Subgroups — #23772196

Order 10112.

492 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#41×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19099031	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098672	C₂	1	2	yes	maximal, complemented
5	#19098667	C₂	1	2	yes	maximal, complemented
6	#19098666	C₂	1	2	yes	maximal, complemented
7	#19098660	C₂	1	2	yes	maximal, complemented
8	#19098599	C₂	1	1	yes	maximal, characteristic, complemented
9	#19098560	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579071	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11929624	C₂²	1	1	yes	characteristic
13	#14579181	C₄	1	8	yes	complemented
14	C₂×#11929614	C₂²	1	1	yes	characteristic
15	C₂×#11929612	C₂²	1	1	yes	characteristic
16	C₂×#11929611	C₂²	1	1	yes	characteristic
17	C₂×#11929609	C₂²	1	1	yes	characteristic
18	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×Dic₇₉	C₂²	1	2	yes	—
20	#14579126	C₂²	1	2	yes	—
21	D₄×Dic₇₉	—	2	2	no	—
22	D₄×Dic₇₉	C₂²	1	2	yes	—
23	#14579123	C₂²	1	2	yes	—
24	(C₂²⋊D₇₉)⋊C₄	—	2	2	no	—
25	#14579120	—	2	1	no	—
26	#14579120	C₂²	1	1	yes	characteristic
27	#14579118	—	2	1	no	—
28	C₂×C₄×Dic₇₉	—	2	1	no	—
29	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
30	C₄²×D₇₉	—	2	1	no	—
31	#14579094	C₂²	1	2	yes	—
32	#14579092	C₂²	1	2	yes	—
33	#14579086	C₂²	1	2	yes	—
34	#14579083	C₂²	1	2	yes	—
35	#14579071	C₂²	1	2	yes	—
36	#14579071	C₂²	1	2	yes	—
37	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
39	#117×C₇₉	C₂²	1	1	yes	characteristic
40	#103×C₇₉	C₂²	1	2	yes	—
41	#102×C₇₉	C₂²	1	2	yes	—
42	#11929624	—	2	2	no	—
43	#11929624	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₁₅₈	C₂³	1	1	yes	characteristic
45	C₂×D₃₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₇₉	—	2	1	no	—
47	C₂²×Dic₇₉	—	2	2	no	—
48	C₂²×Dic₇₉	D₄	1	2	yes	—
49	C₂×C₄×D₇₉	—	2	2	no	—
50	C₂×C₄×D₇₉	—	2	1	no	—
51	#11929614	C₂×C₄	1	4	yes	complemented
52	#11929612	C₂×C₄	1	4	yes	complemented
53	#11929611	C₂×C₄	1	4	yes	complemented
54	#11929609	C₂×C₄	1	4	yes	complemented
55	C₂×C₇₉⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₇₉	—	2	2	no	—
59	C₄⋊Dic₇₉	C₂³	1	2	yes	—
60	Dic₇₉⋊C₄	—	2	2	no	—
61	C₂²⋊Dic₇₉	—	2	2	no	—
62	C₂²⋊Dic₇₉	—	2	1	no	—
63	(C₂×D₇₉)⋊C₄	—	2	2	no	—
64	C₄×Dic₇₉	—	2	2	no	—
65	C₄×Dic₇₉	—	2	2	no	—
66	C₄×Dic₇₉	—	2	1	no	—
67	C₄×Dic₇₉	D₄	1	1	yes	characteristic
68	C₄×Dic₇₉	D₄	1	1	yes	characteristic
69	C₄×Dic₇₉	C₂³	1	2	yes	—
70	#41×C₇₉	—	2	2	no	—
71	#41×C₇₉	C₂×C₄	1	4	yes	complemented
72	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₇₉	—	2	1	no	—
74	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₇₉	—	2	2	no	—
78	C₂×C₈×C₇₉	C₂³	1	2	yes	—
79	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₇₉	—	2	1	no	—
81	C₄²×C₇₉	—	2	1	no	—
82	C₂³×C₇₉	—	2	1	no	—
83	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₇₉	—	2	1	no	—
85	Q₈×C₇₉	—	2	1	no	—
86	Q₈×C₇₉	C₂²×C₄	1	2	yes	—
87	D₄×C₇₉	—	2	4	no	—
88	D₄×C₇₉	—	2	1	no	—
89	D₄×C₇₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	4	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#41	1	2	yes	—
96	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₇₉	—	2	4	no	—
98	C₂×Dic₇₉	—	2	1	no	—
99	C₂×Dic₇₉	—	2	2	no	—
100	C₂×Dic₇₉	—	4	2	no	—
101	C₂×Dic₇₉	—	4	4	no	—
102	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
103	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
104	D₃₁₆	—	2	1	no	—
105	D₃₁₆	C₂²×C₄	1	2	yes	—
106	C₄×D₇₉	—	2	4	no	—
107	C₄×D₇₉	—	4	4	no	—
108	Dic₁₅₈	—	2	1	no	—
109	Dic₁₅₈	C₂²×C₄	1	2	yes	—
110	C₈×C₇₉	—	2	2	no	—
111	C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
112	C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂²×C₇₉	—	2	2	no	—
114	C₂²×C₇₉	—	2	2	no	—
115	C₂²×C₇₉	—	4	2	no	—
116	C₂²×C₇₉	C₄×D₄	1	2	yes	—
117	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₇₉	—	2	2	no	—
119	C₂×D₇₉	—	4	2	no	—
120	C₄×C₇₉	—	2	2	no	—
121	C₄×C₇₉	—	4	2	no	—
122	C₄×C₇₉	C₄×D₄	1	2	yes	—
123	C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
125	Dic₇₉	—	2	2	no	—
126	Dic₇₉	—	2	2	no	—
127	Dic₇₉	—	2	2	no	—
128	Dic₇₉	—	4	2	no	—
129	C₂×C₇₉	—	2	2	no	—
130	C₂×C₇₉	—	4	2	no	—
131	C₂×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₇₉	#575	1	2	yes	—
133	D₇₉	—	4	2	no	—
134	#2943	—	79	1	no	maximal, Sylow
135	C₇₉	#2943	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	79	1	no	—
137	C₄×#41	—	79	1	no	—
138	C₄×#41	—	79	1	no	—
139	#442	—	79	2	no	—
140	#441	—	79	2	no	—
141	#440	—	79	2	no	—
142	#440	—	79	2	no	—
143	#427	—	79	1	no	—
144	#416	—	79	1	no	—
145	#416	D₇₉	1	1	yes	characteristic, complemented
146	C₂×#97	—	79	1	no	—
147	C₂×#41	—	79	1	no	—
148	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
149	#135	—	79	8	no	—
150	C₂×Q₁₆	—	79	1	no	—
151	C₂×SD₁₆	—	79	1	no	—
152	C₂×SD₁₆	—	79	1	no	—
153	C₂×D₈	—	79	1	no	—
154	C₂²×C₈	—	79	1	no	—
155	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
156	C₄×Q₈	—	79	2	no	—
157	C₄×Q₈	—	79	2	no	—
158	C₄×D₄	—	79	2	no	—
159	C₄×D₄	—	79	2	no	—
160	C₄×D₄	—	158	2	no	—
161	C₄×D₄	—	158	2	no	—
162	#117	—	79	1	no	—
163	#117	—	158	1	no	—
164	#117	—	158	1	no	—
165	#117	C₂×D₇₉	1	1	yes	characteristic
166	C₂×C₄²	—	79	1	no	—
167	C₂×C₄²	—	158	1	no	—
168	C₂×C₄²	—	158	1	no	—
169	#107	—	79	2	no	—
170	#106	—	79	2	no	—
171	#103	—	79	2	no	—
172	#103	C₂×D₇₉	1	2	yes	—
173	#102	—	79	2	no	—
174	#102	C₂×D₇₉	1	2	yes	—
175	#97	—	79	2	no	—
176	#97	—	79	2	no	—
177	#41	—	2	2	no	—
178	#41	—	79	4	no	—
179	#41	—	158	2	no	—
180	#41	C₄×D₇₉	1	4	yes	complemented
181	C₂×Q₈	—	79	1	no	—
182	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	79	1	no	—
185	C₂×D₄	—	158	1	no	—
186	C₂×D₄	C₂²×D₇₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	79	2	no	—
189	C₂²×C₄	—	158	2	no	—
190	C₂²×C₄	—	158	2	no	—
191	C₂²×C₄	—	158	1	no	—
192	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
193	Q₁₆	—	79	4	no	—
194	SD₁₆	—	79	4	no	—
195	SD₁₆	—	79	4	no	—
196	D₈	—	79	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	79	4	no	—
199	C₂×C₈	—	79	1	no	—
200	C₂×C₈	—	79	1	no	—
201	C₂×C₈	C₂²×D₇₉	1	2	yes	—
202	C₄⋊C₄	—	79	2	no	—
203	C₄⋊C₄	—	158	2	no	—
204	C₄⋊C₄	—	158	2	no	—
205	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	158	2	no	—
208	C₂²⋊C₄	—	158	2	no	—
209	C₂²⋊C₄	—	158	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	79	1	no	—
212	C₄²	—	79	1	no	—
213	C₄²	—	79	2	no	—
214	C₄²	—	158	2	no	—
215	C₄²	—	158	2	no	—
216	C₄²	—	158	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	158	1	no	—
219	C₂³	D₄×D₇₉	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	79	2	no	—
222	Q₈	—	158	1	no	—
223	Q₈	C₂×C₄×D₇₉	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	79	2	no	—
227	D₄	—	158	4	no	—
228	D₄	—	158	1	no	—
229	D₄	C₂×C₄×D₇₉	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	316	4	no	—
234	C₂×C₄	—	316	4	no	—
235	C₂×C₄	—	316	2	no	—
236	C₂×C₄	—	79	2	no	—
237	C₂×C₄	—	79	2	no	—
238	C₂×C₄	—	158	2	no	—
239	C₂×C₄	—	158	1	no	—
240	C₂×C₄	—	158	4	no	—
241	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
242	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
243	C₂×C₄	#11929623	1	2	yes	—
244	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	79	2	no	—
247	C₈	—	79	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	316	2	no	—
252	C₂²	—	158	2	no	—
253	C₂²	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
254	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	316	2	no	—
258	C₄	—	158	2	no	—
259	C₄	—	158	2	no	—
260	C₄	—	158	2	no	—
261	C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
262	C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
263	C₄	C₂×#11929623	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	316	2	no	—
267	C₂	C₂×(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
268	C₂	#19099360	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₉
commutator	C₄×C₇₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#416×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×#14579181	C₂
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	#416	D₇₉
hypocenter	C₇₉	#2943
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₇₉	C₂	semidirect
3. #19099031	C₂	semidirect
4. #19098672	C₂	semidirect
5. #19098667	C₂	semidirect
6. #19098666	C₂	semidirect
7. #19098660	C₂	semidirect
8. #19098599	C₂	semidirect
9. #19098560	C₂	semidirect
10. C₂×#14579071	C₂	semidirect
11. #416×C₇₉	C₂	semidirect
13. #14579181	C₄	semidirect
43. #11929624	C₂×C₄	semidirect
51. #11929614	C₂×C₄	semidirect
52. #11929612	C₂×C₄	semidirect
53. #11929611	C₂×C₄	semidirect
54. #11929609	C₂×C₄	semidirect
55. C₂×C₇₉⋊C₈	C₂×C₄	semidirect
71. #41×C₇₉	C₂×C₄	semidirect
135. C₇₉	#2943	semidirect
145. #416	D₇₉	semidirect
180. #41	C₄×D₇₉	semidirect