Subgroups — C₂²×#16810676 #28247799

Order 13568.

3086 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊#88322	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊#88322	C₂	1	2	yes	maximal
4	C₂²×C₄×#88322	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₂²	1	4	yes	—
7	#16810676	C₂²	1	64	yes	direct product factor
8	C₂×C₄⋊#88322	C₂²	1	24	yes	complemented
9	C₂×C₂²⋊#88322	C₂²	1	24	yes	—
10	C₂×C₄×#88322	C₂²	1	24	yes	complemented
11	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
12	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₄	1	12	yes	—
16	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₅₃	C₂²	1	12	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	16	no	—
21	C₂²×#88322	C₂³	1	24	yes	—
22	C₄⋊#88322	C₂³	1	32	yes	complemented
23	C₂²⋊#88322	C₂³	1	32	yes	—
24	C₄×#88322	C₂³	1	32	yes	complemented
25	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	—	2	8	no	—
27	C₂³×C₄×C₅₃	—	2	8	no	—
28	C₂³×C₄×C₅₃	C₈	1	16	yes	complemented
29	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	48	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	16	yes	—
47	C₂⁴×C₅₃	—	2	4	no	—
48	C₂⁴×C₅₃	—	2	4	no	—
49	C₂⁴×C₅₃	C₂×C₈	1	8	yes	—
50	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	24	no	—
55	C₂²×C₄×C₅₃	—	2	48	no	—
56	C₂²×C₄×C₅₃	—	2	24	no	—
57	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
58	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	96	yes	complemented
60	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
70	C₂²×C₄×C₅₃	#41	1	8	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
76	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
77	C₄²×C₅₃	C₂⁴	1	4	yes	—
78	C₄²×C₅₃	C₂⁴	1	4	yes	—
79	C₂×#88322	—	2	48	no	—
80	C₂×#88322	C₂⁴	1	16	yes	—
81	C₂³×C₅₃	—	2	12	no	—
82	C₂³×C₅₃	—	2	12	no	—
83	C₂³×C₅₃	—	2	32	no	—
84	C₂³×C₅₃	#117	1	4	yes	—
85	C₂³×C₅₃	C₂²×C₈	1	24	yes	—
86	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
87	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
88	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
89	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₅₃	—	2	12	no	—
91	C₂×C₄×C₅₃	—	2	12	no	—
92	C₂×C₄×C₅₃	—	2	24	no	—
93	C₂×C₄×C₅₃	—	2	32	no	—
94	C₂×C₄×C₅₃	—	2	8	no	—
95	C₂×C₄×C₅₃	#117	1	12	yes	—
96	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	—
97	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	—
98	C₂×C₄×C₅₃	C₂²×C₈	1	64	yes	complemented
99	C₂×C₄×C₅₃	C₂×M₁₆	1	24	yes	—
100	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
107	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
108	#88322	—	2	16	no	—
109	C₂²×#431	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	4	no	—
111	C₂²×C₅₃	—	2	48	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	#431	1	16	yes	complemented
114	C₂²×C₅₃	C₂×#117	1	6	yes	—
115	C₂²×C₅₃	C₂³×C₈	1	8	yes	—
116	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
117	C₂²×C₅₃	C₂²×M₁₆	1	6	yes	—
118	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	2	4	no	—
121	C₄×C₅₃	C₂×#117	1	6	yes	—
122	C₄×C₅₃	C₂³×C₈	1	8	yes	—
123	C₄×C₅₃	C₂³×C₈	1	8	yes	—
124	C₄×C₅₃	C₂²×M₁₆	1	8	yes	—
125	C₄×C₅₃	C₂²×#41	1	2	yes	—
126	C₂³×C₄²	D₅₃	1	1	yes	characteristic
127	C₂×#431	—	53	24	no	—
128	C₂²×C₄⋊C₈	—	53	2	no	—
129	C₂²×C₂²⋊C₈	—	53	2	no	—
130	C₂²×C₄×C₈	—	53	2	no	—
131	C₂×C₅₃	—	2	8	no	—
132	C₂×C₅₃	C₂×#431	1	12	yes	complemented
133	C₂×C₅₃	C₂²×#117	1	1	yes	characteristic
134	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
135	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
136	C₂⁴×C₄	Dic₅₃	1	1	yes	characteristic
137	C₂⁴×C₄	Dic₅₃	1	1	yes	characteristic
138	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
139	C₂³×C₈	—	53	4	no	—
140	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
141	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
142	C₂²×C₄²	Dic₅₃	1	12	yes	—
143	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
144	C₂²×C₄²	C₂×D₅₃	1	12	yes	—
145	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
146	#431	—	53	64	no	—
147	C₂×C₄⋊C₈	—	53	24	no	—
148	C₂×C₂²⋊C₈	—	53	24	no	—
149	C₂×C₄×C₈	—	53	24	no	—
150	C₅₃	C₂²×#431	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂³×C₄	—	2	8	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	#88322	1	16	yes	complemented
155	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₅₃	1	12	yes	—
157	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
159	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₅₃	1	12	yes	—
162	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
164	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
165	C₂²×C₈	—	53	24	no	—
166	C₂²×C₈	—	106	16	no	—
167	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
168	C₂×C₄²	C₂×Dic₅₃	1	48	yes	—
169	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
170	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
171	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
172	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
173	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
174	C₂×C₄²	C₂²×D₅₃	1	16	yes	—
175	C₄⋊C₈	—	53	32	no	—
176	C₂²⋊C₈	—	53	32	no	—
177	C₄×C₈	—	53	32	no	—
178	C₂⁴	—	2	4	no	—
179	C₂⁴	—	2	4	no	—
180	C₂⁴	C₂×#88322	1	8	yes	—
181	C₂⁴	C₂²×Dic₅₃	1	6	yes	—
182	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	2	48	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	C₂×#88322	1	8	yes	—
189	C₂²×C₄	C₂×#88322	1	8	yes	—
190	C₂²×C₄	C₂×#88322	1	96	yes	complemented
191	C₂²×C₄	#11719917	1	8	yes	—
192	C₂²×C₄	#11719945	1	8	yes	—
193	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
194	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
195	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
196	C₂²×C₄	C₂²×Dic₅₃	1	16	yes	—
197	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
198	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
199	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
200	C₂²×C₄	C₂²×Dic₅₃	1	16	yes	—
201	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
202	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
203	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
204	C₂×C₈	—	53	16	no	—
205	C₂×C₈	—	106	48	no	—
206	C₄²	C₂²×Dic₅₃	1	16	yes	—
207	C₄²	C₂²×Dic₅₃	1	16	yes	—
208	C₄²	C₂²×Dic₅₃	1	12	yes	—
209	C₄²	C₂²×Dic₅₃	1	12	yes	—
210	C₄²	C₂³×D₅₃	1	4	yes	—
211	C₄²	C₂³×D₅₃	1	4	yes	—
212	C₂³	—	2	12	no	—
213	C₂³	—	2	12	no	—
214	C₂³	—	2	32	no	—
215	C₂³	#13133176	1	4	yes	—
216	C₂³	C₂²×#88322	1	24	yes	—
217	C₂³	C₂²×#88322	1	4	yes	—
218	C₂³	C₂×#11719917	1	4	yes	—
219	C₂³	C₂³×Dic₅₃	1	4	yes	—
220	C₂³	C₂³×Dic₅₃	1	3	yes	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	12	no	—
223	C₂×C₄	—	2	24	no	—
224	C₂×C₄	—	2	32	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	#13133176	1	12	yes	—
227	C₂×C₄	C₂²×#88322	1	24	yes	—
228	C₂×C₄	C₂²×#88322	1	24	yes	—
229	C₂×C₄	C₂²×#88322	1	64	yes	complemented
230	C₂×C₄	C₂×#11719917	1	24	yes	—
231	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
232	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
233	C₂×C₄	C₂³×Dic₅₃	1	3	yes	—
234	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
235	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
236	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
237	C₂×C₄	C₂×#11719945	1	12	yes	—
238	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
239	C₈	—	106	16	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	48	no	—
243	C₂²	#16810676	1	16	yes	direct product factor
244	C₂²	C₂×#13133176	1	6	yes	—
245	C₂²	C₂³×#88322	1	8	yes	—
246	C₂²	C₂³×#88322	1	6	yes	—
247	C₂²	C₂²×#11719917	1	6	yes	—
248	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	2	4	no	—
251	C₄	C₂×#13133176	1	6	yes	—
252	C₄	C₂³×#88322	1	8	yes	—
253	C₄	C₂³×#88322	1	8	yes	—
254	C₄	C₂²×#11719917	1	8	yes	—
255	C₄	C₂²×#11719945	1	2	yes	—
256	C₂	—	2	8	no	—
257	C₂	C₂×#16810676	1	12	yes	direct product factor
258	C₂	C₂²×#13133176	1	1	yes	characteristic
259	C₂	C₂⁴×#88322	1	1	yes	characteristic
260	C₂	C₂³×#11719917	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#431
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#16810676	C₂	direct (N × H)
2. C₂²×C₄⋊#88322	C₂	semidirect
4. C₂²×C₄×#88322	C₂	semidirect
7. #16810676	C₂²	direct (N × H)
8. C₂×C₄⋊#88322	C₂²	semidirect
10. C₂×C₄×#88322	C₂²	semidirect
22. C₄⋊#88322	C₂³	semidirect
24. C₄×#88322	C₂³	semidirect
28. C₂³×C₄×C₅₃	C₈	semidirect
59. C₂²×C₄×C₅₃	C₂×C₈	semidirect
98. C₂×C₄×C₅₃	C₂²×C₈	semidirect
113. C₂²×C₅₃	#431	semidirect
132. C₂×C₅₃	C₂×#431	semidirect
150. C₅₃	C₂²×#431	semidirect
154. C₂³×C₄	#88322	semidirect
190. C₂²×C₄	C₂×#88322	semidirect
229. C₂×C₄	C₂²×#88322	semidirect
243. C₂²	#16810676	direct (N × H)
257. C₂	C₂×#16810676	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×#16810676	2
C₂²×C₄⋊#88322	2
C₂²×C₂²⋊#88322	2
C₂²×C₄×#88322	2
C₂³×C₄²×C₅₃	2
C₂²×#431	53

Subgroups — C22×#16810676 #28247799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810676 #28247799