Subgroups — #23777202

Order 10112.

1100 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	#19099274	C₂	1	4	yes	maximal, complemented
2	#19099269	C₂	1	4	yes	maximal, complemented
3	#19099269	C₂	1	4	yes	maximal, complemented
4	#19099228	C₂	1	4	yes	maximal, complemented
5	#19099194	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099188	C₂	1	4	yes	maximal, complemented
7	#19099116	C₂	1	2	yes	maximal, complemented
8	C₂²⋊Q₈×D₇₉	C₂	1	4	yes	maximal, complemented
9	#19099037	C₂	1	2	yes	maximal
10	C₂×#14579123	C₂	1	1	yes	maximal, characteristic
11	#544×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₇₉	C₂²	1	4	yes	complemented
13	C₂²×D₃₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₇₉	C₂²	1	2	yes	—
15	#14579160	C₂²	1	4	yes	complemented
16	#14579160	C₂²	1	2	yes	complemented
17	#14579160	C₂²	1	2	yes	complemented
18	#14579152	C₂²	1	4	yes	complemented
19	#14579152	C₂²	1	2	yes	complemented
20	#14579152	C₂²	1	2	yes	complemented
21	#14579141	C₂²	1	8	yes	complemented
22	#14579141	C₂²	1	4	yes	complemented
23	#14579140	C₂²	1	4	yes	—
24	#14579139	C₂²	1	8	yes	complemented
25	#14579139	C₂²	1	8	yes	complemented
26	#14579139	C₂²	1	8	yes	complemented
27	#14579138	C₂²	1	4	yes	—
28	#14579138	C₂²	1	4	yes	—
29	#14579137	C₂²	1	8	yes	complemented
30	#14579137	C₂²	1	4	yes	complemented
31	C₄×Dic₁₅₈	C₂²	1	4	yes	complemented
32	C₄×Dic₁₅₈	C₂²	1	4	yes	complemented
33	C₄×D₃₁₆	C₂²	1	4	yes	—
34	C₄×D₃₁₆	C₂²	1	4	yes	—
35	#14579123	C₂²	1	4	yes	—
36	#14579123	C₂²	1	4	yes	—
37	#14579123	C₂²	1	4	yes	—
38	#14579123	C₂²	1	4	yes	—
39	#14579120	C₂²	1	2	yes	—
40	C₄⋊C₄×D₇₉	C₂²	1	4	yes	complemented
41	C₄⋊C₄×D₇₉	C₂²	1	8	yes	complemented
42	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
43	C₂²⋊C₄×D₇₉	C₂²	1	8	yes	—
44	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
45	C₄⋊Q₈×C₇₉	C₂²	1	4	yes	complemented
46	#125×C₇₉	C₂²	1	4	yes	complemented
47	C₂²⋊Q₈×C₇₉	C₂²	1	4	yes	—
48	#117×C₇₉	C₂²	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
50	C₂³×D₇₉	C₂³	1	2	yes	—
51	C₂×Dic₁₅₈	C₂³	1	4	yes	—
52	C₂×Dic₁₅₈	C₂³	1	4	yes	—
53	C₂×Dic₁₅₈	C₂³	1	4	yes	—
54	Q₈×D₇₉	—	4	8	no	—
55	C₂×D₃₁₆	Q₈	1	8	yes	complemented
56	C₂×D₃₁₆	C₂³	1	4	yes	—
57	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₇₉	—	4	4	no	—
60	C₂×C₄×D₇₉	C₂³	1	4	yes	—
61	C₂×C₄×D₇₉	C₂³	1	8	yes	—
62	C₂×C₄×D₇₉	C₂³	1	8	yes	—
63	C₂×C₄×D₇₉	C₂³	1	4	yes	—
64	C₄⋊Dic₇₉	C₂³	1	4	yes	—
65	C₄⋊Dic₇₉	C₂³	1	4	yes	—
66	C₄⋊Dic₇₉	C₂³	1	2	yes	—
67	C₄⋊Dic₇₉	C₂³	1	2	yes	—
68	C₄⋊Dic₇₉	C₂³	1	4	yes	—
69	Dic₇₉⋊C₄	C₂³	1	4	yes	—
70	Dic₇₉⋊C₄	C₂³	1	4	yes	—
71	Dic₇₉⋊C₄	C₂³	1	8	yes	—
72	Dic₇₉⋊C₄	C₂³	1	4	yes	—
73	Dic₇₉⋊C₄	C₂³	1	4	yes	—
74	C₂²⋊Dic₇₉	C₂³	1	4	yes	—
75	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
76	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
78	C₄×Dic₇₉	C₂³	1	2	yes	—
79	C₄×Dic₇₉	C₂³	1	2	yes	—
80	C₄×Dic₇₉	C₂³	1	4	yes	—
81	C₂×Q₈×C₇₉	C₂³	1	4	yes	—
82	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
85	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
86	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
87	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
88	C₄⋊C₄×C₇₉	C₂³	1	8	yes	—
89	C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
90	C₄²×C₇₉	C₂³	1	4	yes	—
91	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
92	C₂²×D₇₉	—	4	2	no	—
93	C₂²×D₇₉	—	4	2	no	—
94	C₂²×D₇₉	C₂×Q₈	1	8	yes	complemented
95	C₂²×D₇₉	C₂⁴	1	4	yes	—
96	Q₈×C₇₉	—	4	4	no	—
97	C₂×C₄×C₇₉	—	4	2	no	—
98	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	complemented
99	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
100	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
101	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
102	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
103	C₂×Dic₇₉	—	2	2	no	—
104	C₂×Dic₇₉	—	4	1	no	—
105	C₂×Dic₇₉	C₂⁴	1	2	yes	—
106	C₂×Dic₇₉	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₇₉	C₂⁴	1	4	yes	—
108	C₂×Dic₇₉	C₂⁴	1	4	yes	—
109	C₂×Dic₇₉	C₂⁴	1	1	yes	characteristic
110	D₃₁₆	C₂×Q₈	1	16	yes	complemented
111	C₄×D₇₉	—	2	8	no	—
112	C₄×D₇₉	—	4	4	no	—
113	C₄×D₇₉	—	4	8	no	—
114	C₄×D₇₉	—	4	8	no	—
115	Dic₁₅₈	—	2	8	no	—
116	Dic₁₅₈	—	4	4	no	—
117	Dic₁₅₈	—	4	4	no	—
118	C₂²×C₇₉	—	2	1	no	—
119	C₂²×C₇₉	—	2	1	no	—
120	C₂²×C₇₉	C₂²×Q₈	1	2	yes	—
121	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
122	C₂×D₇₉	—	2	4	no	—
123	C₂×D₇₉	—	2	4	no	—
124	C₂×D₇₉	—	4	8	no	—
125	C₂×D₇₉	C₂²×Q₈	1	8	yes	—
126	C₄×C₇₉	—	2	4	no	—
127	C₄×C₇₉	—	2	4	no	—
128	C₄×C₇₉	—	2	4	no	—
129	C₄×C₇₉	C₂²×Q₈	1	4	yes	—
130	Dic₇₉	—	2	4	no	—
131	Dic₇₉	—	2	1	no	—
132	Dic₇₉	—	2	2	no	—
133	Dic₇₉	—	2	4	no	—
134	Dic₇₉	H(5,𝔽₂)	1	2	yes	—
135	C₂×C₇₉	—	2	2	no	—
136	C₂×C₇₉	C₂³×Q₈	1	1	yes	characteristic
137	C₂×C₇₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₇₉	#585	1	1	yes	characteristic
139	D₇₉	—	2	8	no	—
140	#3501	—	79	1	no	maximal, Sylow
141	C₇₉	#3501	1	1	yes	characteristic, complemented, Sylow
142	#554	—	79	4	no	—
143	#554	—	79	4	no	—
144	#554	—	79	4	no	—
145	D₄×Q₈	—	79	4	no	—
146	#544	—	79	1	no	—
147	#544	D₇₉	1	1	yes	characteristic, complemented
148	#543	—	79	4	no	—
149	#533	—	79	2	no	—
150	C₂×C₂²⋊Q₈	—	79	4	no	—
151	#518	—	79	2	no	—
152	C₂×C₄×D₄	—	79	1	no	—
153	C₂²×Q₈	—	79	4	no	—
154	C₂²×D₄	—	79	1	no	—
155	C₂³×C₄	—	79	2	no	—
156	C₄⋊Q₈	—	79	2	no	—
157	C₄⋊Q₈	—	79	2	no	—
158	C₄⋊Q₈	—	79	4	no	—
159	C₄⋊Q₈	C₂×D₇₉	1	4	yes	complemented
160	#125	—	79	2	no	—
161	#125	—	79	2	no	—
162	#125	—	79	4	no	—
163	#125	C₂×D₇₉	1	4	yes	complemented
164	C₂²⋊Q₈	—	79	4	no	—
165	C₂²⋊Q₈	—	79	8	no	—
166	C₂²⋊Q₈	—	79	4	no	—
167	C₂²⋊Q₈	—	79	8	no	—
168	C₂²⋊Q₈	—	79	4	no	—
169	C₂²⋊Q₈	—	79	8	no	—
170	C₂²⋊Q₈	—	79	8	no	—
171	C₂²⋊Q₈	—	79	8	no	—
172	C₂²⋊Q₈	—	79	4	no	—
173	C₂²⋊Q₈	—	79	4	no	—
174	C₂²⋊Q₈	C₂×D₇₉	1	4	yes	—
175	C₄×Q₈	—	79	4	no	—
176	C₄×Q₈	—	79	4	no	—
177	C₄×D₄	—	79	4	no	—
178	C₄×D₄	—	79	4	no	—
179	C₄×D₄	—	79	4	no	—
180	C₄×D₄	—	79	4	no	—
181	C₄×D₄	—	79	4	no	—
182	C₄×D₄	—	79	4	no	—
183	#117	—	79	2	no	—
184	#117	C₂×D₇₉	1	2	yes	—
185	C₂×C₄⋊C₄	—	79	8	no	—
186	C₂×C₄⋊C₄	—	79	4	no	—
187	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	—	79	2	no	—
189	C₂×C₂²⋊C₄	—	79	8	no	—
190	C₂×C₄²	—	79	1	no	—
191	C₂⁴	—	79	2	no	—
192	C₂×Q₈	—	316	8	no	—
193	C₂×Q₈	—	79	4	no	—
194	C₂×Q₈	—	79	4	no	—
195	C₂×Q₈	—	79	4	no	—
196	C₂×Q₈	C₂²×D₇₉	1	4	yes	—
197	C₂×D₄	—	79	8	no	—
198	C₂×D₄	—	79	4	no	—
199	C₂²×C₄	—	316	4	no	—
200	C₂²×C₄	—	79	1	no	—
201	C₂²×C₄	—	79	8	no	—
202	C₂²×C₄	—	79	8	no	—
203	C₂²×C₄	—	79	4	no	—
204	C₂²×C₄	—	79	1	no	—
205	C₂²×C₄	—	79	4	no	—
206	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
207	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
208	C₄⋊C₄	—	79	4	no	—
209	C₄⋊C₄	—	79	4	no	—
210	C₄⋊C₄	—	79	8	no	—
211	C₄⋊C₄	—	79	4	no	—
212	C₄⋊C₄	—	79	4	no	—
213	C₄⋊C₄	—	79	4	no	—
214	C₄⋊C₄	—	79	4	no	—
215	C₄⋊C₄	—	79	2	no	—
216	C₄⋊C₄	—	79	2	no	—
217	C₄⋊C₄	—	79	4	no	—
218	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
219	C₄⋊C₄	C₂²×D₇₉	1	8	yes	—
220	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
221	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
222	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
223	C₂²⋊C₄	—	79	4	no	—
224	C₂²⋊C₄	—	79	8	no	—
225	C₂²⋊C₄	—	79	8	no	—
226	C₂²⋊C₄	—	79	8	no	—
227	C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
228	C₄²	—	79	2	no	—
229	C₄²	—	79	2	no	—
230	C₄²	—	79	4	no	—
231	C₄²	C₂²×D₇₉	1	4	yes	—
232	C₂³	—	316	2	no	—
233	C₂³	—	316	2	no	—
234	C₂³	—	79	8	no	—
235	C₂³	—	79	4	no	—
236	C₂³	C₂³×D₇₉	1	1	yes	characteristic
237	Q₈	—	4	4	no	—
238	Q₈	—	316	4	no	—
239	Q₈	—	316	4	no	—
240	Q₈	—	158	8	no	—
241	D₄	—	79	16	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	316	4	no	—
244	C₂×C₄	—	316	8	no	—
245	C₂×C₄	—	316	8	no	—
246	C₂×C₄	—	316	1	no	—
247	C₂×C₄	—	79	1	no	—
248	C₂×C₄	—	79	2	no	—
249	C₂×C₄	—	79	1	no	—
250	C₂×C₄	—	79	4	no	—
251	C₂×C₄	—	79	4	no	—
252	C₂×C₄	—	158	8	no	—
253	C₂×C₄	—	158	2	no	—
254	C₂×C₄	Q₈×D₇₉	1	4	yes	complemented
255	C₂×C₄	C₂³×D₇₉	1	4	yes	—
256	C₂×C₄	C₂³×D₇₉	1	4	yes	—
257	C₂×C₄	C₂³×D₇₉	1	4	yes	—
258	C₂×C₄	C₂³×D₇₉	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	316	8	no	—
262	C₂²	—	79	8	no	—
263	C₂²	—	158	4	no	—
264	C₂²	—	158	4	no	—
265	C₂²	C₂×Q₈×D₇₉	1	2	yes	—
266	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	2	4	no	—
270	C₄	—	79	2	no	—
271	C₄	—	158	1	no	—
272	C₄	—	158	4	no	—
273	C₄	—	158	4	no	—
274	C₄	—	158	2	no	—
275	C₄	C₂×Q₈×D₇₉	1	4	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	158	8	no	—
278	C₂	C₂²×Q₈×D₇₉	1	1	yes	characteristic
279	C₂	H(5,𝔽₂)×D₇₉	1	1	yes	characteristic
280	C₂	#19099400	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁵
Fitting	#544×C₇₉	C₂
prime radical	C₂²×D₃₁₆	C₂²
hypercenter	#544	D₇₉
hypocenter	C₇₉	#3501
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #544

Complements

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

N	H ≅ G/N	type
1. #19099274	C₂	semidirect
2. #19099269	C₂	semidirect
3. #19099269	C₂	semidirect
4. #19099228	C₂	semidirect
5. #19099194	C₂	semidirect
6. #19099188	C₂	semidirect
7. #19099116	C₂	semidirect
8. C₂²⋊Q₈×D₇₉	C₂	semidirect
11. #544×C₇₉	C₂	semidirect
12. C₂×Q₈×D₇₉	C₂²	semidirect
15. #14579160	C₂²	semidirect
16. #14579160	C₂²	semidirect
17. #14579160	C₂²	semidirect
18. #14579152	C₂²	semidirect
19. #14579152	C₂²	semidirect
20. #14579152	C₂²	semidirect
21. #14579141	C₂²	semidirect
22. #14579141	C₂²	semidirect
24. #14579139	C₂²	semidirect
25. #14579139	C₂²	semidirect
26. #14579139	C₂²	semidirect
29. #14579137	C₂²	semidirect
30. #14579137	C₂²	semidirect
31. C₄×Dic₁₅₈	C₂²	semidirect
32. C₄×Dic₁₅₈	C₂²	semidirect
40. C₄⋊C₄×D₇₉	C₂²	semidirect
41. C₄⋊C₄×D₇₉	C₂²	semidirect
45. C₄⋊Q₈×C₇₉	C₂²	semidirect
46. #125×C₇₉	C₂²	semidirect
55. C₂×D₃₁₆	Q₈	semidirect
94. C₂²×D₇₉	C₂×Q₈	semidirect
98. C₂×C₄×C₇₉	C₂×Q₈	semidirect
110. D₃₁₆	C₂×Q₈	semidirect
141. C₇₉	#3501	semidirect
147. #544	D₇₉	semidirect
159. C₄⋊Q₈	C₂×D₇₉	semidirect
163. #125	C₂×D₇₉	semidirect
254. C₂×C₄	Q₈×D₇₉	semidirect