Subgroups — #27275226

Order 13568.

494 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	C₄×C₄⋊#88322	C₂	1	2	yes	maximal
2	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	#20272542	C₂	1	2	yes	maximal
5	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₄	1	4	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	C₄	1	4	yes	—
10	#16810344	—	2	2	no	—
11	C₈×#88322	—	2	2	no	—
12	C₂×#97×C₅₃	C₄	1	2	yes	—
13	C₂×#97×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	4	yes	—
18	C₂²×#88322	C₂×C₄	1	4	yes	—
19	C₄⋊#88322	—	2	8	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	—	2	2	no	—
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#97×C₅₃	—	2	4	no	—
32	#97×C₅₃	C₈	1	8	yes	complemented
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×C₈×C₅₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	4	no	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	C₂×C₈	1	4	yes	—
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	M₁₆	1	8	yes	—
60	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₅₃	#131	1	4	yes	—
76	C₈×C₅₃	—	2	4	no	—
77	C₈×C₅₃	C₄⋊C₈	1	8	yes	complemented
78	#88322	—	2	8	no	—
79	#88322	—	4	4	no	—
80	#8262	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
84	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₅₃	#405	1	2	yes	—
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	—	4	1	no	—
88	C₄×C₅₃	—	4	1	no	—
89	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
90	C₄×C₅₃	#431	1	2	yes	—
91	C₄×C₅₃	#435	1	4	yes	—
92	C₄×C₅₃	#446	1	4	yes	—
93	C₄×C₄⋊C₈	—	53	2	no	—
94	C₄×#97	D₅₃	1	1	yes	characteristic
95	C₂×#322	—	53	1	no	—
96	C₂×C₈²	—	53	1	no	—
97	#1273	—	53	2	no	—
98	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
99	C₂×C₅₃	#1453	1	1	yes	characteristic
100	C₂×C₅₃	#1455	1	1	yes	characteristic
101	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₅₃	#1771	1	2	yes	—
103	C₂×C₄⋊C₈	—	53	4	no	—
104	C₂×#97	Dic₅₃	1	2	yes	—
105	C₂×#97	Dic₅₃	1	2	yes	—
106	C₂×C₄×C₈	—	53	2	no	—
107	C₂×C₄×C₈	—	53	2	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₄³	C₂×D₅₃	1	1	yes	characteristic
111	#336	—	53	4	no	—
112	#322	—	106	2	no	—
113	C₈²	—	106	2	no	—
114	C₅₃	#8262	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	53	4	no	—
116	C₂²×C₈	—	53	4	no	—
117	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
118	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
119	C₂×C₄²	C₄×D₅₃	1	2	yes	—
120	C₂×C₄²	C₄×D₅₃	1	2	yes	—
121	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊C₈	—	106	8	no	—
125	#97	—	2	4	no	—
126	#97	#88322	1	8	yes	complemented
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	—	106	2	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	—	106	2	no	—
132	C₄×C₈	Dic₁₀₆	1	2	yes	—
133	C₄×C₈	D₂₁₂	1	2	yes	—
134	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	53	8	no	—
142	C₂×C₈	—	106	4	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	—	106	4	no	—
145	C₂×C₈	—	106	4	no	—
146	C₂×C₈	—	106	4	no	—
147	C₂×C₈	C₂×#88322	1	4	yes	—
148	C₂×C₈	C₂×#88322	1	4	yes	—
149	C₂×C₈	#11719917	1	4	yes	—
150	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	C₂×#88322	1	4	yes	—
155	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
156	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
157	C₄²	#11719945	1	2	yes	—
158	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	C₄×#88322	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₄	#13133176	1	1	yes	characteristic
168	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
169	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
170	C₂×C₄	C₂²×#88322	1	2	yes	—
171	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
172	C₂×C₄	C₂×#11719917	1	2	yes	—
173	C₂×C₄	#13133225	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	106	8	no	—
176	C₈	—	212	4	no	—
177	C₈	C₄⋊#88322	1	8	yes	complemented
178	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
180	C₂²	M₁₆×Dic₅₃	1	2	yes	—
181	C₂²	C₄×#11719917	1	1	yes	characteristic
182	C₂²	#16810581	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	C₂×C₄⋊#88322	1	2	yes	—
187	C₄	#16810676	1	2	yes	—
188	C₄	#16810696	1	4	yes	—
189	C₄	#16810751	1	4	yes	—
190	C₂	M₁₆×#88322	1	2	yes	—
191	C₂	#20273012	1	1	yes	characteristic
192	C₂	#20273018	1	1	yes	characteristic
193	C₂	C₄×C₄⋊#88322	1	1	yes	characteristic
194	C₂	#20274587	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄×#97×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#8262
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
32. #97×C₅₃	C₈	semidirect
77. C₈×C₅₃	C₄⋊C₈	semidirect
114. C₅₃	#8262	semidirect
126. #97	#88322	semidirect
177. C₈	C₄⋊#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
C₂×#16810344	2
C₂×C₈×#88322	2
#20272542	2
C₄×#97×C₅₃	2
#8262	53