₈	—	106	4	no	—
100	C₄×C₈	—	2	8	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	212	2	no	—
103	C₄×C₈	—	212	2	no	—
104	C₄×C₈	#88322	1	4	yes	complemented
105	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
106	C₂²×C₄	#11719919	1	2	yes	—
107	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
108	C₂²×C₄	#11719921	1	2	yes	—
109	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	106	8	no	—
113	C₂×C₈	—	212	4	no	—
114	C₂×C₈	—	212	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	C₈×D₅₃	1	4	yes	—
118	C₄²	#11719912	1	4	yes	—
119	C₄²	#11719913	1	2	yes	—
120	C₄²	D₄₂₄	1	1	yes	characteristic
121	C₄²	Dic₂₁₂	1	1	yes	characteristic
122	C₄²	C₂×#88322	1	2	yes	—
123	C₄²	#11719917	1	2	yes	—
124	C₂³	#13133124	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
129	C₂×C₄	C₄×#88322	1	1	yes	characteristic
130	C₂×C₄	#13133127	1	2	yes	—
131	C₂×C₄	#13133128	1	1	yes	characteristic
132	C₂×C₄	#13133129	1	4	yes	—
133	C₂×C₄	C₂²⋊#88322	1	2	yes	—
134	C₂×C₄	#13133139	1	2	yes	—
135	C₂×C₄	#13133142	1	2	yes	—
136	C₂×C₄	#13133145	1	4	yes	—
137	C₂×C₄	#13133146	1	4	yes	—
138	C₂×C₄	C₄⋊#88322	1	2	yes	—
139	C₂×C₄	#13133149	1	1	yes	characteristic
140	C₂×C₄	#13133151	1	1	yes	characteristic
141	C₂×C₄	#13133153	1	2	yes	—
142	C₈	—	2	8	no	—
143	C₈	—	212	8	no	—
144	C₂²	#16810377	1	2	yes	—
145	C₂²	#16810378	1	1	yes	characteristic
146	C₂²	#16810383	1	2	yes	—
147	C₂²	#16810386	1	1	yes	characteristic
148	C₂²	#16810388	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	4	no	—
151	C₄	C₈×#88322	1	1	yes	characteristic
152	C₄	#16810344	1	1	yes	characteristic
153	C₄	#16810345	1	2	yes	—
154	C₄	#16810352	1	4	yes	—
155	C₄	#16810355	1	4	yes	—
156	C₄	#16810374	1	2	yes	—
157	C₄	#16810376	1	2	yes	—
158	C₂	#20272542	1	1	yes	characteristic
159	C₂	#20272548	1	1	yes	characteristic
160	C₂	#20272550	1	1	yes	characteristic
161	C₂	#20272552	1	2	yes	—
162	C₂	#20272555	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₂₁₂
commutator	C₄×C₅₃	C₈²
Frattini	C₄³	C₂×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#7015
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. C₄⋊#88322	C₈	semidirect
22. C₄×C₈×C₅₃	C₈	semidirect
90. C₅₃	#7015	semidirect
104. C₄×C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88322	2
C₄²×C₈×C₅₃	2
#7015	53