Subgroups — C₂²×#16811179 #28290797

Order 13568.

8238 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811179	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133216	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133209	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133195	C₂	1	4	yes	maximal
5	C₂²×#13133182	C₂	1	4	yes	maximal, complemented
6	C₂²×#13133175	C₂	1	2	yes	maximal
7	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Dic₁₀₆	C₂²	1	4	yes	—
10	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
11	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
12	#16811179	C₂²	1	256	yes	direct product factor
13	C₂×#13133216	C₂²	1	48	yes	complemented
14	C₂×#13133209	C₂²	1	24	yes	complemented
15	C₂×#13133195	C₂²	1	96	yes	—
16	C₂×#13133182	C₂²	1	96	yes	complemented
17	C₂×#13133175	C₂²	1	48	yes	—
18	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
19	C₂²×#11719919	C₂²	1	4	yes	—
20	C₂²×#11719919	C₂²	1	4	yes	—
21	C₂²×#11719921	C₂²	1	4	yes	—
22	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
23	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
25	C₂×C₄²×D₅₃	C₂²	1	24	yes	—
26	C₂×#125×C₅₃	C₂²	1	24	yes	complemented
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₀₆	—	2	32	no	—
32	C₂²×Dic₁₀₆	C₂³	1	48	yes	—
33	C₂³×Dic₅₃	C₂³	1	4	yes	—
34	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
35	C₂³×Dic₅₃	C₂³	1	2	yes	—
36	C₂²×C₄×D₅₃	—	2	16	no	—
37	C₂²×C₄×D₅₃	Q₈	1	16	yes	complemented
38	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
39	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
40	#13133216	C₂³	1	128	yes	complemented
41	#13133209	C₂³	1	64	yes	complemented
42	#13133195	C₂³	1	256	yes	—
43	#13133182	C₂³	1	256	yes	complemented
44	#13133175	C₂³	1	128	yes	—
45	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
46	C₂×#11719919	C₂³	1	48	yes	—
47	C₂×#11719919	C₂³	1	48	yes	—
48	C₂×#11719921	C₂³	1	48	yes	—
49	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
50	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
51	C₂×C₄×Dic₅₃	C₂³	1	48	yes	—
52	C₄²×D₅₃	C₂³	1	64	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	#125×C₅₃	C₂³	1	64	yes	complemented
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
60	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂³×D₅₃	—	2	8	no	—
62	C₂³×D₅₃	C₂×Q₈	1	8	yes	—
63	C₂³×D₅₃	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₅₃	—	2	8	no	—
65	C₂²×C₄×C₅₃	—	2	16	no	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
70	C₂²×Dic₅₃	—	2	16	no	—
71	C₂²×Dic₅₃	C₂×Q₈	1	8	yes	—
72	C₂²×Dic₅₃	#41	1	16	yes	—
73	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
74	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
75	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
76	C₂×C₄×D₅₃	—	2	96	no	—
77	C₂×C₄×D₅₃	C₂×Q₈	1	96	yes	complemented
78	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
79	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
80	C₂×Dic₁₀₆	—	2	192	no	—
81	C₂×Dic₁₀₆	C₂⁴	1	64	yes	—
82	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
83	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
84	C₄²×C₅₃	C₂⁴	1	16	yes	—
85	#11719921	C₂⁴	1	64	yes	—
86	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
87	#11719919	C₂⁴	1	64	yes	—
88	#11719919	C₂⁴	1	64	yes	—
89	C₄×Dic₅₃	C₂⁴	1	32	yes	—
90	C₄×Dic₅₃	C₂⁴	1	16	yes	—
91	C₄×Dic₅₃	C₂⁴	1	64	yes	—
92	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
93	C₂³×C₅₃	C₂×#41	1	8	yes	—
94	C₂³×C₅₃	C₂⁵	1	3	yes	—
95	C₂²×D₅₃	—	2	32	no	—
96	C₂²×D₅₃	—	2	24	no	—
97	C₂²×D₅₃	C₂²×Q₈	1	24	yes	—
98	C₂²×D₅₃	C₂⁵	1	4	yes	—
99	C₂×C₄×C₅₃	—	2	48	no	—
100	C₂×C₄×C₅₃	—	2	24	no	—
101	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	—
102	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
103	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
104	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
105	C₂×Dic₅₃	—	2	48	no	—
106	C₂×Dic₅₃	C₂²×Q₈	1	24	yes	—
107	C₂×Dic₅₃	C₂×#41	1	48	yes	—
108	C₂×Dic₅₃	C₂⁵	1	16	yes	—
109	C₂×Dic₅₃	C₂⁵	1	4	yes	—
110	C₂×Dic₅₃	C₂⁵	1	8	yes	—
111	C₄×D₅₃	—	2	64	no	—
112	C₄×D₅₃	C₂²×Q₈	1	64	yes	complemented
113	Dic₁₀₆	—	2	128	no	—
114	C₂²×#533	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	#533	1	16	yes	complemented
116	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
117	C₂²×C₅₃	C₂²×#41	1	12	yes	—
118	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
119	C₂×D₅₃	—	2	8	no	—
120	C₂×D₅₃	—	2	48	no	—
121	C₂×D₅₃	C₂³×Q₈	1	8	yes	—
122	C₄×C₅₃	—	2	16	no	—
123	C₄×C₅₃	—	2	8	no	—
124	C₄×C₅₃	C₂³×Q₈	1	8	yes	—
125	Dic₅₃	—	2	16	no	—
126	Dic₅₃	C₂³×Q₈	1	8	yes	—
127	Dic₅₃	C₂²×#41	1	16	yes	—
128	C₂×#533	—	53	48	no	—
129	C₂²×C₄⋊Q₈	—	53	2	no	—
130	C₂²×#125	—	53	1	no	—
131	C₂²×#125	D₅₃	1	1	yes	characteristic, complemented
132	C₂²×C₂²⋊Q₈	—	53	4	no	—
133	C₂²×C₄×Q₈	—	53	4	no	—
134	C₂²×#117	—	53	2	no	—
135	C₂³×C₄²	—	53	1	no	—
136	C₂×C₅₃	C₂×#533	1	12	yes	complemented
137	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
138	C₂×C₅₃	C₂³×#41	1	2	yes	—
139	D₅₃	—	2	8	no	—
140	C₂³×Q₈	—	53	4	no	—
141	C₂⁴×C₄	—	53	1	no	—
142	C₂⁴×C₄	—	53	2	no	—
143	#533	—	53	256	no	—
144	C₂×C₄⋊Q₈	—	53	48	no	—
145	C₂×#125	—	53	24	no	—
146	C₂×#125	C₂×D₅₃	1	24	yes	complemented
147	C₂×C₂²⋊Q₈	—	53	96	no	—
148	C₂×C₄×Q₈	—	53	96	no	—
149	C₂×#117	—	53	48	no	—
150	C₂²×C₄⋊C₄	—	53	4	no	—
151	C₂²×C₄⋊C₄	—	53	4	no	—
152	C₂²×C₄⋊C₄	—	53	2	no	—
153	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
154	C₂²×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
155	C₂²×C₂²⋊C₄	—	53	4	no	—
156	C₂²×C₄²	—	53	2	no	—
157	C₂²×C₄²	—	53	1	no	—
158	C₂²×C₄²	—	53	4	no	—
159	C₂²×C₄²	—	53	24	no	—
160	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
161	C₅₃	C₂²×#533	1	1	yes	characteristic, complemented, Sylow
162	C₂⁵	—	53	1	no	—
163	C₂²×Q₈	—	53	48	no	—
164	C₂²×Q₈	—	106	32	no	—
165	C₂³×C₄	—	53	2	no	—
166	C₂³×C₄	—	53	1	no	—
167	C₂³×C₄	—	53	16	no	—
168	C₂³×C₄	—	53	12	no	—
169	C₂³×C₄	—	53	4	no	—
170	C₂³×C₄	—	53	24	no	—
171	C₂³×C₄	—	106	16	no	—
172	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
173	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
174	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
175	C₄⋊Q₈	—	53	128	no	—
176	#125	—	53	64	no	—
177	#125	C₂²×D₅₃	1	64	yes	complemented
178	C₂²⋊Q₈	—	53	256	no	—
179	C₄×Q₈	—	53	256	no	—
180	#117	—	53	128	no	—
181	C₂×C₄⋊C₄	—	53	24	no	—
182	C₂×C₄⋊C₄	—	53	48	no	—
183	C₂×C₄⋊C₄	—	53	48	no	—
184	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	—
185	C₂×C₄⋊C₄	C₂²×D₅₃	1	48	yes	—
186	C₂×C₂²⋊C₄	—	53	48	no	—
187	C₂×C₄²	—	53	48	no	—
188	C₂×C₄²	—	53	24	no	—
189	C₂×C₄²	—	53	12	no	—
190	C₂×C₄²	—	53	64	no	—
191	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
192	C₂⁴	—	53	8	no	—
193	C₂⁴	—	53	6	no	—
194	C₂⁴	—	106	8	no	—
195	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
196	C₂×Q₈	—	53	64	no	—
197	C₂×Q₈	—	106	192	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	16	no	—
200	C₂²×C₄	—	53	8	no	—
201	C₂²×C₄	—	53	16	no	—
202	C₂²×C₄	—	53	6	no	—
203	C₂²×C₄	—	53	12	no	—
204	C₂²×C₄	—	53	96	no	—
205	C₂²×C₄	—	53	16	no	—
206	C₂²×C₄	—	53	24	no	—
207	C₂²×C₄	—	53	32	no	—
208	C₂²×C₄	—	106	96	no	—
209	C₂²×C₄	—	106	16	no	—
210	C₂²×C₄	Q₈×D₅₃	1	8	yes	—
211	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
212	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
213	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
214	C₄⋊C₄	—	53	32	no	—
215	C₄⋊C₄	—	53	64	no	—
216	C₄⋊C₄	—	53	64	no	—
217	C₄⋊C₄	C₂³×D₅₃	1	32	yes	—
218	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
219	C₂²⋊C₄	—	53	64	no	—
220	C₄²	—	53	32	no	—
221	C₄²	—	53	16	no	—
222	C₄²	—	53	64	no	—
223	C₄²	C₂³×D₅₃	1	16	yes	—
224	C₂³	—	53	24	no	—
225	C₂³	—	53	4	no	—
226	C₂³	—	106	32	no	—
227	C₂³	—	106	24	no	—
228	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
229	C₂³	#41×D₅₃	1	8	yes	—
230	C₂³	C₂⁴×D₅₃	1	3	yes	—
231	Q₈	—	106	128	no	—
232	C₂×C₄	—	2	24	no	—
233	C₂×C₄	—	2	48	no	—
234	C₂×C₄	—	53	24	no	—
235	C₂×C₄	—	53	48	no	—
236	C₂×C₄	—	53	4	no	—
237	C₂×C₄	—	53	8	no	—
238	C₂×C₄	—	53	64	no	—
239	C₂×C₄	—	53	16	no	—
240	C₂×C₄	—	106	64	no	—
241	C₂×C₄	—	106	48	no	—
242	C₂×C₄	C₂×Q₈×D₅₃	1	24	yes	—
243	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
244	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
245	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
246	C₂²	—	53	8	no	—
247	C₂²	—	106	48	no	—
248	C₂²	—	106	8	no	—
249	C₂²	#16811179	1	16	yes	direct product factor
250	C₂²	C₂²×Q₈×D₅₃	1	6	yes	—
251	C₂²	C₂×#41×D₅₃	1	12	yes	—
252	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
253	C₄	—	2	8	no	—
254	C₄	—	2	16	no	—
255	C₄	—	53	8	no	—
256	C₄	—	53	16	no	—
257	C₄	—	106	16	no	—
258	C₄	C₂²×Q₈×D₅₃	1	8	yes	—
259	C₂	—	106	8	no	—
260	C₂	C₂×#16811179	1	12	yes	direct product factor
261	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
262	C₂	C₂²×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#125×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁴×D₅₃	C₂³
hypercenter	C₂²×#125	D₅₃
hypocenter	C₅₃	C₂²×#533
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
1. C₂×#16811179	C₂	direct (N × H)
2. C₂²×#13133216	C₂	semidirect
3. C₂²×#13133209	C₂	semidirect
5. C₂²×#13133182	C₂	semidirect
8. C₂²×#125×C₅₃	C₂	semidirect
12. #16811179	C₂²	direct (N × H)
13. C₂×#13133216	C₂²	semidirect
14. C₂×#13133209	C₂²	semidirect
16. C₂×#13133182	C₂²	semidirect
26. C₂×#125×C₅₃	C₂²	semidirect
37. C₂²×C₄×D₅₃	Q₈	semidirect
40. #13133216	C₂³	semidirect
41. #13133209	C₂³	semidirect
43. #13133182	C₂³	semidirect
56. #125×C₅₃	C₂³	semidirect
77. C₂×C₄×D₅₃	C₂×Q₈	semidirect
112. C₄×D₅₃	C₂²×Q₈	semidirect
115. C₂²×C₅₃	#533	semidirect
131. C₂²×#125	D₅₃	semidirect
136. C₂×C₅₃	C₂×#533	semidirect
146. C₂×#125	C₂×D₅₃	semidirect
161. C₅₃	C₂²×#533	semidirect
177. #125	C₂²×D₅₃	semidirect
249. C₂²	#16811179	direct (N × H)
260. C₂	C₂×#16811179	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×#16811179	2
C₂²×#13133216	2
C₂²×#13133209	2
C₂²×#13133195	2
C₂²×#13133182	2
C₂²×#13133175	2
C₂²×C₄²×D₅₃	2
C₂²×#125×C₅₃	2
C₂²×#533	53

Subgroups — C22×#16811179 #28290797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811179 #28290797