Subgroups — #20288533

Order 6784.

1060 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811349	C₂	1	2	yes	maximal, complemented
3	#16811334	C₂	1	3	yes	maximal, complemented
4	#16811333	C₂	1	3	yes	maximal, complemented
5	#16811288	C₂	1	6	yes	maximal, complemented
6	#16811250	C₂	1	6	yes	maximal, complemented
7	#16811177	C₂	1	3	yes	maximal, complemented
8	C₄⋊Q₈×D₅₃	C₂	1	3	yes	maximal, complemented
9	#16811105	C₂	1	2	yes	maximal
10	C₄×#11719949	C₂	1	1	yes	maximal, characteristic
11	Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719949	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₅₃	C₂²	1	6	yes	—
14	#13133217	C₂²	1	3	yes	—
15	#13133216	C₂²	1	6	yes	—
16	#13133216	C₂²	1	6	yes	—
17	#13133216	C₂²	1	3	yes	—
18	#13133215	C₂²	1	6	yes	—
19	#13133215	C₂²	1	1	yes	characteristic
20	#13133209	C₂²	1	6	yes	—
21	#13133208	C₂²	1	6	yes	—
22	#13133207	C₂²	1	6	yes	—
23	#13133197	C₂²	1	3	yes	—
24	#13133195	C₂²	1	6	yes	—
25	#13133195	C₂²	1	12	yes	—
26	#13133193	C₂²	1	12	yes	—
27	#13133193	C₂²	1	3	yes	—
28	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	Q₈×Dic₅₃	C₂²	1	2	yes	—
30	C₄×Dic₁₀₆	C₂²	1	6	yes	—
31	C₄×Dic₁₀₆	C₂²	1	3	yes	—
32	C₄×D₂₁₂	C₂²	1	6	yes	—
33	C₄×D₂₁₂	C₂²	1	3	yes	—
34	#13133179	C₂²	1	3	yes	—
35	#13133179	C₂²	1	6	yes	—
36	#13133175	C₂²	1	3	yes	—
37	#13133174	C₂²	1	6	yes	—
38	C₄⋊C₄×D₅₃	C₂²	1	6	yes	—
39	C₄⋊C₄×D₅₃	C₂²	1	6	yes	—
40	C₄²×D₅₃	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₅₃	C₂²	1	6	yes	—
42	C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
43	C₄×Q₈×C₅₃	C₂²	1	3	yes	—
44	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
46	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₅₃	C₂³	1	6	yes	—
48	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
49	#11719949	Q₈	1	8	yes	complemented
50	Q₈×D₅₃	—	4	12	no	—
51	C₂×D₂₁₂	C₂³	1	3	yes	—
52	C₂×C₄×D₅₃	C₂³	1	3	yes	—
53	C₂×C₄×D₅₃	C₂³	1	3	yes	—
54	C₂×C₄×D₅₃	C₂³	1	12	yes	—
55	C₂×C₄×D₅₃	C₂³	1	3	yes	—
56	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₀₆	C₂³	1	6	yes	—
58	C₂×Dic₁₀₆	C₂³	1	6	yes	—
59	C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
60	C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
61	C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
62	C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
63	C₄²×C₅₃	C₂³	1	3	yes	—
64	C₄²×C₅₃	C₂³	1	6	yes	—
65	#11719921	C₂³	1	6	yes	—
66	#11719921	C₂³	1	6	yes	—
67	#11719921	C₂³	1	6	yes	—
68	C₄⋊Dic₅₃	C₂³	1	3	yes	—
69	C₄⋊Dic₅₃	C₂³	1	6	yes	—
70	C₄⋊Dic₅₃	C₂³	1	3	yes	—
71	C₄⋊Dic₅₃	C₂³	1	6	yes	—
72	#11719919	C₂³	1	6	yes	—
73	#11719919	C₂³	1	12	yes	—
74	#11719919	C₂³	1	2	yes	—
75	#11719919	C₂³	1	2	yes	—
76	#11719919	C₂³	1	6	yes	—
77	#11719919	C₂³	1	6	yes	—
78	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₄×Dic₅₃	C₂³	1	3	yes	—
80	C₄×Dic₅₃	C₂³	1	3	yes	—
81	C₄×Dic₅₃	C₂³	1	6	yes	—
82	C₂²×D₅₃	C₂⁴	1	3	yes	—
83	Q₈×C₅₃	—	2	2	no	—
84	Q₈×C₅₃	—	4	6	no	—
85	Q₈×C₅₃	C₂×Q₈	1	4	yes	complemented
86	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
87	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
88	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
89	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
90	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₅₃	C₂⁴	1	6	yes	—
92	C₂×Dic₅₃	C₂⁴	1	2	yes	—
93	C₂×Dic₅₃	C₂⁴	1	3	yes	—
94	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
96	D₂₁₂	C₂×Q₈	1	12	yes	complemented
97	C₄×D₅₃	—	2	6	no	—
98	C₄×D₅₃	—	2	6	no	—
99	C₄×D₅₃	—	4	12	no	—
100	C₄×D₅₃	C₂×Q₈	1	12	yes	complemented
101	Dic₁₀₆	—	2	2	no	—
102	Dic₁₀₆	—	2	12	no	—
103	Dic₁₀₆	—	4	6	no	—
104	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
105	C₂×D₅₃	—	2	3	no	—
106	C₂×D₅₃	—	2	3	no	—
107	C₂×D₅₃	C₂²×Q₈	1	6	yes	—
108	C₄×C₅₃	—	2	6	no	—
109	C₄×C₅₃	—	2	3	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	C₂²×Q₈	1	6	yes	—
112	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
113	Dic₅₃	—	2	6	no	—
114	Dic₅₃	—	2	2	no	—
115	Dic₅₃	—	2	3	no	—
116	Dic₅₃	C₂²×Q₈	1	2	yes	—
117	Dic₅₃	H(5,𝔽₂)	1	2	yes	—
118	#3503	—	53	1	no	maximal, Sylow
119	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
120	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
122	D₅₃	—	2	6	no	—
123	Q₈²	—	53	1	no	—
124	Q₈²	D₅₃	1	1	yes	characteristic, complemented
125	#557	—	53	2	no	—
126	#554	—	53	3	no	—
127	#554	—	53	3	no	—
128	D₄×Q₈	—	53	6	no	—
129	#544	—	53	6	no	—
130	#533	—	53	3	no	—
131	C₂×C₄⋊Q₈	—	53	3	no	—
132	#520	—	53	2	no	—
133	C₄×#41	—	53	1	no	—
134	C₅₃	#3503	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	53	1	no	—
136	C₂²×Q₈	—	53	6	no	—
137	C₄⋊Q₈	—	53	6	no	—
138	C₄⋊Q₈	—	53	1	no	—
139	C₄⋊Q₈	—	53	3	no	—
140	C₄⋊Q₈	—	53	6	no	—
141	C₄⋊Q₈	—	53	6	no	—
142	C₄⋊Q₈	—	53	3	no	—
143	C₄⋊Q₈	C₂×D₅₃	1	3	yes	—
144	C₄⋊Q₈	C₂×D₅₃	1	6	yes	—
145	#125	—	53	6	no	—
146	#125	—	53	6	no	—
147	#125	—	53	6	no	—
148	C₂²⋊Q₈	—	53	3	no	—
149	C₂²⋊Q₈	—	53	12	no	—
150	C₂²⋊Q₈	—	53	6	no	—
151	C₂²⋊Q₈	—	53	3	no	—
152	C₂²⋊Q₈	—	53	12	no	—
153	C₄×Q₈	—	53	1	no	—
154	C₄×Q₈	—	53	2	no	—
155	C₄×Q₈	—	53	6	no	—
156	C₄×Q₈	—	53	3	no	—
157	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
158	C₄×Q₈	C₂×D₅₃	1	2	yes	—
159	C₄×Q₈	C₂×D₅₃	1	3	yes	—
160	C₄×D₄	—	53	3	no	—
161	C₄×D₄	—	53	6	no	—
162	C₄×D₄	—	53	3	no	—
163	C₄×D₄	—	53	6	no	—
164	#117	—	53	3	no	—
165	#117	—	53	6	no	—
166	C₂×C₄⋊C₄	—	53	6	no	—
167	C₂×C₄⋊C₄	—	53	6	no	—
168	C₂×C₄²	—	53	3	no	—
169	#41	—	53	8	no	—
170	C₂×Q₈	—	53	1	no	—
171	C₂×Q₈	—	53	6	no	—
172	C₂×Q₈	—	53	6	no	—
173	C₂×Q₈	—	212	12	no	—
174	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
175	C₂×Q₈	C₂²×D₅₃	1	6	yes	—
176	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×D₄	—	53	3	no	—
178	C₂²×C₄	—	53	3	no	—
179	C₂²×C₄	—	53	3	no	—
180	C₂²×C₄	—	53	3	no	—
181	C₂²×C₄	—	53	12	no	—
182	C₄⋊C₄	—	53	6	no	—
183	C₄⋊C₄	—	53	3	no	—
184	C₄⋊C₄	—	53	6	no	—
185	C₄⋊C₄	—	53	12	no	—
186	C₄⋊C₄	—	53	2	no	—
187	C₄⋊C₄	—	53	3	no	—
188	C₄⋊C₄	—	53	2	no	—
189	C₄⋊C₄	—	53	6	no	—
190	C₄⋊C₄	—	53	6	no	—
191	C₄⋊C₄	—	53	6	no	—
192	C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
193	C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
194	C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
195	C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
196	C₂²⋊C₄	—	53	6	no	—
197	C₂²⋊C₄	—	53	6	no	—
198	C₂²⋊C₄	—	53	6	no	—
199	C₄²	—	53	1	no	—
200	C₄²	—	53	3	no	—
201	C₄²	—	53	3	no	—
202	C₄²	—	53	6	no	—
203	C₄²	C₂²×D₅₃	1	3	yes	—
204	C₄²	C₂²×D₅₃	1	6	yes	—
205	C₂³	—	53	3	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	4	6	no	—
208	Q₈	—	106	12	no	—
209	Q₈	—	106	2	no	—
210	Q₈	—	212	6	no	—
211	Q₈	Q₈×D₅₃	1	4	yes	complemented
212	D₄	—	53	12	no	—
213	C₂×C₄	—	53	1	no	—
214	C₂×C₄	—	53	1	no	—
215	C₂×C₄	—	53	12	no	—
216	C₂×C₄	—	53	2	no	—
217	C₂×C₄	—	53	3	no	—
218	C₂×C₄	—	53	6	no	—
219	C₂×C₄	—	106	6	no	—
220	C₂×C₄	—	106	6	no	—
221	C₂×C₄	—	212	12	no	—
222	C₂×C₄	C₂³×D₅₃	1	3	yes	—
223	C₂×C₄	C₂³×D₅₃	1	2	yes	—
224	C₂×C₄	C₂³×D₅₃	1	6	yes	—
225	C₂×C₄	C₂³×D₅₃	1	3	yes	—
226	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
227	C₂²	—	53	6	no	—
228	C₂²	—	106	3	no	—
229	C₂²	—	106	3	no	—
230	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
231	C₄	—	2	3	no	—
232	C₄	—	2	6	no	—
233	C₄	—	2	2	no	—
234	C₄	—	53	2	no	—
235	C₄	—	53	2	no	—
236	C₄	—	106	6	no	—
237	C₄	—	106	3	no	—
238	C₄	—	106	2	no	—
239	C₄	C₂×Q₈×D₅₃	1	6	yes	—
240	C₄	#13133265	1	2	yes	—
241	C₂	—	106	6	no	—
242	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
243	C₂	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
244	C₂	C₂×#13133265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	Q₈²×C₅₃	C₂
hypercenter	Q₈²	D₅₃
hypocenter	C₅₃	#3503
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₀₆	C₂	semidirect
2. #16811349	C₂	semidirect
3. #16811334	C₂	semidirect
4. #16811333	C₂	semidirect
5. #16811288	C₂	semidirect
6. #16811250	C₂	semidirect
7. #16811177	C₂	semidirect
8. C₄⋊Q₈×D₅₃	C₂	semidirect
11. Q₈²×C₅₃	C₂	semidirect
49. #11719949	Q₈	semidirect
85. Q₈×C₅₃	C₂×Q₈	semidirect
96. D₂₁₂	C₂×Q₈	semidirect
100. C₄×D₅₃	C₂×Q₈	semidirect
124. Q₈²	D₅₃	semidirect
134. C₅₃	#3503	semidirect
211. Q₈	Q₈×D₅₃	semidirect