Subgroups — #27299422

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	2	yes	maximal
2	C₂×#16810401	C₂	1	2	yes	maximal
3	#20272795	C₂	1	2	yes	maximal
4	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	C₄⋊#11719908	C₄	1	8	yes	complemented
7	#16810401	—	2	4	no	—
8	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
9	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄³×C₅₃	C₄	1	1	yes	characteristic
15	#13133157	—	2	4	no	—
16	#13133157	—	2	4	no	—
17	C₂×#11719908	C₂×C₄	1	8	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	#97×C₅₃	—	2	2	no	—
27	#97×C₅₃	—	2	2	no	—
28	#97×C₅₃	—	2	2	no	—
29	#97×C₅₃	D₄	1	2	yes	—
30	#97×C₅₃	Q₈	1	2	yes	—
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₅₃	—	2	1	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	C₄²	1	2	yes	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	#41	1	2	yes	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	—	4	4	no	—
54	C₄²×C₅₃	C₄²	1	2	yes	—
55	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₅₃	M₁₆	1	2	yes	—
57	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
58	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
59	#11719908	—	2	8	no	—
60	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	#97	1	2	yes	—
67	C₂×C₄×C₅₃	#97	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₅₃	—	2	2	no	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	—	2	2	no	—
77	C₈×C₅₃	C₄×D₄	1	2	yes	—
78	C₈×C₅₃	C₄×Q₈	1	2	yes	—
79	#12658	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
82	C₂²×C₅₃	#423	1	1	yes	characteristic
83	C₂²×C₅₃	#432	1	1	yes	characteristic
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	4	2	no	—
86	C₄×C₅₃	—	4	2	no	—
87	C₄×C₅₃	#347	1	4	yes	complemented
88	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
90	C₄×C₅₃	#423	1	1	yes	characteristic
91	C₄×C₅₃	#432	1	1	yes	characteristic
92	#2150	—	53	2	no	—
93	C₂×#347	—	53	2	no	—
94	C₄×#97	D₅₃	1	1	yes	characteristic
95	#1375	—	53	2	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	#1767	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#347	1	2	yes	—
100	C₂×M₃₂	—	53	4	no	—
101	C₂×#97	Dic₅₃	1	1	yes	characteristic
102	C₂×#97	Dic₅₃	1	1	yes	characteristic
103	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
104	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
105	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₄³	Dic₅₃	1	1	yes	characteristic
108	C₄⋊C₁₆	—	53	8	no	—
109	#347	—	106	4	no	—
110	C₅₃	#12658	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
119	M₃₂	—	106	4	no	—
120	M₃₂	—	106	4	no	—
121	C₂×C₁₆	—	53	8	no	—
122	#97	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	Dic₁₀₆	1	2	yes	—
126	#97	D₂₁₂	1	2	yes	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₄×D₅₃	1	4	yes	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	#11719917	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	#11719917	1	2	yes	—
141	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
142	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
143	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
145	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
146	C₂×C₈	#11719945	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	#11719917	1	2	yes	—
151	C₄²	#11719917	1	1	yes	characteristic
152	C₄²	#11719917	1	1	yes	characteristic
153	C₄²	C₄×Dic₅₃	1	2	yes	—
154	C₄²	C₄⋊Dic₅₃	1	2	yes	—
155	C₁₆	—	106	8	no	—
156	C₂³	C₂×#11719917	1	1	yes	characteristic
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	#13133128	1	2	yes	—
163	C₂×C₄	#13133128	1	2	yes	—
164	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
166	C₂×C₄	#13133176	1	1	yes	characteristic
167	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
168	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
169	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	C₄×D₂₁₂	1	2	yes	—
174	C₈	C₄×Dic₁₀₆	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	C₂×#13133128	1	1	yes	characteristic
177	C₂²	#16810647	1	1	yes	characteristic
178	C₂²	#16810680	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	#16810401	1	4	yes	complemented
183	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
184	C₄	C₄×#11719917	1	1	yes	characteristic
185	C₄	#16810647	1	1	yes	characteristic
186	C₄	#16810680	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#20274568	1	1	yes	characteristic
190	C₂	C₂×#16810401	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#12658
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
6. C₄⋊#11719908	C₄	semidirect
87. C₄×C₅₃	#347	semidirect
110. C₅₃	#12658	semidirect
182. C₄	#16810401	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×#16810401	2
#20272795	2
C₄×#97×C₅₃	2
#12658	53