Subgroups — #23777295

Order 10112.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099299	C₂	1	4	yes	maximal, complemented
2	#19099282	C₂	1	4	yes	maximal, complemented
3	#19099244	C₂	1	8	yes	maximal, complemented
4	#19099180	C₂	1	4	yes	maximal, complemented
5	#19099134	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099116	C₂	1	1	yes	maximal, characteristic, complemented
7	#19099094	C₂	1	4	yes	maximal, complemented
8	C₂×#14579148	C₂	1	1	yes	maximal, characteristic, complemented
9	#19099040	C₂	1	1	yes	maximal, characteristic, complemented
10	#19099038	C₂	1	1	yes	maximal, characteristic, complemented
11	#41×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#535×C₇₉	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
14	#14579160	C₂²	1	2	yes	complemented
15	#14579159	C₂²	1	2	yes	complemented
16	#14579154	C₂²	1	8	yes	complemented
17	#14579154	C₂²	1	8	yes	complemented
18	#14579152	C₂²	1	2	yes	complemented
19	#14579148	C₂²	1	2	yes	—
20	#14579148	C₂²	1	4	yes	—
21	#14579148	C₂²	1	2	yes	—
22	#14579148	C₂²	1	8	yes	complemented
23	#14579148	C₂²	1	2	yes	complemented
24	#14579142	C₂²	1	4	yes	—
25	#14579142	C₂²	1	8	yes	complemented
26	#14579142	C₂²	1	8	yes	complemented
27	#14579140	C₂²	1	4	yes	complemented
28	#14579138	C₂²	1	4	yes	—
29	#14579138	C₂²	1	8	yes	complemented
30	Q₈×Dic₇₉	C₂²	1	2	yes	complemented
31	C₄×Dic₁₅₈	C₂²	1	4	yes	complemented
32	C₄×Dic₁₅₈	C₂²	1	4	yes	complemented
33	#14579126	C₂²	1	4	yes	complemented
34	D₄×Dic₇₉	C₂²	1	4	yes	—
35	D₄×Dic₇₉	C₂²	1	4	yes	—
36	D₄×Dic₇₉	C₂²	1	4	yes	—
37	D₄×Dic₇₉	C₂²	1	8	yes	complemented
38	D₄×Dic₇₉	C₂²	1	2	yes	complemented
39	#14579120	C₂²	1	1	yes	characteristic
40	#14579120	C₂²	1	1	yes	characteristic
41	#14579120	C₂²	1	2	yes	complemented
42	#14579117	C₂²	1	4	yes	—
43	#14579117	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
45	C₂×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
46	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
47	C₂×C₄×Dic₇₉	C₂²	1	2	yes	complemented
48	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
49	C₂²×D₄×C₇₉	C₂²	1	1	yes	characteristic
50	#127×C₇₉	C₂²	1	2	yes	—
51	#124×C₇₉	C₂²	1	2	yes	—
52	#121×C₇₉	C₂²	1	4	yes	—
53	#120×C₇₉	C₂²	1	4	yes	—
54	#117×C₇₉	C₂²	1	1	yes	characteristic
55	C₂×Dic₁₅₈	—	2	4	no	—
56	C₂×Dic₁₅₈	C₂³	1	2	yes	—
57	C₂×Dic₁₅₈	C₂³	1	2	yes	—
58	C₂×Dic₁₅₈	C₂³	1	4	yes	—
59	C₂²×Dic₇₉	C₂³	1	2	yes	—
60	C₂²×Dic₇₉	C₂³	1	2	yes	—
61	C₂²×Dic₇₉	C₂³	1	4	yes	—
62	C₂²×Dic₇₉	C₂³	1	8	yes	—
63	C₄⋊Dic₇₉	C₂³	1	2	yes	—
64	C₄⋊Dic₇₉	C₂³	1	2	yes	—
65	C₄⋊Dic₇₉	C₂³	1	4	yes	—
66	C₄⋊Dic₇₉	C₂³	1	4	yes	—
67	Dic₇₉⋊C₄	C₂³	1	8	yes	—
68	Dic₇₉⋊C₄	C₂³	1	4	yes	—
69	Dic₇₉⋊C₄	C₂³	1	8	yes	—
70	Dic₇₉⋊C₄	C₂³	1	4	yes	—
71	C₂²⋊Dic₇₉	C₂³	1	2	yes	—
72	C₂²⋊Dic₇₉	C₂³	1	2	yes	—
73	C₂²⋊Dic₇₉	C₂³	1	8	yes	—
74	C₂²⋊Dic₇₉	C₂³	1	8	yes	—
75	C₂²⋊Dic₇₉	C₂³	1	8	yes	—
76	C₂²⋊Dic₇₉	C₂³	1	4	yes	—
77	C₂²⋊Dic₇₉	C₂³	1	8	yes	—
78	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
79	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
80	C₄×Dic₇₉	C₂³	1	2	yes	—
81	C₄×Dic₇₉	C₂³	1	4	yes	—
82	C₄×Dic₇₉	C₂³	1	2	yes	—
83	C₄×Dic₇₉	C₂³	1	8	yes	—
84	C₄×Dic₇₉	C₂³	1	2	yes	—
85	C₂⁴×C₇₉	C₂³	1	2	yes	—
86	#41×C₇₉	—	2	4	no	—
87	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
88	C₂×D₄×C₇₉	—	2	4	no	—
89	C₂×D₄×C₇₉	C₂³	1	2	yes	—
90	C₂×D₄×C₇₉	C₂³	1	2	yes	—
91	C₂×D₄×C₇₉	C₂³	1	2	yes	—
92	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₇₉	C₂³	1	8	yes	—
94	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
95	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
97	C₂²⋊C₄×C₇₉	C₂³	1	2	yes	—
98	C₂²⋊C₄×C₇₉	C₂³	1	8	yes	—
99	C₄²×C₇₉	C₂³	1	2	yes	—
100	C₂³×C₇₉	—	2	4	no	—
101	C₂³×C₇₉	—	4	4	no	—
102	C₂³×C₇₉	C₂⁴	1	4	yes	—
103	C₂³×C₇₉	C₂⁴	1	2	yes	—
104	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
105	Q₈×C₇₉	—	2	2	no	—
106	D₄×C₇₉	—	2	4	no	—
107	D₄×C₇₉	—	2	4	no	—
108	D₄×C₇₉	—	2	4	no	—
109	D₄×C₇₉	—	2	2	no	—
110	D₄×C₇₉	—	4	8	no	—
111	C₂×C₄×C₇₉	—	2	4	no	—
112	C₂×C₄×C₇₉	#41	1	4	yes	—
113	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
114	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
115	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₇₉	—	2	4	no	—
118	C₂×Dic₇₉	—	4	2	no	—
119	C₂×Dic₇₉	—	4	8	no	—
120	C₂×Dic₇₉	—	4	4	no	—
121	C₂×Dic₇₉	C₂⁴	1	2	yes	—
122	C₂×Dic₇₉	C₂⁴	1	4	yes	—
123	C₂×Dic₇₉	C₂⁴	1	4	yes	—
124	C₂×Dic₇₉	C₂⁴	1	2	yes	—
125	C₂×Dic₇₉	C₂⁴	1	4	yes	—
126	Dic₁₅₈	—	2	4	no	—
127	Dic₁₅₈	—	2	4	no	—
128	Dic₁₅₈	—	4	4	no	—
129	C₂²×C₇₉	—	2	2	no	—
130	C₂²×C₇₉	—	2	2	no	—
131	C₂²×C₇₉	—	2	4	no	—
132	C₂²×C₇₉	—	2	4	no	—
133	C₂²×C₇₉	—	2	8	no	—
134	C₂²×C₇₉	—	8	4	no	—
135	C₂²×C₇₉	C₂×#41	1	2	yes	—
136	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
137	C₄×C₇₉	—	2	2	no	—
138	C₄×C₇₉	—	2	4	no	—
139	C₄×C₇₉	C₂×#41	1	2	yes	—
140	C₄×C₇₉	C₂×#41	1	2	yes	—
141	Dic₇₉	—	2	4	no	—
142	Dic₇₉	—	2	2	no	—
143	Dic₇₉	—	2	4	no	—
144	Dic₇₉	—	2	2	no	—
145	Dic₇₉	—	2	4	no	—
146	C₂×C₇₉	—	2	2	no	—
147	C₂×C₇₉	—	4	2	no	—
148	C₂×C₇₉	—	4	4	no	—
149	C₂×C₇₉	C₂²×#41	1	1	yes	characteristic
150	C₂×C₇₉	#585	1	2	yes	—
151	#3510	—	79	1	no	maximal, Sylow
152	C₇₉	#3510	1	1	yes	characteristic, complemented, Sylow
153	#559	—	79	4	no	—
154	#556	—	79	4	no	—
155	#551	—	79	8	no	—
156	#542	—	79	4	no	—
157	#536	—	79	1	no	—
158	#535	D₇₉	1	1	yes	characteristic
159	#533	—	79	1	no	—
160	#529	—	79	4	no	—
161	C₂×#124	—	79	1	no	—
162	#519	—	79	1	no	—
163	#518	—	79	1	no	—
164	C₄×#41	—	79	1	no	—
165	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
166	C₂²×Q₈	—	79	1	no	—
167	C₂²×D₄	C₂×D₇₉	1	1	yes	characteristic
168	C₄⋊Q₈	—	79	2	no	—
169	C₄⋊Q₈	—	79	2	no	—
170	#127	C₂×D₇₉	1	2	yes	—
171	#126	—	79	8	no	—
172	#126	—	79	8	no	—
173	#125	—	79	2	no	—
174	#124	—	79	2	no	—
175	#124	—	79	2	no	—
176	#124	—	79	4	no	—
177	#124	—	79	2	no	—
178	#124	—	79	8	no	—
179	#124	C₂×D₇₉	1	2	yes	—
180	#123	—	79	8	no	—
181	#123	—	79	4	no	—
182	#123	—	79	8	no	—
183	C₂²⋊Q₈	—	79	4	no	—
184	C₂²⋊Q₈	—	79	4	no	—
185	C₂²⋊Q₈	—	79	8	no	—
186	#121	C₂×D₇₉	1	4	yes	—
187	#120	C₂×D₇₉	1	4	yes	—
188	C₄×Q₈	—	79	2	no	—
189	C₄×Q₈	—	79	4	no	—
190	C₄×Q₈	—	79	4	no	—
191	C₄×Q₈	—	79	4	no	—
192	C₄×D₄	—	79	4	no	—
193	C₄×D₄	—	79	2	no	—
194	C₄×D₄	—	79	4	no	—
195	C₄×D₄	—	79	4	no	—
196	C₄×D₄	—	79	8	no	—
197	#117	—	79	1	no	—
198	#117	—	79	2	no	—
199	#117	—	79	1	no	—
200	#117	—	79	4	no	—
201	#117	—	79	8	no	—
202	#117	C₂×D₇₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	79	4	no	—
204	C₂×C₂²⋊C₄	—	79	4	no	—
205	C₂×C₄²	—	79	1	no	—
206	C₂×C₄²	—	79	2	no	—
207	C₂⁴	C₂²×D₇₉	1	2	yes	—
208	#41	—	2	4	no	—
209	C₂×Q₈	—	79	4	no	—
210	C₂×Q₈	—	79	2	no	—
211	C₂×Q₈	—	79	2	no	—
212	C₂×Q₈	—	158	4	no	—
213	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	C₂²×D₇₉	1	2	yes	—
216	C₂×D₄	C₂²×D₇₉	1	2	yes	—
217	C₂×D₄	C₂²×D₇₉	1	2	yes	—
218	C₂×D₄	C₂²×D₇₉	1	1	yes	characteristic
219	C₂×D₄	C₂²×D₇₉	1	8	yes	—
220	C₂²×C₄	—	79	2	no	—
221	C₂²×C₄	—	79	4	no	—
222	C₂²×C₄	—	79	8	no	—
223	C₂²×C₄	—	79	2	no	—
224	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
225	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
226	C₄⋊C₄	—	79	4	no	—
227	C₄⋊C₄	—	79	8	no	—
228	C₄⋊C₄	—	79	8	no	—
229	C₄⋊C₄	—	79	4	no	—
230	C₄⋊C₄	—	79	2	no	—
231	C₄⋊C₄	—	79	2	no	—
232	C₄⋊C₄	—	79	4	no	—
233	C₄⋊C₄	—	79	4	no	—
234	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
235	C₂²⋊C₄	—	79	2	no	—
236	C₂²⋊C₄	—	79	2	no	—
237	C₂²⋊C₄	—	79	8	no	—
238	C₂²⋊C₄	—	79	8	no	—
239	C₂²⋊C₄	—	79	8	no	—
240	C₂²⋊C₄	—	79	4	no	—
241	C₂²⋊C₄	—	79	8	no	—
242	C₂²⋊C₄	C₂²×D₇₉	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₇₉	1	8	yes	—
244	C₄²	—	79	1	no	—
245	C₄²	—	79	2	no	—
246	C₄²	—	79	4	no	—
247	C₄²	—	79	1	no	—
248	C₄²	—	79	2	no	—
249	C₄²	—	79	8	no	—
250	C₄²	—	79	2	no	—
251	C₄²	C₂²×D₇₉	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	C₂³×D₇₉	1	4	yes	—
255	C₂³	C₂³×D₇₉	1	2	yes	—
256	C₂³	C₂³×D₇₉	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	Q₈	—	316	4	no	—
259	Q₈	—	158	4	no	—
260	Q₈	—	158	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	2	2	no	—
265	D₄	—	4	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	316	4	no	—
268	C₂×C₄	—	316	8	no	—
269	C₂×C₄	—	316	2	no	—
270	C₂×C₄	—	79	2	no	—
271	C₂×C₄	—	79	4	no	—
272	C₂×C₄	—	79	2	no	—
273	C₂×C₄	—	79	4	no	—
274	C₂×C₄	—	79	4	no	—
275	C₂×C₄	—	158	4	no	—
276	C₂×C₄	#11929623	1	4	yes	—
277	C₂×C₄	C₂³×D₇₉	1	4	yes	—
278	C₂×C₄	C₂³×D₇₉	1	2	yes	—
279	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×#11929623	1	2	yes	—
288	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	158	2	no	—
292	C₄	—	158	4	no	—
293	C₄	—	158	4	no	—
294	C₄	—	158	2	no	—
295	C₄	—	158	4	no	—
296	C₄	C₂×#11929623	1	2	yes	—
297	C₄	C₂×#11929623	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	4	2	no	—
301	C₂	C₂²×#11929623	1	1	yes	characteristic
302	C₂	#19099398	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁵
Fitting, prime radical	#535×C₇₉	C₂
hypercenter	#535	D₇₉
hypocenter	C₇₉	#3510
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #535

Complements

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

N	H ≅ G/N	type
1. #19099299	C₂	semidirect
2. #19099282	C₂	semidirect
3. #19099244	C₂	semidirect
4. #19099180	C₂	semidirect
5. #19099134	C₂	semidirect
6. #19099116	C₂	semidirect
7. #19099094	C₂	semidirect
8. C₂×#14579148	C₂	semidirect
9. #19099040	C₂	semidirect
10. #19099038	C₂	semidirect
11. #41×Dic₇₉	C₂	semidirect
14. #14579160	C₂²	semidirect
15. #14579159	C₂²	semidirect
16. #14579154	C₂²	semidirect
17. #14579154	C₂²	semidirect
18. #14579152	C₂²	semidirect
22. #14579148	C₂²	semidirect
23. #14579148	C₂²	semidirect
25. #14579142	C₂²	semidirect
26. #14579142	C₂²	semidirect
27. #14579140	C₂²	semidirect
29. #14579138	C₂²	semidirect
30. Q₈×Dic₇₉	C₂²	semidirect
31. C₄×Dic₁₅₈	C₂²	semidirect
32. C₄×Dic₁₅₈	C₂²	semidirect
33. #14579126	C₂²	semidirect
37. D₄×Dic₇₉	C₂²	semidirect
38. D₄×Dic₇₉	C₂²	semidirect
41. #14579120	C₂²	semidirect
43. #14579117	C₂²	semidirect
47. C₂×C₄×Dic₇₉	C₂²	semidirect
152. C₇₉	#3510	semidirect