Subgroups — C₂×#20274856 #27384322

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	2	yes	maximal
2	#20274856	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810388	C₂	1	4	yes	maximal
4	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	4	yes	—
6	#16810647	—	2	4	no	—
7	#16810647	C₂²	1	8	yes	—
8	C₂×C₄⋊#88322	—	2	4	no	—
9	#16810388	C₂²	1	16	yes	—
10	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#97×C₅₃	C₄	1	4	yes	—
13	C₂×#97×C₅₃	C₄	1	4	yes	—
14	C₂×#97×C₅₃	C₂²	1	4	yes	—
15	C₂×#11719917	—	2	4	no	—
16	C₂×#11719917	—	2	16	no	—
17	C₂×#11719917	C₂³	1	8	yes	—
18	C₂²×#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	16	no	—
20	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	2	8	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
32	#97×C₅₃	C₂×C₄	1	16	yes	—
33	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	4	4	no	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	8	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₅₃	—	2	8	no	—
46	C₂×C₈×C₅₃	—	2	8	no	—
47	C₂×C₈×C₅₃	—	2	8	no	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
49	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
52	C₄²×C₅₃	C₂×D₄	1	4	yes	—
53	#11719917	—	2	16	no	—
54	#11719917	—	4	8	no	—
55	C₂×#88322	—	2	8	no	—
56	C₂×#88322	—	4	8	no	—
57	C₂³×C₅₃	—	2	4	no	—
58	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₅₃	#125	1	4	yes	—
61	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	8	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#117	1	4	yes	—
69	C₂×C₄×C₅₃	#117	1	4	yes	—
70	C₂×C₄×C₅₃	#124	1	4	yes	—
71	C₂×C₄×C₅₃	#124	1	4	yes	—
72	C₂×C₄×C₅₃	#125	1	4	yes	—
73	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
76	C₈×C₅₃	—	2	8	no	—
77	#88322	—	4	8	no	—
78	C₂×#1838	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	8	no	—
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	#382	1	2	yes	—
83	C₂²×C₅₃	#382	1	2	yes	—
84	C₂²×C₅₃	#430	1	4	yes	—
85	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×#125	1	2	yes	—
87	C₄×C₅₃	—	2	8	no	—
88	C₄×C₅₃	#382	1	2	yes	—
89	C₄×C₅₃	#382	1	2	yes	—
90	C₄×C₅₃	C₂×#117	1	2	yes	—
91	C₄×C₅₃	C₂×#124	1	2	yes	—
92	C₂×#423	—	53	2	no	—
93	C₂²×#97	D₅₃	1	1	yes	characteristic
94	#1838	—	53	8	no	—
95	C₂×#341	—	53	4	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	—	2	2	no	—
98	C₂×C₅₃	#1838	1	4	yes	complemented
99	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
100	C₂×C₅₃	C₂×#430	1	2	yes	—
101	C₂²×M₁₆	—	53	4	no	—
102	C₂³×C₈	C₂×D₅₃	1	2	yes	—
103	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
104	#423	—	53	8	no	—
105	#423	—	106	4	no	—
106	C₂×C₄⋊C₈	—	106	4	no	—
107	C₂×#97	Dic₅₃	1	4	yes	—
108	C₂×#97	Dic₅₃	1	4	yes	—
109	C₂×#97	C₂×D₅₃	1	4	yes	—
110	#341	—	53	16	no	—
111	C₅₃	C₂×#1838	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	Dic₁₀₆	1	2	yes	—
113	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×M₁₆	—	53	8	no	—
115	C₂×M₁₆	—	106	4	no	—
116	C₂×M₁₆	—	106	16	no	—
117	C₂²×C₈	—	2	8	no	—
118	C₂²×C₈	—	106	4	no	—
119	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
120	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
121	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
122	C₂×C₄²	Dic₁₀₆	1	2	yes	—
123	C₂×C₄²	D₂₁₂	1	2	yes	—
124	C₂×C₄²	D₂₁₂	1	2	yes	—
125	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
126	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
128	C₄⋊C₈	—	106	16	no	—
129	#97	C₂×Dic₅₃	1	16	yes	—
130	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
133	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
134	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
135	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
138	C₂²×C₄	#11719945	1	8	yes	—
139	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
142	M₁₆	—	106	16	no	—
143	M₁₆	—	212	8	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	106	8	no	—
148	C₂×C₈	—	212	8	no	—
149	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
150	C₄²	C₄⋊Dic₅₃	1	4	yes	—
151	C₄²	C₄⋊Dic₅₃	1	4	yes	—
152	C₄²	C₂×D₂₁₂	1	4	yes	—
153	C₄²	C₂²×Dic₅₃	1	4	yes	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
156	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
157	C₂³	#13133209	1	4	yes	—
158	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133176	1	4	yes	—
166	C₂×C₄	#13133176	1	4	yes	—
167	C₂×C₄	#13133205	1	4	yes	—
168	C₂×C₄	#13133205	1	4	yes	—
169	C₂×C₄	#13133209	1	4	yes	—
170	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719945	1	4	yes	—
173	C₈	—	2	8	no	—
174	C₈	—	212	8	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	#16810489	1	2	yes	—
179	C₂²	#16810489	1	2	yes	—
180	C₂²	#16810672	1	4	yes	—
181	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂²	C₂×#13133209	1	2	yes	—
183	C₄	—	2	8	no	—
184	C₄	#16810489	1	2	yes	—
185	C₄	#16810489	1	2	yes	—
186	C₄	C₂×#13133176	1	2	yes	—
187	C₄	C₂×#13133205	1	2	yes	—
188	C₂	—	2	2	no	—
189	C₂	—	2	2	no	—
190	C₂	#20274856	1	4	yes	direct product factor
191	C₂	C₂×#16810489	1	1	yes	characteristic
192	C₂	C₂×#16810672	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	C₂×#1838
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #20274856	C₂	direct (N × H)
98. C₂×C₅₃	#1838	semidirect
111. C₅₃	C₂×#1838	semidirect
190. C₂	#20274856	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×#16810647	2
#20274856	2
C₂×#16810388	2
C₂²×#97×C₅₃	2
C₂×#1838	53

Subgroups — C2×#20274856 #27384322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274856 #27384322