Subgroups — #2240×D₇₉ #34443550

Order 20224.

726 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765625	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765624	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765623	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765622	C₂	1	4	yes	maximal, complemented
7	#357×D₇₉	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×D₇₉	—	2	1	no	—
11	C₂×D₈×D₇₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×D₇₉	—	2	1	no	—
13	#127×D₇₉	—	2	1	no	—
14	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic, complemented
16	#19098967	—	2	1	no	—
17	#19098967	C₂²	1	2	yes	complemented
18	#19098966	—	2	1	no	—
19	#19098940	C₂²	1	1	yes	characteristic
20	#19098939	—	2	1	no	—
21	#19098938	—	2	1	no	—
22	#19098938	C₂²	1	2	yes	—
23	#107×D₇₉	—	2	1	no	—
24	#107×D₇₉	C₂²	1	2	yes	—
25	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
26	#19098365	C₂²	1	4	yes	complemented
27	#19098364	C₂²	1	4	yes	complemented
28	#19098363	C₂²	1	4	yes	complemented
29	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₁₆×Dic₇₉	C₂²	1	2	yes	complemented
31	#500×C₇₉	C₂²	1	1	yes	characteristic, complemented
32	#493×C₇₉	C₂²	1	1	yes	characteristic
33	#357×C₇₉	C₂²	1	4	yes	complemented
34	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
35	C₂×Q₈×D₇₉	—	4	1	no	—
36	C₂×D₄×D₇₉	—	2	2	no	—
37	C₂×D₄×D₇₉	—	4	1	no	—
38	C₂×D₄×D₇₉	—	4	1	no	—
39	C₂×D₆₃₂	—	2	1	no	—
40	C₂×D₆₃₂	C₂³	1	2	yes	—
41	C₂×#11929609	—	2	2	no	—
42	C₂×#11929609	—	2	2	no	—
43	D₈×D₇₉	—	2	8	no	—
44	D₈×D₇₉	—	4	4	no	—
45	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
46	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
47	#14579161	—	2	1	no	—
48	#14579160	—	2	2	no	—
49	#14579160	—	4	1	no	—
50	#14579159	—	4	1	no	—
51	#14579158	—	2	1	no	—
52	#14579157	—	2	2	no	—
53	#14579157	—	4	1	no	—
54	#14579157	—	4	1	no	—
55	C₄⋊C₄×D₇₉	—	2	2	no	—
56	C₄⋊C₄×D₇₉	—	4	1	no	—
57	C₄²×D₇₉	D₄	1	1	yes	characteristic
58	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
59	C₁₆×D₇₉	—	2	4	no	—
60	#14579095	—	2	1	no	—
61	#14579095	C₂³	1	2	yes	—
62	#14579094	—	2	2	no	—
63	#14579094	—	2	2	no	—
64	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₇₉	D₄	1	1	yes	characteristic
66	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₇₉	—	2	1	no	—
68	C₂×D₈×C₇₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₉	—	2	1	no	—
70	#127×C₇₉	—	2	1	no	—
71	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
72	#107×C₇₉	—	2	1	no	—
73	#107×C₇₉	C₂³	1	2	yes	—
74	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₂³×D₇₉	—	4	2	no	—
76	C₂×Dic₁₅₈	—	4	2	no	—
77	C₂×Dic₁₅₈	—	4	1	no	—
78	Q₈×D₇₉	—	8	4	no	—
79	C₂×D₃₁₆	—	2	2	no	—
80	C₂×D₃₁₆	—	4	1	no	—
81	C₂×D₃₁₆	—	4	1	no	—
82	C₂×C₂²⋊D₇₉	—	4	2	no	—
83	C₂×C₂²⋊D₇₉	—	4	2	no	—
84	C₂×C₂²⋊D₇₉	—	4	2	no	—
85	D₄×D₇₉	—	4	4	no	—
86	D₄×D₇₉	—	4	4	no	—
87	D₄×D₇₉	—	8	4	no	—
88	D₄×D₇₉	—	8	4	no	—
89	C₂×C₄×D₇₉	—	4	2	no	—
90	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
91	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
92	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
93	D₆₃₂	—	2	4	no	—
94	D₆₃₂	—	4	2	no	—
95	#11929609	—	4	4	no	—
96	#11929609	—	4	4	no	—
97	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
99	C₈×D₇₉	SD₁₆	1	4	yes	—
100	C₈×D₇₉	#41	1	4	yes	—
101	C₄⋊Dic₇₉	—	2	2	no	—
102	C₄⋊Dic₇₉	—	4	1	no	—
103	Dic₇₉⋊C₄	—	4	2	no	—
104	Dic₇₉⋊C₄	—	4	2	no	—
105	C₄×Dic₇₉	D₈	1	1	yes	characteristic
106	C₄×Dic₇₉	D₈	1	1	yes	characteristic
107	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
108	C₇₉⋊C₁₆	—	2	2	no	—
109	C₂×Q₈×C₇₉	—	4	1	no	—
110	C₂×D₄×C₇₉	—	2	2	no	—
111	C₂×D₄×C₇₉	—	4	1	no	—
112	C₂×D₄×C₇₉	—	4	1	no	—
113	D₈×C₇₉	—	2	4	no	—
114	D₈×C₇₉	—	4	2	no	—
115	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₇₉	—	2	2	no	—
118	C₄⋊C₄×C₇₉	—	4	1	no	—
119	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₁₆×C₇₉	—	2	2	no	—
121	C₂³×C₇₉	—	4	2	no	—
122	C₂²×D₇₉	—	4	2	no	—
123	C₂²×D₇₉	—	8	4	no	—
124	C₂²×D₇₉	—	8	4	no	—
125	C₂²×D₇₉	—	8	4	no	—
126	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₇₉	—	8	2	no	—
128	D₄×C₇₉	—	4	2	no	—
129	D₄×C₇₉	—	4	2	no	—
130	D₄×C₇₉	—	8	2	no	—
131	D₄×C₇₉	—	8	2	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₇₉	—	8	4	no	—
137	C₂²⋊D₇₉	—	8	4	no	—
138	C₂²⋊D₇₉	—	8	4	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
143	D₃₁₆	—	4	2	no	—
144	D₃₁₆	—	4	2	no	—
145	D₃₁₆	—	8	2	no	—
146	D₃₁₆	—	8	2	no	—
147	C₄×D₇₉	—	8	4	no	—
148	C₄×D₇₉	D₁₆	1	4	yes	—
149	C₄×D₇₉	SD₃₂	1	4	yes	—
150	C₄×D₇₉	#124	1	2	yes	—
151	C₄×D₇₉	C₂×SD₁₆	1	2	yes	—
152	Dic₁₅₈	—	8	2	no	—
153	Dic₁₅₈	—	8	4	no	—
154	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
155	C₈×C₇₉	C₂×#41	1	2	yes	—
156	C₇₉⋊C₈	#124	1	2	yes	—
157	C₇₉⋊C₈	C₂×SD₁₆	1	2	yes	—
158	C₂²×C₇₉	—	8	2	no	—
159	C₂²×C₇₉	—	8	2	no	—
160	C₂²×C₇₉	—	8	2	no	—
161	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₇₉	—	8	2	no	—
163	C₂×D₇₉	—	8	2	no	—
164	C₂×D₇₉	—	8	2	no	—
165	C₂×D₇₉	—	16	8	no	—
166	C₂×D₇₉	#486	1	2	yes	—
167	C₂×D₇₉	C₂×D₁₆	1	2	yes	—
168	C₂×D₇₉	C₂×SD₃₂	1	2	yes	—
169	C₄×C₇₉	—	8	2	no	—
170	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
171	C₄×C₇₉	C₂×SD₃₂	1	2	yes	—
172	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
173	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
174	Dic₇₉	—	8	2	no	—
175	Dic₇₉	#486	1	2	yes	—
176	Dic₇₉	C₂×D₁₆	1	2	yes	—
177	Dic₇₉	C₂×SD₃₂	1	2	yes	—
178	C₂×#2240	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	16	2	no	—
180	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
181	C₂×C₇₉	C₂²×D₁₆	1	1	yes	characteristic
182	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
183	D₇₉	—	16	2	no	—
184	D₇₉	#2240	1	4	yes	direct product factor
185	C₂×#500	—	79	1	no	—
186	C₂×#493	—	79	1	no	—
187	#2240	—	79	1	no	—
188	#2240	—	79	1	no	—
189	#2240	—	79	4	no	—
190	#2240	—	79	1	no	—
191	#2240	D₇₉	1	1	yes	characteristic, direct product factor
192	C₂×#357	—	79	4	no	—
193	C₂×C₄×C₁₆	—	79	1	no	—
194	C₇₉	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
195	C₂²×D₈	—	79	2	no	—
196	C₂²×D₈	—	158	1	no	—
197	C₂×C₄⋊Q₈	—	158	1	no	—
198	C₂×#127	—	158	1	no	—
199	C₂²×C₁₆	—	79	2	no	—
200	#500	—	79	2	no	—
201	#500	—	79	1	no	—
202	#500	—	158	1	no	—
203	#500	—	158	1	no	—
204	#500	C₂×D₇₉	1	1	yes	characteristic, complemented
205	#493	—	79	2	no	—
206	#493	—	79	1	no	—
207	#493	—	158	1	no	—
208	#493	—	158	1	no	—
209	#493	C₂×D₇₉	1	1	yes	characteristic
210	C₂×#107	—	79	2	no	—
211	C₂×#107	—	158	1	no	—
212	C₂×C₄×C₈	—	79	1	no	—
213	#357	—	79	4	no	—
214	#357	—	79	4	no	—
215	#357	—	79	4	no	—
216	#357	C₂×D₇₉	1	4	yes	complemented
217	C₄×C₁₆	—	79	2	no	—
218	C₄×C₁₆	—	79	1	no	—
219	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
220	C₂²×Q₈	—	316	1	no	—
221	C₂²×D₄	—	316	1	no	—
222	C₂²×D₄	—	316	1	no	—
223	C₂²×D₄	—	158	2	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	316	4	no	—
226	C₂×D₈	—	79	2	no	—
227	C₂×D₈	—	158	8	no	—
228	C₂×D₈	—	158	2	no	—
229	C₂×D₈	—	158	2	no	—
230	C₂×D₈	—	158	1	no	—
231	C₂×D₈	C₂²×D₇₉	1	2	yes	—
232	C₂²×C₈	—	79	1	no	—
233	C₂²×C₈	—	79	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	316	1	no	—
236	C₄⋊Q₈	—	316	1	no	—
237	C₄⋊Q₈	—	158	2	no	—
238	C₄⋊Q₈	—	158	1	no	—
239	#127	—	2	1	no	—
240	#127	—	316	1	no	—
241	#127	—	316	1	no	—
242	#127	—	158	2	no	—
243	#127	—	158	1	no	—
244	C₂×C₄⋊C₄	—	316	1	no	—
245	C₂×C₄⋊C₄	—	158	2	no	—
246	C₂×C₄²	—	79	1	no	—
247	C₂×C₁₆	—	79	2	no	—
248	C₂×C₁₆	—	158	4	no	—
249	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
250	#107	—	2	1	no	—
251	#107	—	79	2	no	—
252	#107	—	158	2	no	—
253	#107	—	158	2	no	—
254	#107	—	158	1	no	—
255	#107	C₂²×D₇₉	1	2	yes	—
256	C₄×C₈	—	79	1	no	—
257	C₄×C₈	—	79	1	no	—
258	C₄×C₈	—	79	1	no	—
259	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
260	C₂⁴	—	316	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	316	2	no	—
263	C₂×Q₈	—	316	1	no	—
264	C₂×Q₈	—	632	4	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	316	4	no	—
269	C₂×D₄	—	316	4	no	—
270	C₂×D₄	—	316	2	no	—
271	C₂×D₄	—	316	2	no	—
272	C₂×D₄	—	316	2	no	—
273	C₂×D₄	—	316	1	no	—
274	C₂×D₄	—	316	1	no	—
275	C₂×D₄	—	632	4	no	—
276	C₂×D₄	—	632	4	no	—
277	C₂×D₄	—	158	2	no	—
278	C₂²×C₄	—	316	2	no	—
279	C₂²×C₄	—	79	1	no	—
280	C₂²×C₄	—	79	1	no	—
281	C₂²×C₄	—	79	1	no	—
282	D₈	—	2	4	no	—
283	D₈	—	4	2	no	—
284	D₈	—	316	4	no	—
285	D₈	—	316	4	no	—
286	D₈	—	316	2	no	—
287	D₈	—	158	4	no	—
288	C₂×C₈	—	79	1	no	—
289	C₂×C₈	—	79	1	no	—
290	C₂×C₈	—	79	4	no	—
291	C₂×C₈	—	79	4	no	—
292	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
293	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	316	2	no	—
297	C₄⋊C₄	—	316	2	no	—
298	C₄⋊C₄	—	316	1	no	—
299	C₄⋊C₄	—	158	2	no	—
300	C₄²	—	79	1	no	—
301	C₄²	—	79	1	no	—
302	C₄²	—	79	1	no	—
303	C₄²	D₄×D₇₉	1	1	yes	characteristic
304	C₁₆	—	2	2	no	—
305	C₁₆	—	158	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	316	2	no	—
308	C₂³	—	79	1	no	—
309	C₂³	—	632	4	no	—
310	C₂³	—	632	4	no	—
311	C₂³	—	632	4	no	—
312	Q₈	—	8	2	no	—
313	Q₈	—	632	4	no	—
314	Q₈	—	632	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	316	2	no	—
320	D₄	—	316	2	no	—
321	D₄	—	632	4	no	—
322	D₄	—	632	4	no	—
323	D₄	—	632	4	no	—
324	D₄	—	632	2	no	—
325	D₄	—	632	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	316	2	no	—
328	C₂×C₄	—	79	1	no	—
329	C₂×C₄	—	79	1	no	—
330	C₂×C₄	—	79	1	no	—
331	C₂×C₄	—	79	4	no	—
332	C₂×C₄	—	79	4	no	—
333	C₂×C₄	—	79	2	no	—
334	C₂×C₄	—	79	2	no	—
335	C₂×C₄	—	632	4	no	—
336	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
337	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
338	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
339	C₈	—	79	2	no	—
340	C₈	—	79	2	no	—
341	C₈	SD₁₆×D₇₉	1	2	yes	—
342	C₈	#41×D₇₉	1	2	yes	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	79	2	no	—
347	C₂²	—	79	2	no	—
348	C₂²	—	79	2	no	—
349	C₂²	—	632	2	no	—
350	C₂²	—	632	2	no	—
351	C₂²	—	632	2	no	—
352	C₂²	—	1264	8	no	—
353	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
354	C₄	—	8	2	no	—
355	C₄	—	79	2	no	—
356	C₄	—	79	2	no	—
357	C₄	—	79	2	no	—
358	C₄	—	632	2	no	—
359	C₄	D₁₆×D₇₉	1	2	yes	—
360	C₄	SD₃₂×D₇₉	1	2	yes	—
361	C₄	#124×D₇₉	1	1	yes	characteristic
362	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
363	C₂	—	16	2	no	—
364	C₂	—	79	4	no	—
365	C₂	—	1264	2	no	—
366	C₂	#486×D₇₉	1	1	yes	characteristic
367	C₂	C₂×D₁₆×D₇₉	1	1	yes	characteristic
368	C₂	C₂×SD₃₂×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2240	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#493×D₇₉	C₂
hypercenter	#2240	D₇₉
hypocenter	C₇₉	C₂×#2240
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #500×D₇₉	C₂	semidirect
3. #23765625	C₂	semidirect
4. #23765624	C₂	semidirect
5. #23765623	C₂	semidirect
6. #23765622	C₂	semidirect
7. #357×D₇₉	C₂	semidirect
8. C₄×C₁₆×D₇₉	C₂	semidirect
9. C₇₉×#2240	C₂	semidirect
15. #19098968	C₂²	semidirect
17. #19098967	C₂²	semidirect
26. #19098365	C₂²	semidirect
27. #19098364	C₂²	semidirect
28. #19098363	C₂²	semidirect
29. C₄×C₇₉⋊C₁₆	C₂²	semidirect
30. C₁₆×Dic₇₉	C₂²	semidirect
31. #500×C₇₉	C₂²	semidirect
33. #357×C₇₉	C₂²	semidirect
34. C₄×C₁₆×C₇₉	C₂²	semidirect
184. D₇₉	#2240	direct (N × H)
191. #2240	D₇₉	direct (N × H)
194. C₇₉	C₂×#2240	semidirect
204. #500	C₂×D₇₉	semidirect
216. #357	C₂×D₇₉	semidirect
219. C₄×C₁₆	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#500×D₇₉	2
#493×D₇₉	2
#23765625	2
#23765624	2
#23765623	2
#23765622	2
#357×D₇₉	2
C₄×C₁₆×D₇₉	2
C₇₉×#2240	2
C₂×#2240	79

Subgroups — #2240×D79 #34443550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2240×D₇₉ #34443550