Subgroups — C₂×#20276944 #27539382

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	#20276944	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810426	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	C₂²×Dic₂₁₂	—	2	1	no	—
7	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
8	C₂×#13133217	—	2	1	no	—
9	C₂²×#11719908	C₂²	1	2	yes	—
10	#16811003	—	2	2	no	—
11	#16811003	C₂²	1	4	yes	—
12	#16810426	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	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	4	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	8	no	—
24	C₂×Dic₂₁₂	—	4	4	no	—
25	C₂×Dic₂₁₂	C₂³	1	4	yes	—
26	#13133217	—	2	4	no	—
27	#13133217	—	4	2	no	—
28	C₂×C₄⋊Dic₅₃	—	4	1	no	—
29	C₂×C₄⋊Dic₅₃	—	4	1	no	—
30	C₂×#11719908	—	2	4	no	—
31	C₂×#11719908	C₂³	1	4	yes	—
32	C₂²×Q₈×C₅₃	—	4	1	no	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	—	2	4	no	—
36	C₄⋊Q₈×C₅₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	2	no	—
41	#107×C₅₃	—	2	4	no	—
42	#107×C₅₃	C₂³	1	4	yes	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	8	4	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	2	no	—
52	C₂×Dic₁₀₆	—	2	4	no	—
53	C₂×Dic₁₀₆	—	4	4	no	—
54	C₂×Dic₁₀₆	—	4	4	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₅₃	#41	1	4	yes	—
60	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	4	no	—
64	C₄²×C₅₃	D₈	1	2	yes	—
65	C₄²×C₅₃	C₂×D₄	1	2	yes	—
66	C₄⋊Dic₅₃	—	4	4	no	—
67	C₄⋊Dic₅₃	—	4	4	no	—
68	Dic₂₁₂	—	2	8	no	—
69	Dic₂₁₂	—	4	4	no	—
70	Dic₂₁₂	—	4	8	no	—
71	#11719908	—	2	4	no	—
72	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₅₃	—	8	8	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	8	4	no	—
76	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
77	C₂×C₄×C₅₃	Q₃₂	1	4	yes	—
78	C₂×C₄×C₅₃	#124	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₅₃	—	4	4	no	—
85	C₂×Dic₅₃	—	8	4	no	—
86	Dic₁₀₆	—	4	4	no	—
87	Dic₁₀₆	—	4	4	no	—
88	Dic₁₀₆	—	8	4	no	—
89	C₈×C₅₃	#124	1	2	yes	—
90	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
91	C₈×C₅₃	C₂×#41	1	2	yes	—
92	C₂×#2241	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	#486	1	2	yes	—
94	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
95	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
96	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	#486	1	2	yes	—
99	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
100	C₄×C₅₃	C₂×Q₃₂	1	4	yes	—
101	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
102	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
103	Dic₅₃	—	8	4	no	—
104	C₂×#500	D₅₃	1	1	yes	characteristic
105	C₂×#494	—	53	1	no	—
106	#2241	—	53	8	no	—
107	C₂×#358	—	53	4	no	—
108	C₂×C₄×C₁₆	—	53	1	no	—
109	C₂×C₅₃	#2241	1	4	yes	complemented
110	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
111	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
112	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
113	C₂²×Q₁₆	—	53	2	no	—
114	C₂²×Q₁₆	—	106	1	no	—
115	C₂×C₄⋊Q₈	—	2	1	no	—
116	C₂×C₄⋊Q₈	—	106	1	no	—
117	C₂²×C₁₆	—	53	2	no	—
118	#500	—	2	2	no	—
119	#500	C₂×D₅₃	1	4	yes	—
120	#494	—	53	4	no	—
121	#494	—	106	2	no	—
122	C₂×#107	—	2	1	no	—
123	C₂×#107	C₂×D₅₃	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	#358	—	53	16	no	—
126	C₄×C₁₆	—	53	4	no	—
127	C₅₃	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×Q₈	—	106	2	no	—
130	C₂×Q₁₆	—	53	4	no	—
131	C₂×Q₁₆	—	106	8	no	—
132	C₂×Q₁₆	—	106	2	no	—
133	C₂×Q₁₆	—	106	4	no	—
134	C₂×Q₁₆	—	212	4	no	—
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	4	2	no	—
139	C₄⋊Q₈	—	106	4	no	—
140	C₄⋊Q₈	—	212	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	212	1	no	—
144	C₂×C₄⋊C₄	—	212	1	no	—
145	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂×C₁₆	—	53	4	no	—
147	C₂×C₁₆	—	106	4	no	—
148	#107	—	2	4	no	—
149	#107	—	2	2	no	—
150	#107	C₂²×D₅₃	1	4	yes	—
151	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
152	C₄×C₈	C₂²×D₅₃	1	2	yes	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂×Q₈	—	106	4	no	—
156	C₂×Q₈	—	212	4	no	—
157	C₂×Q₈	—	212	4	no	—
158	C₂×Q₈	—	212	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	212	2	no	—
161	C₂²×C₄	#11719922	1	1	yes	characteristic
162	C₂²×C₄	#11719922	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	Q₁₆	—	106	8	no	—
165	Q₁₆	—	212	4	no	—
166	Q₁₆	—	212	8	no	—
167	C₂×C₈	#11719924	1	4	yes	—
168	C₂×C₈	#11719949	1	4	yes	—
169	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
171	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	212	4	no	—
176	C₄⋊C₄	—	212	4	no	—
177	C₄²	#11719922	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
179	C₁₆	—	106	4	no	—
180	C₂³	C₂×#11719922	1	1	yes	characteristic
181	Q₈	—	8	8	no	—
182	Q₈	—	424	4	no	—
183	Q₈	—	212	4	no	—
184	Q₈	—	212	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	424	4	no	—
188	C₂×C₄	—	212	4	no	—
189	C₂×C₄	#13133161	1	4	yes	—
190	C₂×C₄	#13133163	1	4	yes	—
191	C₂×C₄	#13133206	1	2	yes	—
192	C₂×C₄	C₂×#11719922	1	2	yes	—
193	C₂×C₄	C₂×#11719922	1	2	yes	—
194	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
195	C₂×C₄	C₂×#11719924	1	2	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₈	#13133206	1	2	yes	—
198	C₈	C₂×#11719924	1	4	yes	—
199	C₈	C₂×#11719949	1	2	yes	—
200	C₂²	#16810971	1	2	yes	—
201	C₂²	C₂×#13133161	1	2	yes	—
202	C₂²	C₂×#13133163	1	2	yes	—
203	C₂²	C₂²×#11719922	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	424	4	no	—
206	C₄	#16810971	1	2	yes	—
207	C₄	C₂×#13133161	1	4	yes	—
208	C₄	C₂×#13133163	1	4	yes	—
209	C₄	C₂×#13133206	1	1	yes	characteristic
210	C₄	C₂²×#11719924	1	1	yes	characteristic
211	C₂	#20276944	1	4	yes	direct product factor
212	C₂	C₂×#16810971	1	1	yes	characteristic
213	C₂	C₂²×#13133161	1	1	yes	characteristic
214	C₂	C₂²×#13133163	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×#2241
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. #20276944	C₂	direct (N × H)
109. C₂×C₅₃	#2241	semidirect
127. C₅₃	C₂×#2241	semidirect
211. C₂	#20276944	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×#16811003	2
#20276944	2
C₂×#16810426	2
C₂×C₄×#11719908	2
C₂×C₅₃×#500	2
C₂×#2241	53

Subgroups — C2×#20276944 #27539382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276944 #27539382