Subgroups — C₂²×#16811516 #28327921

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133291	C₂	1	2	yes	maximal
2	C₂×#16811516	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133291	C₄	1	12	yes	complemented
5	C₂×#13133291	C₂²	1	12	yes	—
6	C₂²×#11719910	C₄	1	4	yes	complemented
7	#16811516	C₂²	1	16	yes	direct product factor
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×#13133127	C₄	1	6	yes	—
11	C₂×#13133127	C₂²	1	6	yes	—
12	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
13	#13133291	C₂×C₄	1	24	yes	complemented
14	#13133291	C₂³	1	8	yes	—
15	C₂×#11719910	C₂×C₄	1	24	yes	complemented
16	C₂³×Dic₅₃	—	2	1	no	—
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₂²×#88322	C₂×C₄	1	6	yes	—
19	C₂²×#88322	C₂×C₄	1	3	yes	—
20	C₂²×#88322	C₂³	1	3	yes	—
21	C₂×C₄×Dic₅₃	—	2	3	no	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
23	#13133127	C₂×C₄	1	12	yes	—
24	#13133127	C₂³	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	3	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
29	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
31	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	M₁₆	1	3	yes	—
33	C₂²×C₄×C₅₃	M₁₆	1	3	yes	—
34	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
36	C₂²×Dic₅₃	—	2	6	no	—
37	C₂²×Dic₅₃	—	4	4	no	—
38	C₂×C₈×C₅₃	—	2	6	no	—
39	C₂×C₈×C₅₃	—	2	6	no	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
41	C₄×Dic₅₃	—	2	6	no	—
42	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
43	C₂×#88322	C₄²	1	6	yes	—
44	C₂×#88322	C₄²	1	6	yes	—
45	C₂×#88322	C₂²×C₄	1	3	yes	—
46	C₂×#88322	C₂²×C₄	1	6	yes	—
47	C₂×#88322	C₂²×C₄	1	6	yes	—
48	C₂×#88322	C₂⁴	1	1	yes	characteristic
49	#11719910	C₂²×C₄	1	16	yes	complemented
50	C₂³×C₅₃	—	2	4	no	—
51	C₂³×C₅₃	#97	1	3	yes	—
52	C₂³×C₅₃	#97	1	1	yes	characteristic
53	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
54	C₂×C₄×C₅₃	#97	1	6	yes	—
55	C₂×C₄×C₅₃	#97	1	3	yes	—
56	C₂×C₄×C₅₃	#97	1	3	yes	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
58	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
59	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
60	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
61	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
62	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
63	C₂×Dic₅₃	—	2	4	no	—
64	C₂×Dic₅₃	—	4	12	no	—
65	C₈×C₅₃	—	2	4	no	—
66	#88322	C₂×C₄²	1	6	yes	—
67	#88322	C₂³×C₄	1	2	yes	—
68	C₂²×#347	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	2	6	no	—
70	C₂²×C₅₃	—	2	6	no	—
71	C₂²×C₅₃	#347	1	4	yes	complemented
72	C₂²×C₅₃	C₂×#97	1	3	yes	—
73	C₂²×C₅₃	C₂×#97	1	3	yes	—
74	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
75	C₄×C₅₃	C₂×#97	1	3	yes	—
76	C₄×C₅₃	C₂×#97	1	3	yes	—
77	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
78	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
79	Dic₅₃	—	4	4	no	—
80	C₂²×M₃₂	—	53	2	no	—
81	C₂²×#97	—	53	1	no	—
82	C₂×#347	—	53	12	no	—
83	C₂×C₅₃	—	2	1	no	—
84	C₂×C₅₃	—	2	3	no	—
85	C₂×C₅₃	C₂×#347	1	6	yes	complemented
86	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
87	C₂³×C₈	—	53	1	no	—
88	C₂³×C₈	#6157	1	1	yes	characteristic
89	C₂²×C₄²	—	53	1	no	—
90	C₂×M₃₂	—	53	12	no	—
91	C₂×M₃₂	—	53	12	no	—
92	C₂²×C₁₆	—	53	4	no	—
93	C₂×#97	—	53	6	no	—
94	C₂×#97	—	53	6	no	—
95	#347	—	53	16	no	—
96	C₅₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	—	106	1	no	—
98	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	2	3	no	—
101	C₂²×C₈	—	53	6	no	—
102	C₂²×C₈	—	53	2	no	—
103	C₂²×C₈	—	53	3	no	—
104	C₂²×C₈	—	53	3	no	—
105	C₂²×C₈	C₂×#6157	1	6	yes	—
106	C₂×C₄²	—	53	6	no	—
107	C₂×C₄²	—	106	3	no	—
108	M₃₂	—	53	8	no	—
109	M₃₂	—	53	24	no	—
110	C₂×C₁₆	—	53	24	no	—
111	#97	—	53	12	no	—
112	#97	—	53	4	no	—
113	C₂⁴	#11719936	1	1	yes	characteristic
114	C₂²×C₄	—	106	6	no	—
115	C₂²×C₄	—	212	4	no	—
116	C₂²×C₄	#11719936	1	3	yes	—
117	C₂²×C₄	C₄×#6157	1	3	yes	—
118	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
119	C₂²×C₄	#11719940	1	1	yes	characteristic
120	C₂²×C₄	#11719940	1	3	yes	—
121	C₂²×C₄	C₂²×#6157	1	3	yes	—
122	C₂×C₈	—	2	6	no	—
123	C₂×C₈	—	2	6	no	—
124	C₂×C₈	—	53	6	no	—
125	C₂×C₈	—	53	6	no	—
126	C₂×C₈	—	53	6	no	—
127	C₂×C₈	—	53	6	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	53	3	no	—
130	C₂×C₈	C₂²×#6157	1	4	yes	—
131	C₄²	—	53	4	no	—
132	C₄²	—	106	6	no	—
133	C₁₆	—	53	16	no	—
134	C₂³	—	2	4	no	—
135	C₂³	#13133272	1	3	yes	—
136	C₂³	#13133272	1	1	yes	characteristic
137	C₂³	C₂×#11719936	1	3	yes	—
138	C₂×C₄	—	106	4	no	—
139	C₂×C₄	—	212	12	no	—
140	C₂×C₄	#13133272	1	6	yes	—
141	C₂×C₄	#13133272	1	3	yes	—
142	C₂×C₄	#13133272	1	3	yes	—
143	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
144	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
145	C₂×C₄	C₂×#11719940	1	3	yes	—
146	C₂×C₄	C₂×#11719940	1	3	yes	—
147	C₂×C₄	C₂×#11719936	1	3	yes	—
148	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
149	C₈	—	2	4	no	—
150	C₈	—	53	6	no	—
151	C₈	—	53	2	no	—
152	C₂²	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	#16811516	1	4	yes	direct product factor
155	C₂²	C₂×#13133272	1	3	yes	—
156	C₂²	C₂×#13133272	1	3	yes	—
157	C₂²	C₂²×#11719936	1	1	yes	characteristic
158	C₄	—	212	4	no	—
159	C₄	C₂×#13133272	1	3	yes	—
160	C₄	C₂×#13133272	1	3	yes	—
161	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
162	C₄	C₂²×#11719940	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×#16811516	1	6	yes	direct product factor
166	C₂	C₂²×#13133272	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle	C₂³×C₅₃	#97
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₂²×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16811516	C₂	direct (N × H)
4. C₂×#13133291	C₄	semidirect
6. C₂²×#11719910	C₄	semidirect
7. #16811516	C₂²	direct (N × H)
13. #13133291	C₂×C₄	semidirect
15. C₂×#11719910	C₂×C₄	semidirect
49. #11719910	C₂²×C₄	semidirect
71. C₂²×C₅₃	#347	semidirect
85. C₂×C₅₃	C₂×#347	semidirect
96. C₅₃	C₂²×#347	semidirect
154. C₂²	#16811516	direct (N × H)
165. C₂	C₂×#16811516	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×#13133291	2
C₂×#16811516	2
C₂²×#13133127	2
C₂²×#347	53

Subgroups — C22×#16811516 #28327921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811516 #28327921