Subgroups — #1277×D₇₉ #34177723

Order 20224.

830 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×C₄⋊C₈×D₇₉	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23761235	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761234	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₉	—	2	2	no	—
7	C₂²×C₈×D₇₉	C₄	1	2	yes	—
8	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×Dic₇₉⋊C₈	—	2	2	no	—
11	C₄⋊C₈×D₇₉	—	2	4	no	—
12	C₄⋊C₈×D₇₉	C₄	1	8	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₉	C₄	1	2	yes	—
15	C₄×C₈×D₇₉	—	2	4	no	—
16	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×D₇₉	Q₈	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₈×D₇₉	—	2	4	no	—
24	C₂×C₈×D₇₉	—	2	4	no	—
25	C₂×C₈×D₇₉	—	2	8	no	—
26	C₂×C₈×D₇₉	—	4	4	no	—
27	C₂×C₈×D₇₉	—	4	4	no	—
28	C₂×C₈×D₇₉	C₈	1	8	yes	complemented
29	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₉	Q₈	1	1	yes	characteristic
32	C₄²×D₇₉	C₂×C₄	1	4	yes	—
33	C₄²×D₇₉	D₄	1	4	yes	—
34	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
36	Dic₇₉⋊C₈	—	2	8	no	—
37	C₄×C₇₉⋊C₈	—	2	2	no	—
38	C₈×Dic₇₉	—	2	4	no	—
39	C₂²×C₈×C₇₉	—	2	2	no	—
40	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₉	—	2	2	no	—
43	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₇₉	—	2	2	no	—
45	C₂³×D₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×D₇₉	—	2	4	no	—
50	C₂×C₄×D₇₉	C₄²	1	2	yes	—
51	C₂×C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×D₇₉	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×D₇₉	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×D₇₉	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₇₉	M₁₆	1	4	yes	—
56	C₂×C₄×D₇₉	D₈	1	2	yes	—
57	C₂×C₄×D₇₉	SD₁₆	1	4	yes	—
58	C₂×C₄×D₇₉	Q₁₆	1	2	yes	—
59	C₂×C₇₉⋊C₈	—	2	2	no	—
60	C₂×C₇₉⋊C₈	—	2	4	no	—
61	C₂×C₇₉⋊C₈	—	2	2	no	—
62	C₂×C₇₉⋊C₈	—	4	2	no	—
63	C₂×C₇₉⋊C₈	—	4	2	no	—
64	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	complemented
65	C₈×D₇₉	—	2	16	no	—
66	C₈×D₇₉	—	4	16	no	—
67	C₄×Dic₇₉	C₄²	1	2	yes	—
68	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
69	C₄×Dic₇₉	C₂²⋊C₄	1	4	yes	—
70	C₄×Dic₇₉	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₇₉	—	2	2	no	—
75	C₂×C₈×C₇₉	—	2	4	no	—
76	C₂×C₈×C₇₉	—	2	2	no	—
77	C₂×C₈×C₇₉	—	4	2	no	—
78	C₂×C₈×C₇₉	—	4	2	no	—
79	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	complemented
80	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
81	C₄²×C₇₉	C₂×D₄	1	2	yes	—
82	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×D₇₉	#95	1	2	yes	—
84	C₂²×D₇₉	C₄⋊C₈	1	4	yes	—
85	C₂²×D₇₉	#106	1	2	yes	—
86	C₂²×D₇₉	#107	1	2	yes	—
87	C₂²×D₇₉	#108	1	4	yes	—
88	C₂×C₄×C₇₉	—	2	2	no	—
89	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×Dic₇₉	—	2	2	no	—
99	C₂×Dic₇₉	#95	1	2	yes	—
100	C₂×Dic₇₉	#95	1	2	yes	—
101	C₂×Dic₇₉	#95	1	2	yes	—
102	C₂×Dic₇₉	C₄⋊C₈	1	4	yes	—
103	C₂×Dic₇₉	#106	1	2	yes	—
104	C₂×Dic₇₉	#107	1	2	yes	—
105	C₄×D₇₉	—	2	8	no	—
106	C₄×D₇₉	C₄×C₈	1	4	yes	—
107	C₄×D₇₉	#97	1	4	yes	—
108	C₄×D₇₉	C₂²⋊C₈	1	8	yes	—
109	C₄×D₇₉	#102	1	8	yes	—
110	C₄×D₇₉	#103	1	8	yes	—
111	C₈×C₇₉	—	2	4	no	—
112	C₈×C₇₉	—	4	4	no	—
113	C₇₉⋊C₈	—	2	4	no	—
114	C₇₉⋊C₈	—	4	4	no	—
115	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
116	C₂²×C₇₉	C₂×C₄⋊C₈	1	2	yes	—
117	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
118	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
119	C₂²×C₇₉	C₂×#108	1	2	yes	—
120	C₂×D₇₉	#334	1	8	yes	—
121	C₂×D₇₉	#335	1	8	yes	—
122	C₂×D₇₉	#336	1	4	yes	—
123	C₂×D₇₉	#340	1	4	yes	—
124	C₂×D₇₉	#341	1	4	yes	—
125	C₄×C₇₉	—	2	2	no	—
126	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
127	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
128	C₄×C₇₉	C₂×C₂²⋊C₈	1	2	yes	—
129	C₄×C₇₉	C₂×#102	1	2	yes	—
130	C₄×C₇₉	C₂×#103	1	2	yes	—
131	Dic₇₉	—	2	2	no	—
132	Dic₇₉	#336	1	4	yes	—
133	Dic₇₉	#340	1	4	yes	—
134	C₂×#1277	—	79	1	no	maximal, Sylow
135	C₂×C₇₉	C₂×#334	1	2	yes	—
136	C₂×C₇₉	C₂×#335	1	2	yes	—
137	C₂×C₇₉	C₂×#336	1	1	yes	characteristic
138	C₂×C₇₉	C₂×#340	1	1	yes	characteristic
139	C₂×C₇₉	C₂×#341	1	1	yes	characteristic
140	D₇₉	#1277	1	8	yes	direct product factor
141	C₂²×C₄⋊C₈	—	79	2	no	—
142	C₂²×C₄×C₈	—	79	1	no	—
143	#1277	—	79	2	no	—
144	#1277	—	79	1	no	—
145	#1277	D₇₉	1	1	yes	characteristic, direct product factor
146	C₇₉	C₂×#1277	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₈	—	79	2	no	—
148	C₂³×C₈	—	158	2	no	—
149	C₂²×C₄²	—	79	1	no	—
150	C₂×C₄⋊C₈	—	79	2	no	—
151	C₂×C₄⋊C₈	—	79	8	no	—
152	C₂×C₄⋊C₈	—	158	4	no	—
153	C₂×C₄⋊C₈	—	158	2	no	—
154	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
155	C₂×C₄×C₈	—	79	2	no	—
156	C₂×C₄×C₈	—	79	1	no	—
157	C₂×C₄×C₈	—	158	4	no	—
158	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
159	C₂³×C₄	—	79	1	no	—
160	C₂³×C₄	—	79	1	no	—
161	C₂³×C₄	—	79	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	316	4	no	—
164	C₂²×C₈	—	316	4	no	—
165	C₂²×C₈	—	79	2	no	—
166	C₂²×C₈	—	79	8	no	—
167	C₂²×C₈	—	158	8	no	—
168	C₂²×C₈	—	158	4	no	—
169	C₂²×C₈	—	158	4	no	—
170	C₂²×C₈	—	158	2	no	—
171	C₂²×C₈	C₄×D₇₉	1	2	yes	—
172	C₂×C₄²	—	79	1	no	—
173	C₂×C₄²	—	79	1	no	—
174	C₂×C₄²	—	79	1	no	—
175	C₂×C₄²	—	79	4	no	—
176	C₂×C₄²	—	79	4	no	—
177	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	79	4	no	—
180	C₄⋊C₈	—	158	8	no	—
181	C₄⋊C₈	—	158	2	no	—
182	C₄⋊C₈	C₄×D₇₉	1	4	yes	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	158	4	no	—
185	C₄×C₈	—	158	2	no	—
186	C₂⁴	—	79	1	no	—
187	C₂²×C₄	—	79	1	no	—
188	C₂²×C₄	—	79	1	no	—
189	C₂²×C₄	—	79	2	no	—
190	C₂²×C₄	—	79	2	no	—
191	C₂²×C₄	—	79	4	no	—
192	C₂²×C₄	—	79	4	no	—
193	C₂²×C₄	—	79	4	no	—
194	C₂²×C₄	—	79	2	no	—
195	C₂²×C₄	—	79	4	no	—
196	C₂²×C₄	—	79	2	no	—
197	C₂²×C₄	—	79	1	no	—
198	C₂²×C₄	—	79	4	no	—
199	C₂²×C₄	—	158	4	no	—
200	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
201	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
202	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	316	16	no	—
209	C₂×C₈	—	316	2	no	—
210	C₂×C₈	—	316	2	no	—
211	C₂×C₈	—	79	4	no	—
212	C₂×C₈	—	158	16	no	—
213	C₂×C₈	—	158	4	no	—
214	C₂×C₈	—	158	2	no	—
215	C₂×C₈	—	158	2	no	—
216	C₂×C₈	C₈×D₇₉	1	4	yes	complemented
217	C₄²	—	79	2	no	—
218	C₄²	—	79	2	no	—
219	C₄²	—	79	4	no	—
220	C₄²	—	79	4	no	—
221	C₄²	C₂×C₄×D₇₉	1	2	yes	—
222	C₄²	D₄×D₇₉	1	2	yes	—
223	C₂³	—	79	2	no	—
224	C₂³	—	79	4	no	—
225	C₂³	—	79	2	no	—
226	C₂³	—	79	2	no	—
227	C₂³	—	79	4	no	—
228	C₂³	C₄⋊C₄×D₇₉	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	79	2	no	—
231	C₂×C₄	—	79	2	no	—
232	C₂×C₄	—	79	4	no	—
233	C₂×C₄	—	79	2	no	—
234	C₂×C₄	—	79	2	no	—
235	C₂×C₄	—	79	4	no	—
236	C₂×C₄	—	79	4	no	—
237	C₂×C₄	—	79	8	no	—
238	C₂×C₄	—	79	8	no	—
239	C₂×C₄	—	79	8	no	—
240	C₂×C₄	—	79	2	no	—
241	C₂×C₄	—	158	8	no	—
242	C₂×C₄	—	158	2	no	—
243	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊C₄×D₇₉	1	2	yes	—
246	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
248	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
249	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
250	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
251	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	4	4	no	—
254	C₈	—	316	4	no	—
255	C₈	—	158	4	no	—
256	C₂²	—	79	4	no	—
257	C₂²	—	79	4	no	—
258	C₂²	—	79	4	no	—
259	C₂²	—	79	8	no	—
260	C₂²	—	79	8	no	—
261	C₂²	#95×D₇₉	1	1	yes	characteristic
262	C₂²	C₄⋊C₈×D₇₉	1	2	yes	—
263	C₂²	#106×D₇₉	1	1	yes	characteristic
264	C₂²	#107×D₇₉	1	1	yes	characteristic
265	C₂²	#108×D₇₉	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	79	4	no	—
268	C₄	—	79	4	no	—
269	C₄	—	158	2	no	—
270	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
271	C₄	#97×D₇₉	1	1	yes	characteristic
272	C₄	C₂²⋊C₈×D₇₉	1	2	yes	—
273	C₄	#102×D₇₉	1	2	yes	—
274	C₄	#103×D₇₉	1	2	yes	—
275	C₂	—	79	8	no	—
276	C₂	#334×D₇₉	1	2	yes	—
277	C₂	#335×D₇₉	1	2	yes	—
278	C₂	#336×D₇₉	1	1	yes	characteristic
279	C₂	#340×D₇₉	1	1	yes	characteristic
280	C₂	#341×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1277	C₂
socle	C₂³×C₇₉	C₂×C₄⋊C₄
prime radical	C₂³×D₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1277	D₇₉
hypocenter	C₇₉	C₂×#1277
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
3. #23761235	C₂	semidirect
4. #23761234	C₂	semidirect
5. C₇₉×#1277	C₂	semidirect
28. C₂×C₈×D₇₉	C₈	semidirect
64. C₂×C₇₉⋊C₈	C₂×C₈	semidirect
79. C₂×C₈×C₇₉	C₂×C₈	semidirect
140. D₇₉	#1277	direct (N × H)
145. #1277	D₇₉	direct (N × H)
146. C₇₉	C₂×#1277	semidirect
216. C₂×C₈	C₈×D₇₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊C₈×D₇₉	2
C₂×C₄×C₈×D₇₉	2
#23761235	2
#23761234	2
C₇₉×#1277	2
C₂×#1277	79

Subgroups — #1277×D79 #34177723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1277×D₇₉ #34177723