Subgroups — #34286506

Order 20224.

906 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	C₂×#19098624	C₂	1	1	yes	maximal, characteristic
2	#23765898	C₂	1	1	yes	maximal, characteristic
3	#23763272	C₂	1	2	yes	maximal, complemented
4	#23763267	C₂	1	2	yes	maximal, complemented
5	#23763181	C₂	1	2	yes	maximal
6	#23763176	C₂	1	2	yes	maximal
7	#23763153	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763148	C₂	1	1	yes	maximal, characteristic, complemented
9	#23763086	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14579071	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
12	#19098624	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×#14579072	C₂²	1	1	yes	characteristic
16	C₂×#14579071	C₂²	1	2	yes	—
17	C₂×#14579071	C₂²	1	2	yes	—
18	C₂×#14579071	C₂²	1	2	yes	—
19	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
21	#19098456	C₄	1	4	yes	—
22	#19098449	C₄	1	4	yes	—
23	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
24	C₄×C₂²⋊Dic₇₉	C₂²	1	2	yes	—
25	#19098409	C₂²	1	1	yes	characteristic
26	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
27	#19098317	C₂²	1	2	yes	—
28	#19098317	C₂²	1	2	yes	—
29	#19098317	C₂²	1	2	yes	—
30	#19098317	C₂²	1	2	yes	—
31	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
32	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
34	#405×C₇₉	C₄	1	8	yes	complemented
35	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊Dic₇₉	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
45	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
46	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
47	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
48	#14579072	C₂×C₄	1	4	yes	—
49	#14579071	—	2	4	no	—
50	#14579071	—	2	4	no	—
51	#14579071	—	2	4	no	—
52	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
53	C₈×Dic₇₉	—	2	4	no	—
54	C₇₉⋊#95	C₂×C₄	1	4	yes	—
55	C₇₉⋊#95	C₂×C₄	1	4	yes	—
56	C₇₉⋊#95	C₂×C₄	1	4	yes	—
57	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₇₉	—	2	4	no	—
59	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₇₉	—	2	4	no	—
61	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
62	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	#117×C₇₉	C₂×C₄	1	4	yes	—
67	#117×C₇₉	C₂×C₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
72	#97×C₇₉	C₂×C₄	1	4	yes	—
73	#97×C₇₉	C₂×C₄	1	4	yes	—
74	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
75	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
76	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
77	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
78	C₂×C₇₉⋊C₈	—	2	2	no	—
79	C₂×C₇₉⋊C₈	—	2	2	no	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₂×C₇₉⋊C₈	—	2	2	no	—
82	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
83	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
84	C₄⋊Dic₇₉	—	4	4	no	—
85	C₂²⋊Dic₇₉	—	4	4	no	—
86	C₄×Dic₇₉	—	2	2	no	—
87	C₄×Dic₇₉	—	2	2	no	—
88	C₄×Dic₇₉	—	2	2	no	—
89	C₄×Dic₇₉	—	2	2	no	—
90	C₄×Dic₇₉	—	4	4	no	—
91	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	—	2	2	no	—
93	C₂²×C₄×C₇₉	—	2	2	no	—
94	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
100	M₁₆×C₇₉	—	2	8	no	—
101	C₂×C₈×C₇₉	—	2	8	no	—
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	—	2	2	no	—
104	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₇₉	#41	1	4	yes	—
109	C₂×C₈×C₇₉	#41	1	4	yes	—
110	C₄⋊C₄×C₇₉	C₄²	1	8	yes	—
111	C₂²⋊C₄×C₇₉	C₄²	1	8	yes	—
112	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
113	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
114	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
115	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
116	C₂³×C₇₉	—	2	2	no	—
117	C₂³×C₇₉	—	2	4	no	—
118	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
119	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	—	2	2	no	—
121	C₂×C₄×C₇₉	—	2	4	no	—
122	C₂×C₄×C₇₉	—	2	4	no	—
123	C₂×C₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	2	2	no	—
125	C₂×C₄×C₇₉	—	2	4	no	—
126	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
127	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
128	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
132	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
133	C₂×Dic₇₉	—	2	4	no	—
134	C₂×Dic₇₉	—	2	4	no	—
135	C₂×Dic₇₉	—	2	4	no	—
136	C₂×Dic₇₉	—	2	4	no	—
137	C₂×Dic₇₉	—	2	4	no	—
138	C₂×Dic₇₉	—	2	4	no	—
139	C₈×C₇₉	—	2	2	no	—
140	C₈×C₇₉	—	2	2	no	—
141	C₈×C₇₉	#117	1	4	yes	—
142	C₈×C₇₉	#117	1	4	yes	—
143	C₇₉⋊C₈	—	2	4	no	—
144	C₇₉⋊C₈	—	2	4	no	—
145	C₂²×C₇₉	—	2	4	no	—
146	C₂²×C₇₉	—	2	2	no	—
147	C₂²×C₇₉	—	4	4	no	—
148	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₇₉	C₄×#41	1	2	yes	—
150	C₂²×C₇₉	#568	1	4	yes	—
151	C₄×C₇₉	—	2	4	no	—
152	C₄×C₇₉	—	2	4	no	—
153	C₄×C₇₉	—	2	2	no	—
154	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
155	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
156	C₄×C₇₉	C₄×#41	1	2	yes	—
157	Dic₇₉	—	4	4	no	—
158	Dic₇₉	—	4	4	no	—
159	#20475	—	79	1	no	maximal, Sylow
160	C₂×C₇₉	—	4	2	no	—
161	C₂×C₇₉	C₄×#117	1	1	yes	characteristic
162	C₂×C₇₉	#2875	1	2	yes	—
163	C₂×C₇₉	#2967	1	4	yes	—
164	C₂×#433	—	79	1	no	—
165	C₂×#405	D₇₉	1	1	yes	characteristic
166	#2298	—	79	1	no	—
167	#1771	—	79	2	no	—
168	#1770	—	79	2	no	—
169	#1753	—	79	2	no	—
170	#1752	—	79	2	no	—
171	#1746	—	79	1	no	—
172	#1745	—	79	1	no	—
173	#1726	—	79	1	no	—
174	C₄×#97	—	79	1	no	—
175	C₇₉	#20475	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
177	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
178	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
179	#433	—	79	8	no	—
180	C₂×C₄⋊C₈	—	79	2	no	—
181	C₂×C₂²⋊C₈	—	79	2	no	—
182	#405	Dic₇₉	1	8	yes	complemented
183	C₂×#97	—	79	1	no	—
184	C₂×#97	—	79	2	no	—
185	C₂×#97	—	79	2	no	—
186	C₂×#97	—	79	2	no	—
187	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
188	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
189	C₂×C₄×C₈	—	79	1	no	—
190	C₂×C₄×C₈	—	79	2	no	—
191	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
193	#388	—	79	4	no	—
194	#387	—	79	4	no	—
195	C₄×C₄⋊C₄	—	79	2	no	—
196	C₄×C₂²⋊C₄	—	79	2	no	—
197	#376	—	79	1	no	—
198	C₄³	—	79	1	no	—
199	#336	—	79	2	no	—
200	#336	—	79	2	no	—
201	#336	—	79	2	no	—
202	#336	—	79	2	no	—
203	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
204	C₂×M₁₆	—	2	4	no	—
205	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	—	79	2	no	—
208	C₂²×C₈	—	79	2	no	—
209	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
210	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
214	#117	C₄×D₇₉	1	4	yes	—
215	#117	C₂×Dic₇₉	1	4	yes	—
216	C₂×C₄⋊C₄	—	79	4	no	—
217	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
218	C₂×C₂²⋊C₄	—	79	4	no	—
219	C₂×C₂²⋊C₄	C₄×D₇₉	1	2	yes	—
220	C₂×C₄²	—	79	4	no	—
221	C₂×C₄²	—	79	2	no	—
222	C₂×C₄²	—	79	2	no	—
223	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
225	C₄⋊C₈	—	79	8	no	—
226	C₂²⋊C₈	—	79	8	no	—
227	#97	—	79	4	no	—
228	#97	—	158	4	no	—
229	#97	—	158	4	no	—
230	#97	—	158	4	no	—
231	#97	C₂×Dic₇₉	1	4	yes	—
232	#97	C₂×Dic₇₉	1	4	yes	—
233	C₄×C₈	—	79	4	no	—
234	C₄×C₈	—	158	4	no	—
235	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
236	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
237	#95	—	79	4	no	—
238	#95	—	79	4	no	—
239	#95	—	79	4	no	—
240	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	79	4	no	—
244	C₂²×C₄	—	79	4	no	—
245	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
248	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
250	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
251	M₁₆	—	2	8	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	79	4	no	—
256	C₂×C₈	—	79	4	no	—
257	C₂×C₈	—	158	2	no	—
258	C₂×C₈	—	158	2	no	—
259	C₂×C₈	—	158	2	no	—
260	C₂×C₈	—	158	2	no	—
261	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
262	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
263	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
264	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
265	C₂×C₈	#11929624	1	4	yes	—
266	C₂×C₈	#11929624	1	4	yes	—
267	C₄⋊C₄	—	316	4	no	—
268	C₄⋊C₄	C₄×Dic₇₉	1	8	yes	—
269	C₂²⋊C₄	—	316	4	no	—
270	C₂²⋊C₄	C₄×Dic₇₉	1	8	yes	—
271	C₄²	—	316	4	no	—
272	C₄²	—	158	2	no	—
273	C₄²	—	158	2	no	—
274	C₄²	—	158	2	no	—
275	C₄²	—	158	2	no	—
276	C₄²	C₂×C₄×D₇₉	1	2	yes	—
277	C₄²	C₂×C₄×D₇₉	1	2	yes	—
278	C₄²	C₂²×Dic₇₉	1	2	yes	—
279	C₄²	C₂²×Dic₇₉	1	2	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	4	no	—
282	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
283	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	158	4	no	—
291	C₂×C₄	—	158	4	no	—
292	C₂×C₄	—	158	4	no	—
293	C₂×C₄	—	158	4	no	—
294	C₂×C₄	—	158	4	no	—
295	C₂×C₄	—	158	4	no	—
296	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
297	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
298	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
301	C₂×C₄	C₂×#11929624	1	2	yes	—
302	C₂×C₄	C₂×#11929624	1	2	yes	—
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₈	—	158	4	no	—
306	C₈	—	158	4	no	—
307	C₈	#14579120	1	4	yes	—
308	C₈	#14579120	1	4	yes	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
313	C₂²	C₄×#11929624	1	2	yes	—
314	C₂²	#19099328	1	4	yes	—
315	C₄	—	2	4	no	—
316	C₄	—	2	4	no	—
317	C₄	—	2	2	no	—
318	C₄	—	316	4	no	—
319	C₄	—	316	4	no	—
320	C₄	C₂×#14579120	1	1	yes	characteristic
321	C₄	C₂×#14579120	1	1	yes	characteristic
322	C₄	C₄×#11929624	1	2	yes	—
323	C₂	—	4	2	no	—
324	C₂	C₄×#14579120	1	1	yes	characteristic
325	C₂	#23771826	1	2	yes	—
326	C₂	#23772381	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₂×#405×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	#20475
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #23763272	C₂	semidirect
4. #23763267	C₂	semidirect
7. #23763153	C₂	semidirect
8. #23763148	C₂	semidirect
9. #23763086	C₂	semidirect
10. C₄×#14579071	C₂	semidirect
12. #19098624	C₄	semidirect
34. #405×C₇₉	C₄	semidirect
175. C₇₉	#20475	semidirect
182. #405	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098624	2
#23765898	2
#23763272	2
#23763267	2
#23763181	2
#23763176	2
#23763153	2
#23763148	2
#23763086	2
C₄×#14579071	2
C₂×#405×C₇₉	2
#20475	79