Subgroups — #27299405

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	2	yes	maximal
2	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
4	#20272793	C₂	1	2	yes	maximal
5	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	C₄⋊#11719908	C₄	1	8	yes	complemented
9	#16810399	—	2	2	no	—
10	C₄×#11719908	—	2	2	no	—
11	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
13	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#376	C₄	1	1	yes	characteristic
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	C₂×C₄	1	4	yes	—
23	C₂×#11719908	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#97×C₅₃	—	2	2	no	—
32	#97×C₅₃	—	2	2	no	—
33	#97×C₅₃	C₈	1	4	yes	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	#95×C₅₃	C₈	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	C₄²	1	2	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	2	yes	—
56	C₄²×C₅₃	—	2	1	no	—
57	C₄²×C₅₃	—	2	1	no	—
58	C₄²×C₅₃	C₄²	1	2	yes	—
59	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₄²×C₅₃	C₂×C₈	1	2	yes	—
61	C₄²×C₅₃	M₁₆	1	2	yes	—
62	#11719908	—	2	4	no	—
63	#11719908	—	2	4	no	—
64	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
70	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₈×C₅₃	—	2	2	no	—
78	C₈×C₅₃	—	2	2	no	—
79	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
80	C₈×C₅₃	C₄×D₄	1	2	yes	—
81	C₈×C₅₃	C₄×Q₈	1	2	yes	—
82	#12654	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₅₃	#431	1	1	yes	characteristic
86	C₂²×C₅₃	#504	1	4	yes	—
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	—	2	4	no	—
89	C₄×C₅₃	—	4	2	no	—
90	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₄×C₅₃	#431	1	1	yes	characteristic
94	C₂×C₄⋊C₁₆	—	53	2	no	—
95	C₂×#346	—	53	1	no	—
96	C₂×C₄×C₁₆	—	53	1	no	—
97	#1726	D₅₃	1	1	yes	characteristic
98	#1374	—	53	2	no	—
99	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₅₃	#2108	1	2	yes	—
101	C₂×C₅₃	#2151	1	2	yes	—
102	C₂×C₅₃	#2164	1	2	yes	—
103	C₂²×C₁₆	—	53	2	no	—
104	C₂²×C₁₆	—	53	2	no	—
105	C₂×#97	Dic₅₃	1	1	yes	characteristic
106	C₂×#97	Dic₅₃	1	1	yes	characteristic
107	C₂×#97	Dic₅₃	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
111	#376	Dic₅₃	1	1	yes	characteristic
112	C₄⋊C₁₆	—	53	8	no	—
113	#346	—	106	2	no	—
114	C₄×C₁₆	—	106	2	no	—
115	C₅₃	#12654	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
123	C₂×C₁₆	—	53	4	no	—
124	C₂×C₁₆	—	53	4	no	—
125	C₂×C₁₆	—	106	2	no	—
126	C₂×C₁₆	—	106	2	no	—
127	C₂×C₁₆	—	106	2	no	—
128	C₂×C₁₆	—	106	2	no	—
129	#97	—	2	2	no	—
130	#97	—	2	2	no	—
131	#97	#88322	1	4	yes	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	Dic₁₀₆	1	2	yes	—
134	C₄×C₈	C₄×D₅₃	1	4	yes	—
135	C₄×C₈	D₂₁₂	1	2	yes	—
136	#95	#88322	1	4	yes	—
137	C₂²×C₄	C₂×#88322	1	2	yes	—
138	C₂²×C₄	C₂×#88322	1	2	yes	—
139	C₂²×C₄	C₂×#88322	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	C₂×#88322	1	2	yes	—
146	C₂×C₈	C₂×#88322	1	2	yes	—
147	C₂×C₈	#11719917	1	2	yes	—
148	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
150	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
151	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
152	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
153	C₂×C₈	#11719945	1	2	yes	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	C₂×#88322	1	2	yes	—
157	C₄²	#11719917	1	2	yes	—
158	C₄²	C₄×Dic₅₃	1	2	yes	—
159	C₄²	C₄⋊Dic₅₃	1	2	yes	—
160	C₁₆	—	106	4	no	—
161	C₁₆	—	106	4	no	—
162	C₂³	C₂²×#88322	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	C₄×#88322	1	4	yes	—
168	C₂×C₄	C₄⋊#88322	1	4	yes	—
169	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂×C₄	#13133176	1	1	yes	characteristic
172	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
174	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	C₄⋊#88322	1	4	yes	—
178	C₈	C₄×D₂₁₂	1	2	yes	—
179	C₈	C₄×Dic₁₀₆	1	2	yes	—
180	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
182	C₂²	#16810676	1	1	yes	characteristic
183	C₂²	#16811040	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
188	C₄	C₄×#11719917	1	1	yes	characteristic
189	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
190	C₄	#16810676	1	1	yes	characteristic
191	C₂	C₄×C₄⋊#88322	1	1	yes	characteristic
192	C₂	#20276312	1	2	yes	—
193	C₂	#20276474	1	2	yes	—
194	C₂	#20276522	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1726	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12654
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
8. C₄⋊#11719908	C₄	semidirect
115. C₅₃	#12654	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×#16810399	2
C₂×C₄×#11719908	2
#20272793	2
C₅₃×#1726	2
#12654	53