Subgroups — C₂×#20274890 #27384552

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	4	yes	maximal
2	C₂×#16810647	C₂	1	2	yes	maximal
3	#20274890	C₂	1	8	yes	maximal, direct product factor
4	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	4	yes	—
6	C₂×#13133153	—	2	16	no	—
7	C₂×#13133153	C₂²	1	16	yes	—
8	#16810647	—	2	4	no	—
9	#16810647	C₂²	1	8	yes	—
10	C₂×C₄⋊#88322	—	2	4	no	—
11	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#97×C₅₃	C₄	1	4	yes	—
14	C₂×#97×C₅₃	C₄	1	4	yes	—
15	C₂×#97×C₅₃	C₂²	1	4	yes	—
16	C₂×#11719917	—	2	4	no	—
17	C₂×#11719917	—	2	16	no	—
18	C₂×#11719917	C₂³	1	8	yes	—
19	C₂²×#88322	—	2	4	no	—
20	#13133153	—	2	32	no	—
21	C₄⋊#88322	—	2	16	no	—
22	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	D₄	1	8	yes	—
27	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
35	#97×C₅₃	C₂×C₄	1	16	yes	—
36	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	4	4	no	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
52	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
53	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
54	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
57	C₄²×C₅₃	C₂×D₄	1	4	yes	—
58	#11719917	—	2	16	no	—
59	#11719917	—	4	8	no	—
60	C₂×#88322	—	2	8	no	—
61	C₂×#88322	—	4	8	no	—
62	C₂³×C₅₃	—	2	4	no	—
63	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
66	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	8	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#127	1	2	yes	—
76	C₂×C₄×C₅₃	#127	1	2	yes	—
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
84	C₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
85	#88322	—	4	8	no	—
86	C₂×#1849	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	8	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	#384	1	2	yes	—
91	C₂²×C₅₃	#384	1	2	yes	—
92	C₂²×C₅₃	#430	1	4	yes	—
93	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
95	C₄×C₅₃	—	2	8	no	—
96	C₄×C₅₃	#384	1	2	yes	—
97	C₄×C₅₃	#384	1	2	yes	—
98	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
99	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
100	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂²×#108	—	53	4	no	—
102	C₂×#423	—	53	2	no	—
103	C₂²×#97	D₅₃	1	1	yes	characteristic
104	#1849	—	53	8	no	—
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	—	2	2	no	—
107	C₂×C₅₃	#1849	1	4	yes	complemented
108	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#430	1	2	yes	—
110	C₂²×M₁₆	—	53	4	no	—
111	C₂³×C₈	C₂×D₅₃	1	2	yes	—
112	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
113	C₂×#108	—	53	16	no	—
114	C₂×#108	—	106	16	no	—
115	#423	—	53	8	no	—
116	#423	—	106	4	no	—
117	C₂×C₄⋊C₈	—	106	4	no	—
118	C₂×#97	Dic₅₃	1	4	yes	—
119	C₂×#97	Dic₅₃	1	4	yes	—
120	C₂×#97	C₂×D₅₃	1	4	yes	—
121	C₅₃	C₂×#1849	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	Dic₁₀₆	1	2	yes	—
123	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×M₁₆	—	53	8	no	—
125	C₂×M₁₆	—	106	4	no	—
126	C₂×M₁₆	—	106	16	no	—
127	C₂²×C₈	—	106	4	no	—
128	C₂²×C₈	Dic₁₀₆	1	8	yes	—
129	C₂²×C₈	D₂₁₂	1	8	yes	—
130	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
131	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
132	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
133	C₂×C₄²	Dic₁₀₆	1	2	yes	—
134	C₂×C₄²	D₂₁₂	1	2	yes	—
135	C₂×C₄²	D₂₁₂	1	2	yes	—
136	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
137	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
138	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
139	#108	—	106	32	no	—
140	C₄⋊C₈	—	106	16	no	—
141	#97	C₂×Dic₅₃	1	16	yes	—
142	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
145	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
146	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
147	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
149	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
150	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
152	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
155	M₁₆	—	106	16	no	—
156	M₁₆	—	212	8	no	—
157	C₂×C₈	—	106	8	no	—
158	C₂×C₈	—	212	8	no	—
159	C₂×C₈	C₄⋊Dic₅₃	1	16	yes	—
160	C₂×C₈	C₄⋊Dic₅₃	1	16	yes	—
161	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
162	C₂×C₈	C₂×D₂₁₂	1	8	yes	—
163	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
164	C₄²	C₄⋊Dic₅₃	1	4	yes	—
165	C₄²	C₄⋊Dic₅₃	1	4	yes	—
166	C₄²	C₂×D₂₁₂	1	4	yes	—
167	C₄²	C₂²×Dic₅₃	1	4	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂³	#13133217	1	4	yes	—
172	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
179	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	#13133214	1	2	yes	—
182	C₂×C₄	#13133214	1	2	yes	—
183	C₂×C₄	#13133217	1	2	yes	—
184	C₂×C₄	#13133217	1	2	yes	—
185	C₂×C₄	#13133217	1	4	yes	—
186	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
188	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₁₀₆	1	2	yes	—
190	C₈	—	212	8	no	—
191	C₈	C₂×C₄⋊Dic₅₃	1	16	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	8	no	—
195	C₂²	#16810493	1	2	yes	—
196	C₂²	#16810493	1	2	yes	—
197	C₂²	#16810672	1	4	yes	—
198	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
199	C₂²	C₂×#13133217	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	#16810493	1	2	yes	—
202	C₄	#16810493	1	2	yes	—
203	C₄	C₂²×C₄⋊Dic₅₃	1	2	yes	—
204	C₄	C₂×#13133214	1	1	yes	characteristic
205	C₄	C₂×#13133217	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#20274890	1	4	yes	direct product factor
209	C₂	C₂×#16810493	1	1	yes	characteristic
210	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×#1849
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
3. #20274890	C₂	direct (N × H)
107. C₂×C₅₃	#1849	semidirect
121. C₅₃	C₂×#1849	semidirect
208. C₂	#20274890	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×#13133153	2
C₂×#16810647	2
#20274890	2
C₂²×#97×C₅₃	2
C₂×#1849	53

Subgroups — C2×#20274890 #27384552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274890 #27384552