Subgroups — #34176634

Order 20224.

482 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763380	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#23761226	C₂	1	2	yes	maximal, complemented
4	#23761223	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2855	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
9	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
10	#19098318	C₂²	1	2	yes	—
11	#19098318	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₇₉	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
14	#391×C₇₉	C₄	1	1	yes	characteristic
15	#391×C₇₉	C₂²	1	1	yes	characteristic
16	#389×C₇₉	C₄	1	1	yes	characteristic
17	#389×C₇₉	C₂²	1	1	yes	characteristic
18	#387×C₇₉	—	2	1	no	—
19	#387×C₇₉	—	2	1	no	—
20	#385×C₇₉	—	2	1	no	—
21	#381×C₇₉	—	2	1	no	—
22	#380×C₇₉	C₄	1	1	yes	characteristic
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
26	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
27	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
28	C₄×C₇₉⋊C₈	—	4	2	no	—
29	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
33	C₂²⋊Q₈×C₇₉	—	4	2	no	—
34	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
42	C₂×C₄²×C₇₉	—	4	2	no	—
43	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	#95×C₇₉	—	2	1	no	—
48	#95×C₇₉	—	2	1	no	—
49	#95×C₇₉	—	2	1	no	—
50	#95×C₇₉	—	2	1	no	—
51	#95×C₇₉	—	2	1	no	—
52	#95×C₇₉	—	2	1	no	—
53	#95×C₇₉	—	2	1	no	—
54	C₂×C₇₉⋊C₈	—	2	4	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₂×C₇₉⋊C₈	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	4	2	no	—
59	C₂×C₇₉⋊C₈	—	4	2	no	—
60	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Q₈×C₇₉	—	2	2	no	—
62	C₂×Q₈×C₇₉	—	4	1	no	—
63	C₂×Q₈×C₇₉	—	4	1	no	—
64	C₂²×C₄×C₇₉	—	2	2	no	—
65	C₂²×C₄×C₇₉	—	2	2	no	—
66	C₂²×C₄×C₇₉	—	2	2	no	—
67	C₂²×C₄×C₇₉	—	2	2	no	—
68	C₂²×C₄×C₇₉	—	2	1	no	—
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	2	1	no	—
71	C₂²×C₄×C₇₉	—	2	1	no	—
72	C₂²×C₄×C₇₉	—	4	1	no	—
73	C₂²×C₄×C₇₉	—	4	1	no	—
74	C₂²×C₄×C₇₉	—	4	1	no	—
75	C₂²×C₄×C₇₉	—	4	1	no	—
76	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₄⋊C₄×C₇₉	—	2	2	no	—
85	C₄⋊C₄×C₇₉	—	4	1	no	—
86	C₄⋊C₄×C₇₉	—	4	1	no	—
87	C₄⋊C₄×C₇₉	—	4	2	no	—
88	C₄⋊C₄×C₇₉	—	4	2	no	—
89	C₄⋊C₄×C₇₉	—	4	2	no	—
90	C₂²⋊C₄×C₇₉	—	4	2	no	—
91	C₂²⋊C₄×C₇₉	—	4	2	no	—
92	C₄²×C₇₉	—	2	2	no	—
93	C₄²×C₇₉	—	2	2	no	—
94	C₄²×C₇₉	—	4	1	no	—
95	C₄²×C₇₉	—	4	1	no	—
96	C₂³×C₇₉	—	2	1	no	—
97	C₂³×C₇₉	—	2	1	no	—
98	C₂³×C₇₉	—	2	1	no	—
99	C₂³×C₇₉	—	2	1	no	—
100	C₂³×C₇₉	—	2	1	no	—
101	C₂³×C₇₉	—	2	1	no	—
102	C₂³×C₇₉	#103	1	2	yes	—
103	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₇₉	—	8	2	no	—
105	C₂×C₄×C₇₉	—	2	1	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	4	2	no	—
118	C₂×C₄×C₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	—	4	4	no	—
120	C₂×C₄×C₇₉	—	4	4	no	—
121	C₂×C₄×C₇₉	—	4	1	no	—
122	C₂×C₄×C₇₉	—	4	1	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	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	1	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	1	no	—
133	C₂×C₄×C₇₉	#99	1	2	yes	—
134	C₂×C₄×C₇₉	#103	1	2	yes	—
135	C₂×C₄×C₇₉	#104	1	2	yes	—
136	C₂×C₄×C₇₉	#131	1	2	yes	—
137	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×C₄×C₇₉	#135	1	2	yes	—
140	C₇₉⋊C₈	—	4	4	no	—
141	C₇₉⋊C₈	—	8	2	no	—
142	C₂²×C₇₉	—	2	2	no	—
143	C₂²×C₇₉	—	2	2	no	—
144	C₂²×C₇₉	—	2	2	no	—
145	C₂²×C₇₉	—	4	1	no	—
146	C₂²×C₇₉	—	4	1	no	—
147	C₂²×C₇₉	—	4	1	no	—
148	C₂²×C₇₉	—	4	1	no	—
149	C₂²×C₇₉	#408	1	1	yes	characteristic
150	C₂²×C₇₉	C₂×#99	1	1	yes	characteristic
151	C₂²×C₇₉	#413	1	1	yes	characteristic
152	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
153	C₂²×C₇₉	#416	1	1	yes	characteristic
154	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
155	C₂²×C₇₉	#421	1	1	yes	characteristic
156	C₄×C₇₉	—	2	2	no	—
157	C₄×C₇₉	—	2	2	no	—
158	C₄×C₇₉	—	2	2	no	—
159	C₄×C₇₉	—	2	2	no	—
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	2	1	no	—
163	C₄×C₇₉	—	8	1	no	—
164	C₄×C₇₉	—	8	1	no	—
165	C₄×C₇₉	—	8	1	no	—
166	C₄×C₇₉	—	8	1	no	—
167	#7661	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	4	2	no	—
169	C₂×C₇₉	#1468	1	1	yes	characteristic
170	C₂×C₇₉	#1482	1	1	yes	characteristic
171	C₂×C₇₉	#1493	1	1	yes	characteristic
172	C₂×C₇₉	#1501	1	1	yes	characteristic
173	C₂×C₇₉	#1520	1	1	yes	characteristic
174	C₂×C₇₉	#1529	1	1	yes	characteristic
175	C₂×C₇₉	#1542	1	1	yes	characteristic
176	#2855	D₇₉	1	1	yes	characteristic
177	#1797	—	79	1	no	—
178	C₄×C₂²⋊C₈	—	79	1	no	—
179	#1272	—	79	2	no	—
180	#1271	—	79	2	no	—
181	C₇₉	#7661	1	1	yes	characteristic, complemented, Sylow
182	C₂×C₂²⋊Q₈	Dic₇₉	1	1	yes	characteristic
183	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
184	C₂×C₄⋊C₈	—	158	1	no	—
185	C₂×C₂²⋊C₈	—	158	1	no	—
186	C₂×C₂²⋊C₈	—	158	1	no	—
187	C₂×C₄×C₈	—	158	1	no	—
188	#391	Dic₇₉	1	1	yes	characteristic
189	#391	C₂×D₇₉	1	1	yes	characteristic
190	#389	Dic₇₉	1	1	yes	characteristic
191	#389	C₂×D₇₉	1	1	yes	characteristic
192	#387	—	2	1	no	—
193	#387	—	2	1	no	—
194	#385	—	2	1	no	—
195	#381	—	2	1	no	—
196	#380	Dic₇₉	1	1	yes	characteristic
197	#336	—	79	2	no	—
198	#336	—	79	2	no	—
199	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
200	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
201	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
203	C₂²×C₈	—	158	2	no	—
204	C₂²×C₈	—	158	2	no	—
205	C₂²⋊Q₈	—	4	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	4	2	no	—
215	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
219	C₄⋊C₈	—	316	2	no	—
220	C₂²⋊C₈	—	316	2	no	—
221	C₂²⋊C₈	—	158	4	no	—
222	C₄×C₈	—	316	2	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	#95	—	2	1	no	—
227	#95	—	2	1	no	—
228	#95	—	2	1	no	—
229	#95	—	2	1	no	—
230	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
249	C₂²×C₄	#11929611	1	2	yes	—
250	C₂²×C₄	#11929614	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
254	C₂×C₈	—	316	2	no	—
255	C₂×C₈	—	316	2	no	—
256	C₂×C₈	—	316	2	no	—
257	C₂×C₈	—	316	2	no	—
258	C₂×C₈	—	316	2	no	—
259	C₂×C₈	—	158	4	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	#14579085	1	2	yes	—
279	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
280	Q₈	—	8	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	#14579076	1	2	yes	—
310	C₂×C₄	#14579085	1	2	yes	—
311	C₂×C₄	#14579088	1	2	yes	—
312	C₂×C₄	#14579168	1	2	yes	—
313	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
314	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
315	C₂×C₄	#14579181	1	2	yes	—
316	C₈	—	316	4	no	—
317	C₈	—	632	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#19098530	1	1	yes	characteristic
326	C₂²	C₂×#14579076	1	1	yes	characteristic
327	C₂²	#19098547	1	1	yes	characteristic
328	C₂²	C₂×#14579085	1	1	yes	characteristic
329	C₂²	#19098559	1	1	yes	characteristic
330	C₂²	C₂×#14579088	1	1	yes	characteristic
331	C₂²	#19098580	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	8	1	no	—
340	C₄	—	8	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₂	—	4	2	no	—
344	C₂	#23761748	1	1	yes	characteristic
345	C₂	#23761808	1	1	yes	characteristic
346	C₂	#23761855	1	1	yes	characteristic
347	C₂	#23761891	1	1	yes	characteristic
348	C₂	#23761972	1	1	yes	characteristic
349	C₂	#23762007	1	1	yes	characteristic
350	C₂	#23762057	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2855	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2855	D₇₉
hypocenter	C₇₉	#7661
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2855

Complements

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

N	H ≅ G/N	type
3. #23761226	C₂	semidirect
4. #23761223	C₂	semidirect
181. C₇₉	#7661	semidirect

Maximal subgroups

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

subgroup	index
#23763380	2
C₄×C₂²⋊(C₇₉⋊C₈)	2
#23761226	2
#23761223	2
C₇₉×#2855	2
#7661	79