Subgroups — C₇₉×#13126 #34124815

Order 20224.

300 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₉×#2117	C₂	1	1	yes	maximal, characteristic
3	C₂×#350×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1757	C₂	1	1	yes	maximal, characteristic
6	C₇₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#413×C₇₉	C₄	1	1	yes	characteristic
14	#410×C₇₉	C₄	1	1	yes	characteristic
15	#405×C₇₉	C₂²	1	1	yes	characteristic
16	#405×C₇₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₇₉	—	2	2	no	—
18	#350×C₇₉	—	2	2	no	—
19	#350×C₇₉	—	2	2	no	—
20	#347×C₇₉	—	2	2	no	—
21	#344×C₇₉	C₄	1	1	yes	characteristic
22	#344×C₇₉	C₄	1	1	yes	characteristic
23	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
24	#131×C₇₉	—	2	1	no	—
25	#131×C₇₉	—	2	1	no	—
26	#131×C₇₉	—	2	1	no	—
27	#131×C₇₉	—	2	1	no	—
28	C₂×M₁₆×C₇₉	—	2	1	no	—
29	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	—	2	1	no	—
33	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
35	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₇₉	—	2	1	no	—
37	M₃₂×C₇₉	—	2	1	no	—
38	M₃₂×C₇₉	—	2	1	no	—
39	M₃₂×C₇₉	—	2	1	no	—
40	M₃₂×C₇₉	—	2	1	no	—
41	M₃₂×C₇₉	D₄	1	2	yes	—
42	C₂×C₁₆×C₇₉	—	2	1	no	—
43	C₂×C₁₆×C₇₉	—	2	1	no	—
44	C₂×C₁₆×C₇₉	—	2	1	no	—
45	C₂×C₁₆×C₇₉	—	2	1	no	—
46	C₂×C₁₆×C₇₉	—	2	1	no	—
47	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
48	#101×C₇₉	—	2	1	no	—
49	#100×C₇₉	—	2	1	no	—
50	#99×C₇₉	—	2	2	no	—
51	#97×C₇₉	—	2	1	no	—
52	#97×C₇₉	—	2	1	no	—
53	C₄×C₈×C₇₉	—	2	1	no	—
54	C₄×C₈×C₇₉	—	2	1	no	—
55	#41×C₇₉	—	2	1	no	—
56	#41×C₇₉	—	2	1	no	—
57	#41×C₇₉	—	2	1	no	—
58	#41×C₇₉	—	2	1	no	—
59	C₂×Q₈×C₇₉	M₁₆	1	1	yes	characteristic
60	C₂×D₄×C₇₉	—	2	1	no	—
61	C₂×D₄×C₇₉	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	—	2	1	no	—
63	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
64	M₁₆×C₇₉	—	2	1	no	—
65	M₁₆×C₇₉	—	2	1	no	—
66	M₁₆×C₇₉	—	2	1	no	—
67	M₁₆×C₇₉	—	2	1	no	—
68	M₁₆×C₇₉	—	2	1	no	—
69	M₁₆×C₇₉	—	2	1	no	—
70	M₁₆×C₇₉	—	4	1	no	—
71	M₁₆×C₇₉	—	4	1	no	—
72	C₂×C₈×C₇₉	—	2	1	no	—
73	C₂×C₈×C₇₉	—	2	1	no	—
74	C₂×C₈×C₇₉	—	2	1	no	—
75	C₂×C₈×C₇₉	—	4	1	no	—
76	C₂×C₈×C₇₉	—	4	1	no	—
77	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₇₉	—	2	1	no	—
84	C₄⋊C₄×C₇₉	—	2	1	no	—
85	C₂²⋊C₄×C₇₉	—	2	1	no	—
86	C₂²⋊C₄×C₇₉	M₁₆	1	2	yes	—
87	C₄²×C₇₉	—	2	1	no	—
88	C₄²×C₇₉	—	2	1	no	—
89	C₁₆×C₇₉	—	2	2	no	—
90	C₁₆×C₇₉	—	2	2	no	—
91	C₁₆×C₇₉	—	4	1	no	—
92	C₁₆×C₇₉	—	4	1	no	—
93	C₂³×C₇₉	—	2	1	no	—
94	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
95	Q₈×C₇₉	—	2	1	no	—
96	Q₈×C₇₉	—	2	1	no	—
97	D₄×C₇₉	—	2	1	no	—
98	D₄×C₇₉	—	2	1	no	—
99	D₄×C₇₉	—	4	1	no	—
100	D₄×C₇₉	—	4	1	no	—
101	C₂×C₄×C₇₉	—	2	1	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	4	1	no	—
104	C₂×C₄×C₇₉	—	4	1	no	—
105	C₂×C₄×C₇₉	—	4	1	no	—
106	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
112	C₈×C₇₉	—	4	1	no	—
113	C₈×C₇₉	—	4	1	no	—
114	C₈×C₇₉	—	4	1	no	—
115	C₈×C₇₉	#121	1	1	yes	characteristic
116	C₈×C₇₉	#123	1	1	yes	characteristic
117	C₈×C₇₉	#124	1	1	yes	characteristic
118	C₈×C₇₉	#126	1	1	yes	characteristic
119	C₂²×C₇₉	—	4	1	no	—
120	C₂²×C₇₉	—	4	1	no	—
121	C₂²×C₇₉	—	8	1	no	—
122	C₂²×C₇₉	#432	1	1	yes	characteristic
123	C₂²×C₇₉	#435	1	1	yes	characteristic
124	C₂²×C₇₉	#436	1	1	yes	characteristic
125	C₄×C₇₉	—	4	2	no	—
126	C₄×C₇₉	—	4	1	no	—
127	C₄×C₇₉	—	8	1	no	—
128	C₄×C₇₉	#388	1	1	yes	characteristic
129	C₄×C₇₉	#433	1	1	yes	characteristic
130	C₄×C₇₉	#435	1	1	yes	characteristic
131	C₄×C₇₉	#435	1	1	yes	characteristic
132	#13126	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₇₉	—	2	1	no	—
134	C₂×C₇₉	—	2	1	no	—
135	C₂×C₇₉	—	2	1	no	—
136	C₂×C₇₉	—	8	1	no	—
137	C₂×C₇₉	#1924	1	1	yes	characteristic
138	#2151	C₂×C₇₉	1	1	yes	characteristic, complemented
139	#2117	C₂×C₇₉	1	1	yes	characteristic
140	C₂×#350	C₂×C₇₉	1	1	yes	characteristic
141	#2110	C₂×C₇₉	1	1	yes	characteristic, complemented
142	#1757	C₂×C₇₉	1	1	yes	characteristic
143	#1379	C₂×C₇₉	1	1	yes	characteristic, complemented
144	#1378	C₂×C₇₉	1	1	yes	characteristic, complemented
145	C₇₉	#13126	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#131	C₂²×C₇₉	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₇₉	1	1	yes	characteristic
148	C₂×M₃₂	C₂²×C₇₉	1	1	yes	characteristic
149	C₂×M₃₂	C₂²×C₇₉	1	1	yes	characteristic
150	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
151	#413	C₄×C₇₉	1	1	yes	characteristic
152	#410	C₄×C₇₉	1	1	yes	characteristic
153	#405	C₂²×C₇₉	1	1	yes	characteristic
154	#405	C₂²×C₇₉	1	1	yes	characteristic
155	C₄⋊C₁₆	—	2	2	no	—
156	#350	—	2	2	no	—
157	#350	—	2	2	no	—
158	#347	—	2	2	no	—
159	#344	C₄×C₇₉	1	1	yes	characteristic
160	#344	C₄×C₇₉	1	1	yes	characteristic
161	C₂×#41	C₂×C₄×C₇₉	1	1	yes	characteristic
162	#131	—	2	1	no	—
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	#131	—	2	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
169	C₂×M₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
172	#117	C₂×C₄×C₇₉	1	1	yes	characteristic
173	#117	C₂×C₄×C₇₉	1	1	yes	characteristic
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	D₄×C₇₉	1	2	yes	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	D₄×C₇₉	1	2	yes	—
186	#101	—	2	1	no	—
187	#100	—	2	1	no	—
188	#99	—	2	2	no	—
189	#97	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	#41	—	2	1	no	—
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	—	2	1	no	—
197	C₂×Q₈	M₁₆×C₇₉	1	1	yes	characteristic
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	M₁₆×C₇₉	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
202	M₁₆	—	2	1	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	2	1	no	—
207	M₁₆	—	2	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
217	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
218	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
219	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
220	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	M₁₆×C₇₉	1	2	yes	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₁₆	—	2	2	no	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	4	1	no	—
230	C₁₆	—	4	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	C₂×M₁₆×C₇₉	1	1	yes	characteristic
233	Q₈	—	2	1	no	—
234	Q₈	—	2	1	no	—
235	D₄	—	2	1	no	—
236	D₄	—	2	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	#117×C₇₉	1	1	yes	characteristic
245	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
247	C₂×C₄	C₂×M₁₆×C₇₉	1	1	yes	characteristic
248	C₂×C₄	#131×C₇₉	1	1	yes	characteristic
249	C₂×C₄	#131×C₇₉	1	1	yes	characteristic
250	C₈	—	4	1	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₈	#121×C₇₉	1	1	yes	characteristic
254	C₈	#123×C₇₉	1	1	yes	characteristic
255	C₈	#124×C₇₉	1	1	yes	characteristic
256	C₈	#126×C₇₉	1	1	yes	characteristic
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	1	no	—
260	C₂²	#432×C₇₉	1	1	yes	characteristic
261	C₂²	#435×C₇₉	1	1	yes	characteristic
262	C₂²	#436×C₇₉	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	1	no	—
266	C₄	#388×C₇₉	1	1	yes	characteristic
267	C₄	#433×C₇₉	1	1	yes	characteristic
268	C₄	#435×C₇₉	1	1	yes	characteristic
269	C₄	#435×C₇₉	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	8	1	no	—
274	C₂	C₇₉×#1924	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₇₉	#124
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
socle	C₂×C₇₉	#1924
prime radical	C₂×#41×C₇₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₇₉ ◁ C₂²×C₈×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₉×#2151	C₂	semidirect
4. C₇₉×#2110	C₂	semidirect
6. C₇₉×#1379	C₂	semidirect
7. C₇₉×#1378	C₂	semidirect
132. #13126	C₇₉	direct (N × H)
138. #2151	C₂×C₇₉	semidirect
141. #2110	C₂×C₇₉	semidirect
143. #1379	C₂×C₇₉	semidirect
144. #1378	C₂×C₇₉	semidirect
145. C₇₉	#13126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₉×#2151	2
C₇₉×#2117	2
C₂×#350×C₇₉	2
C₇₉×#2110	2
C₇₉×#1757	2
C₇₉×#1379	2
C₇₉×#1378	2
#13126	79

Subgroups — C79×#13126 #34124815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#13126 #34124815