Subgroups — #544×D₇₉ #23776796

Order 10112.

1080 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099194	C₂	1	1	yes	maximal, characteristic, complemented
2	#19099193	C₂	1	4	yes	maximal, complemented
3	#19099192	C₂	1	4	yes	maximal, complemented
4	#19099191	C₂	1	2	yes	maximal, complemented
5	#19099190	C₂	1	4	yes	maximal, complemented
6	C₄⋊Q₈×D₇₉	C₂	1	4	yes	maximal, complemented
7	#125×D₇₉	C₂	1	4	yes	maximal, complemented
8	C₂²⋊Q₈×D₇₉	C₂	1	4	yes	maximal
9	#117×D₇₉	C₂	1	2	yes	maximal
10	C₂×C₄⋊C₄×D₇₉	C₂	1	1	yes	maximal, characteristic
11	#544×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₇₉	C₂²	1	4	yes	—
13	C₂²×C₄×D₇₉	C₂²	1	2	yes	—
14	C₂²×C₄×D₇₉	C₂²	1	1	yes	characteristic
15	#14579161	C₂²	1	4	yes	complemented
16	#14579160	C₂²	1	4	yes	complemented
17	#14579160	C₂²	1	8	yes	complemented
18	#14579160	C₂²	1	4	yes	complemented
19	#14579159	C₂²	1	8	yes	complemented
20	#14579153	C₂²	1	4	yes	complemented
21	#14579152	C₂²	1	4	yes	complemented
22	#14579152	C₂²	1	4	yes	complemented
23	#14579151	C₂²	1	8	yes	complemented
24	#14579151	C₂²	1	8	yes	complemented
25	#14579141	C₂²	1	4	yes	—
26	#14579140	C₂²	1	4	yes	—
27	#14579139	C₂²	1	8	yes	—
28	#14579138	C₂²	1	8	yes	—
29	#14579137	C₂²	1	4	yes	—
30	#14579120	C₂²	1	2	yes	—
31	#14579119	C₂²	1	2	yes	—
32	#14579119	C₂²	1	2	yes	—
33	#14579118	C₂²	1	4	yes	—
34	#14579117	C₂²	1	4	yes	—
35	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
36	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
37	C₄⋊C₄×D₇₉	C₂²	1	4	yes	—
38	C₄⋊C₄×D₇₉	C₂²	1	4	yes	—
39	C₄⋊C₄×D₇₉	C₂²	1	2	yes	—
40	C₄⋊C₄×D₇₉	C₂²	1	2	yes	—
41	C₄⋊C₄×D₇₉	C₂²	1	4	yes	—
42	C₄⋊C₄×D₇₉	C₂²	1	8	yes	—
43	C₂²⋊C₄×D₇₉	C₂²	1	4	yes	—
44	C₄²×D₇₉	C₂²	1	4	yes	—
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	1	yes	characteristic
51	C₂×Dic₁₅₈	C₂³	1	8	yes	—
52	C₂×Dic₁₅₈	C₂³	1	4	yes	—
53	Q₈×D₇₉	—	4	8	no	—
54	C₂²×Dic₇₉	C₂³	1	2	yes	—
55	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₇₉	—	4	4	no	—
57	C₂×C₄×D₇₉	Q₈	1	8	yes	complemented
58	C₂×C₄×D₇₉	C₂³	1	4	yes	—
59	C₂×C₄×D₇₉	C₂³	1	4	yes	—
60	C₂×C₄×D₇₉	C₂³	1	4	yes	—
61	C₂×C₄×D₇₉	C₂³	1	4	yes	—
62	C₂×C₄×D₇₉	C₂³	1	2	yes	—
63	C₂×C₄×D₇₉	C₂³	1	2	yes	—
64	C₄⋊Dic₇₉	C₂³	1	4	yes	—
65	C₄⋊Dic₇₉	C₂³	1	2	yes	—
66	C₄⋊Dic₇₉	C₂³	1	2	yes	—
67	C₄⋊Dic₇₉	C₂³	1	4	yes	—
68	C₄⋊Dic₇₉	C₂³	1	8	yes	—
69	Dic₇₉⋊C₄	C₂³	1	8	yes	—
70	Dic₇₉⋊C₄	C₂³	1	4	yes	—
71	Dic₇₉⋊C₄	C₂³	1	4	yes	—
72	Dic₇₉⋊C₄	C₂³	1	8	yes	—
73	Dic₇₉⋊C₄	C₂³	1	8	yes	—
74	Dic₇₉⋊C₄	C₂³	1	8	yes	—
75	C₂²⋊Dic₇₉	C₂³	1	4	yes	—
76	(C₂×D₇₉)⋊C₄	C₂³	1	4	yes	—
77	(C₂×D₇₉)⋊C₄	C₂³	1	4	yes	—
78	C₄×Dic₇₉	C₂³	1	4	yes	—
79	C₄×Dic₇₉	C₂³	1	4	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	—	2	2	no	—
93	C₂²×D₇₉	—	2	2	no	—
94	C₂²×D₇₉	C₂×Q₈	1	4	yes	—
95	C₂²×D₇₉	C₂⁴	1	1	yes	characteristic
96	C₂²×D₇₉	C₂⁴	1	1	yes	characteristic
97	Q₈×C₇₉	—	4	4	no	—
98	C₂×C₄×C₇₉	—	4	2	no	—
99	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	complemented
100	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
101	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
102	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
103	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
104	C₂×Dic₇₉	—	4	2	no	—
105	C₂×Dic₇₉	C₂×Q₈	1	4	yes	complemented
106	C₂×Dic₇₉	C₂⁴	1	4	yes	—
107	C₂×Dic₇₉	C₂⁴	1	4	yes	—
108	C₂×Dic₇₉	C₂⁴	1	4	yes	—
109	C₂×Dic₇₉	C₂⁴	1	2	yes	—
110	C₄×D₇₉	—	2	8	no	—
111	C₄×D₇₉	—	2	8	no	—
112	C₄×D₇₉	—	2	8	no	—
113	C₄×D₇₉	—	4	4	no	—
114	C₄×D₇₉	C₂×Q₈	1	8	yes	—
115	C₄×D₇₉	C₂×Q₈	1	8	yes	complemented
116	Dic₁₅₈	—	4	4	no	—
117	Dic₁₅₈	—	4	8	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	1	no	—
123	C₂×D₇₉	—	2	1	no	—
124	C₂×D₇₉	—	2	8	no	—
125	C₂×D₇₉	C₂²×Q₈	1	2	yes	—
126	C₂×D₇₉	C₂²×Q₈	1	2	yes	—
127	C₂×D₇₉	H(5,𝔽₂)	1	2	yes	—
128	C₂×D₇₉	2₋¹⁺⁴	1	2	yes	—
129	C₄×C₇₉	—	2	4	no	—
130	C₄×C₇₉	—	2	4	no	—
131	C₄×C₇₉	—	2	4	no	—
132	C₄×C₇₉	C₂²×Q₈	1	4	yes	—
133	Dic₇₉	—	2	4	no	—
134	Dic₇₉	—	2	4	no	—
135	Dic₇₉	—	2	4	no	—
136	Dic₇₉	C₂²×Q₈	1	4	yes	—
137	C₂×C₇₉	—	2	2	no	—
138	C₂×C₇₉	C₂³×Q₈	1	1	yes	characteristic
139	C₂×C₇₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₇₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
141	D₇₉	—	2	2	no	—
142	D₇₉	#544	1	4	yes	direct product factor
143	C₂×#544	—	79	1	no	maximal, Sylow
144	C₇₉	C₂×#544	1	1	yes	characteristic, complemented, Sylow
145	#544	—	79	4	no	—
146	#544	—	79	4	no	—
147	#544	—	79	4	no	—
148	#544	—	79	2	no	—
149	#544	—	79	1	no	—
150	#544	D₇₉	1	1	yes	characteristic, direct product factor
151	C₂×C₄⋊Q₈	—	79	4	no	—
152	C₂×#125	—	79	4	no	—
153	C₂×C₂²⋊Q₈	—	79	4	no	—
154	C₂×#117	—	79	2	no	—
155	C₂²×C₄⋊C₄	—	79	1	no	—
156	C₂²×Q₈	—	79	4	no	—
157	C₂³×C₄	—	79	2	no	—
158	C₂³×C₄	—	79	1	no	—
159	C₄⋊Q₈	—	79	8	no	—
160	C₄⋊Q₈	—	79	8	no	—
161	C₄⋊Q₈	—	79	4	no	—
162	C₄⋊Q₈	—	79	4	no	—
163	C₄⋊Q₈	—	79	4	no	—
164	C₄⋊Q₈	C₂×D₇₉	1	4	yes	complemented
165	#125	—	79	8	no	—
166	#125	—	79	8	no	—
167	#125	—	79	4	no	—
168	#125	—	79	4	no	—
169	#125	—	79	4	no	—
170	#125	C₂×D₇₉	1	4	yes	complemented
171	C₂²⋊Q₈	—	79	4	no	—
172	C₂²⋊Q₈	—	79	4	no	—
173	C₂²⋊Q₈	—	79	8	no	—
174	C₂²⋊Q₈	—	79	4	no	—
175	C₂²⋊Q₈	—	79	8	no	—
176	C₂²⋊Q₈	C₂×D₇₉	1	4	yes	—
177	#117	—	79	2	no	—
178	#117	—	79	2	no	—
179	#117	—	79	2	no	—
180	#117	—	79	4	no	—
181	#117	—	79	4	no	—
182	#117	C₂×D₇₉	1	2	yes	—
183	C₂×C₄⋊C₄	—	79	2	no	—
184	C₂×C₄⋊C₄	—	79	1	no	—
185	C₂×C₄⋊C₄	—	79	4	no	—
186	C₂×C₄⋊C₄	—	79	4	no	—
187	C₂×C₄⋊C₄	—	79	2	no	—
188	C₂×C₄⋊C₄	—	79	2	no	—
189	C₂×C₄⋊C₄	—	79	4	no	—
190	C₂×C₄⋊C₄	—	79	8	no	—
191	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	79	4	no	—
193	C₂×C₄²	—	79	4	no	—
194	C₂⁴	—	79	1	no	—
195	C₂×Q₈	—	316	8	no	—
196	C₂×Q₈	—	79	4	no	—
197	C₂×Q₈	—	79	8	no	—
198	C₂×Q₈	C₂²×D₇₉	1	4	yes	—
199	C₂²×C₄	—	316	4	no	—
200	C₂²×C₄	—	79	2	no	—
201	C₂²×C₄	—	79	4	no	—
202	C₂²×C₄	—	79	1	no	—
203	C₂²×C₄	—	79	8	no	—
204	C₂²×C₄	—	79	2	no	—
205	C₂²×C₄	—	79	2	no	—
206	C₂²×C₄	—	79	4	no	—
207	C₂²×C₄	—	79	4	no	—
208	C₂²×C₄	—	79	4	no	—
209	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
210	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
211	C₄⋊C₄	—	79	8	no	—
212	C₄⋊C₄	—	79	4	no	—
213	C₄⋊C₄	—	79	4	no	—
214	C₄⋊C₄	—	79	8	no	—
215	C₄⋊C₄	—	79	8	no	—
216	C₄⋊C₄	—	79	8	no	—
217	C₄⋊C₄	—	79	4	no	—
218	C₄⋊C₄	—	79	2	no	—
219	C₄⋊C₄	—	79	2	no	—
220	C₄⋊C₄	—	79	4	no	—
221	C₄⋊C₄	—	79	8	no	—
222	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
223	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
224	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
225	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
226	C₄⋊C₄	C₂²×D₇₉	1	8	yes	—
227	C₂²⋊C₄	—	79	4	no	—
228	C₂²⋊C₄	—	79	4	no	—
229	C₂²⋊C₄	—	79	4	no	—
230	C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
231	C₄²	—	79	4	no	—
232	C₄²	—	79	4	no	—
233	C₄²	—	79	4	no	—
234	C₄²	C₂²×D₇₉	1	4	yes	—
235	C₂³	—	79	1	no	—
236	C₂³	—	79	4	no	—
237	C₂³	—	79	1	no	—
238	C₂³	—	158	2	no	—
239	C₂³	—	158	2	no	—
240	C₂³	C₂³×D₇₉	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	316	4	no	—
243	Q₈	—	316	8	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	316	4	no	—
246	C₂×C₄	—	316	2	no	—
247	C₂×C₄	—	79	4	no	—
248	C₂×C₄	—	79	2	no	—
249	C₂×C₄	—	79	8	no	—
250	C₂×C₄	—	79	8	no	—
251	C₂×C₄	—	79	4	no	—
252	C₂×C₄	—	79	4	no	—
253	C₂×C₄	—	79	4	no	—
254	C₂×C₄	—	158	8	no	—
255	C₂×C₄	—	158	8	no	—
256	C₂×C₄	—	158	8	no	—
257	C₂×C₄	Q₈×D₇₉	1	4	yes	complemented
258	C₂×C₄	C₂³×D₇₉	1	4	yes	—
259	C₂×C₄	C₂³×D₇₉	1	4	yes	—
260	C₂×C₄	C₂³×D₇₉	1	4	yes	—
261	C₂×C₄	C₂³×D₇₉	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	79	2	no	—
265	C₂²	—	79	2	no	—
266	C₂²	—	79	2	no	—
267	C₂²	—	79	2	no	—
268	C₂²	—	158	8	no	—
269	C₂²	—	158	1	no	—
270	C₂²	—	158	1	no	—
271	C₂²	C₂×Q₈×D₇₉	1	2	yes	—
272	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	79	4	no	—
277	C₄	—	158	4	no	—
278	C₄	—	158	4	no	—
279	C₄	—	158	4	no	—
280	C₄	C₂×Q₈×D₇₉	1	4	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	79	4	no	—
283	C₂	—	158	2	no	—
284	C₂	C₂²×Q₈×D₇₉	1	1	yes	characteristic
285	C₂	H(5,𝔽₂)×D₇₉	1	1	yes	characteristic
286	C₂	2₋¹⁺⁴×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁵
Fitting	#544×C₇₉	C₂
prime radical	C₂³×D₇₉	C₂³
hypercenter	#544	D₇₉
hypocenter	C₇₉	C₂×#544
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. #19099194	C₂	semidirect
2. #19099193	C₂	semidirect
3. #19099192	C₂	semidirect
4. #19099191	C₂	semidirect
5. #19099190	C₂	semidirect
6. C₄⋊Q₈×D₇₉	C₂	semidirect
7. #125×D₇₉	C₂	semidirect
11. #544×C₇₉	C₂	semidirect
15. #14579161	C₂²	semidirect
16. #14579160	C₂²	semidirect
17. #14579160	C₂²	semidirect
18. #14579160	C₂²	semidirect
19. #14579159	C₂²	semidirect
20. #14579153	C₂²	semidirect
21. #14579152	C₂²	semidirect
22. #14579152	C₂²	semidirect
23. #14579151	C₂²	semidirect
24. #14579151	C₂²	semidirect
45. C₄⋊Q₈×C₇₉	C₂²	semidirect
46. #125×C₇₉	C₂²	semidirect
57. C₂×C₄×D₇₉	Q₈	semidirect
99. C₂×C₄×C₇₉	C₂×Q₈	semidirect
105. C₂×Dic₇₉	C₂×Q₈	semidirect
115. C₄×D₇₉	C₂×Q₈	semidirect
142. D₇₉	#544	direct (N × H)
144. C₇₉	C₂×#544	semidirect
150. #544	D₇₉	direct (N × H)
164. C₄⋊Q₈	C₂×D₇₉	semidirect
170. #125	C₂×D₇₉	semidirect
257. C₂×C₄	Q₈×D₇₉	semidirect

Subgroups — #544×D79 #23776796

Conjugacy classes of subgroups

Complements

Subgroups — #544×D₇₉ #23776796