Subgroups — #27537080

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	2	yes	maximal
2	#20276474	C₂	1	8	yes	maximal, complemented
3	#20276471	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	8	no	—
6	C₂×#13133157	C₂²	1	4	yes	—
7	C₂×#13133157	C₂²	1	4	yes	—
8	C₂²×#11719908	C₂²	1	4	yes	—
9	C₄⋊#11719908	C₂²	1	16	yes	complemented
10	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#405	C₄	1	4	yes	—
14	C₅₃×#405	C₂²	1	4	yes	—
15	C₂×#97×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	#13133157	—	2	8	no	—
18	#13133157	—	2	8	no	—
19	C₂×#11719908	—	2	8	no	—
20	C₂×#11719908	C₂³	1	8	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	4	no	—
23	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	D₄	1	4	yes	—
28	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	#117×C₅₃	C₈	1	4	yes	—
32	#117×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
34	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	#97×C₅₃	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₃	C₂³	1	4	yes	—
39	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	M₁₆×C₅₃	—	2	8	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
60	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
61	C₄²×C₅₃	C₂×C₈	1	4	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
63	#11719908	—	2	8	no	—
64	C₂³×C₅₃	—	4	2	no	—
65	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
66	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
78	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₅₃	—	2	4	no	—
80	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
82	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
84	#33397	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	4	4	no	—
88	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
89	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
90	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₅₃	—	2	8	no	—
92	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
93	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
94	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
96	C₂²×M₃₂	—	53	2	no	—
97	C₂×#405	D₅₃	1	1	yes	characteristic
98	#2151	—	53	8	no	—
99	#2150	—	53	4	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	—	2	4	no	—
102	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₅₃	#3407	1	2	yes	—
104	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
105	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
106	C₂×#117	Dic₅₃	1	1	yes	characteristic
107	C₂×M₃₂	—	53	4	no	—
108	C₂×M₃₂	—	53	4	no	—
109	C₂×M₃₂	—	106	8	no	—
110	C₂²×C₁₆	—	53	4	no	—
111	#405	Dic₅₃	1	4	yes	—
112	#405	C₂×D₅₃	1	4	yes	—
113	C₂×#97	Dic₅₃	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
115	C₄⋊C₁₆	—	53	16	no	—
116	C₅₃	#33397	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂×M₁₆	—	2	4	no	—
119	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
120	C₂²×C₈	Dic₁₀₆	1	4	yes	—
121	C₂²×C₈	D₂₁₂	1	4	yes	—
122	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
123	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
125	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
127	#117	#88322	1	4	yes	—
128	#117	C₂×Dic₅₃	1	4	yes	—
129	C₂×C₄⋊C₄	#88322	1	2	yes	—
130	C₂×C₂²⋊C₄	#88322	1	2	yes	—
131	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
132	M₃₂	—	106	8	no	—
133	M₃₂	—	106	8	no	—
134	C₂×C₁₆	—	53	8	no	—
135	C₂×C₁₆	—	106	8	no	—
136	#97	C₂×Dic₅₃	1	8	yes	—
137	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
138	C₄×C₈	C₂²×D₅₃	1	4	yes	—
139	C₂⁴	C₂×#88322	1	1	yes	characteristic
140	C₂²×C₄	C₂×#88322	1	4	yes	—
141	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
142	C₂²×C₄	C₂×#88322	1	2	yes	—
143	C₂²×C₄	#11719917	1	4	yes	—
144	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
145	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
147	M₁₆	—	2	8	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
150	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
151	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
152	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
153	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
154	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
155	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
156	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
158	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
159	C₄⋊C₄	C₂×#88322	1	8	yes	—
160	C₂²⋊C₄	C₂×#88322	1	8	yes	—
161	C₄²	C₂×#88322	1	4	yes	—
162	C₄²	C₂²×Dic₅₃	1	4	yes	—
163	C₁₆	—	106	8	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₄⋊#88322	1	4	yes	—
166	C₂³	C₂²×#88322	1	2	yes	—
167	C₂³	C₂²×#88322	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₄⋊#88322	1	4	yes	—
170	C₂×C₄	C₄⋊#88322	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
173	C₂×C₄	C₂²×#88322	1	8	yes	—
174	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
175	C₂×C₄	C₂²×#88322	1	2	yes	—
176	C₂×C₄	C₂×#11719917	1	2	yes	—
177	C₂×C₄	C₂×#11719917	1	4	yes	—
178	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
181	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
182	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
183	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₄⋊#88322	1	2	yes	—
188	C₂²	C₂×C₄⋊#88322	1	4	yes	—
189	C₂²	C₂³×#88322	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₄⋊#88322	1	2	yes	—
192	C₄	C₂×C₄⋊#88322	1	4	yes	—
193	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
194	C₄	C₂²×#11719917	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
198	C₂	#20287818	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×#405	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#33397
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20276474	C₂	semidirect
3. #20276471	C₂	semidirect
9. C₄⋊#11719908	C₂²	semidirect
116. C₅₃	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
#20276474	2
#20276471	2
C₂×C₅₃×#405	2
#33397	53