Subgroups — #34178804

Order 20224.

400 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098625	C₂	1	1	yes	maximal, characteristic
2	#23761712	C₂	1	1	yes	maximal, characteristic
3	#23761712	C₂	1	1	yes	maximal, characteristic
4	#23761693	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761684	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761245	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579118	C₂²	1	1	yes	characteristic
9	#19098625	—	2	1	no	—
10	#19098625	—	2	1	no	—
11	#19098625	—	2	1	no	—
12	#19098625	—	2	1	no	—
13	#19098616	C₂²	1	1	yes	characteristic
14	#19098616	C₂²	1	1	yes	characteristic
15	C₂×Dic₇₉⋊C₈	C₄	1	1	yes	characteristic
16	C₂×Dic₇₉⋊C₈	C₄	1	1	yes	characteristic
17	C₂×(C₂×D₇₉)⋊C₈	C₄	1	1	yes	characteristic
18	C₂×(C₂×D₇₉)⋊C₈	C₄	1	1	yes	characteristic
19	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	#19098288	C₄	1	2	yes	—
21	#19098288	C₄	1	2	yes	—
22	#19098286	C₄	1	2	yes	—
23	#19098286	C₄	1	2	yes	—
24	#19098285	—	2	1	no	—
25	#19098285	—	2	1	no	—
26	#19098284	—	2	1	no	—
27	C₈×C₇₉⋊C₈	—	2	1	no	—
28	#431×C₇₉	C₂²	1	1	yes	characteristic
29	#431×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
34	#14579118	—	2	1	no	—
35	#14579118	—	2	1	no	—
36	#14579118	—	2	1	no	—
37	#14579118	—	2	1	no	—
38	C₂×Dic₇₉⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₇₉⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
43	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	Dic₇₉⋊C₈	—	2	1	no	—
46	Dic₇₉⋊C₈	—	2	1	no	—
47	Dic₇₉⋊C₈	—	2	1	no	—
48	Dic₇₉⋊C₈	—	2	1	no	—
49	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
50	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
51	(C₂×D₇₉)⋊C₈	—	2	1	no	—
52	(C₂×D₇₉)⋊C₈	—	2	1	no	—
53	(C₂×D₇₉)⋊C₈	—	2	1	no	—
54	(C₂×D₇₉)⋊C₈	—	2	1	no	—
55	C₄×C₇₉⋊C₈	—	2	1	no	—
56	C₄×C₇₉⋊C₈	—	2	1	no	—
57	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
59	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₇₉	—	2	1	no	—
65	C₄⋊C₈×C₇₉	—	2	1	no	—
66	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
67	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
68	#97×C₇₉	—	2	1	no	—
69	#97×C₇₉	—	2	1	no	—
70	C₄×C₈×C₇₉	—	2	1	no	—
71	C₄×C₈×C₇₉	—	2	1	no	—
72	C₂³×D₇₉	C₄²	1	1	yes	characteristic
73	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
74	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
75	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₇₉	—	2	1	no	—
77	C₂×C₄×D₇₉	—	2	1	no	—
78	C₂×C₄×D₇₉	—	2	1	no	—
79	C₂×C₄×D₇₉	—	2	1	no	—
80	C₂×C₄×D₇₉	—	2	1	no	—
81	C₂×C₄×D₇₉	—	2	1	no	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	—	2	1	no	—
85	C₂×C₇₉⋊C₈	—	2	1	no	—
86	C₂×C₇₉⋊C₈	—	2	1	no	—
87	C₂×C₇₉⋊C₈	—	2	1	no	—
88	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
92	Dic₇₉⋊C₄	—	2	1	no	—
93	Dic₇₉⋊C₄	—	2	1	no	—
94	Dic₇₉⋊C₄	—	2	1	no	—
95	Dic₇₉⋊C₄	—	2	1	no	—
96	(C₂×D₇₉)⋊C₄	—	2	1	no	—
97	(C₂×D₇₉)⋊C₄	—	2	1	no	—
98	(C₂×D₇₉)⋊C₄	—	2	1	no	—
99	(C₂×D₇₉)⋊C₄	—	2	1	no	—
100	C₄×Dic₇₉	—	2	1	no	—
101	C₄×Dic₇₉	—	2	1	no	—
102	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	—	2	1	no	—
106	C₂×C₈×C₇₉	—	2	1	no	—
107	C₂×C₈×C₇₉	—	2	2	no	—
108	C₂×C₈×C₇₉	—	2	2	no	—
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	—	2	1	no	—
111	C₂×C₈×C₇₉	—	2	1	no	—
112	C₂×C₈×C₇₉	—	2	1	no	—
113	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₉	#41	1	1	yes	characteristic
116	C₄²×C₇₉	#41	1	1	yes	characteristic
117	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
118	C₂²×D₇₉	—	2	1	no	—
119	C₂²×D₇₉	—	2	1	no	—
120	C₂²×D₇₉	—	2	1	no	—
121	C₂²×D₇₉	—	4	1	no	—
122	C₂²×D₇₉	—	4	1	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	1	no	—
125	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
134	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
135	C₂×Dic₇₉	—	2	1	no	—
136	C₂×Dic₇₉	—	2	1	no	—
137	C₂×Dic₇₉	—	2	1	no	—
138	C₂×Dic₇₉	—	2	1	no	—
139	C₂×Dic₇₉	—	2	1	no	—
140	C₂×Dic₇₉	—	4	1	no	—
141	C₂×Dic₇₉	—	4	1	no	—
142	C₄×D₇₉	—	4	1	no	—
143	C₄×D₇₉	—	4	1	no	—
144	C₄×D₇₉	—	4	1	no	—
145	C₄×D₇₉	—	4	1	no	—
146	C₈×C₇₉	—	4	2	no	—
147	C₈×C₇₉	—	4	2	no	—
148	C₈×C₇₉	—	4	1	no	—
149	C₈×C₇₉	—	4	1	no	—
150	C₇₉⋊C₈	—	2	1	no	—
151	C₇₉⋊C₈	—	2	1	no	—
152	C₇₉⋊C₈	—	2	1	no	—
153	C₇₉⋊C₈	—	2	1	no	—
154	C₇₉⋊C₈	—	4	2	no	—
155	C₇₉⋊C₈	—	4	2	no	—
156	C₂²×C₇₉	—	2	1	no	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₇₉	#405	1	1	yes	characteristic
160	C₂²×C₇₉	#405	1	1	yes	characteristic
161	C₂×D₇₉	—	4	1	no	—
162	C₂×D₇₉	—	4	1	no	—
163	C₂×D₇₉	—	4	1	no	—
164	C₂×D₇₉	—	8	1	no	—
165	C₂×D₇₉	—	8	1	no	—
166	C₄×C₇₉	—	2	1	no	—
167	C₄×C₇₉	—	2	1	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	#408	1	1	yes	characteristic
171	C₄×C₇₉	#408	1	1	yes	characteristic
172	C₄×C₇₉	#433	1	1	yes	characteristic
173	C₄×C₇₉	#433	1	1	yes	characteristic
174	Dic₇₉	—	4	1	no	—
175	Dic₇₉	—	4	1	no	—
176	Dic₇₉	—	4	1	no	—
177	#8158	—	79	1	no	maximal, Sylow
178	C₂×C₇₉	—	2	1	no	—
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	#1750	1	1	yes	characteristic
181	C₂×C₇₉	#1757	1	1	yes	characteristic
182	C₂×C₇₉	#1757	1	1	yes	characteristic
183	D₇₉	—	8	1	no	—
184	C₂×#433	—	79	1	no	—
185	#1458	—	79	1	no	—
186	#1458	—	79	1	no	—
187	#1452	—	79	1	no	—
188	#1450	—	79	1	no	—
189	#1281	—	79	1	no	—
190	#1281	D₇₉	1	1	yes	characteristic, complemented
191	C₇₉	#8158	1	1	yes	characteristic, complemented, Sylow
192	C₂×#117	—	79	1	no	—
193	#433	—	158	1	no	—
194	#433	—	158	1	no	—
195	#433	—	158	1	no	—
196	#433	—	158	1	no	—
197	#431	—	79	1	no	—
198	#431	—	79	1	no	—
199	#431	C₂×D₇₉	1	1	yes	characteristic
200	#431	C₂×D₇₉	1	1	yes	characteristic
201	C₂×C₄⋊C₈	—	79	1	no	—
202	C₂×C₄⋊C₈	—	79	1	no	—
203	C₂×C₂²⋊C₈	—	79	1	no	—
204	C₂×C₂²⋊C₈	—	79	1	no	—
205	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
206	C₂×C₄×C₈	—	79	1	no	—
207	#324	—	79	2	no	—
208	#324	—	79	2	no	—
209	#323	—	79	2	no	—
210	#323	—	79	2	no	—
211	#322	—	158	1	no	—
212	#322	—	158	1	no	—
213	#322	—	158	1	no	—
214	C₈²	—	158	1	no	—
215	C₂³×C₄	—	79	1	no	—
216	C₂²×C₈	—	79	1	no	—
217	C₂²×C₈	—	79	1	no	—
218	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
219	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
220	#117	—	158	1	no	—
221	#117	—	158	1	no	—
222	#117	—	158	1	no	—
223	#117	—	158	1	no	—
224	C₂×C₄⋊C₄	—	79	1	no	—
225	C₂×C₄⋊C₄	—	79	1	no	—
226	C₂×C₂²⋊C₄	—	79	1	no	—
227	C₂×C₂²⋊C₄	—	79	1	no	—
228	C₂×C₄²	—	79	1	no	—
229	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	158	1	no	—
233	C₄⋊C₈	—	158	1	no	—
234	C₄⋊C₈	—	158	1	no	—
235	C₄⋊C₈	—	158	1	no	—
236	C₄⋊C₈	—	158	1	no	—
237	C₄⋊C₈	—	158	1	no	—
238	C₂²⋊C₈	—	79	2	no	—
239	C₂²⋊C₈	—	79	2	no	—
240	C₂²⋊C₈	—	158	1	no	—
241	C₂²⋊C₈	—	158	1	no	—
242	C₂²⋊C₈	—	158	1	no	—
243	C₂²⋊C₈	—	158	1	no	—
244	C₂²⋊C₈	C₄×D₇₉	1	2	yes	—
245	C₂²⋊C₈	C₄×D₇₉	1	2	yes	—
246	#97	—	2	1	no	—
247	#97	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	79	1	no	—
251	C₄×C₈	—	79	1	no	—
252	C₄×C₈	—	79	1	no	—
253	C₄×C₈	—	79	1	no	—
254	C₄×C₈	—	158	1	no	—
255	C₄×C₈	—	158	1	no	—
256	C₂⁴	—	79	1	no	—
257	C₂²×C₄	—	79	1	no	—
258	C₂²×C₄	—	79	1	no	—
259	C₂²×C₄	—	79	1	no	—
260	C₂²×C₄	—	158	1	no	—
261	C₂²×C₄	—	158	1	no	—
262	C₂²×C₄	—	158	1	no	—
263	C₂²×C₄	—	158	1	no	—
264	C₂²×C₄	—	158	1	no	—
265	C₂²×C₄	—	158	1	no	—
266	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	79	1	no	—
278	C₂×C₈	—	79	1	no	—
279	C₂×C₈	—	79	1	no	—
280	C₂×C₈	—	79	1	no	—
281	C₂×C₈	—	158	2	no	—
282	C₂×C₈	—	158	2	no	—
283	C₂×C₈	—	158	1	no	—
284	C₂×C₈	—	158	1	no	—
285	C₂×C₈	—	158	1	no	—
286	C₂×C₈	—	158	1	no	—
287	C₄⋊C₄	—	158	1	no	—
288	C₄⋊C₄	—	158	1	no	—
289	C₄⋊C₄	—	158	1	no	—
290	C₄⋊C₄	—	158	1	no	—
291	C₂²⋊C₄	—	158	1	no	—
292	C₂²⋊C₄	—	158	1	no	—
293	C₂²⋊C₄	—	158	1	no	—
294	C₂²⋊C₄	—	158	1	no	—
295	C₄²	—	158	1	no	—
296	C₄²	—	158	1	no	—
297	C₄²	D₄×D₇₉	1	1	yes	characteristic
298	C₄²	D₄×D₇₉	1	1	yes	characteristic
299	C₄²	#11929623	1	1	yes	characteristic
300	C₄²	#11929625	1	1	yes	characteristic
301	C₂³	—	316	1	no	—
302	C₂³	—	316	1	no	—
303	C₂³	—	158	1	no	—
304	C₂³	—	158	1	no	—
305	C₂³	—	158	1	no	—
306	C₂³	C₄²×D₇₉	1	1	yes	characteristic
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	316	1	no	—
310	C₂×C₄	—	316	1	no	—
311	C₂×C₄	—	316	1	no	—
312	C₂×C₄	—	316	1	no	—
313	C₂×C₄	—	316	1	no	—
314	C₂×C₄	—	316	1	no	—
315	C₂×C₄	—	158	1	no	—
316	C₂×C₄	—	158	1	no	—
317	C₂×C₄	—	158	1	no	—
318	C₂×C₄	—	158	1	no	—
319	C₂×C₄	—	158	1	no	—
320	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
321	C₂×C₄	#14579119	1	1	yes	characteristic
322	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
323	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
324	C₂×C₄	#14579123	1	1	yes	characteristic
325	C₂×C₄	#14579123	1	1	yes	characteristic
326	C₂×C₄	#14579167	1	1	yes	characteristic
327	C₂×C₄	#14579167	1	1	yes	characteristic
328	C₂×C₄	#14579169	1	1	yes	characteristic
329	C₂×C₄	#14579169	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	1	no	—
334	C₈	—	316	2	no	—
335	C₈	—	316	2	no	—
336	C₈	—	158	1	no	—
337	C₈	—	158	1	no	—
338	C₈	—	158	1	no	—
339	C₈	—	158	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	316	1	no	—
343	C₂²	—	316	1	no	—
344	C₂²	—	316	1	no	—
345	C₂²	—	632	1	no	—
346	C₂²	—	632	1	no	—
347	C₂²	#19098410	1	1	yes	characteristic
348	C₂²	#19098517	1	1	yes	characteristic
349	C₂²	#19098520	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	316	1	no	—
355	C₄	—	316	1	no	—
356	C₄	—	316	1	no	—
357	C₄	#19098528	1	1	yes	characteristic
358	C₄	#19098528	1	1	yes	characteristic
359	C₄	#19098623	1	1	yes	characteristic
360	C₄	#19098623	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	632	1	no	—
364	C₂	#23763165	1	1	yes	characteristic
365	C₂	#23763199	1	1	yes	characteristic
366	C₂	#23763199	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₉
commutator	C₂³×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1281	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	C₂³×D₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1281	D₇₉
hypocenter	C₇₉	#8158
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

Complements

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

N	H ≅ G/N	type
4. #23761693	C₂	semidirect
5. #23761684	C₂	semidirect
6. #23761245	C₂	semidirect
7. C₇₉×#1281	C₂	semidirect
190. #1281	D₇₉	semidirect
191. C₇₉	#8158	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098625	2
#23761712	2
#23761693	2
#23761684	2
#23761245	2
C₇₉×#1281	2
#8158	79