Subgroups — C₂×#20276995 #27539576

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	#20276995	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810450	C₂	1	4	yes	maximal
4	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	—	2	1	no	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	#16811028	—	2	2	no	—
9	#16811028	C₂²	1	4	yes	—
10	C₂×#13133151	—	2	1	no	—
11	C₂×#13133151	C₂²	1	2	yes	—
12	#16810450	C₂²	1	16	yes	—
13	C₄×#11719908	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
15	C₅₃×#500	—	2	2	no	—
16	C₅₃×#500	C₂²	1	4	yes	—
17	C₂×#107×C₅₃	—	2	1	no	—
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₀₆	—	4	1	no	—
21	#13133217	—	2	4	no	—
22	#13133217	—	4	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	C₂×C₄⋊Dic₅₃	—	4	1	no	—
25	C₂×#11719908	Q₈	1	8	yes	complemented
26	C₂×#11719908	C₂³	1	4	yes	—
27	#13133151	—	2	2	no	—
28	#13133151	—	2	4	no	—
29	#13133151	C₂³	1	4	yes	—
30	C₂²×Q₈×C₅₃	—	4	1	no	—
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₅₃	—	2	4	no	—
34	C₄⋊Q₈×C₅₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	#107×C₅₃	—	2	2	no	—
39	#107×C₅₃	—	2	4	no	—
40	#107×C₅₃	C₂³	1	4	yes	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂×Q₈×C₅₃	—	4	4	no	—
44	C₂×Q₈×C₅₃	—	8	4	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	4	2	no	—
50	C₂×Dic₁₀₆	—	4	4	no	—
51	C₂×Dic₁₀₆	—	8	4	no	—
52	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
56	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	4	no	—
60	C₄²×C₅₃	D₈	1	2	yes	—
61	C₄²×C₅₃	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₅₃	—	2	4	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₄⋊Dic₅₃	—	4	4	no	—
65	#11719908	C₂×Q₈	1	8	yes	complemented
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₅₃	—	8	8	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
71	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
72	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₅₃	—	4	4	no	—
79	C₂×Dic₅₃	—	8	4	no	—
80	Dic₁₀₆	—	8	8	no	—
81	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
82	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
83	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
84	C₂×#2254	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	#500	1	2	yes	—
86	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
87	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
88	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	#500	1	2	yes	—
91	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
92	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
93	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
95	Dic₅₃	—	8	4	no	—
96	C₂×#500	—	53	1	no	—
97	C₂×#500	D₅₃	1	1	yes	characteristic
98	#2254	—	53	8	no	—
99	C₂×#367	—	53	4	no	—
100	C₂×C₄×C₁₆	—	53	1	no	—
101	C₂×C₅₃	#2254	1	4	yes	complemented
102	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
103	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
104	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	—	2	1	no	—
106	C₂×C₄⋊Q₈	—	106	1	no	—
107	C₂²×C₁₆	—	53	2	no	—
108	#500	—	2	2	no	—
109	#500	—	53	4	no	—
110	#500	—	106	2	no	—
111	#500	C₂×D₅₃	1	4	yes	—
112	C₂×#107	—	2	1	no	—
113	C₂×#107	—	53	2	no	—
114	C₂×#107	—	106	1	no	—
115	C₂×#107	C₂×D₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	#367	—	53	16	no	—
118	C₄×C₁₆	—	53	4	no	—
119	C₅₃	C₂×#2254	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	4	1	no	—
121	C₂²×Q₈	—	212	1	no	—
122	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	C₄⋊Q₈	—	106	4	no	—
127	C₄⋊Q₈	—	212	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	106	2	no	—
131	C₂×C₄⋊C₄	—	212	1	no	—
132	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂×C₁₆	—	53	8	no	—
134	C₂×C₁₆	—	53	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	2	no	—
137	#107	—	53	4	no	—
138	#107	—	106	2	no	—
139	#107	—	106	4	no	—
140	#107	C₂²×D₅₃	1	4	yes	—
141	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
142	C₄×C₈	C₂²×D₅₃	1	2	yes	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	8	4	no	—
145	C₂×Q₈	—	424	4	no	—
146	C₂×Q₈	—	212	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	212	2	no	—
149	C₂²×C₄	#11719922	1	1	yes	characteristic
150	C₂²×C₄	#11719922	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂×C₈	#11719925	1	4	yes	—
153	C₂×C₈	Q₈×D₅₃	1	4	yes	—
154	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
156	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	106	4	no	—
161	C₄⋊C₄	—	212	2	no	—
162	C₄⋊C₄	—	212	4	no	—
163	C₄²	#11719922	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
165	C₁₆	—	53	8	no	—
166	C₂³	C₂×#11719922	1	1	yes	characteristic
167	Q₈	—	8	8	no	—
168	Q₈	—	424	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	424	4	no	—
172	C₂×C₄	—	212	4	no	—
173	C₂×C₄	#13133160	1	4	yes	—
174	C₂×C₄	#13133161	1	4	yes	—
175	C₂×C₄	#13133215	1	2	yes	—
176	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
177	C₂×C₄	C₂×#11719922	1	2	yes	—
178	C₂×C₄	C₂×#11719922	1	2	yes	—
179	C₂×C₄	C₂×#11719925	1	2	yes	—
180	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₈	#13133215	1	2	yes	—
182	C₈	C₂×#11719925	1	4	yes	—
183	C₈	C₂×Q₈×D₅₃	1	2	yes	—
184	C₂²	#16811026	1	2	yes	—
185	C₂²	C₂×#13133160	1	2	yes	—
186	C₂²	C₂×#13133161	1	2	yes	—
187	C₂²	C₂²×#11719922	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	424	4	no	—
190	C₄	#16811026	1	2	yes	—
191	C₄	C₂×#13133160	1	4	yes	—
192	C₄	C₂×#13133161	1	4	yes	—
193	C₄	C₂×#13133215	1	1	yes	characteristic
194	C₄	C₂²×#11719925	1	1	yes	characteristic
195	C₂	#20276995	1	4	yes	direct product factor
196	C₂	C₂×#16811026	1	1	yes	characteristic
197	C₂	C₂²×#13133160	1	1	yes	characteristic
198	C₂	C₂²×#13133161	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	C₂×#2254
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #20276995	C₂	direct (N × H)
25. C₂×#11719908	Q₈	semidirect
65. #11719908	C₂×Q₈	semidirect
101. C₂×C₅₃	#2254	semidirect
119. C₅₃	C₂×#2254	semidirect
195. C₂	#20276995	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×#16811028	2
#20276995	2
C₂×#16810450	2
C₂×C₄×#11719908	2
C₂×C₅₃×#500	2
C₂×#2254	53

Subgroups — C2×#20276995 #27539576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276995 #27539576