Subgroups — C₂²×#16811365 #28290964

Order 13568.

10246 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811365	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133206	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133205	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133203	C₂	1	4	yes	maximal, complemented
5	C₂²×#13133187	C₂	1	4	yes	maximal, complemented
6	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
7	C₂²×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
9	C₂³×Dic₁₀₆	C₂²	1	2	yes	complemented
10	C₂³×D₂₁₂	C₂²	1	2	yes	—
11	C₂³×D₂₁₂	C₂²	1	2	yes	—
12	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	—
13	#16811365	C₂²	1	256	yes	direct product factor
14	C₂×#13133206	C₂²	1	48	yes	complemented
15	C₂×#13133205	C₂²	1	48	yes	complemented
16	C₂×#13133203	C₂²	1	96	yes	complemented
17	C₂×#13133187	C₂²	1	96	yes	complemented
18	C₂×#13133185	C₂²	1	48	yes	complemented
19	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
20	C₂²×#11719921	C₂²	1	4	yes	complemented
21	C₂²×#11719921	C₂²	1	4	yes	complemented
22	C₂²×#11719921	C₂²	1	4	yes	complemented
23	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
24	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#124×C₅₃	C₂²	1	24	yes	complemented
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂⁴×D₅₃	—	4	16	no	—
30	C₂⁴×D₅₃	—	4	8	no	—
31	C₂⁴×D₅₃	C₂³	1	4	yes	—
32	C₂⁴×D₅₃	C₂³	1	24	yes	complemented
33	C₂²×Dic₁₀₆	—	4	8	no	—
34	C₂²×Dic₁₀₆	C₂³	1	24	yes	complemented
35	C₂²×D₂₁₂	—	4	8	no	—
36	C₂²×D₂₁₂	—	4	8	no	—
37	C₂²×D₂₁₂	C₂³	1	24	yes	—
38	C₂²×D₂₁₂	C₂³	1	24	yes	—
39	C₂²×C₂²⋊D₅₃	—	4	16	no	—
40	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	—
41	C₂³×Dic₅₃	C₂³	1	4	yes	—
42	#13133206	C₂³	1	128	yes	complemented
43	#13133205	C₂³	1	128	yes	complemented
44	#13133203	C₂³	1	256	yes	complemented
45	#13133187	C₂³	1	256	yes	complemented
46	#13133185	C₂³	1	128	yes	complemented
47	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
48	C₂×#11719921	C₂³	1	48	yes	complemented
49	C₂×#11719921	C₂³	1	48	yes	complemented
50	C₂×#11719921	C₂³	1	48	yes	complemented
51	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
52	C₂⁵×C₅₃	C₂³	1	2	yes	—
53	C₂²×Q₈×C₅₃	—	4	4	no	—
54	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
55	C₂²×D₄×C₅₃	—	4	4	no	—
56	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
60	#124×C₅₃	C₂³	1	64	yes	complemented
61	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	complemented
62	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
63	C₂⁴×C₅₃	—	2	8	no	—
64	C₂⁴×C₅₃	—	2	16	no	—
65	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
66	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂³×D₅₃	—	2	16	no	—
68	C₂³×D₅₃	—	2	16	no	—
69	C₂³×D₅₃	—	2	16	no	—
70	C₂³×D₅₃	—	4	48	no	—
71	C₂³×D₅₃	—	4	96	no	—
72	C₂³×D₅₃	—	4	128	no	—
73	C₂³×D₅₃	C₂⁴	1	24	yes	—
74	C₂³×D₅₃	C₂⁴	1	32	yes	complemented
75	C₂×Q₈×C₅₃	—	4	24	no	—
76	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
77	C₂×D₄×C₅₃	—	4	24	no	—
78	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
79	C₂²×C₄×C₅₃	—	2	4	no	—
80	C₂²×C₄×C₅₃	—	2	8	no	—
81	C₂²×C₄×C₅₃	—	2	8	no	—
82	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
83	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
85	C₂×C₂²⋊D₅₃	—	4	96	no	—
86	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
87	C₂²×Dic₅₃	—	2	16	no	—
88	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
89	C₂×D₂₁₂	—	4	48	no	—
90	C₂×D₂₁₂	—	4	48	no	—
91	C₂×D₂₁₂	C₂⁴	1	32	yes	—
92	C₂×D₂₁₂	C₂⁴	1	32	yes	—
93	C₂×Dic₁₀₆	—	4	48	no	—
94	C₂×Dic₁₀₆	C₂⁴	1	32	yes	complemented
95	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	complemented
96	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
97	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
98	#11719921	C₂⁴	1	64	yes	complemented
99	#11719921	C₂⁴	1	64	yes	complemented
100	#11719921	C₂⁴	1	64	yes	complemented
101	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
102	C₂³×C₅₃	—	2	48	no	—
103	C₂³×C₅₃	—	2	24	no	—
104	C₂³×C₅₃	—	4	32	no	—
105	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
106	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
107	C₂³×C₅₃	C₂⁵	1	8	yes	—
108	C₂³×C₅₃	C₂⁵	1	3	yes	—
109	C₂²×D₅₃	—	2	48	no	—
110	C₂²×D₅₃	—	2	48	no	—
111	C₂²×D₅₃	—	2	48	no	—
112	C₂²×D₅₃	—	4	32	no	—
113	C₂²×D₅₃	—	4	64	no	—
114	C₂²×D₅₃	—	4	384	no	—
115	C₂²×D₅₃	—	4	64	no	—
116	C₂²×D₅₃	C₂⁵	1	16	yes	—
117	Q₈×C₅₃	—	4	16	no	—
118	D₄×C₅₃	—	4	16	no	—
119	C₂×C₄×C₅₃	—	2	24	no	—
120	C₂×C₄×C₅₃	—	2	24	no	—
121	C₂×C₄×C₅₃	—	2	12	no	—
122	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
123	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
124	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
125	C₂²⋊D₅₃	—	4	64	no	—
126	C₂×Dic₅₃	—	2	48	no	—
127	C₂×Dic₅₃	C₂⁵	1	16	yes	—
128	D₂₁₂	—	4	32	no	—
129	D₂₁₂	—	4	32	no	—
130	Dic₁₀₆	—	4	32	no	—
131	C₂²×H(3,𝔽₄)	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	16	no	—
133	C₂²×C₅₃	—	2	8	no	—
134	C₂²×C₅₃	—	4	48	no	—
135	C₂²×C₅₃	H(3,𝔽₄)	1	16	yes	complemented
136	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
137	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
138	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
139	C₂×D₅₃	—	2	16	no	—
140	C₂×D₅₃	—	2	16	no	—
141	C₂×D₅₃	—	2	16	no	—
142	C₂×D₅₃	—	4	128	no	—
143	C₂×D₅₃	—	4	96	no	—
144	C₄×C₅₃	—	2	8	no	—
145	C₄×C₅₃	—	2	8	no	—
146	C₄×C₅₃	—	2	4	no	—
147	Dic₅₃	—	2	16	no	—
148	C₂×H(3,𝔽₄)	—	53	48	no	—
149	C₂²×#124	—	53	2	no	—
150	C₂²×#124	—	53	2	no	—
151	C₂²×#124	—	53	4	no	—
152	C₂²×#124	D₅₃	1	1	yes	characteristic, complemented
153	C₂²×#120	—	53	2	no	—
154	C₂²×#120	—	53	4	no	—
155	C₂×C₅₃	—	4	8	no	—
156	C₂×C₅₃	C₂×H(3,𝔽₄)	1	12	yes	complemented
157	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
159	D₅₃	—	4	16	no	—
160	C₂⁶	—	53	2	no	—
161	C₂³×Q₈	—	53	2	no	—
162	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
163	C₂³×D₄	—	53	4	no	—
164	C₂³×D₄	—	53	2	no	—
165	C₂³×D₄	—	53	2	no	—
166	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
167	H(3,𝔽₄)	—	53	256	no	—
168	C₂×#124	—	53	48	no	—
169	C₂×#124	—	53	96	no	—
170	C₂×#124	—	53	48	no	—
171	C₂×#124	C₂×D₅₃	1	24	yes	complemented
172	C₂×#120	—	53	48	no	—
173	C₂×#120	—	53	96	no	—
174	C₂²×C₂²⋊C₄	—	53	2	no	—
175	C₂²×C₂²⋊C₄	—	53	4	no	—
176	C₂²×C₂²⋊C₄	—	53	4	no	—
177	C₂²×C₂²⋊C₄	—	53	4	no	—
178	C₂²×C₂²⋊C₄	C₂×D₅₃	1	4	yes	complemented
179	C₂²×C₄²	—	53	2	no	—
180	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
181	C₅₃	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
182	C₂⁵	—	53	4	no	—
183	C₂⁵	—	53	24	no	—
184	C₂⁵	—	212	8	no	—
185	C₂⁵	—	212	16	no	—
186	C₂⁵	C₂²×D₅₃	1	2	yes	—
187	C₂²×Q₈	—	4	4	no	—
188	C₂²×Q₈	—	53	24	no	—
189	C₂²×Q₈	—	212	8	no	—
190	C₂²×Q₈	C₂²×D₅₃	1	12	yes	complemented
191	C₂²×D₄	—	4	4	no	—
192	C₂²×D₄	—	53	48	no	—
193	C₂²×D₄	—	53	24	no	—
194	C₂²×D₄	—	53	24	no	—
195	C₂²×D₄	—	212	8	no	—
196	C₂²×D₄	—	212	8	no	—
197	C₂²×D₄	—	212	16	no	—
198	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
199	C₂³×C₄	—	53	4	no	—
200	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
202	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
203	#124	—	53	128	no	—
204	#124	—	53	256	no	—
205	#124	—	53	128	no	—
206	#124	C₂²×D₅₃	1	64	yes	complemented
207	#120	—	53	128	no	—
208	#120	—	53	256	no	—
209	C₂×C₂²⋊C₄	—	53	24	no	—
210	C₂×C₂²⋊C₄	—	53	48	no	—
211	C₂×C₂²⋊C₄	—	53	48	no	—
212	C₂×C₂²⋊C₄	—	53	48	no	—
213	C₂×C₂²⋊C₄	C₂²×D₅₃	1	48	yes	complemented
214	C₂×C₄²	—	53	24	no	—
215	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
216	C₂⁴	—	2	8	no	—
217	C₂⁴	—	2	16	no	—
218	C₂⁴	—	53	24	no	—
219	C₂⁴	—	53	32	no	—
220	C₂⁴	—	106	16	no	—
221	C₂⁴	—	106	16	no	—
222	C₂⁴	—	106	16	no	—
223	C₂⁴	—	212	48	no	—
224	C₂⁴	—	212	96	no	—
225	C₂⁴	—	212	128	no	—
226	C₂⁴	C₂³×D₅₃	1	12	yes	—
227	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
228	C₂×Q₈	—	4	24	no	—
229	C₂×Q₈	—	53	32	no	—
230	C₂×Q₈	—	212	48	no	—
231	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
232	C₂×D₄	—	4	24	no	—
233	C₂×D₄	—	53	64	no	—
234	C₂×D₄	—	53	32	no	—
235	C₂×D₄	—	53	32	no	—
236	C₂×D₄	—	212	48	no	—
237	C₂×D₄	—	212	48	no	—
238	C₂×D₄	—	212	96	no	—
239	C₂×D₄	C₂³×D₅₃	1	16	yes	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	53	24	no	—
244	C₂²×C₄	—	106	16	no	—
245	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
246	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
247	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
248	C₂²⋊C₄	—	53	32	no	—
249	C₂²⋊C₄	—	53	64	no	—
250	C₂²⋊C₄	—	53	64	no	—
251	C₂²⋊C₄	—	53	64	no	—
252	C₂²⋊C₄	C₂³×D₅₃	1	64	yes	complemented
253	C₄²	—	53	32	no	—
254	C₄²	C₂³×D₅₃	1	16	yes	complemented
255	C₂³	—	2	24	no	—
256	C₂³	—	2	48	no	—
257	C₂³	—	4	32	no	—
258	C₂³	—	53	16	no	—
259	C₂³	—	106	48	no	—
260	C₂³	—	106	48	no	—
261	C₂³	—	106	48	no	—
262	C₂³	—	212	32	no	—
263	C₂³	—	212	64	no	—
264	C₂³	—	212	384	no	—
265	C₂³	—	212	64	no	—
266	C₂³	#13133262	1	4	yes	—
267	C₂³	#13133263	1	8	yes	—
268	C₂³	C₂⁴×D₅₃	1	8	yes	—
269	C₂³	C₂⁴×D₅₃	1	3	yes	—
270	Q₈	—	4	16	no	—
271	Q₈	—	212	32	no	—
272	D₄	—	4	16	no	—
273	D₄	—	212	32	no	—
274	D₄	—	212	32	no	—
275	D₄	—	212	64	no	—
276	C₂×C₄	—	2	12	no	—
277	C₂×C₄	—	2	24	no	—
278	C₂×C₄	—	2	24	no	—
279	C₂×C₄	—	53	16	no	—
280	C₂×C₄	—	106	48	no	—
281	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
282	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
283	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
284	C₂²	—	2	8	no	—
285	C₂²	—	2	16	no	—
286	C₂²	—	4	48	no	—
287	C₂²	—	106	16	no	—
288	C₂²	—	106	16	no	—
289	C₂²	—	106	16	no	—
290	C₂²	—	212	128	no	—
291	C₂²	—	212	96	no	—
292	C₂²	#16811365	1	16	yes	direct product factor
293	C₂²	C₂×#13133262	1	6	yes	—
294	C₂²	C₂×#13133263	1	12	yes	—
295	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
296	C₄	—	2	4	no	—
297	C₄	—	2	8	no	—
298	C₄	—	2	8	no	—
299	C₄	—	106	16	no	—
300	C₂	—	4	8	no	—
301	C₂	—	212	16	no	—
302	C₂	C₂×#16811365	1	12	yes	direct product factor
303	C₂	C₂²×#13133262	1	1	yes	characteristic
304	C₂	C₂²×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#124×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#124	D₅₃
hypocenter	C₅₃	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. C₂×#16811365	C₂	direct (N × H)
2. C₂²×#13133206	C₂	semidirect
3. C₂²×#13133205	C₂	semidirect
4. C₂²×#13133203	C₂	semidirect
5. C₂²×#13133187	C₂	semidirect
6. C₂²×#13133185	C₂	semidirect
7. C₂²×#124×C₅₃	C₂	semidirect
8. C₂⁵×D₅₃	C₂²	semidirect
9. C₂³×Dic₁₀₆	C₂²	semidirect
13. #16811365	C₂²	direct (N × H)
14. C₂×#13133206	C₂²	semidirect
15. C₂×#13133205	C₂²	semidirect
16. C₂×#13133203	C₂²	semidirect
17. C₂×#13133187	C₂²	semidirect
18. C₂×#13133185	C₂²	semidirect
19. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
20. C₂²×#11719921	C₂²	semidirect
21. C₂²×#11719921	C₂²	semidirect
22. C₂²×#11719921	C₂²	semidirect
23. C₂²×C₄×Dic₅₃	C₂²	semidirect
24. C₂³×Q₈×C₅₃	C₂²	semidirect
26. C₂×#124×C₅₃	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
28. C₂²×C₄²×C₅₃	C₂²	semidirect
32. C₂⁴×D₅₃	C₂³	semidirect
34. C₂²×Dic₁₀₆	C₂³	semidirect
42. #13133206	C₂³	semidirect
43. #13133205	C₂³	semidirect
44. #13133203	C₂³	semidirect
45. #13133187	C₂³	semidirect
46. #13133185	C₂³	semidirect
47. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
48. C₂×#11719921	C₂³	semidirect
49. C₂×#11719921	C₂³	semidirect
50. C₂×#11719921	C₂³	semidirect
51. C₂×C₄×Dic₅₃	C₂³	semidirect
54. C₂²×Q₈×C₅₃	C₂³	semidirect
60. #124×C₅₃	C₂³	semidirect
61. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
62. C₂×C₄²×C₅₃	C₂³	semidirect
74. C₂³×D₅₃	C₂⁴	semidirect
76. C₂×Q₈×C₅₃	C₂⁴	semidirect
94. C₂×Dic₁₀₆	C₂⁴	semidirect
95. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
96. C₄²×C₅₃	C₂⁴	semidirect
97. C₂²⋊Dic₅₃	C₂⁴	semidirect
98. #11719921	C₂⁴	semidirect
99. #11719921	C₂⁴	semidirect
100. #11719921	C₂⁴	semidirect
101. C₄×Dic₅₃	C₂⁴	semidirect
135. C₂²×C₅₃	H(3,𝔽₄)	semidirect
152. C₂²×#124	D₅₃	semidirect
156. C₂×C₅₃	C₂×H(3,𝔽₄)	semidirect
162. C₂³×Q₈	C₂×D₅₃	semidirect
171. C₂×#124	C₂×D₅₃	semidirect
178. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
180. C₂²×C₄²	C₂×D₅₃	semidirect
181. C₅₃	C₂²×H(3,𝔽₄)	semidirect
190. C₂²×Q₈	C₂²×D₅₃	semidirect
206. #124	C₂²×D₅₃	semidirect
213. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
215. C₂×C₄²	C₂²×D₅₃	semidirect
231. C₂×Q₈	C₂³×D₅₃	semidirect
252. C₂²⋊C₄	C₂³×D₅₃	semidirect
254. C₄²	C₂³×D₅₃	semidirect
292. C₂²	#16811365	direct (N × H)
302. C₂	C₂×#16811365	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×#16811365	2
C₂²×#13133206	2
C₂²×#13133205	2
C₂²×#13133203	2
C₂²×#13133187	2
C₂²×#13133185	2
C₂²×#124×C₅₃	2
C₂²×H(3,𝔽₄)	53

Subgroups — C22×#16811365 #28290964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811365 #28290964