Subgroups — C₇₉×#8717 #34120406

Order 20224.

588 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉×#2285	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#1817	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1434	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1294	C₂	1	4	yes	maximal
5	C₂×#117×C₇₉	—	2	1	no	—
6	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
8	#384×C₇₉	C₄	1	1	yes	characteristic, complemented
9	#384×C₇₉	C₄	1	1	yes	characteristic, complemented
10	#382×C₇₉	—	2	1	no	—
11	#380×C₇₉	—	2	1	no	—
12	#380×C₇₉	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
14	C₄×C₄⋊C₄×C₇₉	C₄	1	2	yes	complemented
15	C₂×#95×C₇₉	—	2	1	no	—
16	C₂×#95×C₇₉	—	2	1	no	—
17	C₂×#95×C₇₉	C₂²	1	2	yes	—
18	#336×C₇₉	—	2	1	no	—
19	#336×C₇₉	—	2	1	no	—
20	C₂³×C₄×C₇₉	—	2	2	no	—
21	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	—	2	2	no	—
25	#117×C₇₉	—	4	4	no	—
26	C₂×C₄⋊C₄×C₇₉	—	2	1	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	2	yes	—
31	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
33	C₂×C₄²×C₇₉	—	2	4	no	—
34	C₂×C₄²×C₇₉	—	2	2	no	—
35	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₂²⋊C₈×C₇₉	—	2	4	no	—
40	#95×C₇₉	—	2	4	no	—
41	#95×C₇₉	—	2	2	no	—
42	#95×C₇₉	—	2	2	no	—
43	#95×C₇₉	—	2	4	no	—
44	#95×C₇₉	—	4	2	no	—
45	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	—	2	2	no	—
47	C₂²×C₄×C₇₉	—	2	2	no	—
48	C₂²×C₄×C₇₉	—	2	2	no	—
49	C₂²×C₄×C₇₉	—	2	2	no	—
50	C₂²×C₄×C₇₉	—	2	4	no	—
51	C₂²×C₄×C₇₉	—	2	2	no	—
52	C₂²×C₄×C₇₉	—	2	2	no	—
53	C₂²×C₄×C₇₉	—	4	1	no	—
54	C₂²×C₄×C₇₉	—	4	1	no	—
55	C₂²×C₄×C₇₉	—	4	1	no	—
56	C₂²×C₄×C₇₉	—	4	2	no	—
57	C₂²×C₄×C₇₉	—	4	2	no	—
58	C₂²×C₄×C₇₉	—	4	2	no	—
59	C₂²×C₄×C₇₉	—	4	2	no	—
60	C₂²×C₄×C₇₉	—	4	2	no	—
61	C₂²×C₄×C₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	#41	1	4	yes	—
71	C₂×C₈×C₇₉	—	4	2	no	—
72	C₂×C₈×C₇₉	—	4	2	no	—
73	C₂×C₈×C₇₉	—	4	2	no	—
74	C₄⋊C₄×C₇₉	—	4	2	no	—
75	C₄⋊C₄×C₇₉	—	4	2	no	—
76	C₄⋊C₄×C₇₉	—	4	2	no	—
77	C₄⋊C₄×C₇₉	—	4	2	no	—
78	C₄⋊C₄×C₇₉	—	4	2	no	—
79	C₄⋊C₄×C₇₉	—	4	2	no	—
80	C₂²⋊C₄×C₇₉	—	4	4	no	—
81	C₄²×C₇₉	—	2	2	no	—
82	C₄²×C₇₉	—	2	2	no	—
83	C₄²×C₇₉	—	2	2	no	—
84	C₄²×C₇₉	—	2	2	no	—
85	C₄²×C₇₉	—	4	4	no	—
86	C₂³×C₇₉	—	2	1	no	—
87	C₂³×C₇₉	—	2	1	no	—
88	C₂³×C₇₉	—	2	1	no	—
89	C₂³×C₇₉	—	2	1	no	—
90	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₇₉	#123	1	2	yes	—
92	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₂³×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
94	C₂³×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×C₄×C₇₉	—	2	1	no	—
96	C₂×C₄×C₇₉	—	2	1	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	4	no	—
106	C₂×C₄×C₇₉	—	2	4	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	—	4	4	no	—
111	C₂×C₄×C₇₉	—	4	4	no	—
112	C₂×C₄×C₇₉	—	4	4	no	—
113	C₂×C₄×C₇₉	—	4	4	no	—
114	C₂×C₄×C₇₉	—	4	4	no	—
115	C₂×C₄×C₇₉	—	4	2	no	—
116	C₂×C₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	—	4	2	no	—
118	C₂×C₄×C₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	—	4	2	no	—
120	C₂×C₄×C₇₉	—	4	2	no	—
121	C₂×C₄×C₇₉	—	8	2	no	—
122	C₂×C₄×C₇₉	—	8	2	no	—
123	C₂×C₄×C₇₉	—	8	2	no	—
124	C₂×C₄×C₇₉	—	8	2	no	—
125	C₂×C₄×C₇₉	#103	1	2	yes	—
126	C₂×C₄×C₇₉	#103	1	2	yes	—
127	C₂×C₄×C₇₉	#104	1	2	yes	—
128	C₂×C₄×C₇₉	#117	1	2	yes	—
129	C₂×C₄×C₇₉	#123	1	2	yes	—
130	C₈×C₇₉	—	8	2	no	—
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	—	2	2	no	—
133	C₂²×C₇₉	—	2	1	no	—
134	C₂²×C₇₉	—	2	1	no	—
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	—	2	2	no	—
137	C₂²×C₇₉	—	2	1	no	—
138	C₂²×C₇₉	—	2	1	no	—
139	C₂²×C₇₉	#381	1	1	yes	characteristic
140	C₂²×C₇₉	#410	1	1	yes	characteristic
141	C₂²×C₇₉	#410	1	1	yes	characteristic
142	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
143	C₂²×C₇₉	#418	1	1	yes	characteristic
144	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
145	C₂²×C₇₉	#421	1	1	yes	characteristic
146	C₂²×C₇₉	#481	1	2	yes	—
147	C₂²×C₇₉	#484	1	2	yes	—
148	C₄×C₇₉	—	2	2	no	—
149	C₄×C₇₉	—	2	2	no	—
150	C₄×C₇₉	—	2	2	no	—
151	C₄×C₇₉	—	2	2	no	—
152	C₄×C₇₉	—	2	2	no	—
153	C₄×C₇₉	—	4	1	no	—
154	C₄×C₇₉	—	8	4	no	—
155	C₄×C₇₉	—	8	2	no	—
156	C₄×C₇₉	—	8	2	no	—
157	#8717	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₇₉	—	2	2	no	—
159	C₂×C₇₉	—	2	2	no	—
160	C₂×C₇₉	#1503	1	1	yes	characteristic
161	C₂×C₇₉	#1505	1	1	yes	characteristic
162	C₂×C₇₉	#1514	1	1	yes	characteristic
163	C₂×C₇₉	#1518	1	1	yes	characteristic
164	C₂×C₇₉	#1820	1	1	yes	characteristic
165	C₂×C₇₉	#1824	1	1	yes	characteristic
166	C₂×C₇₉	#1826	1	1	yes	characteristic
167	#2285	C₂×C₇₉	1	1	yes	characteristic
168	#1817	C₂×C₇₉	1	1	yes	characteristic
169	#1434	C₂×C₇₉	1	1	yes	characteristic
170	#1294	C₂×C₇₉	1	4	yes	—
171	C₇₉	#8717	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#117	—	2	1	no	—
173	C₂²×C₄²	C₂²×C₇₉	1	1	yes	characteristic
174	C₂×C₂²⋊C₈	C₂²×C₇₉	1	2	yes	—
175	#384	C₄×C₇₉	1	1	yes	characteristic, complemented
176	#384	C₄×C₇₉	1	1	yes	characteristic, complemented
177	#382	—	2	1	no	—
178	#380	—	2	1	no	—
179	#380	C₂²×C₇₉	1	2	yes	—
180	C₄×C₄⋊C₄	—	2	1	no	—
181	C₄×C₄⋊C₄	C₄×C₇₉	1	2	yes	complemented
182	C₂×#95	—	2	1	no	—
183	C₂×#95	—	2	1	no	—
184	C₂×#95	C₂²×C₇₉	1	2	yes	—
185	#336	—	2	1	no	—
186	#336	—	2	1	no	—
187	C₂³×C₄	—	2	2	no	—
188	C₂³×C₄	D₄×C₇₉	1	1	yes	characteristic
189	C₂³×C₄	D₄×C₇₉	1	1	yes	characteristic
190	C₂³×C₄	C₂³×C₇₉	1	1	yes	characteristic
191	C₂²×C₈	—	2	2	no	—
192	#117	—	4	4	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₇₉	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₇₉	1	2	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂×C₄×C₇₉	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₇₉	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
206	C₂²⋊C₈	—	2	4	no	—
207	#95	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	—	4	2	no	—
212	C₂⁴	C₂×D₄×C₇₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
233	C₂²×C₄	SD₁₆×C₇₉	1	2	yes	—
234	C₂²×C₄	Q₁₆×C₇₉	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
237	C₂²×C₄	#41×C₇₉	1	4	yes	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	4	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₂×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
258	C₂³	#123×C₇₉	1	2	yes	—
259	C₂³	C₂×SD₁₆×C₇₉	1	1	yes	characteristic
260	C₂³	C₂×Q₁₆×C₇₉	1	1	yes	characteristic
261	C₂³	AGL(1,ℤ/8)×C₇₉	1	2	yes	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	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	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	#103×C₇₉	1	2	yes	—
293	C₂×C₄	#103×C₇₉	1	2	yes	—
294	C₂×C₄	#104×C₇₉	1	2	yes	—
295	C₂×C₄	#117×C₇₉	1	2	yes	—
296	C₂×C₄	#123×C₇₉	1	2	yes	—
297	C₈	—	8	2	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	#381×C₇₉	1	1	yes	characteristic
307	C₂²	#410×C₇₉	1	1	yes	characteristic
308	C₂²	#410×C₇₉	1	1	yes	characteristic
309	C₂²	C₂×#103×C₇₉	1	1	yes	characteristic
310	C₂²	#418×C₇₉	1	1	yes	characteristic
311	C₂²	C₂×#104×C₇₉	1	1	yes	characteristic
312	C₂²	#421×C₇₉	1	1	yes	characteristic
313	C₂²	#481×C₇₉	1	2	yes	—
314	C₂²	#484×C₇₉	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	8	4	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₂	—	2	2	no	—
325	C₂	—	2	2	no	—
326	C₂	C₇₉×#1503	1	1	yes	characteristic
327	C₂	C₇₉×#1505	1	1	yes	characteristic
328	C₂	C₇₉×#1514	1	1	yes	characteristic
329	C₂	C₇₉×#1518	1	1	yes	characteristic
330	C₂	C₇₉×#1820	1	1	yes	characteristic
331	C₂	C₇₉×#1824	1	1	yes	characteristic
332	C₂	C₇₉×#1826	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂³×C₄	C₂³×C₇₉
prime radical	C₂⁴×C₇₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₂²×C₄²×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. #384×C₇₉	C₄	semidirect
9. #384×C₇₉	C₄	semidirect
14. C₄×C₄⋊C₄×C₇₉	C₄	semidirect
157. #8717	C₇₉	direct (N × H)
171. C₇₉	#8717	direct (N × H)
175. #384	C₄×C₇₉	semidirect
176. #384	C₄×C₇₉	semidirect
181. C₄×C₄⋊C₄	C₄×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₇₉×#2285	2
C₇₉×#1817	2
C₇₉×#1434	2
C₇₉×#1294	2
#8717	79

Subgroups — C79×#8717 #34120406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#8717 #34120406