Subgroups — #20291478

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811692	C₂	1	2	yes	maximal, complemented
2	#16811664	C₂	1	2	yes	maximal, complemented
3	#16811663	C₂	1	2	yes	maximal, complemented
4	SD₁₆×#6157	C₂	1	4	yes	maximal, complemented
5	C₂×#13133285	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811614	C₂	1	2	yes	maximal, complemented
7	#16811586	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₅₃	C₂	1	1	yes	maximal, characteristic
9	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
10	Q₈×#6157	C₂²	1	4	yes	—
11	D₄×#6157	—	2	4	no	—
12	D₄×#6157	C₂²	1	4	yes	—
13	#13133296	—	2	2	no	—
14	#13133296	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#6157	—	2	2	no	—
16	C₂×C₄⋊#6157	C₂²	1	2	yes	—
17	#13133285	C₂²	1	2	yes	—
18	#13133285	C₂²	1	2	yes	—
19	#13133280	C₂²	1	4	yes	—
20	#13133279	C₂²	1	4	yes	—
21	#13133271	C₂²	1	2	yes	—
22	C₈×#6157	C₂²	1	2	yes	—
23	#41×D₅₃	C₂²	1	2	yes	—
24	#13133240	C₄	1	1	yes	characteristic, complemented
25	#13133239	C₄	1	1	yes	characteristic, complemented
26	#13133238	C₄	1	2	yes	complemented
27	#13133237	C₄	1	2	yes	complemented
28	#13133236	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₅₃	C₂²	1	1	yes	characteristic
30	SD₁₆×D₅₃	C₂²	1	2	yes	—
31	D₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	#135×C₅₃	C₄	1	1	yes	characteristic, complemented
34	C₂²×#6157	—	2	2	no	—
35	C₂²×#6157	—	2	4	no	—
36	#41×C₅₃	C₂×C₄	1	2	yes	complemented
37	#11719949	C₂×C₄	1	2	yes	complemented
38	Q₈×D₅₃	C₂³	1	2	yes	—
39	#11719947	—	2	2	no	—
40	#11719947	C₂×C₄	1	2	yes	complemented
41	D₄×D₅₃	—	2	2	no	—
42	D₄×D₅₃	C₂³	1	2	yes	—
43	#11719945	—	2	2	no	—
44	#11719945	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₅₃	—	2	2	no	—
46	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
47	C₂²⋊#6157	—	2	4	no	—
48	C₂²⋊#6157	D₄	1	2	yes	—
49	#11719940	—	2	1	no	—
50	C₄⋊#6157	—	2	4	no	—
51	C₄⋊#6157	—	2	2	no	—
52	C₄⋊#6157	D₄	1	2	yes	—
53	C₄⋊#6157	C₂³	1	4	yes	—
54	C₄×#6157	—	2	4	no	—
55	C₄×#6157	C₂³	1	2	yes	—
56	#11719936	—	2	1	no	—
57	D₅₃⋊C₈	C₂³	1	2	yes	—
58	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
59	SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
60	D₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
61	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
62	#11719925	C₂×C₄	1	2	yes	complemented
63	#11719924	C₂×C₄	1	2	yes	complemented
64	#11719923	C₂×C₄	1	2	yes	complemented
65	#11719922	C₂×C₄	1	2	yes	complemented
66	C₂×#88322	C₂×C₄	1	1	yes	characteristic, complemented
67	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic, complemented
68	D₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
69	#11719913	C₂×C₄	1	2	yes	complemented
70	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
71	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₅₃	C₂³	1	1	yes	characteristic
73	C₈×D₅₃	C₂³	1	1	yes	characteristic
74	C₂²×D₅₃	—	2	2	no	—
75	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×#6157	—	2	4	no	—
77	C₂×#6157	—	8	4	no	—
78	C₂×#6157	—	8	2	no	—
79	C₂×#6157	C₂×D₄	1	4	yes	—
80	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
81	D₄×C₅₃	—	2	2	no	—
82	D₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₅₃	—	2	2	no	—
86	C₂²⋊D₅₃	—	2	2	no	—
87	C₂²⋊D₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	2	2	no	—
89	C₂×Dic₅₃	#41	1	1	yes	characteristic
90	D₂₁₂	—	2	2	no	—
91	D₂₁₂	C₂²×C₄	1	2	yes	—
92	C₄×D₅₃	—	2	2	no	—
93	C₄×D₅₃	—	2	2	no	—
94	C₄×D₅₃	—	2	2	no	—
95	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄×D₅₃	#41	1	1	yes	characteristic
98	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
99	Dic₁₀₆	—	2	2	no	—
100	Dic₁₀₆	C₂²×C₄	1	2	yes	—
101	#88324	—	2	2	no	—
102	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	#88322	C₂²×C₄	1	1	yes	characteristic
105	#88322	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×D₅₃	—	2	2	no	—
109	C₂×D₅₃	—	2	2	no	—
110	C₂×D₅₃	—	4	2	no	—
111	C₂×D₅₃	—	4	2	no	—
112	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×D₅₃	C₂²×D₄	1	1	yes	characteristic
114	#6157	—	2	4	no	—
115	#6157	—	4	4	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	Dic₅₃	—	2	2	no	—
120	Dic₅₃	C₄×D₄	1	1	yes	characteristic
121	Dic₅₃	C₂×#41	1	1	yes	characteristic
122	#2952	—	53	1	no	maximal, Sylow
123	C₂×C₅₃	—	2	1	no	—
124	C₂×C₅₃	—	4	2	no	—
125	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
126	D₅₃	—	2	1	no	—
127	D₅₃	—	4	2	no	—
128	D₅₃	#577	1	2	yes	—
129	C₂×#135	—	53	1	no	—
130	#520	—	53	2	no	—
131	#442	—	53	2	no	—
132	#441	—	53	2	no	—
133	C₄×SD₁₆	—	53	4	no	—
134	C₂×#106	—	53	1	no	—
135	#417	—	53	2	no	—
136	#405	—	53	1	no	—
137	C₅₃	#2952	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	53	2	no	—
139	#135	—	53	2	no	—
140	#135	—	53	1	no	—
141	#135	—	53	2	no	—
142	#135	—	53	1	no	—
143	#135	—	53	1	no	—
144	#135	#6157	1	1	yes	characteristic, complemented
145	C₂×Q₁₆	—	53	1	no	—
146	C₂×SD₁₆	—	53	2	no	—
147	C₂×D₈	—	53	1	no	—
148	C₂×M₁₆	—	53	1	no	—
149	C₂²×C₈	—	53	1	no	—
150	C₄×Q₈	—	53	4	no	—
151	C₄×D₄	—	53	4	no	—
152	C₄×D₄	—	106	4	no	—
153	#117	—	53	1	no	—
154	#117	—	106	2	no	—
155	C₂×C₄⋊C₄	—	53	2	no	—
156	C₂×C₄⋊C₄	—	106	2	no	—
157	#106	—	53	2	no	—
158	#106	—	53	2	no	—
159	#103	—	53	4	no	—
160	#102	—	53	4	no	—
161	#97	—	53	2	no	—
162	C₄×C₈	—	53	2	no	—
163	#41	—	53	2	no	—
164	#41	—	53	2	no	—
165	#41	—	53	2	no	—
166	#41	—	106	2	no	—
167	#41	—	106	2	no	—
168	#41	C₂×#6157	1	2	yes	complemented
169	C₂×Q₈	—	53	2	no	—
170	C₂×D₄	—	53	2	no	—
171	C₂×D₄	—	106	2	no	—
172	C₂²×C₄	—	53	1	no	—
173	C₂²×C₄	—	106	2	no	—
174	C₂²×C₄	—	106	4	no	—
175	C₂²×C₄	—	106	2	no	—
176	Q₁₆	—	53	2	no	—
177	Q₁₆	—	53	1	no	—
178	Q₁₆	C₂×#6157	1	1	yes	characteristic, complemented
179	SD₁₆	—	53	2	no	—
180	SD₁₆	—	53	2	no	—
181	SD₁₆	—	53	2	no	—
182	SD₁₆	C₂×#6157	1	2	yes	complemented
183	D₈	—	53	2	no	—
184	D₈	—	53	1	no	—
185	D₈	C₂×#6157	1	1	yes	characteristic, complemented
186	M₁₆	—	106	1	no	—
187	M₁₆	—	106	1	no	—
188	C₂×C₈	—	53	1	no	—
189	C₂×C₈	—	53	1	no	—
190	C₂×C₈	—	53	1	no	—
191	C₂×C₈	—	53	1	no	—
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	C₂×#6157	1	1	yes	characteristic, complemented
195	C₄⋊C₄	—	53	2	no	—
196	C₄⋊C₄	—	53	4	no	—
197	C₄⋊C₄	—	106	2	no	—
198	C₄⋊C₄	—	106	4	no	—
199	C₂²⋊C₄	—	53	2	no	—
200	C₂²⋊C₄	—	106	4	no	—
201	C₄²	—	53	2	no	—
202	C₄²	—	106	4	no	—
203	C₂³	—	53	1	no	—
204	C₂³	—	106	2	no	—
205	Q₈	—	53	2	no	—
206	Q₈	—	106	2	no	—
207	Q₈	C₂²×#6157	1	2	yes	—
208	D₄	—	2	2	no	—
209	D₄	—	53	2	no	—
210	D₄	—	106	2	no	—
211	D₄	—	106	2	no	—
212	D₄	—	106	2	no	—
213	D₄	—	106	2	no	—
214	D₄	C₂²×#6157	1	2	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	53	1	no	—
217	C₂×C₄	—	53	1	no	—
218	C₂×C₄	—	53	1	no	—
219	C₂×C₄	—	53	1	no	—
220	C₂×C₄	—	53	1	no	—
221	C₂×C₄	—	53	4	no	—
222	C₂×C₄	—	106	2	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	106	2	no	—
226	C₂×C₄	—	106	4	no	—
227	C₂×C₄	—	424	4	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
230	C₈	—	53	1	no	—
231	C₈	—	53	1	no	—
232	C₈	—	106	2	no	—
233	C₈	C₂²×#6157	1	1	yes	characteristic
234	C₈	C₂²×#6157	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	53	1	no	—
237	C₂²	—	53	1	no	—
238	C₂²	—	106	2	no	—
239	C₂²	—	106	2	no	—
240	C₂²	—	212	2	no	—
241	C₂²	—	212	2	no	—
242	C₂²	D₄×#6157	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	53	1	no	—
245	C₄	—	53	1	no	—
246	C₄	—	106	4	no	—
247	C₄	—	106	2	no	—
248	C₄	—	212	4	no	—
249	C₄	D₄×#6157	1	1	yes	characteristic
250	C₄	C₂³×#6157	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	53	2	no	—
254	C₂	—	106	1	no	—
255	C₂	—	212	2	no	—
256	C₂	C₂×D₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	#135×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#135	#6157
hypocenter	C₅₃	#2952
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #16811692	C₂	semidirect
2. #16811664	C₂	semidirect
3. #16811663	C₂	semidirect
4. SD₁₆×#6157	C₂	semidirect
5. C₂×#13133285	C₂	semidirect
6. #16811614	C₂	semidirect
7. #16811586	C₂	semidirect
24. #13133240	C₄	semidirect
25. #13133239	C₄	semidirect
26. #13133238	C₄	semidirect
27. #13133237	C₄	semidirect
28. #13133236	C₄	semidirect
33. #135×C₅₃	C₄	semidirect
36. #41×C₅₃	C₂×C₄	semidirect
37. #11719949	C₂×C₄	semidirect
40. #11719947	C₂×C₄	semidirect
44. #11719945	C₂×C₄	semidirect
58. Q₁₆×C₅₃	C₂×C₄	semidirect
59. SD₁₆×C₅₃	C₂×C₄	semidirect
60. D₈×C₅₃	C₂×C₄	semidirect
61. C₂×C₈×C₅₃	C₂×C₄	semidirect
62. #11719925	C₂×C₄	semidirect
63. #11719924	C₂×C₄	semidirect
64. #11719923	C₂×C₄	semidirect
65. #11719922	C₂×C₄	semidirect
66. C₂×#88322	C₂×C₄	semidirect
67. Dic₂₁₂	C₂×C₄	semidirect
68. D₄₂₄	C₂×C₄	semidirect
69. #11719913	C₂×C₄	semidirect
70. C₈×D₅₃	C₂×C₄	semidirect
71. C₈×D₅₃	C₂×C₄	semidirect
137. C₅₃	#2952	semidirect
144. #135	#6157	semidirect
168. #41	C₂×#6157	semidirect
178. Q₁₆	C₂×#6157	semidirect
182. SD₁₆	C₂×#6157	semidirect
185. D₈	C₂×#6157	semidirect
194. C₂×C₈	C₂×#6157	semidirect