Subgroups — #2242×D₇₉ #34443560

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765634	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765633	C₂	1	2	yes	maximal, complemented
5	#23765632	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765631	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765630	C₂	1	2	yes	maximal, complemented
8	#358×D₇₉	C₂	1	2	yes	maximal, complemented
9	#357×D₇₉	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2242	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×D₇₉	—	2	1	no	—
14	C₂×D₈×D₇₉	C₂²	1	1	yes	characteristic
15	#125×D₇₉	—	2	1	no	—
16	#124×D₇₉	—	2	1	no	—
17	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
18	#19098965	C₂²	1	1	yes	characteristic, complemented
19	#19098964	C₂²	1	2	yes	complemented
20	#19098963	—	2	1	no	—
21	#19098962	—	2	1	no	—
22	#19098948	C₂²	1	1	yes	characteristic
23	#19098947	C₂²	1	1	yes	characteristic
24	#19098946	C₂²	1	1	yes	characteristic
25	#19098945	—	2	1	no	—
26	#19098944	—	2	1	no	—
27	#107×D₇₉	C₂²	1	2	yes	—
28	#106×D₇₉	—	2	1	no	—
29	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
30	#19098368	C₂²	1	2	yes	complemented
31	#19098367	C₂²	1	2	yes	complemented
32	#19098366	C₂²	1	2	yes	complemented
33	#19098365	C₂²	1	2	yes	—
34	#19098364	C₂²	1	2	yes	—
35	#19098363	C₂²	1	2	yes	—
36	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₇₉	C₂²	1	2	yes	complemented
38	#499×C₇₉	C₂²	1	1	yes	characteristic, complemented
39	#495×C₇₉	C₂²	1	1	yes	characteristic
40	#358×C₇₉	C₂²	1	2	yes	complemented
41	#357×C₇₉	C₂²	1	2	yes	—
42	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₇₉	—	2	1	no	—
44	C₂×D₄×D₇₉	—	2	1	no	—
45	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
46	C₂×#11929614	—	2	1	no	—
47	Q₁₆×D₇₉	—	2	4	no	—
48	C₂×#11929613	—	2	1	no	—
49	C₂×#11929612	—	2	1	no	—
50	C₂×#11929611	—	2	1	no	—
51	SD₁₆×D₇₉	—	4	4	no	—
52	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
53	C₂×#11929609	—	2	1	no	—
54	D₈×D₇₉	—	2	4	no	—
55	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
56	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
57	#14579153	—	2	1	no	—
58	#14579152	—	2	2	no	—
59	#14579151	—	4	2	no	—
60	#14579150	—	2	1	no	—
61	#14579149	—	2	1	no	—
62	#14579148	—	2	1	no	—
63	#14579147	—	4	2	no	—
64	C₄⋊C₄×D₇₉	—	2	2	no	—
65	C₄⋊C₄×D₇₉	—	4	2	no	—
66	C₂²⋊C₄×D₇₉	—	4	2	no	—
67	C₄²×D₇₉	D₄	1	1	yes	characteristic
68	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
69	C₁₆×D₇₉	—	2	4	no	—
70	#14579095	C₂³	1	2	yes	—
71	#14579094	—	2	2	no	—
72	#14579093	—	2	1	no	—
73	#14579092	—	2	2	no	—
74	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
75	C₈×Dic₇₉	D₄	1	1	yes	characteristic
76	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₇₉	—	2	1	no	—
79	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
80	#125×C₇₉	—	2	1	no	—
81	#124×C₇₉	—	2	1	no	—
82	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
83	#107×C₇₉	C₂³	1	2	yes	—
84	#106×C₇₉	—	2	1	no	—
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₂³×D₇₉	—	4	1	no	—
87	C₂×Dic₁₅₈	—	2	1	no	—
88	C₂×Dic₁₅₈	—	4	1	no	—
89	Q₈×D₇₉	—	4	2	no	—
90	Q₈×D₇₉	—	4	2	no	—
91	C₂×D₃₁₆	—	2	1	no	—
92	C₂×C₂²⋊D₇₉	—	4	1	no	—
93	D₄×D₇₉	—	4	2	no	—
94	D₄×D₇₉	—	4	2	no	—
95	C₂×C₄×D₇₉	—	4	1	no	—
96	C₂×C₄×D₇₉	—	4	2	no	—
97	C₂×C₄×D₇₉	D₈	1	2	yes	—
98	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
99	Dic₃₁₆	—	2	2	no	—
100	#11929614	—	4	2	no	—
101	#11929613	—	4	2	no	—
102	#11929612	—	4	2	no	—
103	#11929611	—	4	2	no	—
104	D₆₃₂	—	2	2	no	—
105	#11929609	—	4	2	no	—
106	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₈×D₇₉	SD₁₆	1	4	yes	—
109	C₈×D₇₉	#41	1	4	yes	—
110	C₄⋊Dic₇₉	—	2	2	no	—
111	C₄⋊Dic₇₉	—	4	2	no	—
112	Dic₇₉⋊C₄	—	4	2	no	—
113	Dic₇₉⋊C₄	—	4	2	no	—
114	Dic₇₉⋊C₄	—	4	2	no	—
115	C₂²⋊Dic₇₉	—	4	2	no	—
116	(C₂×D₇₉)⋊C₄	—	4	2	no	—
117	(C₂×D₇₉)⋊C₄	—	4	2	no	—
118	C₄×Dic₇₉	D₈	1	2	yes	—
119	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
120	C₇₉⋊C₁₆	—	2	2	no	—
121	C₂×Q₈×C₇₉	—	2	1	no	—
122	C₂×D₄×C₇₉	—	2	1	no	—
123	Q₁₆×C₇₉	—	2	2	no	—
124	SD₁₆×C₇₉	—	4	2	no	—
125	D₈×C₇₉	—	2	2	no	—
126	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₇₉	—	2	2	no	—
129	C₄⋊C₄×C₇₉	—	4	2	no	—
130	C₂²⋊C₄×C₇₉	—	4	2	no	—
131	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
132	C₁₆×C₇₉	—	2	2	no	—
133	C₂³×C₇₉	—	4	1	no	—
134	C₂²×D₇₉	—	4	1	no	—
135	C₂²×D₇₉	—	8	2	no	—
136	C₂²×D₇₉	—	8	4	no	—
137	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₇₉	—	4	1	no	—
139	Q₈×C₇₉	—	4	1	no	—
140	D₄×C₇₉	—	4	1	no	—
141	D₄×C₇₉	—	4	1	no	—
142	C₂×C₄×C₇₉	—	4	1	no	—
143	C₂×C₄×C₇₉	—	4	2	no	—
144	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₇₉	—	8	2	no	—
147	C₂×Dic₇₉	—	4	1	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
150	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
151	D₃₁₆	—	4	1	no	—
152	D₃₁₆	—	4	1	no	—
153	C₄×D₇₉	—	2	4	no	—
154	C₄×D₇₉	—	8	2	no	—
155	C₄×D₇₉	—	8	4	no	—
156	C₄×D₇₉	#124	1	2	yes	—
157	C₄×D₇₉	C₂×SD₁₆	1	2	yes	—
158	Dic₁₅₈	—	4	1	no	—
159	Dic₁₅₈	—	4	1	no	—
160	Dic₁₅₈	—	8	2	no	—
161	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
162	C₈×C₇₉	C₂×#41	1	2	yes	—
163	C₇₉⋊C₈	#124	1	2	yes	—
164	C₇₉⋊C₈	C₂×SD₁₆	1	2	yes	—
165	C₂²×C₇₉	—	8	1	no	—
166	C₂²×C₇₉	—	8	2	no	—
167	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₇₉	—	8	1	no	—
169	C₂×D₇₉	—	8	2	no	—
170	C₂×D₇₉	—	16	4	no	—
171	C₂×D₇₉	#486	1	2	yes	—
172	C₂×D₇₉	#508	1	4	yes	—
173	C₄×C₇₉	—	2	2	no	—
174	C₄×C₇₉	—	8	1	no	—
175	C₄×C₇₉	—	8	2	no	—
176	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
177	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
178	Dic₇₉	—	2	2	no	—
179	Dic₇₉	—	8	1	no	—
180	Dic₇₉	—	8	2	no	—
181	Dic₇₉	#486	1	2	yes	—
182	C₂×#2242	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	16	1	no	—
184	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
185	C₂×C₇₉	C₂×#508	1	2	yes	—
186	D₇₉	—	16	1	no	—
187	D₇₉	#2242	1	4	yes	direct product factor
188	C₂×#499	—	79	1	no	—
189	C₂×#495	—	79	1	no	—
190	#2242	—	79	1	no	—
191	#2242	—	79	1	no	—
192	#2242	—	79	2	no	—
193	#2242	—	79	2	no	—
194	#2242	—	79	1	no	—
195	#2242	D₇₉	1	1	yes	characteristic, direct product factor
196	C₂×#358	—	79	2	no	—
197	C₂×#357	—	79	2	no	—
198	C₂×C₄×C₁₆	—	79	1	no	—
199	C₇₉	C₂×#2242	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₁₆	—	79	1	no	—
201	C₂²×SD₁₆	—	158	1	no	—
202	C₂²×D₈	—	79	1	no	—
203	C₂×#125	—	158	1	no	—
204	C₂×#124	—	158	1	no	—
205	C₂²×C₁₆	—	79	2	no	—
206	#499	—	79	2	no	—
207	#499	—	79	1	no	—
208	#499	—	158	1	no	—
209	#499	—	158	1	no	—
210	#499	C₂×D₇₉	1	1	yes	characteristic, complemented
211	#495	—	79	1	no	—
212	#495	—	79	1	no	—
213	#495	—	79	1	no	—
214	#495	—	158	1	no	—
215	#495	—	158	1	no	—
216	#495	C₂×D₇₉	1	1	yes	characteristic
217	C₂×#107	—	79	2	no	—
218	C₂×#106	—	158	1	no	—
219	C₂×C₄×C₈	—	79	1	no	—
220	#358	—	79	2	no	—
221	#358	—	79	2	no	—
222	#358	—	79	2	no	—
223	#358	C₂×D₇₉	1	2	yes	complemented
224	#357	—	79	2	no	—
225	#357	—	79	2	no	—
226	#357	—	79	2	no	—
227	#357	C₂×D₇₉	1	2	yes	—
228	C₄×C₁₆	—	79	2	no	—
229	C₄×C₁₆	—	79	1	no	—
230	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
231	C₂²×Q₈	—	158	1	no	—
232	C₂²×D₄	—	158	1	no	—
233	C₂×Q₁₆	—	79	1	no	—
234	C₂×Q₁₆	—	158	4	no	—
235	C₂×Q₁₆	—	158	1	no	—
236	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×SD₁₆	—	316	4	no	—
239	C₂×SD₁₆	—	158	1	no	—
240	C₂×SD₁₆	—	158	1	no	—
241	C₂×SD₁₆	—	158	1	no	—
242	C₂×D₈	—	79	1	no	—
243	C₂×D₈	—	158	4	no	—
244	C₂×D₈	—	158	1	no	—
245	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
246	C₂²×C₈	—	79	1	no	—
247	C₂²×C₈	—	79	1	no	—
248	#125	—	2	1	no	—
249	#125	—	316	2	no	—
250	#125	—	158	2	no	—
251	#125	—	158	1	no	—
252	#124	—	2	1	no	—
253	#124	—	316	2	no	—
254	#124	—	158	1	no	—
255	#124	—	158	1	no	—
256	#124	—	158	1	no	—
257	C₂×C₄⋊C₄	—	316	2	no	—
258	C₂×C₄⋊C₄	—	158	2	no	—
259	C₂×C₂²⋊C₄	—	316	2	no	—
260	C₂×C₄²	—	79	1	no	—
261	C₂×C₁₆	—	79	2	no	—
262	C₂×C₁₆	—	158	4	no	—
263	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
264	#107	—	79	2	no	—
265	#107	—	158	2	no	—
266	#107	C₂²×D₇₉	1	2	yes	—
267	#106	—	2	1	no	—
268	#106	—	158	2	no	—
269	#106	—	158	1	no	—
270	C₄×C₈	—	79	1	no	—
271	C₄×C₈	—	79	1	no	—
272	C₄×C₈	—	79	1	no	—
273	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
274	C₂⁴	—	316	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	316	1	no	—
277	C₂×Q₈	—	316	2	no	—
278	C₂×Q₈	—	316	2	no	—
279	C₂×Q₈	—	158	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	316	2	no	—
282	C₂×D₄	—	316	2	no	—
283	C₂×D₄	—	316	1	no	—
284	C₂×D₄	—	158	1	no	—
285	C₂²×C₄	—	316	1	no	—
286	C₂²×C₄	—	316	2	no	—
287	C₂²×C₄	—	79	1	no	—
288	C₂²×C₄	—	79	2	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	316	2	no	—
291	Q₁₆	—	158	2	no	—
292	SD₁₆	—	4	2	no	—
293	SD₁₆	—	316	2	no	—
294	SD₁₆	—	316	2	no	—
295	SD₁₆	—	316	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	316	2	no	—
298	D₈	—	158	2	no	—
299	C₂×C₈	—	79	1	no	—
300	C₂×C₈	—	79	1	no	—
301	C₂×C₈	—	79	4	no	—
302	C₂×C₈	—	79	4	no	—
303	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	316	2	no	—
308	C₄⋊C₄	—	316	2	no	—
309	C₄⋊C₄	—	316	2	no	—
310	C₄⋊C₄	—	316	2	no	—
311	C₄⋊C₄	—	158	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	316	2	no	—
314	C₂²⋊C₄	—	316	2	no	—
315	C₂²⋊C₄	—	316	2	no	—
316	C₄²	—	79	2	no	—
317	C₄²	—	79	1	no	—
318	C₄²	D₄×D₇₉	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	158	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	316	1	no	—
323	C₂³	—	79	1	no	—
324	C₂³	—	632	2	no	—
325	C₂³	—	632	4	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	316	1	no	—
329	Q₈	—	316	1	no	—
330	Q₈	—	632	2	no	—
331	D₄	—	4	1	no	—
332	D₄	—	4	1	no	—
333	D₄	—	316	1	no	—
334	D₄	—	316	1	no	—
335	D₄	—	632	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	316	1	no	—
339	C₂×C₄	—	316	2	no	—
340	C₂×C₄	—	79	1	no	—
341	C₂×C₄	—	79	2	no	—
342	C₂×C₄	—	79	2	no	—
343	C₂×C₄	—	79	2	no	—
344	C₂×C₄	—	632	2	no	—
345	C₂×C₄	—	632	4	no	—
346	C₂×C₄	—	158	4	no	—
347	C₂×C₄	D₈×D₇₉	1	2	yes	—
348	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
349	C₈	—	79	2	no	—
350	C₈	—	79	2	no	—
351	C₈	SD₁₆×D₇₉	1	2	yes	—
352	C₈	#41×D₇₉	1	2	yes	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	79	4	no	—
356	C₂²	—	79	2	no	—
357	C₂²	—	632	2	no	—
358	C₂²	—	632	1	no	—
359	C₂²	—	1264	4	no	—
360	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	79	2	no	—
365	C₄	—	632	2	no	—
366	C₄	—	632	1	no	—
367	C₄	—	158	2	no	—
368	C₄	#124×D₇₉	1	1	yes	characteristic
369	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
370	C₂	—	16	1	no	—
371	C₂	—	79	4	no	—
372	C₂	—	1264	1	no	—
373	C₂	#486×D₇₉	1	1	yes	characteristic
374	C₂	#508×D₇₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2242	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₈×D₇₉	C₂²
hypercenter	#2242	D₇₉
hypocenter	C₇₉	C₂×#2242
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2242

Complements

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

N	H ≅ G/N	type
1. #499×D₇₉	C₂	semidirect
3. #23765634	C₂	semidirect
4. #23765633	C₂	semidirect
5. #23765632	C₂	semidirect
6. #23765631	C₂	semidirect
7. #23765630	C₂	semidirect
8. #358×D₇₉	C₂	semidirect
10. C₄×C₁₆×D₇₉	C₂	semidirect
11. C₇₉×#2242	C₂	semidirect
18. #19098965	C₂²	semidirect
19. #19098964	C₂²	semidirect
30. #19098368	C₂²	semidirect
31. #19098367	C₂²	semidirect
32. #19098366	C₂²	semidirect
36. C₄×C₇₉⋊C₁₆	C₂²	semidirect
37. C₁₆×Dic₇₉	C₂²	semidirect
38. #499×C₇₉	C₂²	semidirect
40. #358×C₇₉	C₂²	semidirect
42. C₄×C₁₆×C₇₉	C₂²	semidirect
187. D₇₉	#2242	direct (N × H)
195. #2242	D₇₉	direct (N × H)
199. C₇₉	C₂×#2242	semidirect
210. #499	C₂×D₇₉	semidirect
223. #358	C₂×D₇₉	semidirect
230. 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
#499×D₇₉	2
#495×D₇₉	2
#23765634	2
#23765633	2
#23765632	2
#23765631	2
#23765630	2
#358×D₇₉	2
#357×D₇₉	2
C₄×C₁₆×D₇₉	2
C₇₉×#2242	2
C₂×#2242	79

Subgroups — #2242×D79 #34443560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2242×D₇₉ #34443560