Subgroups — D₄×#11719917 #20283452

Order 6784.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	2	yes	maximal, complemented
2	#16810701	C₂	1	2	yes	maximal, complemented
3	#16810695	C₂	1	4	yes	maximal, complemented
4	D₄×#88322	C₂	1	2	yes	maximal, complemented
5	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810587	C₂	1	2	yes	maximal, complemented
7	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719917	—	2	8	no	—
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₂×#11719917	C₂²	1	2	yes	—
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	C₂×#11719917	C₂²	1	4	yes	complemented
14	C₂²×#88322	C₂²	1	4	yes	complemented
15	C₄⋊#88322	C₂²	1	2	yes	complemented
16	C₄⋊#88322	C₂²	1	2	yes	complemented
17	C₂²⋊#88322	C₂²	1	4	yes	complemented
18	C₂²⋊#88322	C₂²	1	4	yes	complemented
19	#13133128	C₂²	1	2	yes	complemented
20	C₄×#88322	C₂²	1	2	yes	complemented
21	C₂²×D₄×C₅₃	C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
23	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
24	C₄×D₄×C₅₃	C₄	1	2	yes	—
25	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
26	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₅₃	C₂²	1	2	yes	—
28	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂⁴×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₅₃	—	2	4	no	—
34	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	—	2	4	no	—
39	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	#11719917	—	2	4	no	—
59	#11719917	—	2	2	no	—
60	#11719917	—	2	8	no	—
61	#11719917	D₄	1	4	yes	direct product factor
62	C₂×#88322	—	2	8	no	—
63	C₂×#88322	C₂³	1	2	yes	—
64	C₂×#88322	C₂³	1	4	yes	—
65	C₂×#88322	C₂³	1	2	yes	—
66	C₂³×C₅₃	—	2	4	no	—
67	C₂³×C₅₃	—	2	4	no	—
68	C₂³×C₅₃	—	4	2	no	—
69	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
72	D₄×C₅₃	—	2	4	no	—
73	D₄×C₅₃	—	2	2	no	—
74	D₄×C₅₃	M₁₆	1	4	yes	complemented
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₅₃	#41	1	2	yes	—
93	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	#88322	—	2	4	no	—
95	#88322	—	2	2	no	—
96	#88322	C₂×D₄	1	4	yes	complemented
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	4	8	no	—
105	C₂²×C₅₃	C₄×D₄	1	2	yes	—
106	C₂²×C₅₃	C₂×M₁₆	1	4	yes	complemented
107	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	C₄×D₄	1	2	yes	—
114	C₄×C₅₃	C₂×M₁₆	1	2	yes	complemented
115	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	D₄×M₁₆	—	53	1	no	maximal, Sylow
118	C₂×C₅₃	—	2	2	no	—
119	C₂×C₅₃	—	2	4	no	—
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
123	C₂×C₅₃	#568	1	1	yes	characteristic
124	C₂²×M₁₆	—	53	2	no	—
125	C₂×C₄×D₄	D₅₃	1	1	yes	characteristic
126	#436	—	53	2	no	—
127	#435	—	53	4	no	—
128	C₈×D₄	—	53	2	no	—
129	#423	—	53	1	no	—
130	#407	—	53	2	no	—
131	C₄×M₁₆	—	53	1	no	—
132	C₅₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
133	C₂²×D₄	Dic₅₃	1	1	yes	characteristic
134	C₂³×C₄	C₂×D₅₃	1	2	yes	—
135	C₂×M₁₆	—	53	1	no	—
136	C₂×M₁₆	—	53	2	no	—
137	C₂×M₁₆	—	53	4	no	—
138	C₂×M₁₆	—	53	1	no	—
139	C₂×M₁₆	—	106	8	no	—
140	C₂²×C₈	—	53	4	no	—
141	C₄×D₄	Dic₅₃	1	2	yes	—
142	C₄×D₄	Dic₅₃	1	1	yes	characteristic
143	C₄×D₄	Dic₅₃	1	1	yes	characteristic
144	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
145	C₄×D₄	C₂×D₅₃	1	2	yes	—
146	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
149	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
150	C₄⋊C₈	—	53	2	no	—
151	C₄⋊C₈	—	53	2	no	—
152	C₂²⋊C₈	—	53	4	no	—
153	C₂²⋊C₈	—	53	4	no	—
154	#97	—	53	2	no	—
155	C₄×C₈	—	53	2	no	—
156	C₂⁴	C₂×Dic₅₃	1	2	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
159	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
165	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
166	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
167	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
169	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
170	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
171	M₁₆	—	53	4	no	—
172	M₁₆	—	106	8	no	—
173	M₁₆	—	106	4	no	—
174	M₁₆	—	106	2	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	4	no	—
178	C₂×C₈	—	106	8	no	—
179	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
183	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
184	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
186	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
187	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
188	C₄²	C₂²×D₅₃	1	1	yes	characteristic
189	C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂²×Dic₅₃	1	2	yes	—
194	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
195	C₂³	C₂²×Dic₅₃	1	2	yes	—
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	#11719917	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	D₄×D₅₃	1	2	yes	—
209	C₂×C₄	#11719947	1	2	yes	—
210	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
214	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
215	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
218	C₈	—	53	4	no	—
219	C₈	—	106	4	no	—
220	C₈	—	106	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	D₄×Dic₅₃	1	2	yes	—
230	C₂²	C₂×#11719917	1	4	yes	complemented
231	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	D₄×Dic₅₃	1	2	yes	—
238	C₄	C₂×#11719917	1	2	yes	complemented
239	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
240	C₄	C₂×#11719947	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×D₄×Dic₅₃	1	1	yes	characteristic
245	C₂	C₂²×#11719917	1	1	yes	characteristic
246	C₂	#16811390	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄×D₄×C₅₃	C₂
hypercenter	C₂×C₄×D₄	D₅₃
hypocenter	C₅₃	D₄×M₁₆
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#11719917	C₂	semidirect
2. #16810701	C₂	semidirect
3. #16810695	C₂	semidirect
4. D₄×#88322	C₂	semidirect
5. #16810647	C₂	semidirect
6. #16810587	C₂	semidirect
7. C₄×#11719917	C₂	semidirect
13. C₂×#11719917	C₂²	semidirect
14. C₂²×#88322	C₂²	semidirect
15. C₄⋊#88322	C₂²	semidirect
16. C₄⋊#88322	C₂²	semidirect
17. C₂²⋊#88322	C₂²	semidirect
18. C₂²⋊#88322	C₂²	semidirect
19. #13133128	C₂²	semidirect
20. C₄×#88322	C₂²	semidirect
61. #11719917	D₄	direct (N × H)
74. D₄×C₅₃	M₁₆	semidirect
96. #88322	C₂×D₄	semidirect
106. C₂²×C₅₃	C₂×M₁₆	semidirect
114. C₄×C₅₃	C₂×M₁₆	semidirect
132. C₅₃	D₄×M₁₆	semidirect
198. D₄	#11719917	direct (N × H)
230. C₂²	C₂×#11719917	semidirect
238. C₄	C₂×#11719917	semidirect

Subgroups — D4×#11719917 #20283452

Conjugacy classes of subgroups

Complements

Subgroups — D₄×#11719917 #20283452