Subgroups — C₇₉×#7726 #34119415

Order 20224.

400 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉×#2860	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#1271	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1270	C₂	1	2	yes	maximal
6	C₂×#126×C₇₉	—	2	1	no	—
7	#390×C₇₉	C₄	1	1	yes	characteristic
8	#390×C₇₉	C₄	1	1	yes	characteristic
9	#390×C₇₉	C₂²	1	1	yes	characteristic
10	#389×C₇₉	C₄	1	1	yes	characteristic
11	#389×C₇₉	C₄	1	1	yes	characteristic
12	#389×C₇₉	C₂²	1	1	yes	characteristic
13	#388×C₇₉	—	2	1	no	—
14	#388×C₇₉	—	2	1	no	—
15	#388×C₇₉	—	2	1	no	—
16	C₄³×C₇₉	C₂²	1	1	yes	characteristic
17	#336×C₇₉	—	2	1	no	—
18	#336×C₇₉	—	2	1	no	—
19	#336×C₇₉	—	2	1	no	—
20	#336×C₇₉	—	2	1	no	—
21	#336×C₇₉	C₂²	1	2	yes	—
22	#336×C₇₉	C₂²	1	2	yes	—
23	C₂²×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	—	2	2	no	—
25	C₂²×C₈×C₇₉	—	2	2	no	—
26	#126×C₇₉	—	4	4	no	—
27	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
46	#95×C₇₉	—	2	1	no	—
47	#95×C₇₉	—	2	1	no	—
48	#95×C₇₉	—	2	1	no	—
49	C₂⁴×C₇₉	—	2	1	no	—
50	C₂×D₄×C₇₉	—	2	2	no	—
51	C₂×D₄×C₇₉	—	4	1	no	—
52	C₂×D₄×C₇₉	—	4	1	no	—
53	C₂²×C₄×C₇₉	—	2	1	no	—
54	C₂²×C₄×C₇₉	—	2	1	no	—
55	C₂²×C₄×C₇₉	—	2	1	no	—
56	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	—	4	2	no	—
64	C₂×C₈×C₇₉	—	4	2	no	—
65	C₂×C₈×C₇₉	—	4	2	no	—
66	C₂×C₈×C₇₉	—	4	2	no	—
67	C₂×C₈×C₇₉	—	4	2	no	—
68	C₂×C₈×C₇₉	—	4	2	no	—
69	C₄⋊C₄×C₇₉	—	2	2	no	—
70	C₄⋊C₄×C₇₉	—	4	2	no	—
71	C₄⋊C₄×C₇₉	—	4	2	no	—
72	C₄⋊C₄×C₇₉	—	4	1	no	—
73	C₄⋊C₄×C₇₉	—	4	1	no	—
74	C₄⋊C₄×C₇₉	—	4	2	no	—
75	C₂²⋊C₄×C₇₉	—	4	2	no	—
76	C₂²⋊C₄×C₇₉	—	4	2	no	—
77	C₂²⋊C₄×C₇₉	—	4	2	no	—
78	C₂²⋊C₄×C₇₉	—	4	2	no	—
79	C₂²⋊C₄×C₇₉	—	4	2	no	—
80	C₂²⋊C₄×C₇₉	—	4	2	no	—
81	C₄²×C₇₉	—	2	2	no	—
82	C₄²×C₇₉	—	2	2	no	—
83	C₄²×C₇₉	—	2	2	no	—
84	C₄²×C₇₉	—	4	1	no	—
85	C₄²×C₇₉	—	4	1	no	—
86	C₄²×C₇₉	—	4	1	no	—
87	C₄²×C₇₉	—	4	1	no	—
88	C₂³×C₇₉	—	4	1	no	—
89	C₂³×C₇₉	—	4	1	no	—
90	C₂³×C₇₉	—	4	1	no	—
91	C₂³×C₇₉	—	4	1	no	—
92	C₂³×C₇₉	—	4	1	no	—
93	C₂³×C₇₉	—	4	1	no	—
94	C₂³×C₇₉	—	4	1	no	—
95	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	D₄×C₇₉	—	8	2	no	—
97	C₂×C₄×C₇₉	—	2	1	no	—
98	C₂×C₄×C₇₉	—	2	1	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	1	no	—
102	C₂×C₄×C₇₉	—	2	1	no	—
103	C₂×C₄×C₇₉	—	2	1	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	1	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	4	1	no	—
116	C₂×C₄×C₇₉	—	4	1	no	—
117	C₂×C₄×C₇₉	—	4	1	no	—
118	C₂×C₄×C₇₉	—	4	1	no	—
119	C₂×C₄×C₇₉	—	4	1	no	—
120	C₂×C₄×C₇₉	—	4	1	no	—
121	C₂×C₄×C₇₉	—	4	1	no	—
122	C₂×C₄×C₇₉	—	4	1	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	#131	1	2	yes	—
125	C₂×C₄×C₇₉	#135	1	2	yes	—
126	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
127	C₈×C₇₉	—	8	2	no	—
128	C₈×C₇₉	—	8	2	no	—
129	C₂²×C₇₉	—	8	1	no	—
130	C₂²×C₇₉	—	8	1	no	—
131	C₂²×C₇₉	—	8	1	no	—
132	C₂²×C₇₉	—	8	1	no	—
133	C₂²×C₇₉	—	8	1	no	—
134	C₂²×C₇₉	—	8	1	no	—
135	C₂²×C₇₉	—	8	1	no	—
136	C₂²×C₇₉	#408	1	1	yes	characteristic
137	C₂²×C₇₉	#410	1	1	yes	characteristic
138	C₂²×C₇₉	#413	1	1	yes	characteristic
139	C₂²×C₇₉	#416	1	1	yes	characteristic
140	C₂²×C₇₉	#418	1	1	yes	characteristic
141	C₂²×C₇₉	#421	1	1	yes	characteristic
142	C₂²×C₇₉	#421	1	1	yes	characteristic
143	C₄×C₇₉	—	2	2	no	—
144	C₄×C₇₉	—	2	2	no	—
145	C₄×C₇₉	—	2	2	no	—
146	C₄×C₇₉	—	4	1	no	—
147	C₄×C₇₉	—	4	1	no	—
148	C₄×C₇₉	—	4	1	no	—
149	C₄×C₇₉	—	4	1	no	—
150	C₄×C₇₉	—	8	1	no	—
151	C₄×C₇₉	—	8	1	no	—
152	C₄×C₇₉	—	8	1	no	—
153	#7726	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₇₉	—	16	1	no	—
155	C₂×C₇₉	#1473	1	1	yes	characteristic
156	C₂×C₇₉	#1482	1	1	yes	characteristic
157	C₂×C₇₉	#1496	1	1	yes	characteristic
158	C₂×C₇₉	#1514	1	1	yes	characteristic
159	C₂×C₇₉	#1521	1	1	yes	characteristic
160	C₂×C₇₉	#1535	1	1	yes	characteristic
161	C₂×C₇₉	#1553	1	1	yes	characteristic
162	#2860	C₂×C₇₉	1	1	yes	characteristic
163	#1839	C₂×C₇₉	1	1	yes	characteristic, complemented
164	#1839	C₂×C₇₉	1	1	yes	characteristic, complemented
165	#1271	C₂×C₇₉	1	2	yes	complemented
166	#1270	C₂×C₇₉	1	2	yes	—
167	C₇₉	#7726	1	1	yes	characteristic, direct product factor, Sylow
168	C₂×#126	—	2	1	no	—
169	#390	C₄×C₇₉	1	1	yes	characteristic
170	#390	C₄×C₇₉	1	1	yes	characteristic
171	#390	C₂²×C₇₉	1	1	yes	characteristic
172	#389	C₄×C₇₉	1	1	yes	characteristic
173	#389	C₄×C₇₉	1	1	yes	characteristic
174	#389	C₂²×C₇₉	1	1	yes	characteristic
175	#388	—	2	1	no	—
176	#388	—	2	1	no	—
177	#388	—	2	1	no	—
178	C₄³	C₂²×C₇₉	1	1	yes	characteristic
179	#336	—	2	1	no	—
180	#336	—	2	1	no	—
181	#336	—	2	1	no	—
182	#336	—	2	1	no	—
183	#336	C₂²×C₇₉	1	2	yes	—
184	#336	C₂²×C₇₉	1	2	yes	—
185	C₂²×D₄	C₂×C₄×C₇₉	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	#126	—	4	4	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₇₉	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×C₄×C₇₉	1	1	yes	characteristic
194	C₂×C₄⋊C₄	C₂×C₄×C₇₉	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₄²	C₂×C₄×C₇₉	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₇₉	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₇₉	1	1	yes	characteristic
208	#95	—	2	1	no	—
209	#95	—	2	1	no	—
210	#95	—	2	1	no	—
211	C₂⁴	—	2	1	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	4	1	no	—
256	C₂³	—	4	1	no	—
257	C₂³	C₂×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
258	D₄	—	8	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	#131×C₇₉	1	2	yes	—
287	C₂×C₄	#135×C₇₉	1	2	yes	—
288	C₂×C₄	AGL(1,ℤ/8)×C₇₉	1	2	yes	—
289	C₈	—	8	2	no	—
290	C₈	—	8	2	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	—	8	1	no	—
294	C₂²	—	8	1	no	—
295	C₂²	—	8	1	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	#408×C₇₉	1	1	yes	characteristic
299	C₂²	#410×C₇₉	1	1	yes	characteristic
300	C₂²	#413×C₇₉	1	1	yes	characteristic
301	C₂²	#416×C₇₉	1	1	yes	characteristic
302	C₂²	#418×C₇₉	1	1	yes	characteristic
303	C₂²	#421×C₇₉	1	1	yes	characteristic
304	C₂²	#421×C₇₉	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	1	no	—
315	C₂	—	16	1	no	—
316	C₂	C₇₉×#1473	1	1	yes	characteristic
317	C₂	C₇₉×#1482	1	1	yes	characteristic
318	C₂	C₇₉×#1496	1	1	yes	characteristic
319	C₂	C₇₉×#1514	1	1	yes	characteristic
320	C₂	C₇₉×#1521	1	1	yes	characteristic
321	C₂	C₇₉×#1535	1	1	yes	characteristic
322	C₂	C₇₉×#1553	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₂²×D₄×C₇₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₄³×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₉×#1839	C₂	semidirect
3. C₇₉×#1839	C₂	semidirect
4. C₇₉×#1271	C₂	semidirect
153. #7726	C₇₉	direct (N × H)
163. #1839	C₂×C₇₉	semidirect
164. #1839	C₂×C₇₉	semidirect
165. #1271	C₂×C₇₉	semidirect
167. C₇₉	#7726	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×#2860	2
C₇₉×#1839	2
C₇₉×#1271	2
C₇₉×#1270	2
#7726	79

Subgroups — C79×#7726 #34119415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#7726 #34119415