Subgroups — #23772313

Order 10112.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14579181	C₂	1	1	yes	maximal, characteristic
2	#19099330	C₂	1	1	yes	maximal, characteristic, complemented
3	#19099329	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098682	C₂	1	4	yes	maximal, complemented
5	#19098677	C₂	1	4	yes	maximal, complemented
6	C₂×#14579096	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098581	C₂	1	1	yes	maximal, characteristic, complemented
8	#19098519	C₂	1	1	yes	maximal, characteristic, complemented
9	#421×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11929624	C₂²	1	1	yes	characteristic
11	#14579181	C₄	1	2	yes	complemented
12	#14579181	C₄	1	2	yes	complemented
13	#14579181	C₂²	1	2	yes	—
14	#14579181	C₂²	1	2	yes	—
15	C₂×#11929614	C₄	1	1	yes	characteristic, complemented
16	C₂×#11929612	C₄	1	1	yes	characteristic, complemented
17	C₂×#11929611	C₄	1	1	yes	characteristic, complemented
18	C₂×#11929609	C₄	1	1	yes	characteristic, complemented
19	#14579169	C₂²	1	4	yes	complemented
20	#14579168	—	2	2	no	—
21	#14579168	C₂²	1	4	yes	complemented
22	#14579167	—	2	2	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	C₂²	1	1	yes	characteristic
25	C₂×#11929608	C₂²	1	1	yes	characteristic
26	M₁₆×D₇₉	—	2	2	no	—
27	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
28	#14579096	C₂²	1	2	yes	complemented
29	#14579096	C₂²	1	2	yes	complemented
30	#14579089	C₂²	1	4	yes	complemented
31	#14579072	C₂²	1	2	yes	complemented
32	C₄×C₇₉⋊C₈	C₂²	1	2	yes	complemented
33	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	#117×C₇₉	C₂²	1	1	yes	characteristic
36	#104×C₇₉	C₂²	1	4	yes	complemented
37	#11929624	—	2	1	no	—
38	#11929624	—	2	1	no	—
39	#11929624	C₂×C₄	1	2	yes	—
40	#11929624	C₂³	1	2	yes	—
41	C₂×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
42	C₂×D₃₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₇₉	—	2	1	no	—
44	C₂×C₄×D₇₉	—	2	1	no	—
45	#11929614	C₂×C₄	1	2	yes	—
46	#11929614	C₂×C₄	1	2	yes	—
47	#11929612	C₂×C₄	1	2	yes	—
48	#11929612	C₂×C₄	1	2	yes	—
49	#11929611	C₂×C₄	1	2	yes	—
50	#11929611	C₂×C₄	1	2	yes	—
51	#11929609	C₂×C₄	1	2	yes	—
52	#11929609	C₂×C₄	1	2	yes	—
53	#11929608	—	2	4	no	—
54	#11929608	—	2	1	no	—
55	#11929608	D₄	1	2	yes	—
56	#11929608	D₄	1	2	yes	—
57	#11929608	C₂³	1	2	yes	—
58	#11929607	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	2	1	no	—
61	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
68	C₈×D₇₉	—	2	4	no	—
69	#41×C₇₉	—	2	1	no	—
70	#41×C₇₉	—	2	1	no	—
71	#41×C₇₉	C₂×C₄	1	2	yes	—
72	#41×C₇₉	C₂³	1	2	yes	—
73	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
74	C₂×D₄×C₇₉	—	2	1	no	—
75	C₂×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
78	M₁₆×C₇₉	—	2	1	no	—
79	M₁₆×C₇₉	C₂³	1	2	yes	—
80	C₂×C₈×C₇₉	—	2	1	no	—
81	C₄⋊C₄×C₇₉	—	2	1	no	—
82	C₂²⋊C₄×C₇₉	—	2	1	no	—
83	C₄²×C₇₉	C₂³	1	2	yes	—
84	C₂³×C₇₉	—	2	1	no	—
85	C₂³×C₇₉	#41	1	1	yes	characteristic
86	C₂²×D₇₉	—	2	1	no	—
87	Q₈×C₇₉	—	2	1	no	—
88	Q₈×C₇₉	C₂²×C₄	1	2	yes	—
89	D₄×C₇₉	—	2	2	no	—
90	D₄×C₇₉	—	2	2	no	—
91	D₄×C₇₉	—	2	1	no	—
92	D₄×C₇₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₇₉	—	2	1	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	—	2	2	no	—
96	C₂×C₄×C₇₉	—	2	2	no	—
97	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₇₉	—	2	2	no	—
104	C₂²⋊D₇₉	—	2	2	no	—
105	C₂×Dic₇₉	—	2	1	no	—
106	D₃₁₆	—	2	1	no	—
107	D₃₁₆	C₂²×C₄	1	2	yes	—
108	C₄×D₇₉	—	2	2	no	—
109	C₄×D₇₉	—	2	2	no	—
110	Dic₁₅₈	—	2	1	no	—
111	Dic₁₅₈	C₂²×C₄	1	2	yes	—
112	C₈×C₇₉	—	2	2	no	—
113	C₇₉⋊C₈	—	2	2	no	—
114	C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
115	C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
117	C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
118	C₇₉⋊C₈	C₂×D₄	1	4	yes	—
119	C₂²×C₇₉	—	2	2	no	—
120	C₂²×C₇₉	—	4	1	no	—
121	C₂²×C₇₉	—	8	1	no	—
122	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
125	C₂×D₇₉	—	2	2	no	—
126	C₂×D₇₉	—	8	1	no	—
127	C₄×C₇₉	—	2	2	no	—
128	C₄×C₇₉	—	4	2	no	—
129	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
130	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
131	C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
133	Dic₇₉	—	2	2	no	—
134	C₂×C₇₉	—	2	1	no	—
135	C₂×C₇₉	—	2	1	no	—
136	C₂×C₇₉	—	2	1	no	—
137	C₂×C₇₉	—	4	2	no	—
138	C₂×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
139	D₇₉	—	4	2	no	—
140	#2957	—	79	1	no	maximal, Sylow
141	C₇₉	#2957	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	—	79	1	no	—
143	#568	—	79	1	no	—
144	#568	—	79	1	no	—
145	#444	—	79	4	no	—
146	#443	—	79	4	no	—
147	C₂×#108	—	79	1	no	—
148	#421	—	79	1	no	—
149	#421	D₇₉	1	1	yes	characteristic, complemented
150	#405	—	79	1	no	—
151	C₂×#41	—	79	1	no	—
152	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
153	#135	—	79	2	no	—
154	#135	—	79	2	no	—
155	#135	—	79	2	no	—
156	#135	—	79	2	no	—
157	C₂×Q₁₆	—	79	1	no	—
158	C₂×SD₁₆	—	79	1	no	—
159	C₂×SD₁₆	—	79	1	no	—
160	C₂×D₈	—	79	1	no	—
161	#131	—	79	4	no	—
162	#131	—	79	4	no	—
163	#131	—	158	2	no	—
164	#131	—	158	2	no	—
165	C₂×M₁₆	—	79	1	no	—
166	C₂×M₁₆	—	79	1	no	—
167	C₂×M₁₆	—	158	2	no	—
168	C₂×M₁₆	—	158	2	no	—
169	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
170	C₂²×C₈	—	79	1	no	—
171	#117	C₂×D₇₉	1	1	yes	characteristic
172	#108	—	79	2	no	—
173	#108	—	79	2	no	—
174	#104	—	79	4	no	—
175	#104	C₂×D₇₉	1	4	yes	complemented
176	#97	—	79	2	no	—
177	C₄×C₈	—	79	2	no	—
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	79	2	no	—
181	#41	—	79	2	no	—
182	#41	—	158	1	no	—
183	#41	—	158	1	no	—
184	#41	C₄×D₇₉	1	2	yes	—
185	#41	C₂²×D₇₉	1	2	yes	—
186	C₂×Q₈	—	79	1	no	—
187	C₂×Q₈	C₄×D₇₉	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	79	1	no	—
190	C₂×D₄	—	158	1	no	—
191	C₂×D₄	C₄×D₇₉	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	158	1	no	—
194	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
195	Q₁₆	—	79	2	no	—
196	Q₁₆	—	79	2	no	—
197	SD₁₆	—	79	2	no	—
198	SD₁₆	—	79	2	no	—
199	SD₁₆	—	79	2	no	—
200	SD₁₆	—	79	2	no	—
201	D₈	—	79	2	no	—
202	D₈	—	79	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	79	2	no	—
205	M₁₆	—	79	2	no	—
206	M₁₆	—	79	2	no	—
207	M₁₆	—	158	4	no	—
208	M₁₆	—	158	4	no	—
209	M₁₆	—	158	1	no	—
210	M₁₆	C₂²×D₇₉	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	79	1	no	—
213	C₂×C₈	—	79	1	no	—
214	C₂×C₈	—	79	1	no	—
215	C₂×C₈	—	79	1	no	—
216	C₂×C₈	—	79	1	no	—
217	C₂×C₈	—	79	2	no	—
218	C₂×C₈	—	79	1	no	—
219	C₂×C₈	—	158	4	no	—
220	C₂×C₈	—	158	4	no	—
221	C₂×C₈	—	158	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	C₂²×D₇₉	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	158	1	no	—
227	C₂³	#11929623	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	79	2	no	—
230	Q₈	—	158	1	no	—
231	Q₈	C₂×C₄×D₇₉	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	79	2	no	—
236	D₄	—	158	2	no	—
237	D₄	—	158	2	no	—
238	D₄	—	158	1	no	—
239	D₄	C₂×C₄×D₇₉	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	158	1	no	—
245	C₂×C₄	—	158	2	no	—
246	C₂×C₄	—	158	2	no	—
247	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
249	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
250	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
251	C₂×C₄	#11929623	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	79	1	no	—
255	C₈	—	79	1	no	—
256	C₈	—	79	1	no	—
257	C₈	—	79	1	no	—
258	C₈	—	79	4	no	—
259	C₈	—	158	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	632	1	no	—
264	C₂²	—	158	2	no	—
265	C₂²	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
266	C₂²	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
267	C₂²	C₂×#11929623	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	158	2	no	—
271	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
272	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
274	C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	316	2	no	—
280	C₂	C₂×(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₇₉
commutator	C₄×C₇₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#421×C₇₉	C₂
socle	C₂×C₇₉	C₂×C₄×D₄
prime radical	C₂×#14579181	C₂
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	#421	D₇₉
hypocenter	C₇₉	#2957
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. #19099330	C₂	semidirect
3. #19099329	C₂	semidirect
4. #19098682	C₂	semidirect
5. #19098677	C₂	semidirect
6. C₂×#14579096	C₂	semidirect
7. #19098581	C₂	semidirect
8. #19098519	C₂	semidirect
9. #421×C₇₉	C₂	semidirect
11. #14579181	C₄	semidirect
12. #14579181	C₄	semidirect
15. C₂×#11929614	C₄	semidirect
16. C₂×#11929612	C₄	semidirect
17. C₂×#11929611	C₄	semidirect
18. C₂×#11929609	C₄	semidirect
19. #14579169	C₂²	semidirect
21. #14579168	C₂²	semidirect
28. #14579096	C₂²	semidirect
29. #14579096	C₂²	semidirect
30. #14579089	C₂²	semidirect
31. #14579072	C₂²	semidirect
32. C₄×C₇₉⋊C₈	C₂²	semidirect
36. #104×C₇₉	C₂²	semidirect
141. C₇₉	#2957	semidirect
149. #421	D₇₉	semidirect
175. #104	C₂×D₇₉	semidirect