Subgroups — #20291666

Order 6784.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811674	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811657	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811652	C₂	1	2	yes	maximal, complemented
5	#16811651	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811650	C₂	1	2	yes	maximal, complemented
7	#16811648	C₂	1	2	yes	maximal, complemented
8	#16811640	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811630	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811159	C₂	1	1	yes	maximal, characteristic
12	C₂×#11719936	C₂²	1	2	yes	—
13	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
14	C₂²×#88324	C₂²	1	2	yes	—
15	#13133284	C₂²	1	1	yes	characteristic
16	#13133284	C₂²	1	2	yes	—
17	#13133284	C₂²	1	1	yes	characteristic
18	C₄⋊#88324	C₂²	1	2	yes	—
19	C₄⋊#88324	C₂²	1	1	yes	characteristic
20	C₄⋊#88324	C₂²	1	1	yes	characteristic
21	C₄⋊#88324	C₂²	1	2	yes	—
22	#13133274	C₂²	1	2	yes	—
23	#13133274	C₂²	1	2	yes	—
24	C₂²⋊#88324	C₂²	1	1	yes	characteristic
25	C₂²⋊#88324	C₂²	1	1	yes	characteristic
26	#13133272	C₂²	1	2	yes	—
27	C₄×#88324	C₂²	1	1	yes	characteristic
28	C₄×#88324	C₂²	1	1	yes	characteristic
29	C₄×#88324	C₂²	1	2	yes	—
30	#13133212	C₄	1	2	yes	—
31	#13133208	C₄	1	1	yes	characteristic
32	#13133206	C₄	1	1	yes	characteristic
33	#13133201	C₄	1	2	yes	—
34	#13133191	C₄	1	1	yes	characteristic
35	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
36	#13133179	C₂²	1	1	yes	characteristic
37	#13133177	C₂²	1	2	yes	—
38	#13133175	C₂²	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂³	1	2	yes	—
48	C₂×#88324	—	4	2	no	—
49	C₂×#88324	C₂³	1	1	yes	characteristic
50	C₂×#88324	C₂³	1	1	yes	characteristic
51	C₂×#88324	C₂³	1	1	yes	characteristic
52	C₂×#88324	C₂³	1	1	yes	characteristic
53	C₂×#88324	C₂³	1	2	yes	—
54	C₂×#88324	C₂³	1	2	yes	—
55	#11719936	—	4	2	no	—
56	D₅₃⋊C₈	—	4	2	no	—
57	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	#11719921	C₂×C₄	1	2	yes	—
61	#11719921	C₂×C₄	1	2	yes	—
62	#11719921	C₂×C₄	1	2	yes	—
63	#11719921	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
67	#11719919	C₂×C₄	1	2	yes	—
68	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄×Dic₅₃	C₂³	1	2	yes	—
73	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
75	Q₈×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊D₅₃	—	2	4	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
90	D₂₁₂	—	2	2	no	—
91	C₄×D₅₃	—	2	2	no	—
92	C₄×D₅₃	—	2	2	no	—
93	#88324	—	2	1	no	—
94	#88324	—	2	2	no	—
95	#88324	—	2	2	no	—
96	#88324	—	2	1	no	—
97	#88324	—	2	1	no	—
98	#88324	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	#131	1	2	yes	—
102	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₅₃	—	2	2	no	—
104	C₂×D₅₃	—	2	2	no	—
105	C₂×D₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	#131	1	2	yes	—
110	Dic₅₃	—	2	1	no	—
111	Dic₅₃	—	2	1	no	—
112	Dic₅₃	—	2	1	no	—
113	Dic₅₃	—	2	2	no	—
114	Dic₅₃	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₅₃	2₋¹⁺⁴	1	1	yes	characteristic
116	#2994	—	53	1	no	maximal, Sylow
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	#520	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
121	D₅₃	—	4	2	no	—
122	#529	—	53	1	no	—
123	#446	—	53	1	no	—
124	C₈×Q₈	—	53	1	no	—
125	#436	—	53	1	no	—
126	#435	—	53	2	no	—
127	C₈×D₄	—	53	2	no	—
128	C₈×D₄	—	53	1	no	—
129	#433	—	53	2	no	—
130	#431	—	53	1	no	—
131	#424	—	53	2	no	—
132	C₂×C₄⋊C₈	—	53	1	no	—
133	C₅₃	#2994	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	53	2	no	—
135	C₂²×C₈	—	53	2	no	—
136	C₂²×C₈	—	53	2	no	—
137	#126	—	53	2	no	—
138	#125	—	53	1	no	—
139	#124	—	53	1	no	—
140	#123	—	53	2	no	—
141	C₂²⋊Q₈	#6157	1	1	yes	characteristic
142	#121	—	53	1	no	—
143	C₄×Q₈	—	53	1	no	—
144	C₄×D₄	—	53	1	no	—
145	C₄×D₄	—	53	2	no	—
146	#117	—	53	2	no	—
147	C₂×C₄²	—	53	1	no	—
148	C₄⋊C₈	—	53	2	no	—
149	C₄⋊C₈	—	53	1	no	—
150	C₄⋊C₈	—	53	1	no	—
151	C₄⋊C₈	—	53	1	no	—
152	C₄⋊C₈	—	53	2	no	—
153	C₄⋊C₈	—	53	2	no	—
154	C₄⋊C₈	—	53	1	no	—
155	C₂²⋊C₈	—	53	1	no	—
156	C₂²⋊C₈	—	53	1	no	—
157	C₂²⋊C₈	—	53	2	no	—
158	C₂²⋊C₈	—	53	2	no	—
159	#97	—	53	2	no	—
160	C₄×C₈	—	53	1	no	—
161	C₄×C₈	—	53	1	no	—
162	C₄×C₈	—	53	2	no	—
163	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
164	C₂×D₄	—	53	2	no	—
165	C₂×D₄	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	2	no	—
169	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
170	M₁₆	—	212	2	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	53	1	no	—
173	C₂×C₈	—	53	1	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	53	1	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	212	2	no	—
178	C₂×C₈	—	212	2	no	—
179	C₄⋊C₄	—	53	2	no	—
180	C₄⋊C₄	—	53	2	no	—
181	C₄⋊C₄	—	53	1	no	—
182	C₄⋊C₄	—	53	2	no	—
183	C₄⋊C₄	C₂×#6157	1	2	yes	—
184	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
185	C₂²⋊C₄	—	53	2	no	—
186	C₂²⋊C₄	—	53	2	no	—
187	C₂²⋊C₄	—	53	2	no	—
188	C₂²⋊C₄	—	53	2	no	—
189	C₂²⋊C₄	C₂×#6157	1	2	yes	—
190	C₄²	—	53	1	no	—
191	C₄²	—	53	2	no	—
192	C₄²	—	53	1	no	—
193	C₄²	—	53	1	no	—
194	C₄²	—	53	1	no	—
195	C₂³	—	53	2	no	—
196	C₂³	C₂²×#6157	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	106	4	no	—
199	D₄	—	106	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	1	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	106	1	no	—
209	C₂×C₄	—	106	1	no	—
210	C₂×C₄	—	106	2	no	—
211	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
212	C₂×C₄	C₂²×#6157	1	2	yes	—
213	C₂×C₄	C₂²×#6157	1	2	yes	—
214	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
215	C₈	—	106	1	no	—
216	C₈	—	106	1	no	—
217	C₈	—	106	2	no	—
218	C₈	—	106	1	no	—
219	C₈	—	106	2	no	—
220	C₈	—	106	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	106	2	no	—
224	C₂²	—	106	2	no	—
225	C₂²	—	106	2	no	—
226	C₂²	#13133305	1	2	yes	—
227	C₂²	C₂³×#6157	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	53	1	no	—
232	C₄	—	53	1	no	—
233	C₄	—	106	1	no	—
234	C₄	—	106	1	no	—
235	C₄	—	106	1	no	—
236	C₄	—	106	2	no	—
237	C₄	#13133304	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	212	2	no	—
240	C₂	#16811692	1	1	yes	characteristic
241	C₂	C₂×#13133304	1	1	yes	characteristic
242	C₂	C₂×#13133305	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂²⋊Q₈×C₅₃	C₄
hypercenter	C₂²⋊Q₈	#6157
hypocenter	C₅₃	#2994
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16811674	C₂	semidirect
2. Q₈×#88324	C₂	semidirect
3. #16811657	C₂	semidirect
4. #16811652	C₂	semidirect
5. #16811651	C₂	semidirect
6. #16811650	C₂	semidirect
7. #16811648	C₂	semidirect
8. #16811640	C₂	semidirect
9. #16811630	C₂	semidirect
10. C₂×C₄⋊#88324	C₂	semidirect
133. C₅₃	#2994	semidirect