Subgroups — C₂²×#16810543 #27550282

Order 13568.

6046 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719919	C₂	1	2	yes	maximal
3	C₂×#16810543	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133124	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13133124	C₂	1	2	yes	maximal
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	—	2	16	no	—
8	C₂³×Dic₁₀₆	—	2	8	no	—
9	C₂³×Dic₁₀₆	C₂²	1	12	yes	—
10	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
11	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
12	C₂²×#11719919	—	2	16	no	—
13	C₂²×#11719919	C₂²	1	24	yes	—
14	#16810543	C₂²	1	64	yes	direct product factor
15	C₂×#13133124	C₂²	1	12	yes	—
16	C₂×#13133124	C₂²	1	24	yes	—
17	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₅₃	—	2	8	no	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
22	C₂²×Dic₁₀₆	—	2	128	no	—
23	C₂²×Dic₁₀₆	—	2	48	no	—
24	C₂²×Dic₁₀₆	—	2	96	no	—
25	C₂²×Dic₁₀₆	C₂³	1	16	yes	—
26	C₂³×Dic₅₃	—	2	8	no	—
27	C₂³×Dic₅₃	—	2	8	no	—
28	C₂³×Dic₅₃	—	2	8	no	—
29	C₂³×Dic₅₃	—	2	8	no	—
30	C₂³×Dic₅₃	D₄	1	16	yes	—
31	C₂³×Dic₅₃	Q₈	1	16	yes	complemented
32	C₂³×Dic₅₃	C₂³	1	12	yes	—
33	C₂³×Dic₅₃	C₂³	1	12	yes	—
34	C₂×#11719919	—	2	96	no	—
35	C₂×#11719919	C₂³	1	32	yes	—
36	#13133124	C₂³	1	32	yes	—
37	#13133124	C₂³	1	16	yes	—
38	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	—	2	4	no	—
40	C₂³×C₄×C₅₃	—	2	8	no	—
41	C₂³×C₄×C₅₃	—	2	4	no	—
42	C₂³×C₄×C₅₃	—	2	4	no	—
43	C₂³×C₄×C₅₃	—	2	8	no	—
44	C₂³×C₄×C₅₃	D₄	1	8	yes	—
45	C₂³×C₄×C₅₃	Q₈	1	8	yes	complemented
46	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
49	C₂×C₄⋊C₄×C₅₃	—	2	48	no	—
50	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
51	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
52	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
53	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
55	C₂⁴×C₅₃	#41	1	4	yes	—
56	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
57	C₂²×C₄×C₅₃	—	2	32	no	—
58	C₂²×C₄×C₅₃	—	2	32	no	—
59	C₂²×C₄×C₅₃	—	2	12	no	—
60	C₂²×C₄×C₅₃	—	2	24	no	—
61	C₂²×C₄×C₅₃	—	2	12	no	—
62	C₂²×C₄×C₅₃	—	2	12	no	—
63	C₂²×C₄×C₅₃	—	2	24	no	—
64	C₂²×C₄×C₅₃	—	4	16	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	complemented
67	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
70	C₂²×Dic₅₃	—	2	64	no	—
71	C₂²×Dic₅₃	—	2	64	no	—
72	C₂²×Dic₅₃	—	2	24	no	—
73	C₂²×Dic₅₃	—	2	24	no	—
74	C₂²×Dic₅₃	—	2	24	no	—
75	C₂²×Dic₅₃	—	2	24	no	—
76	C₂²×Dic₅₃	C₂×D₄	1	48	yes	—
77	C₂²×Dic₅₃	C₂×Q₈	1	48	yes	complemented
78	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
79	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
80	C₂×Dic₁₀₆	—	2	384	no	—
81	C₂×Dic₁₀₆	—	2	32	no	—
82	C₂×Dic₁₀₆	—	2	64	no	—
83	C₄⋊C₄×C₅₃	—	2	32	no	—
84	#11719919	—	2	64	no	—
85	C₂³×C₅₃	#120	1	16	yes	—
86	C₂³×C₅₃	C₂²⋊Q₈	1	32	yes	—
87	C₂³×C₅₃	C₂²⋊Q₈	1	16	yes	—
88	C₂³×C₅₃	C₄⋊Q₈	1	16	yes	—
89	C₂³×C₅₃	C₄⋊Q₈	1	32	yes	—
90	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
91	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
92	C₂³×C₅₃	C₂²×Q₈	1	12	yes	—
93	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
94	C₂³×C₅₃	C₂×#41	1	6	yes	—
95	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	48	no	—
97	C₂×C₄×C₅₃	—	2	48	no	—
98	C₂×C₄×C₅₃	—	2	8	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	4	24	no	—
104	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
105	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	complemented
106	C₂×Dic₅₃	—	2	96	no	—
107	C₂×Dic₅₃	—	2	96	no	—
108	C₂×Dic₅₃	—	2	8	no	—
109	C₂×Dic₅₃	—	2	8	no	—
110	C₂×Dic₅₃	—	2	8	no	—
111	C₂×Dic₅₃	—	2	8	no	—
112	C₂×Dic₅₃	C₂²×D₄	1	16	yes	—
113	C₂×Dic₅₃	C₂²×Q₈	1	16	yes	complemented
114	Dic₁₀₆	—	2	128	no	—
115	C₂²×#395	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	#395	1	64	yes	complemented
117	C₂²×C₅₃	C₂×#120	1	12	yes	—
118	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	24	yes	—
119	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	12	yes	—
120	C₂²×C₅₃	C₂×C₄⋊Q₈	1	12	yes	—
121	C₂²×C₅₃	C₂×C₄⋊Q₈	1	24	yes	—
122	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
123	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
124	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
125	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
126	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
127	C₄×C₅₃	—	2	8	no	—
128	C₄×C₅₃	—	2	8	no	—
129	C₄×C₅₃	—	4	4	no	—
130	Dic₅₃	—	2	16	no	—
131	Dic₅₃	—	2	16	no	—
132	C₂⁴×Q₈	—	53	1	no	—
133	C₂³×C₄⋊C₄	—	53	2	no	—
134	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
135	C₂×#395	—	53	24	no	—
136	C₂²×#95	—	53	2	no	—
137	C₂²×#95	—	53	1	no	—
138	C₂×C₅₃	C₂×#395	1	24	yes	complemented
139	C₂×C₅₃	C₂²×#120	1	1	yes	characteristic
140	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	2	yes	—
141	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₅₃	C₂²×C₄⋊Q₈	1	2	yes	—
144	C₂³×Q₈	—	53	12	no	—
145	C₂³×Q₈	—	106	16	no	—
146	C₂³×Q₈	—	106	8	no	—
147	C₂⁴×C₄	—	53	2	no	—
148	C₂⁴×C₄	—	53	2	no	—
149	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
150	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
151	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	—	2	8	no	—
153	C₂²×C₄⋊C₄	—	53	24	no	—
154	C₂²×C₄⋊C₄	—	106	16	no	—
155	C₂²×C₄⋊C₄	C₂×D₅₃	1	12	yes	—
156	#395	—	53	64	no	—
157	C₂×#95	—	53	24	no	—
158	C₂×#95	—	53	12	no	—
159	C₅₃	C₂²×#395	1	1	yes	characteristic, complemented, Sylow
160	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
161	C₂²×Q₈	—	53	16	no	—
162	C₂²×Q₈	—	106	96	no	—
163	C₂²×Q₈	—	106	128	no	—
164	C₂²×Q₈	—	106	48	no	—
165	C₂³×C₄	—	2	4	no	—
166	C₂³×C₄	—	2	8	no	—
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	—	2	8	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	53	16	no	—
171	C₂³×C₄	—	53	16	no	—
172	C₂³×C₄	—	53	12	no	—
173	C₂³×C₄	—	53	12	no	—
174	C₂³×C₄	—	106	8	no	—
175	C₂³×C₄	—	106	8	no	—
176	C₂³×C₄	—	106	8	no	—
177	C₂³×C₄	—	106	8	no	—
178	C₂³×C₄	Dic₁₀₆	1	8	yes	complemented
179	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
180	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
181	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
182	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
183	C₂×C₄⋊C₄	—	2	48	no	—
184	C₂×C₄⋊C₄	—	53	32	no	—
185	C₂×C₄⋊C₄	—	106	96	no	—
186	C₂×C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
187	#95	—	53	16	no	—
188	#95	—	53	32	no	—
189	C₂⁴	C₂×Dic₁₀₆	1	4	yes	—
190	C₂⁴	#11719945	1	4	yes	—
191	C₂⁴	D₄×D₅₃	1	8	yes	—
192	C₂⁴	Q₈×D₅₃	1	8	yes	—
193	C₂⁴	C₂×C₂²⋊D₅₃	1	4	yes	—
194	C₂⁴	C₂³×D₅₃	1	3	yes	—
195	C₂×Q₈	—	106	64	no	—
196	C₂×Q₈	—	106	384	no	—
197	C₂×Q₈	—	106	32	no	—
198	C₂²×C₄	—	2	32	no	—
199	C₂²×C₄	—	2	32	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	24	no	—
204	C₂²×C₄	—	2	12	no	—
205	C₂²×C₄	—	4	16	no	—
206	C₂²×C₄	—	53	48	no	—
207	C₂²×C₄	—	53	48	no	—
208	C₂²×C₄	—	53	8	no	—
209	C₂²×C₄	—	53	8	no	—
210	C₂²×C₄	—	106	64	no	—
211	C₂²×C₄	—	106	64	no	—
212	C₂²×C₄	—	106	24	no	—
213	C₂²×C₄	—	106	24	no	—
214	C₂²×C₄	—	106	24	no	—
215	C₂²×C₄	—	106	24	no	—
216	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	complemented
217	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
218	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
219	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
220	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
221	C₄⋊C₄	—	2	32	no	—
222	C₄⋊C₄	—	106	64	no	—
223	C₂³	#13133185	1	16	yes	—
224	C₂³	#13133195	1	32	yes	—
225	C₂³	#13133196	1	16	yes	—
226	C₂³	#13133215	1	16	yes	—
227	C₂³	#13133216	1	32	yes	—
228	C₂³	C₂×D₄×D₅₃	1	12	yes	—
229	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
230	C₂³	C₂×Q₈×D₅₃	1	12	yes	—
231	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
232	C₂³	C₂×#11719945	1	6	yes	—
233	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
234	Q₈	—	106	128	no	—
235	C₂×C₄	—	2	48	no	—
236	C₂×C₄	—	2	48	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	24	no	—
243	C₂×C₄	—	53	16	no	—
244	C₂×C₄	—	53	16	no	—
245	C₂×C₄	—	106	96	no	—
246	C₂×C₄	—	106	96	no	—
247	C₂×C₄	—	106	8	no	—
248	C₂×C₄	—	106	8	no	—
249	C₂×C₄	—	106	8	no	—
250	C₂×C₄	—	106	8	no	—
251	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
252	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	complemented
253	C₂²	#16810543	1	64	yes	direct product factor
254	C₂²	C₂×#13133185	1	12	yes	—
255	C₂²	C₂×#13133195	1	24	yes	—
256	C₂²	C₂×#13133196	1	12	yes	—
257	C₂²	C₂×#13133215	1	12	yes	—
258	C₂²	C₂×#13133216	1	24	yes	—
259	C₂²	C₂²×D₄×D₅₃	1	2	yes	—
260	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂²	C₂²×Q₈×D₅₃	1	2	yes	—
262	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
263	C₂²	C₂²×#11719945	1	1	yes	characteristic
264	C₄	—	2	8	no	—
265	C₄	—	2	8	no	—
266	C₄	—	4	4	no	—
267	C₄	—	106	16	no	—
268	C₄	—	106	16	no	—
269	C₂	C₂×#16810543	1	24	yes	direct product factor
270	C₂	C₂²×#13133185	1	1	yes	characteristic
271	C₂	C₂²×#13133195	1	2	yes	—
272	C₂	C₂²×#13133196	1	1	yes	characteristic
273	C₂	C₂²×#13133215	1	1	yes	characteristic
274	C₂	C₂²×#13133216	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×#395
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#16810543	C₂	direct (N × H)
14. #16810543	C₂²	direct (N × H)
31. C₂³×Dic₅₃	Q₈	semidirect
45. C₂³×C₄×C₅₃	Q₈	semidirect
66. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
77. C₂²×Dic₅₃	C₂×Q₈	semidirect
105. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
113. C₂×Dic₅₃	C₂²×Q₈	semidirect
116. C₂²×C₅₃	#395	semidirect
138. C₂×C₅₃	C₂×#395	semidirect
159. C₅₃	C₂²×#395	semidirect
178. C₂³×C₄	Dic₁₀₆	semidirect
216. C₂²×C₄	C₂×Dic₁₀₆	semidirect
252. C₂×C₄	C₂²×Dic₁₀₆	semidirect
253. C₂²	#16810543	direct (N × H)
269. C₂	C₂×#16810543	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×Dic₁₀₆	2
C₂³×#11719919	2
C₂×#16810543	2
C₂²×#13133124	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#395	53

Subgroups — C22×#16810543 #27550282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810543 #27550282