Subgroups — C₇₉×#17632 #34129321

Order 20224.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#2983	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#2356	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1518	C₂	1	2	yes	maximal
5	C₇₉×#1515	C₂	1	4	yes	maximal
6	C₇₉×#1506	C₂	1	4	yes	maximal
7	C₇₉×#1503	C₂	1	2	yes	maximal
8	Q₈²×C₇₉	C₄	1	4	yes	complemented
9	#557×C₇₉	—	2	4	no	—
10	#557×C₇₉	C₄	1	4	yes	complemented
11	#544×C₇₉	—	2	1	no	—
12	#544×C₇₉	—	2	2	no	—
13	#533×C₇₉	C₂²	1	2	yes	—
14	#533×C₇₉	C₂²	1	4	yes	—
15	#519×C₇₉	—	2	1	no	—
16	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
17	#446×C₇₉	—	2	4	no	—
18	#432×C₇₉	C₂²	1	4	yes	—
19	#431×C₇₉	C₂²	1	2	yes	—
20	#391×C₇₉	—	2	1	no	—
21	#387×C₇₉	—	2	2	no	—
22	#383×C₇₉	C₂²	1	2	yes	—
23	#382×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
25	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
26	#376×C₇₉	C₂²	1	2	yes	—
27	#328×C₇₉	C₂²	1	4	yes	—
28	#328×C₇₉	C₂²	1	4	yes	—
29	#328×C₇₉	C₂²	1	8	yes	—
30	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₇₉	—	2	8	no	—
32	C₄⋊Q₈×C₇₉	—	2	4	no	—
33	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
35	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
36	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
37	#125×C₇₉	—	2	8	no	—
38	#125×C₇₉	—	2	4	no	—
39	#125×C₇₉	—	2	2	no	—
40	#125×C₇₉	—	2	4	no	—
41	C₂²⋊Q₈×C₇₉	—	2	4	no	—
42	C₂²⋊Q₈×C₇₉	—	2	2	no	—
43	C₂²⋊Q₈×C₇₉	C₂³	1	4	yes	—
44	C₄×Q₈×C₇₉	—	2	4	no	—
45	C₄×Q₈×C₇₉	—	2	4	no	—
46	C₄×Q₈×C₇₉	—	2	2	no	—
47	C₄×Q₈×C₇₉	—	2	4	no	—
48	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
49	C₄×Q₈×C₇₉	C₂×C₄	1	8	yes	—
50	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₇₉	D₄	1	4	yes	—
52	#117×C₇₉	—	2	4	no	—
53	#117×C₇₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
58	C₂×C₄²×C₇₉	—	2	2	no	—
59	C₂×C₄²×C₇₉	—	2	1	no	—
60	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
62	C₄⋊C₈×C₇₉	—	2	2	no	—
63	C₄⋊C₈×C₇₉	—	2	4	no	—
64	C₂²⋊C₈×C₇₉	C₂³	1	4	yes	—
65	#97×C₇₉	—	2	4	no	—
66	C₄×C₈×C₇₉	—	2	2	no	—
67	#95×C₇₉	—	2	2	no	—
68	#95×C₇₉	—	2	2	no	—
69	#95×C₇₉	—	2	1	no	—
70	#95×C₇₉	—	2	2	no	—
71	#95×C₇₉	C₂³	1	1	yes	characteristic
72	#95×C₇₉	C₂³	1	2	yes	—
73	#95×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₇₉	—	2	2	no	—
75	C₂×Q₈×C₇₉	—	2	4	no	—
76	C₂×Q₈×C₇₉	—	2	2	no	—
77	C₂×Q₈×C₇₉	—	2	2	no	—
78	C₂×Q₈×C₇₉	C₂²⋊C₄	1	4	yes	—
79	C₂×Q₈×C₇₉	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₇₉	—	2	1	no	—
81	C₂²×C₄×C₇₉	—	2	2	no	—
82	C₂²×C₄×C₇₉	—	2	1	no	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₇₉	—	2	4	no	—
89	C₄⋊C₄×C₇₉	—	2	4	no	—
90	C₄⋊C₄×C₇₉	—	2	2	no	—
91	C₄⋊C₄×C₇₉	—	2	4	no	—
92	C₄⋊C₄×C₇₉	—	2	2	no	—
93	C₄⋊C₄×C₇₉	—	2	2	no	—
94	C₄⋊C₄×C₇₉	—	2	4	no	—
95	C₄⋊C₄×C₇₉	—	2	4	no	—
96	C₄⋊C₄×C₇₉	—	2	8	no	—
97	C₄⋊C₄×C₇₉	—	2	4	no	—
98	C₄⋊C₄×C₇₉	—	4	2	no	—
99	C₄⋊C₄×C₇₉	—	4	2	no	—
100	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	8	yes	—
101	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
102	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
103	C₂²⋊C₄×C₇₉	—	2	4	no	—
104	C₄²×C₇₉	—	2	8	no	—
105	C₄²×C₇₉	—	2	4	no	—
106	C₄²×C₇₉	—	4	4	no	—
107	C₄²×C₇₉	—	4	2	no	—
108	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
109	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
110	C₄²×C₇₉	C₂×D₄	1	2	yes	—
111	C₄²×C₇₉	C₂×D₄	1	4	yes	—
112	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₇₉	—	2	8	no	—
114	Q₈×C₇₉	—	4	4	no	—
115	Q₈×C₇₉	—	8	4	no	—
116	Q₈×C₇₉	—	8	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	2	no	—
119	C₂×C₄×C₇₉	—	2	2	no	—
120	C₂×C₄×C₇₉	—	2	4	no	—
121	C₂×C₄×C₇₉	—	2	4	no	—
122	C₂×C₄×C₇₉	—	4	2	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	1	no	—
125	C₂×C₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	4	yes	—
136	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂×C₄×C₇₉	#137	1	2	yes	—
138	C₂×C₄×C₇₉	#137	1	2	yes	—
139	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₇₉	—	4	4	no	—
142	C₂²×C₇₉	—	2	1	no	—
143	C₂²×C₇₉	—	2	1	no	—
144	C₂²×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₇₉	C₂×#137	1	1	yes	characteristic
146	C₂²×C₇₉	C₂×#137	1	1	yes	characteristic
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	2	1	no	—
149	C₄×C₇₉	—	2	4	no	—
150	C₄×C₇₉	—	2	4	no	—
151	C₄×C₇₉	—	2	2	no	—
152	C₄×C₇₉	—	4	2	no	—
153	C₄×C₇₉	—	4	4	no	—
154	C₄×C₇₉	—	4	4	no	—
155	C₄×C₇₉	—	8	1	no	—
156	C₄×C₇₉	—	8	2	no	—
157	C₄×C₇₉	#419	1	4	yes	—
158	#17632	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₇₉	—	2	1	no	—
160	C₂×C₇₉	—	2	1	no	—
161	C₂×C₇₉	#2884	1	1	yes	characteristic
162	C₂×C₇₉	C₂×#419	1	1	yes	characteristic
163	C₂×C₇₉	#2901	1	1	yes	characteristic
164	#3502	C₂×C₇₉	1	1	yes	characteristic
165	#2983	C₂×C₇₉	1	1	yes	characteristic
166	#2356	C₂×C₇₉	1	1	yes	characteristic
167	#1518	C₂×C₇₉	1	2	yes	—
168	#1515	C₂×C₇₉	1	4	yes	—
169	#1506	C₂×C₇₉	1	4	yes	—
170	#1503	C₂×C₇₉	1	2	yes	—
171	C₇₉	#17632	1	1	yes	characteristic, direct product factor, Sylow
172	Q₈²	C₄×C₇₉	1	4	yes	complemented
173	#557	—	2	4	no	—
174	#557	C₄×C₇₉	1	4	yes	complemented
175	#544	—	2	1	no	—
176	#544	—	2	2	no	—
177	#533	C₂²×C₇₉	1	2	yes	—
178	#533	C₂²×C₇₉	1	4	yes	—
179	#519	—	2	1	no	—
180	C₂×C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
181	#446	—	2	4	no	—
182	#432	C₂²×C₇₉	1	4	yes	—
183	#431	C₂²×C₇₉	1	2	yes	—
184	#391	—	2	1	no	—
185	#387	—	2	2	no	—
186	#383	C₂²×C₇₉	1	2	yes	—
187	#382	C₂²×C₇₉	1	1	yes	characteristic
188	C₄×C₄⋊C₄	—	2	1	no	—
189	C₄×C₄⋊C₄	C₂²×C₇₉	1	1	yes	characteristic
190	#376	C₂²×C₇₉	1	2	yes	—
191	#328	C₂²×C₇₉	1	4	yes	—
192	#328	C₂²×C₇₉	1	4	yes	—
193	#328	C₂²×C₇₉	1	8	yes	—
194	C₂²×Q₈	D₄×C₇₉	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	8	no	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	C₂×C₄×C₇₉	1	2	yes	—
198	C₄⋊Q₈	C₂×C₄×C₇₉	1	4	yes	—
199	C₄⋊Q₈	C₂×C₄×C₇₉	1	2	yes	—
200	C₄⋊Q₈	C₂×C₄×C₇₉	1	4	yes	—
201	#125	—	2	8	no	—
202	#125	—	2	4	no	—
203	#125	—	2	2	no	—
204	#125	—	2	4	no	—
205	C₂²⋊Q₈	—	2	4	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	C₂³×C₇₉	1	4	yes	—
208	C₄×Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	C₂×C₄×C₇₉	1	4	yes	—
213	C₄×Q₈	C₂×C₄×C₇₉	1	8	yes	—
214	C₄×Q₈	C₂×C₄×C₇₉	1	4	yes	—
215	C₄×Q₈	D₄×C₇₉	1	4	yes	—
216	#117	—	2	4	no	—
217	#117	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
221	C₂×C₄⋊C₄	D₄×C₇₉	1	2	yes	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂³×C₇₉	1	2	yes	—
225	C₂×C₄²	C₂³×C₇₉	1	1	yes	characteristic
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	4	no	—
228	C₂²⋊C₈	C₂³×C₇₉	1	4	yes	—
229	#97	—	2	4	no	—
230	C₄×C₈	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	1	no	—
234	#95	—	2	2	no	—
235	#95	C₂³×C₇₉	1	1	yes	characteristic
236	#95	C₂³×C₇₉	1	2	yes	—
237	#95	C₂³×C₇₉	1	1	yes	characteristic
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₂²⋊C₄×C₇₉	1	4	yes	—
243	C₂×Q₈	C₂²×C₄×C₇₉	1	4	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂×D₄×C₇₉	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₇₉	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₂²⋊C₄×C₇₉	1	8	yes	—
265	C₄⋊C₄	C₂²⋊C₄×C₇₉	1	4	yes	—
266	C₄⋊C₄	C₂²×C₄×C₇₉	1	4	yes	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₄²	—	2	8	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	4	4	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂²×C₄×C₇₉	1	4	yes	—
273	C₄²	C₂²×C₄×C₇₉	1	2	yes	—
274	C₄²	C₂×D₄×C₇₉	1	2	yes	—
275	C₄²	C₂×D₄×C₇₉	1	4	yes	—
276	C₂³	C₂²×D₄×C₇₉	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	8	4	no	—
280	Q₈	—	8	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	4	yes	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	2	yes	—
299	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	4	yes	—
300	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	2	yes	—
301	C₂×C₄	#137×C₇₉	1	2	yes	—
302	C₂×C₄	#137×C₇₉	1	2	yes	—
303	C₂×C₄	C₂³×C₄×C₇₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₇₉	1	1	yes	characteristic
305	C₈	—	4	4	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
309	C₂²	C₂×#137×C₇₉	1	1	yes	characteristic
310	C₂²	C₂×#137×C₇₉	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	4	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	#419×C₇₉	1	4	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	C₇₉×#2884	1	1	yes	characteristic
325	C₂	C₂×#419×C₇₉	1	1	yes	characteristic
326	C₂	C₇₉×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₉
prime radical	C₂³×C₇₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄×Q₈×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. Q₈²×C₇₉	C₄	semidirect
10. #557×C₇₉	C₄	semidirect
158. #17632	C₇₉	direct (N × H)
171. C₇₉	#17632	direct (N × H)
172. Q₈²	C₄×C₇₉	semidirect
174. #557	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×#3502	2
C₇₉×#2983	2
C₇₉×#2356	2
C₇₉×#1518	2
C₇₉×#1515	2
C₇₉×#1506	2
C₇₉×#1503	2
#17632	79

Subgroups — C79×#17632 #34129321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#17632 #34129321