Subgroups — C₂×#20276481 #27537092

Order 13568.

698 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	2	yes	maximal
2	#20276481	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810444	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	2	no	—
6	C₂×#13133157	—	2	4	no	—
7	C₂×#13133157	C₂²	1	8	yes	—
8	C₂×#13133157	C₂²	1	4	yes	—
9	C₂²×#11719908	—	2	2	no	—
10	#16810444	C₂²	1	16	yes	complemented
11	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
14	C₅₃×#405	C₄	1	4	yes	—
15	C₅₃×#405	C₂²	1	4	yes	—
16	C₂×#97×C₅₃	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	#13133157	—	2	8	no	—
19	#13133157	—	2	4	no	—
20	#13133157	C₂³	1	8	yes	—
21	C₂×#11719908	—	2	8	no	—
22	C₂×#11719908	—	2	4	no	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
25	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	#117×C₅₃	—	2	2	no	—
37	#117×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	#97×C₅₃	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₅₃	C₂³	1	4	yes	—
44	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	M₁₆×C₅₃	C₂×C₈	1	16	yes	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
59	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₂²⋊C₄×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
75	#11719908	—	2	8	no	—
76	C₂³×C₅₃	—	4	2	no	—
77	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
78	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
79	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
85	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
98	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₈×C₅₃	C₂²×C₈	1	8	yes	—
100	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×#2153	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	4	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	—	4	2	no	—
107	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
109	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
111	C₄×C₅₃	—	4	4	no	—
112	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
115	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
117	C₂²×M₃₂	—	53	2	no	—
118	C₂×#405	D₅₃	1	1	yes	characteristic
119	#2153	—	53	8	no	—
120	C₂×#364	—	53	4	no	—
121	C₂×C₅₃	—	2	1	no	—
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	—	2	4	no	—
124	C₂×C₅₃	#2153	1	2	yes	complemented
125	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
127	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₂×#117	Dic₅₃	1	1	yes	characteristic
129	C₂×M₃₂	—	53	4	no	—
130	C₂×M₃₂	—	53	8	no	—
131	C₂×M₃₂	—	106	4	no	—
132	C₂×M₃₂	—	106	2	no	—
133	C₂²×C₁₆	—	106	2	no	—
134	#405	Dic₅₃	1	4	yes	—
135	#405	C₂×D₅₃	1	4	yes	—
136	C₂×#97	Dic₅₃	1	2	yes	—
137	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
138	#364	—	53	16	no	—
139	C₅₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂×M₁₆	#88322	1	4	yes	—
142	C₂×M₁₆	#88322	1	4	yes	—
143	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
144	C₂²×C₈	Dic₁₀₆	1	2	yes	—
145	C₂²×C₈	Dic₁₀₆	1	2	yes	—
146	C₂²×C₈	D₂₁₂	1	2	yes	—
147	C₂²×C₈	D₂₁₂	1	2	yes	—
148	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
149	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
151	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
152	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
153	#117	—	2	2	no	—
154	#117	C₂×Dic₅₃	1	4	yes	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₂²⋊C₄	—	2	1	no	—
157	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
158	M₃₂	—	53	8	no	—
159	M₃₂	—	106	8	no	—
160	M₃₂	—	106	4	no	—
161	C₂×C₁₆	—	106	8	no	—
162	C₂×C₁₆	—	106	4	no	—
163	#97	C₂×Dic₅₃	1	8	yes	—
164	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
165	C₄×C₈	C₂²×D₅₃	1	4	yes	—
166	C₂⁴	#11719917	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	C₂×#88322	1	2	yes	—
169	C₂²×C₄	C₂×#88322	1	2	yes	—
170	C₂²×C₄	#11719917	1	1	yes	characteristic
171	C₂²×C₄	#11719917	1	2	yes	—
172	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
173	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
176	M₁₆	C₂×#88322	1	16	yes	—
177	C₂×C₈	C₂×#88322	1	4	yes	—
178	C₂×C₈	C₂×#88322	1	4	yes	—
179	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
180	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
181	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
182	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
183	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
184	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
185	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
186	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
187	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
188	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
189	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
191	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
192	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	C₂²×Dic₅₃	1	4	yes	—
197	C₁₆	—	106	8	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄⋊#88322	1	2	yes	—
200	C₂³	C₄⋊#88322	1	2	yes	—
201	C₂³	C₂×#11719917	1	1	yes	characteristic
202	C₂³	C₂×#11719917	1	2	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₄⋊#88322	1	2	yes	—
206	C₂×C₄	C₄⋊#88322	1	4	yes	—
207	C₂×C₄	C₄⋊#88322	1	4	yes	—
208	C₂×C₄	C₄⋊#88322	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
213	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
214	C₂×C₄	C₂²×#88322	1	4	yes	—
215	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719917	1	2	yes	—
217	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
218	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
219	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
220	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
221	C₈	C₂²×#88322	1	8	yes	—
222	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
223	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
229	C₂²	C₂×C₄⋊#88322	1	4	yes	—
230	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
231	C₂²	C₂²×#11719917	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
234	C₄	C₂×C₄⋊#88322	1	4	yes	—
235	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
236	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
237	C₄	C₂³×#88322	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	2	4	no	—
241	C₂	#20276481	1	2	yes	direct product factor
242	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	C₂×#2153
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20276481	C₂	direct (N × H)
3. C₂×#16810444	C₂	semidirect
10. #16810444	C₂²	semidirect
124. C₂×C₅₃	#2153	semidirect
139. C₅₃	C₂×#2153	semidirect
241. C₂	#20276481	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×#13133157	2
#20276481	2
C₂×#16810444	2
C₂×C₅₃×#405	2
C₂×#2153	53

Subgroups — C2×#20276481 #27537092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276481 #27537092