Subgroups — #27295681

Order 13568.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	#20272789	C₂	1	2	yes	maximal, complemented
4	#20272787	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	#16810376	—	2	1	no	—
9	#16810376	C₂²	1	2	yes	—
10	#16810376	C₂²	1	1	yes	characteristic
11	#16810376	C₂²	1	1	yes	characteristic
12	#16810374	—	2	1	no	—
13	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
14	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	C₄	1	2	yes	—
17	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂²×#88322	—	4	1	no	—
19	C₄⋊#88322	—	2	1	no	—
20	C₄⋊#88322	—	2	1	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	4	1	no	—
23	C₂²⋊#88322	—	4	1	no	—
24	C₄×#88322	—	4	1	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	M₃₂×C₅₃	—	2	2	no	—
29	M₃₂×C₅₃	C₈	1	4	yes	complemented
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	D₈	1	1	yes	characteristic
48	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×#88322	—	4	1	no	—
51	C₂×#88322	—	4	1	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	—	8	2	no	—
54	C₁₆×C₅₃	—	2	2	no	—
55	C₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
56	C₂³×C₅₃	#107	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#108	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
71	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	#88322	—	8	1	no	—
74	#88322	—	8	2	no	—
75	#88322	—	8	1	no	—
76	#12009	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	#335	1	2	yes	—
80	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	#335	1	2	yes	—
85	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
87	C₄×C₅₃	#509	1	1	yes	characteristic
88	C₄×C₅₃	#510	1	1	yes	characteristic
89	C₄×M₃₂	D₅₃	1	1	yes	characteristic
90	#1564	—	53	1	no	—
91	C₂×#335	—	53	1	no	—
92	#1372	—	53	2	no	—
93	#1371	—	53	2	no	—
94	C₂×C₅₃	—	2	2	no	—
95	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
96	C₂×C₅₃	#2159	1	1	yes	characteristic
97	C₂×C₅₃	#2161	1	1	yes	characteristic
98	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
99	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
100	#431	—	106	1	no	—
101	C₂×C₄⋊C₈	—	106	1	no	—
102	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	#346	Dic₅₃	1	2	yes	—
104	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
105	#335	—	53	2	no	—
106	#335	—	53	1	no	—
107	#335	—	53	1	no	—
108	#335	—	106	1	no	—
109	#334	—	106	1	no	—
110	C₅₃	#12009	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	212	1	no	—
112	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	#88322	1	4	yes	complemented
117	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
118	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
119	C₄⋊C₈	—	106	1	no	—
120	C₄⋊C₈	—	106	1	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₄⋊C₈	—	212	1	no	—
123	C₂²⋊C₈	—	212	1	no	—
124	C₄×C₈	—	212	1	no	—
125	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
126	C₄×C₈	D₂₁₂	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
128	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
129	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
130	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
131	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	424	2	no	—
134	C₂×C₈	—	212	1	no	—
135	C₂×C₈	—	212	1	no	—
136	C₂×C₈	—	212	2	no	—
137	C₂×C₈	C₂×#88322	1	2	yes	—
138	C₂×C₈	#11719917	1	2	yes	—
139	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
143	C₄²	D₄₂₄	1	1	yes	characteristic
144	C₄²	Dic₂₁₂	1	1	yes	characteristic
145	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
146	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	C₂×#88322	1	4	yes	complemented
149	C₂³	#13133151	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊#88322	1	2	yes	—
154	C₂×C₄	#13133151	1	1	yes	characteristic
155	C₂×C₄	#13133151	1	1	yes	characteristic
156	C₂×C₄	#13133151	1	1	yes	characteristic
157	C₂×C₄	#13133153	1	2	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	424	2	no	—
164	C₈	—	424	1	no	—
165	C₈	—	424	1	no	—
166	C₈	C₄⋊#88322	1	2	yes	—
167	C₈	C₂²×#88322	1	1	yes	characteristic
168	C₈	C₂×#11719917	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16810376	1	2	yes	—
172	C₂²	C₂×#13133151	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#16810376	1	2	yes	—
177	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
178	C₄	C₂×#13133153	1	1	yes	characteristic
179	C₄	#16811061	1	1	yes	characteristic
180	C₄	#16811066	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#16810376	1	1	yes	characteristic
183	C₂	#20276503	1	1	yes	characteristic
184	C₂	#20276510	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	#107
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12009
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272789	C₂	semidirect
4. #20272787	C₂	semidirect
29. M₃₂×C₅₃	C₈	semidirect
55. C₁₆×C₅₃	C₂×C₈	semidirect
110. C₅₃	#12009	semidirect
116. M₃₂	#88322	semidirect
148. C₁₆	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
C₂×#16810376	2
#20272789	2
#20272787	2
C₄×M₃₂×C₅₃	2
#12009	53