Subgroups — C₂²×#16811484 #28316072

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88324	C₂	1	2	yes	maximal
2	C₂×#16811484	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88324	—	2	2	no	—
5	C₂×C₄⋊#88324	—	2	6	no	—
6	C₂×C₄⋊#88324	C₂²	1	12	yes	—
7	#16811484	C₂²	1	16	yes	direct product factor
8	C₂³×#88322	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	6	yes	—
11	C₂×C₈×Dic₅₃	C₂²	1	6	yes	—
12	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×#88324	—	2	12	no	—
14	C₂²×#88324	—	4	8	no	—
15	C₄⋊#88324	—	2	12	no	—
16	C₄⋊#88324	C₂³	1	8	yes	—
17	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
18	C₂³×Dic₅₃	Q₈	1	1	yes	characteristic
19	C₂²×#88322	—	2	4	no	—
20	C₂²×#88322	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
22	C₂×C₄×Dic₅₃	D₄	1	3	yes	—
23	C₂×C₄×Dic₅₃	Q₈	1	3	yes	—
24	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
25	C₈×Dic₅₃	C₂×C₄	1	12	yes	—
26	C₈×Dic₅₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₈	1	8	yes	complemented
29	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
30	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
33	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
35	C₂²×Dic₅₃	—	2	4	no	—
36	C₂²×Dic₅₃	C₄⋊C₄	1	3	yes	—
37	C₂²×Dic₅₃	C₄⋊C₄	1	3	yes	—
38	C₂²×Dic₅₃	SD₁₆	1	8	yes	—
39	C₂²×Dic₅₃	C₂×D₄	1	3	yes	—
40	C₂²×Dic₅₃	C₂×Q₈	1	3	yes	—
41	C₂×#88324	—	2	8	no	—
42	C₂×#88324	—	4	24	no	—
43	C₂×C₈×C₅₃	C₂×C₈	1	24	yes	complemented
44	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
45	C₄×Dic₅₃	C₄⋊C₄	1	6	yes	—
46	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
47	C₄×Dic₅₃	C₂×D₄	1	3	yes	—
48	C₄×Dic₅₃	C₂×Q₈	1	3	yes	—
49	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×#88322	—	2	12	no	—
51	C₂×#88322	C₂²×C₄	1	4	yes	—
52	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
53	C₂³×C₅₃	#106	1	4	yes	—
54	C₂³×C₅₃	#108	1	4	yes	—
55	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
59	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
60	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₅₃	—	2	6	no	—
62	C₂×Dic₅₃	—	2	6	no	—
63	C₂×Dic₅₃	#106	1	12	yes	—
64	C₂×Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×Dic₅₃	C₂×SD₁₆	1	12	yes	—
67	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
69	#88324	—	4	8	no	—
70	C₈×C₅₃	C₂²×C₈	1	8	yes	complemented
71	#88322	—	2	4	no	—
72	C₂²×#334	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	#334	1	16	yes	complemented
74	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
75	C₂²×C₅₃	C₂×#106	1	6	yes	—
76	C₂²×C₅₃	C₂×#108	1	6	yes	—
77	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
79	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
81	Dic₅₃	—	2	3	no	—
82	Dic₅₃	—	2	1	no	—
83	Dic₅₃	C₂×#106	1	6	yes	—
84	Dic₅₃	C₂²×SD₁₆	1	2	yes	—
85	C₂²×C₄⋊C₈	—	53	2	no	—
86	C₂²×C₄×C₈	—	53	1	no	—
87	C₂×#334	—	53	12	no	—
88	C₂×C₅₃	C₂×#334	1	12	yes	complemented
89	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
90	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
91	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
92	C₂³×C₈	—	53	1	no	—
93	C₂³×C₈	—	106	2	no	—
94	C₂³×C₈	#6157	1	1	yes	characteristic
95	C₂²×C₄²	—	53	1	no	—
96	C₂×C₄⋊C₈	—	53	12	no	—
97	C₂×C₄⋊C₈	—	106	6	no	—
98	C₂×C₄×C₈	—	53	6	no	—
99	C₂×C₄×C₈	—	53	6	no	—
100	#334	—	53	16	no	—
101	C₅₃	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₄	—	53	1	no	—
103	C₂³×C₄	—	53	1	no	—
104	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
105	C₂²×C₈	—	53	6	no	—
106	C₂²×C₈	—	106	4	no	—
107	C₂²×C₈	—	106	12	no	—
108	C₂²×C₈	—	212	8	no	—
109	C₂²×C₈	#88324	1	8	yes	complemented
110	C₂²×C₈	C₂×#6157	1	6	yes	—
111	C₂×C₄²	—	53	3	no	—
112	C₂×C₄²	—	53	3	no	—
113	C₂×C₄²	—	53	3	no	—
114	C₂×C₄²	—	53	3	no	—
115	C₄⋊C₈	—	53	8	no	—
116	C₄⋊C₈	—	106	12	no	—
117	C₄×C₈	—	53	12	no	—
118	C₄×C₈	—	53	4	no	—
119	C₂⁴	C₄⋊#6157	1	1	yes	characteristic
120	C₂²×C₄	—	53	8	no	—
121	C₂²×C₄	—	53	3	no	—
122	C₂²×C₄	—	53	3	no	—
123	C₂²×C₄	—	53	3	no	—
124	C₂²×C₄	—	53	3	no	—
125	C₂²×C₄	—	106	4	no	—
126	C₂²×C₄	#11719936	1	4	yes	—
127	C₂²×C₄	C₄⋊#6157	1	3	yes	—
128	C₂²×C₄	C₂×#88324	1	4	yes	—
129	C₂²×C₄	C₂²×#6157	1	3	yes	—
130	C₂×C₈	—	53	4	no	—
131	C₂×C₈	—	106	12	no	—
132	C₂×C₈	—	106	8	no	—
133	C₂×C₈	—	212	24	no	—
134	C₂×C₈	C₂×#88324	1	24	yes	complemented
135	C₂×C₈	C₂²×#6157	1	4	yes	—
136	C₄²	—	53	3	no	—
137	C₄²	—	53	6	no	—
138	C₄²	—	53	3	no	—
139	C₄²	—	53	1	no	—
140	C₄²	—	53	3	no	—
141	C₂³	C₄⋊#88324	1	4	yes	—
142	C₂³	#13133285	1	4	yes	—
143	C₂³	#13133287	1	4	yes	—
144	C₂³	C₂×C₄⋊#6157	1	3	yes	—
145	C₂×C₄	—	53	12	no	—
146	C₂×C₄	—	53	12	no	—
147	C₂×C₄	—	53	3	no	—
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	53	3	no	—
150	C₂×C₄	—	53	1	no	—
151	C₂×C₄	—	106	6	no	—
152	C₂×C₄	—	106	6	no	—
153	C₂×C₄	C₄⋊#88324	1	12	yes	—
154	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
155	C₂×C₄	C₂²×#88324	1	6	yes	—
156	C₂×C₄	C₂×#11719936	1	6	yes	—
157	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
158	C₈	—	106	4	no	—
159	C₈	—	212	8	no	—
160	C₈	C₂²×#88324	1	8	yes	complemented
161	C₂²	#16811484	1	16	yes	direct product factor
162	C₂²	C₂×C₄⋊#88324	1	6	yes	—
163	C₂²	C₂×#13133285	1	6	yes	—
164	C₂²	C₂×#13133287	1	6	yes	—
165	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
166	C₄	—	53	2	no	—
167	C₄	—	53	6	no	—
168	C₄	—	106	3	no	—
169	C₄	—	106	1	no	—
170	C₄	C₂×C₄⋊#88324	1	6	yes	—
171	C₄	C₂³×#88324	1	1	yes	characteristic
172	C₄	C₂²×#11719936	1	1	yes	characteristic
173	C₂	C₂×#16811484	1	12	yes	direct product factor
174	C₂	C₂²×C₄⋊#88324	1	1	yes	characteristic
175	C₂	C₂²×#13133285	1	1	yes	characteristic
176	C₂	C₂²×#13133287	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₄⋊#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle, prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₂²×#334
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16811484	C₂	direct (N × H)
7. #16811484	C₂²	direct (N × H)
28. C₂²×C₈×C₅₃	C₈	semidirect
43. C₂×C₈×C₅₃	C₂×C₈	semidirect
70. C₈×C₅₃	C₂²×C₈	semidirect
73. C₂²×C₅₃	#334	semidirect
88. C₂×C₅₃	C₂×#334	semidirect
101. C₅₃	C₂²×#334	semidirect
109. C₂²×C₈	#88324	semidirect
134. C₂×C₈	C₂×#88324	semidirect
160. C₈	C₂²×#88324	semidirect
161. C₂²	#16811484	direct (N × H)
173. C₂	C₂×#16811484	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×C₄⋊#88324	2
C₂×#16811484	2
C₂²×C₈×Dic₅₃	2
C₂²×#334	53

Subgroups — C22×#16811484 #28316072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811484 #28316072