Subgroups — C₂²×#16810444 #27537088

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 182 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	C₂×#16810444	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	—	2	6	no	—
5	C₂×#13133157	C₂²	1	12	yes	—
6	C₂²×#11719908	—	2	2	no	—
7	#16810444	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	6	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
13	#13133157	—	2	12	no	—
14	#13133157	C₂³	1	8	yes	—
15	C₂×#11719908	—	2	12	no	—
16	C₂³×C₄×C₅₃	—	2	1	no	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₈	1	6	yes	—
19	C₂²×C₈×C₅₃	C₈	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
22	C₂²×C₈×C₅₃	D₄	1	3	yes	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	Q₈	1	3	yes	—
25	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
27	C₂×C₄²×C₅₃	—	2	3	no	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	—
30	C₄×C₈×C₅₃	C₂³	1	4	yes	—
31	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	—	2	8	no	—
33	C₂²×C₄×C₅₃	—	2	6	no	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	M₁₆	1	3	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	6	yes	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
43	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
48	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
49	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
50	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
51	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₄²×C₅₃	—	2	6	no	—
53	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
54	#11719908	—	2	8	no	—
55	C₂³×C₅₃	—	2	4	no	—
56	C₂³×C₅₃	C₄⋊C₈	1	3	yes	—
57	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
58	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
59	C₂×C₄×C₅₃	—	2	24	no	—
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	C₄⋊C₈	1	6	yes	—
62	C₂×C₄×C₅₃	C₄⋊C₈	1	3	yes	—
63	C₂×C₄×C₅₃	C₄⋊C₈	1	3	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
69	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
70	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₈×C₅₃	C₂²×C₈	1	8	yes	—
73	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
74	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×#364	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	6	no	—
77	C₂²×C₅₃	—	2	6	no	—
78	C₂²×C₅₃	#364	1	4	yes	complemented
79	C₂²×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
80	C₂²×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
81	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
82	C₄×C₅₃	—	2	8	no	—
83	C₄×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
84	C₄×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
85	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
87	C₂²×M₃₂	—	53	2	no	—
88	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
89	C₂×#364	—	53	12	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	3	no	—
92	C₂×C₅₃	C₂×#364	1	6	yes	complemented
93	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
94	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
95	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
97	C₂×M₃₂	—	53	12	no	—
98	C₂×M₃₂	—	106	6	no	—
99	C₂²×C₁₆	—	106	2	no	—
100	C₂×C₄×C₈	Dic₅₃	1	6	yes	—
101	C₂×C₄×C₈	C₂×D₅₃	1	6	yes	—
102	#364	—	53	16	no	—
103	C₅₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₄	—	2	1	no	—
105	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂²×C₈	#88322	1	6	yes	—
107	C₂²×C₈	#88322	1	2	yes	—
108	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
109	C₂²×C₈	Dic₁₀₆	1	3	yes	—
110	C₂²×C₈	D₂₁₂	1	3	yes	—
111	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₅₃	1	6	yes	—
113	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
114	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
115	C₂×C₄²	—	2	3	no	—
116	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
117	M₃₂	—	53	8	no	—
118	M₃₂	—	106	12	no	—
119	C₂×C₁₆	—	106	12	no	—
120	C₄×C₈	C₂×Dic₅₃	1	12	yes	—
121	C₄×C₈	C₂²×D₅₃	1	4	yes	—
122	C₂⁴	#11719917	1	1	yes	characteristic
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	6	no	—
125	C₂²×C₄	C₂×#88322	1	3	yes	—
126	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
127	C₂²×C₄	#11719917	1	3	yes	—
128	C₂²×C₄	C₄⋊Dic₅₃	1	3	yes	—
129	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
131	C₂×C₈	C₂×#88322	1	12	yes	—
132	C₂×C₈	C₂×#88322	1	12	yes	—
133	C₂×C₈	C₄⋊Dic₅₃	1	6	yes	—
134	C₂×C₈	C₄⋊Dic₅₃	1	3	yes	—
135	C₂×C₈	C₄⋊Dic₅₃	1	3	yes	—
136	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
137	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
138	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
139	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
140	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
141	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
142	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
143	C₄²	—	2	6	no	—
144	C₄²	C₂²×Dic₅₃	1	4	yes	—
145	C₁₆	—	106	8	no	—
146	C₂³	—	2	4	no	—
147	C₂³	C₄⋊#88322	1	3	yes	—
148	C₂³	C₄⋊#88322	1	1	yes	characteristic
149	C₂³	C₂×#11719917	1	3	yes	—
150	C₂×C₄	—	2	24	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	C₄⋊#88322	1	6	yes	—
153	C₂×C₄	C₄⋊#88322	1	3	yes	—
154	C₂×C₄	C₄⋊#88322	1	3	yes	—
155	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
156	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
157	C₂×C₄	C₂²×#88322	1	3	yes	—
158	C₂×C₄	C₂²×#88322	1	3	yes	—
159	C₂×C₄	C₂×#11719917	1	3	yes	—
160	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
161	C₈	C₂×C₄⋊Dic₅₃	1	3	yes	—
162	C₈	C₂×C₄⋊Dic₅₃	1	3	yes	—
163	C₈	C₂²×#88322	1	8	yes	—
164	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
165	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
166	C₂²	—	2	6	no	—
167	C₂²	—	2	6	no	—
168	C₂²	#16810444	1	4	yes	direct product factor
169	C₂²	C₂×C₄⋊#88322	1	3	yes	—
170	C₂²	C₂×C₄⋊#88322	1	3	yes	—
171	C₂²	C₂²×#11719917	1	1	yes	characteristic
172	C₄	—	2	8	no	—
173	C₄	C₂×C₄⋊#88322	1	3	yes	—
174	C₄	C₂×C₄⋊#88322	1	3	yes	—
175	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
176	C₄	C₂³×#88322	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	3	no	—
179	C₂	C₂×#16810444	1	6	yes	direct product factor
180	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂²×#364
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810444	C₂	direct (N × H)
7. #16810444	C₂²	direct (N × H)
78. C₂²×C₅₃	#364	semidirect
92. C₂×C₅₃	C₂×#364	semidirect
103. C₅₃	C₂²×#364	semidirect
168. C₂²	#16810444	direct (N × H)
179. C₂	C₂×#16810444	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂×#16810444	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#364	53

Subgroups — C22×#16810444 #27537088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810444 #27537088