Subgroups — C₂×#20272784 #27295659

Order 13568.

286 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	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272784	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	—	2	1	no	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₄⋊#11719908	—	2	1	no	—
9	C₄⋊#11719908	C₂²	1	2	yes	—
10	C₈×#88322	C₄	1	2	yes	—
11	C₈×#88322	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₅₃	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	—	2	1	no	—
15	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×#88322	—	2	1	no	—
17	C₂²×#88322	C₈	1	1	yes	characteristic
18	C₂²×#88322	C₈	1	1	yes	characteristic
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	4	2	no	—
21	C₄×#88322	—	2	1	no	—
22	C₄×#88322	C₈	1	2	yes	—
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₄×#88322	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₂×C₁₆×C₅₃	—	4	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	D₈	1	2	yes	—
41	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×#88322	—	2	4	no	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	C₁₆	1	4	yes	complemented
52	C₂×#88322	C₂×C₈	1	2	yes	—
53	C₂×#88322	C₂×C₈	1	2	yes	—
54	C₁₆×C₅₃	—	4	2	no	—
55	#11719908	—	4	2	no	—
56	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#107	1	2	yes	—
62	C₂×C₄×C₅₃	#108	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	#107	1	2	yes	—
71	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
72	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
73	#88322	—	2	4	no	—
74	#88322	—	2	2	no	—
75	#88322	C₂×C₁₆	1	4	yes	complemented
76	C₂×#1370	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#335	1	2	yes	—
78	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
79	C₂²×C₅₃	#364	1	2	yes	—
80	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	#335	1	2	yes	—
84	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
85	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
86	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
87	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
88	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
89	C₂×C₄⋊C₁₆	—	53	1	no	—
90	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
91	C₂×C₈²	—	53	1	no	—
92	#1370	—	53	4	no	—
93	C₂×C₅₃	#1370	1	4	yes	complemented
94	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
95	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
97	C₂²×C₁₆	—	2	1	no	—
98	C₂²×C₁₆	—	106	1	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	C₂×C₄×C₈	—	53	1	no	—
101	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₄⋊C₁₆	—	2	1	no	—
103	C₄⋊C₁₆	—	53	2	no	—
104	C₄⋊C₁₆	—	106	1	no	—
105	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
106	C₈²	—	53	2	no	—
107	C₈²	—	53	2	no	—
108	C₅₃	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	53	1	no	—
110	C₂²×C₈	—	53	1	no	—
111	C₂²×C₈	—	106	1	no	—
112	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	4	2	no	—
117	C₂×C₁₆	—	106	2	no	—
118	C₂×C₁₆	—	212	2	no	—
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	2	no	—
121	C₄×C₈	—	53	2	no	—
122	C₄×C₈	—	106	1	no	—
123	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
124	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	#11719912	1	1	yes	characteristic
129	C₂²×C₄	#11719919	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	53	4	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	106	4	no	—
135	C₂×C₈	—	106	2	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	#11719919	1	1	yes	characteristic
138	C₂×C₈	#11719919	1	1	yes	characteristic
139	C₂×C₈	#11719922	1	2	yes	—
140	C₂×C₈	#11719925	1	2	yes	—
141	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
143	C₄²	C₈×D₅₃	1	1	yes	characteristic
144	C₄²	#11719912	1	1	yes	characteristic
145	C₄²	#11719919	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
147	C₁₆	—	4	2	no	—
148	C₁₆	—	212	2	no	—
149	C₂³	#13133146	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	#13133146	1	1	yes	characteristic
152	C₂×C₄	#13133146	1	1	yes	characteristic
153	C₂×C₄	#13133146	1	1	yes	characteristic
154	C₂×C₄	#13133150	1	2	yes	—
155	C₂×C₄	#13133152	1	2	yes	—
156	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
157	C₂×C₄	#13133156	1	2	yes	—
158	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
159	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
160	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	53	4	no	—
164	C₈	—	106	4	no	—
165	C₈	—	106	2	no	—
166	C₈	#13133150	1	2	yes	—
167	C₈	C₂×#11719922	1	1	yes	characteristic
168	C₈	C₂×#11719925	1	1	yes	characteristic
169	C₂²	#16810375	1	2	yes	—
170	C₂²	#16810441	1	2	yes	—
171	C₂²	#16810443	1	2	yes	—
172	C₂²	C₂×#13133146	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#16810375	1	2	yes	—
176	C₄	#16810441	1	2	yes	—
177	C₄	C₂×#13133150	1	1	yes	characteristic
178	C₄	C₂×#13133152	1	1	yes	characteristic
179	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
180	C₄	C₂×#13133156	1	1	yes	characteristic
181	C₂	#20272784	1	4	yes	direct product factor
182	C₂	C₂×#16810375	1	1	yes	characteristic
183	C₂	C₂×#16810441	1	1	yes	characteristic
184	C₂	C₂×#16810443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1370
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20272784	C₂	direct (N × H)
51. C₂×#88322	C₁₆	semidirect
75. #88322	C₂×C₁₆	semidirect
93. C₂×C₅₃	#1370	semidirect
108. C₅₃	C₂×#1370	semidirect
181. C₂	#20272784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×C₈×#88322	2
#20272784	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#1370	53

Subgroups — C2×#20272784 #27295659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272784 #27295659