Subgroups — #20291548

Order 6784.

370 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	#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	#16811656	C₂	1	2	yes	maximal, complemented
5	#16811655	C₂	1	2	yes	maximal, complemented
6	#16811651	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811648	C₂	1	2	yes	maximal, complemented
8	#16811639	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811588	C₂	1	2	yes	maximal, complemented
10	C₄×#11719940	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811159	C₂	1	1	yes	maximal, characteristic
12	C₂×#11719936	C₂²	1	2	yes	—
13	C₂×#11719940	C₂²	1	1	yes	characteristic
14	C₂×#11719940	C₂²	1	1	yes	characteristic
15	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
16	#13133284	C₂²	1	2	yes	—
17	C₄⋊#88324	C₂²	1	1	yes	characteristic
18	C₄⋊#88324	C₂²	1	1	yes	characteristic
19	C₄⋊#88324	C₂²	1	2	yes	—
20	#13133274	C₂²	1	2	yes	—
21	#13133274	C₂²	1	2	yes	—
22	C₂²⋊#88324	C₂²	1	2	yes	—
23	#13133272	C₂²	1	1	yes	characteristic
24	#13133272	C₂²	1	1	yes	characteristic
25	#13133272	C₂²	1	2	yes	—
26	C₄×#88324	C₂²	1	1	yes	characteristic
27	C₄×#88324	C₂²	1	1	yes	characteristic
28	C₄×#88324	C₂²	1	2	yes	—
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	#11719940	—	2	2	no	—
49	#11719940	—	2	2	no	—
50	C₂×#88324	C₂³	1	2	yes	—
51	C₂×#88324	C₂³	1	2	yes	—
52	C₂×#88324	C₂³	1	1	yes	characteristic
53	C₂×#88324	C₂³	1	1	yes	characteristic
54	C₂×#88324	C₂³	1	1	yes	characteristic
55	C₂×#88324	C₂³	1	1	yes	characteristic
56	#11719936	—	2	4	no	—
57	D₅₃⋊C₈	—	2	4	no	—
58	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₄×C₅₃	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	#11719921	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
68	#11719919	C₂×C₄	1	2	yes	—
69	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
73	C₄×Dic₅₃	C₂³	1	2	yes	—
74	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
76	Q₈×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	—	4	1	no	—
86	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
91	D₂₁₂	—	2	2	no	—
92	C₄×D₅₃	—	2	2	no	—
93	C₄×D₅₃	—	2	2	no	—
94	#88324	—	2	2	no	—
95	#88324	—	2	1	no	—
96	#88324	—	2	1	no	—
97	#88324	—	2	2	no	—
98	#88324	#41	1	2	yes	—
99	#88324	#41	1	2	yes	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₅₃	—	2	2	no	—
105	C₂×D₅₃	—	2	2	no	—
106	C₂×D₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	#131	1	2	yes	—
111	Dic₅₃	—	2	1	no	—
112	Dic₅₃	—	2	1	no	—
113	Dic₅₃	—	2	1	no	—
114	Dic₅₃	—	2	2	no	—
115	Dic₅₃	C₂×#41	1	1	yes	characteristic
116	Dic₅₃	C₂×#41	1	1	yes	characteristic
117	#2969	—	53	1	no	maximal, Sylow
118	C₂×C₅₃	—	4	1	no	—
119	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
121	C₂×C₅₃	#568	1	1	yes	characteristic
122	D₅₃	—	4	2	no	—
123	#529	—	53	1	no	—
124	#446	—	53	1	no	—
125	C₈×Q₈	—	53	1	no	—
126	#436	—	53	2	no	—
127	#436	—	53	1	no	—
128	#435	—	53	2	no	—
129	C₈×D₄	—	53	1	no	—
130	#433	—	53	2	no	—
131	#431	—	53	1	no	—
132	#405	—	53	2	no	—
133	C₄×M₁₆	—	53	1	no	—
134	C₅₃	#2969	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	53	1	no	—
136	C₂×M₁₆	—	53	1	no	—
137	C₂×M₁₆	—	53	2	no	—
138	C₂²×C₈	—	53	2	no	—
139	#126	—	53	2	no	—
140	#125	—	53	1	no	—
141	#124	—	53	1	no	—
142	#123	—	53	2	no	—
143	C₂²⋊Q₈	#6157	1	1	yes	characteristic
144	#121	—	53	1	no	—
145	C₄×Q₈	—	53	1	no	—
146	C₄×D₄	—	53	1	no	—
147	C₄×D₄	—	53	2	no	—
148	#117	—	53	2	no	—
149	C₂×C₄²	—	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	C₂²⋊C₈	—	53	2	no	—
155	C₂²⋊C₈	—	53	2	no	—
156	C₂²⋊C₈	—	53	2	no	—
157	#97	—	53	1	no	—
158	#97	—	53	1	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	C₄×C₈	—	53	2	no	—
164	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
165	C₂×D₄	—	53	2	no	—
166	C₂×D₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	—	53	2	no	—
170	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
171	M₁₆	—	106	4	no	—
172	M₁₆	—	106	2	no	—
173	M₁₆	—	106	2	no	—
174	C₂×C₈	—	53	1	no	—
175	C₂×C₈	—	53	1	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	53	1	no	—
179	C₂×C₈	—	53	1	no	—
180	C₂×C₈	—	106	4	no	—
181	C₄⋊C₄	—	53	2	no	—
182	C₄⋊C₄	—	53	2	no	—
183	C₄⋊C₄	—	53	1	no	—
184	C₄⋊C₄	—	53	2	no	—
185	C₄⋊C₄	C₂×#6157	1	2	yes	—
186	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
187	C₂²⋊C₄	—	53	2	no	—
188	C₂²⋊C₄	—	53	2	no	—
189	C₂²⋊C₄	—	53	2	no	—
190	C₂²⋊C₄	—	53	2	no	—
191	C₂²⋊C₄	C₂×#6157	1	2	yes	—
192	C₄²	—	53	1	no	—
193	C₄²	—	53	1	no	—
194	C₄²	—	53	2	no	—
195	C₄²	—	53	1	no	—
196	C₄²	—	53	1	no	—
197	C₂³	—	53	2	no	—
198	C₂³	C₂²×#6157	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	D₄	—	106	2	no	—
201	D₄	—	212	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	1	no	—
205	C₂×C₄	—	53	1	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	106	2	no	—
209	C₂×C₄	—	106	2	no	—
210	C₂×C₄	—	106	1	no	—
211	C₂×C₄	—	106	1	no	—
212	C₂×C₄	—	212	1	no	—
213	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
214	C₂×C₄	C₂²×#6157	1	2	yes	—
215	C₂×C₄	C₂²×#6157	1	2	yes	—
216	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
217	C₈	—	53	2	no	—
218	C₈	—	53	2	no	—
219	C₈	—	106	2	no	—
220	C₈	—	106	2	no	—
221	C₈	—	106	1	no	—
222	C₈	—	106	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	106	2	no	—
227	C₂²	—	106	2	no	—
228	C₂²	—	106	2	no	—
229	C₂²	C₂³×#6157	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	53	1	no	—
234	C₄	—	53	1	no	—
235	C₄	—	106	1	no	—
236	C₄	—	106	1	no	—
237	C₄	—	106	1	no	—
238	C₄	—	106	2	no	—
239	C₄	#13133304	1	2	yes	—
240	C₂	—	4	1	no	—
241	C₂	—	212	2	no	—
242	C₂	#41×#6157	1	1	yes	characteristic
243	C₂	C₂×#13133304	1	1	yes	characteristic
244	C₂	#16811703	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	#2969
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. #16811656	C₂	semidirect
5. #16811655	C₂	semidirect
6. #16811651	C₂	semidirect
7. #16811648	C₂	semidirect
8. #16811639	C₂	semidirect
9. #16811588	C₂	semidirect
10. C₄×#11719940	C₂	semidirect
134. C₅₃	#2969	semidirect