Subgroups — C₂×#20274621 #27382193

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal
2	C₂×C₄×#11719917	C₂	1	2	yes	maximal
3	#20274621	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810388	C₂	1	2	yes	maximal
5	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	4	yes	—
7	C₂×#13133153	C₄	1	16	yes	complemented
8	C₂×#13133153	C₂²	1	8	yes	—
9	C₄×#11719917	—	2	4	no	—
10	C₄×#11719917	C₂²	1	8	yes	—
11	C₂×#13133128	—	2	2	no	—
12	C₂×C₄×#88322	—	2	2	no	—
13	#16810388	C₂²	1	8	yes	—
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₄	1	4	yes	—
18	C₂×#97×C₅₃	C₄	1	4	yes	—
19	C₂×#97×C₅₃	C₂²	1	4	yes	—
20	C₂×#11719917	—	2	4	no	—
21	C₂×#11719917	—	2	8	no	—
22	C₂×#11719917	C₂×C₄	1	16	yes	—
23	C₂×#11719917	C₂³	1	8	yes	—
24	C₂²×#88322	—	2	4	no	—
25	#13133153	C₂×C₄	1	32	yes	complemented
26	#13133128	—	2	8	no	—
27	C₄×#88322	—	2	8	no	—
28	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
44	#97×C₅₃	C₂×C₄	1	16	yes	—
45	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	4	4	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	C₄²	1	8	yes	—
62	C₂×C₈×C₅₃	C₄²	1	8	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
69	C₄²×C₅₃	C₂×D₄	1	4	yes	—
70	#11719917	—	2	16	no	—
71	#11719917	—	2	8	no	—
72	#11719917	C₂²×C₄	1	16	yes	—
73	C₂×#88322	—	2	16	no	—
74	C₂×#88322	—	2	8	no	—
75	C₂³×C₅₃	—	2	4	no	—
76	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
79	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	8	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#117	1	2	yes	—
89	C₂×C₄×C₅₃	#117	1	2	yes	—
90	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
93	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
97	C₈×C₅₃	—	2	4	no	—
98	C₈×C₅₃	C₂×C₄²	1	8	yes	—
99	#88322	—	2	16	no	—
100	C₂×#1778	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	8	no	—
104	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
105	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
106	C₂²×C₅₃	#430	1	4	yes	—
107	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
109	C₄×C₅₃	—	2	8	no	—
110	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
112	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
113	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
114	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
115	C₂²×#108	—	53	2	no	—
116	C₂×C₄×M₁₆	—	53	2	no	—
117	C₂²×#97	D₅₃	1	1	yes	characteristic
118	#1778	—	53	8	no	—
119	C₂×#341	—	53	2	no	—
120	C₂×C₅₃	—	2	2	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	#1778	1	4	yes	complemented
123	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#430	1	2	yes	—
125	C₂²×M₁₆	—	53	4	no	—
126	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
129	C₂×#108	—	53	16	no	—
130	C₂×#108	—	53	8	no	—
131	C₄×M₁₆	—	53	8	no	—
132	C₄×M₁₆	—	106	4	no	—
133	C₂×#97	—	106	2	no	—
134	C₂×#97	Dic₅₃	1	4	yes	—
135	C₂×#97	Dic₅₃	1	4	yes	—
136	C₂×#97	C₂×D₅₃	1	4	yes	—
137	C₂×C₄×C₈	—	106	2	no	—
138	#341	—	53	8	no	—
139	C₅₃	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	Dic₁₀₆	1	2	yes	—
141	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
142	C₂×M₁₆	—	53	16	no	—
143	C₂×M₁₆	—	53	8	no	—
144	C₂×M₁₆	—	106	4	no	—
145	C₂×M₁₆	—	106	8	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	—	106	4	no	—
148	C₂²×C₈	C₄×D₅₃	1	8	yes	—
149	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
150	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
151	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
152	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
153	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
154	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
155	C₂×C₄²	Dic₁₀₆	1	2	yes	—
156	C₂×C₄²	D₂₁₂	1	2	yes	—
157	C₂×C₄²	D₂₁₂	1	2	yes	—
158	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
159	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
160	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
161	#108	—	53	32	no	—
162	#97	—	106	8	no	—
163	#97	C₂×Dic₅₃	1	16	yes	—
164	C₄×C₈	—	106	8	no	—
165	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
168	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
169	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
170	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
172	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
174	C₂²×C₄	#11719945	1	4	yes	—
175	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
178	M₁₆	—	53	16	no	—
179	M₁₆	—	106	16	no	—
180	M₁₆	—	106	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	106	16	no	—
185	C₂×C₈	—	106	8	no	—
186	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
187	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
188	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
189	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
190	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
191	C₄²	C₄⋊Dic₅₃	1	4	yes	—
192	C₄²	C₄⋊Dic₅₃	1	4	yes	—
193	C₄²	C₂×D₂₁₂	1	4	yes	—
194	C₄²	C₂²×Dic₅₃	1	4	yes	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
198	C₂³	C₄×Dic₁₀₆	1	4	yes	—
199	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
208	C₂×C₄	#13133176	1	2	yes	—
209	C₂×C₄	#13133176	1	2	yes	—
210	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
211	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
212	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
213	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
214	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719945	1	2	yes	—
217	C₈	—	2	4	no	—
218	C₈	—	106	16	no	—
219	C₈	C₂×C₄×Dic₅₃	1	8	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
224	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
225	C₂²	#16810672	1	4	yes	—
226	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
227	C₂²	C₂×C₄×Dic₁₀₆	1	2	yes	—
228	C₄	—	2	8	no	—
229	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
230	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
231	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
232	C₄	C₂×#13133176	1	1	yes	characteristic
233	C₄	C₂×C₄×D₂₁₂	1	2	yes	—
234	C₂	—	2	2	no	—
235	C₂	—	2	2	no	—
236	C₂	#20274621	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
238	C₂	C₂×#16810672	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
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×#1778
LCS	G ▷ 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. #20274621	C₂	direct (N × H)
7. C₂×#13133153	C₄	semidirect
25. #13133153	C₂×C₄	semidirect
122. C₂×C₅₃	#1778	semidirect
139. C₅₃	C₂×#1778	semidirect
236. C₂	#20274621	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×C₄×#11719917	2
#20274621	2
C₂×#16810388	2
C₂²×#97×C₅₃	2
C₂×#1778	53

Subgroups — C2×#20274621 #27382193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274621 #27382193