Subgroups — #34182292

Order 20224.

420 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771962	C₂	1	1	yes	maximal, characteristic
2	#23763381	C₂	1	1	yes	maximal, characteristic
3	#23762170	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761712	C₂	1	1	yes	maximal, characteristic
5	#23761296	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761295	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₇₉	—	2	1	no	—
9	C₂×#14579118	C₂²	1	1	yes	characteristic
10	#19098616	C₂²	1	1	yes	characteristic
11	#19098588	—	2	2	no	—
12	#19098588	—	2	2	no	—
13	#19098587	C₂²	1	1	yes	characteristic
14	#19098586	—	2	1	no	—
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₇₉)⋊C₈	—	2	1	no	—
17	#19098317	—	2	1	no	—
18	#19098313	—	2	2	no	—
19	#19098288	C₄	1	2	yes	—
20	#19098286	C₄	1	2	yes	—
21	#431×C₇₉	C₂²	1	1	yes	characteristic
22	#423×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#334×C₇₉	—	2	2	no	—
25	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
26	C₂×#11929608	—	2	1	no	—
27	C₂×#11929607	—	2	2	no	—
28	M₁₆×D₇₉	—	4	2	no	—
29	M₁₆×D₇₉	—	4	2	no	—
30	C₂²×C₇₉⋊C₈	—	2	1	no	—
31	C₂×C₈×D₇₉	—	2	2	no	—
32	#14579118	—	2	1	no	—
33	#14579118	—	2	1	no	—
34	#14579118	—	2	1	no	—
35	#14579118	—	2	1	no	—
36	C₂×Dic₇₉⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₇₉⋊C₄	Q₈	1	1	yes	characteristic
38	C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×(C₂×D₇₉)⋊C₄	Q₈	1	1	yes	characteristic
40	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
41	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
42	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
43	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
46	Dic₇₉⋊C₈	—	2	4	no	—
47	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
48	(C₂×D₇₉)⋊C₈	—	4	2	no	—
49	C₄×C₇₉⋊C₈	—	2	1	no	—
50	C₂×M₁₆×C₇₉	—	2	1	no	—
51	C₂²×C₈×C₇₉	—	2	1	no	—
52	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₇₉	—	2	1	no	—
54	C₄⋊C₈×C₇₉	—	2	1	no	—
55	C₄⋊C₈×C₇₉	—	2	1	no	—
56	C₄⋊C₈×C₇₉	—	2	1	no	—
57	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
58	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₇₉	—	2	1	no	—
60	C₂³×D₇₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₇₉	—	2	1	no	—
65	C₂×C₄×D₇₉	—	2	1	no	—
66	C₂×C₄×D₇₉	—	2	1	no	—
67	C₂×C₄×D₇₉	—	2	1	no	—
68	C₂×C₄×D₇₉	SD₁₆	1	2	yes	—
69	C₂×C₄×D₇₉	SD₁₆	1	2	yes	—
70	#11929608	—	4	1	no	—
71	#11929608	—	4	1	no	—
72	#11929607	—	4	4	no	—
73	C₂×C₇₉⋊C₈	—	2	2	no	—
74	C₂×C₇₉⋊C₈	—	2	2	no	—
75	C₂×C₇₉⋊C₈	—	4	1	no	—
76	C₂×C₇₉⋊C₈	—	4	1	no	—
77	C₂×C₇₉⋊C₈	—	4	1	no	—
78	C₈×D₇₉	—	4	4	no	—
79	Dic₇₉⋊C₄	—	2	1	no	—
80	Dic₇₉⋊C₄	—	2	1	no	—
81	Dic₇₉⋊C₄	—	2	1	no	—
82	Dic₇₉⋊C₄	—	2	1	no	—
83	(C₂×D₇₉)⋊C₄	—	2	1	no	—
84	(C₂×D₇₉)⋊C₄	—	2	1	no	—
85	(C₂×D₇₉)⋊C₄	—	2	1	no	—
86	(C₂×D₇₉)⋊C₄	—	2	1	no	—
87	C₄×Dic₇₉	—	2	1	no	—
88	C₄×Dic₇₉	—	2	1	no	—
89	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
92	M₁₆×C₇₉	—	4	1	no	—
93	M₁₆×C₇₉	—	4	1	no	—
94	C₂×C₈×C₇₉	—	2	2	no	—
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	—	4	1	no	—
97	C₂×C₈×C₇₉	—	4	1	no	—
98	C₂×C₈×C₇₉	—	4	1	no	—
99	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₉	#41	1	1	yes	characteristic
102	C₄²×C₇₉	#41	1	1	yes	characteristic
103	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×D₇₉	—	2	1	no	—
105	C₂²×D₇₉	—	2	1	no	—
106	C₂²×D₇₉	—	4	1	no	—
107	C₂²×D₇₉	—	4	1	no	—
108	C₂²×D₇₉	#106	1	2	yes	—
109	C₂×C₄×C₇₉	—	4	1	no	—
110	C₂×C₄×C₇₉	—	4	1	no	—
111	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
121	C₂×Dic₇₉	—	2	1	no	—
122	C₂×Dic₇₉	—	2	1	no	—
123	C₂×Dic₇₉	—	2	1	no	—
124	C₂×Dic₇₉	—	2	1	no	—
125	C₂×Dic₇₉	—	4	1	no	—
126	C₂×Dic₇₉	—	4	1	no	—
127	C₂×Dic₇₉	#106	1	2	yes	—
128	C₄×D₇₉	—	2	2	no	—
129	C₄×D₇₉	—	2	2	no	—
130	C₄×D₇₉	—	2	2	no	—
131	C₄×D₇₉	—	2	2	no	—
132	C₈×C₇₉	—	4	2	no	—
133	C₈×C₇₉	—	4	2	no	—
134	C₈×C₇₉	—	8	1	no	—
135	C₇₉⋊C₈	—	4	2	no	—
136	C₇₉⋊C₈	—	4	2	no	—
137	C₇₉⋊C₈	—	8	1	no	—
138	C₂²×C₇₉	—	2	1	no	—
139	C₂²×C₇₉	—	2	1	no	—
140	C₂²×C₇₉	#385	1	1	yes	characteristic
141	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
142	C₂²×C₇₉	#428	1	1	yes	characteristic
143	C₂×D₇₉	—	2	2	no	—
144	C₂×D₇₉	—	4	1	no	—
145	C₂×D₇₉	—	4	1	no	—
146	C₂×D₇₉	—	4	2	no	—
147	C₂×D₇₉	—	8	1	no	—
148	C₄×C₇₉	—	2	1	no	—
149	C₄×C₇₉	—	2	1	no	—
150	C₄×C₇₉	—	2	1	no	—
151	C₄×C₇₉	—	2	1	no	—
152	C₄×C₇₉	#448	1	1	yes	characteristic
153	C₄×C₇₉	#450	1	1	yes	characteristic
154	C₄×C₇₉	#481	1	1	yes	characteristic
155	C₄×C₇₉	#483	1	1	yes	characteristic
156	Dic₇₉	—	2	2	no	—
157	Dic₇₉	—	4	1	no	—
158	Dic₇₉	—	4	1	no	—
159	#8862	—	79	1	no	maximal, Sylow
160	C₂×C₇₉	—	2	1	no	—
161	C₂×C₇₉	—	2	1	no	—
162	C₂×C₇₉	#1853	1	1	yes	characteristic
163	C₂×C₇₉	#1858	1	1	yes	characteristic
164	C₂×C₇₉	#1868	1	1	yes	characteristic
165	D₇₉	—	4	2	no	—
166	#2905	—	79	1	no	—
167	#1798	—	79	1	no	—
168	#1570	—	79	1	no	—
169	#1570	D₇₉	1	1	yes	characteristic, complemented
170	#1458	—	79	1	no	—
171	#1301	—	79	1	no	—
172	#1301	—	79	1	no	—
173	C₇₉	#8862	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	158	1	no	—
175	C₂×#117	—	79	1	no	—
176	#431	—	79	1	no	—
177	#431	C₂×D₇₉	1	1	yes	characteristic
178	#424	—	158	2	no	—
179	#424	—	158	2	no	—
180	#423	—	79	1	no	—
181	#423	—	158	1	no	—
182	#423	C₂×D₇₉	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	79	1	no	—
184	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
185	C₂×C₂²⋊C₈	—	158	1	no	—
186	#336	—	158	1	no	—
187	#334	—	2	2	no	—
188	#334	—	158	2	no	—
189	#324	—	79	2	no	—
190	#323	—	79	2	no	—
191	C₂³×C₄	—	79	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	316	2	no	—
194	C₂×M₁₆	—	316	2	no	—
195	C₂×M₁₆	—	158	2	no	—
196	C₂×M₁₆	—	158	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	158	2	no	—
199	C₂²×C₈	—	158	1	no	—
200	#117	—	158	1	no	—
201	#117	—	158	1	no	—
202	#117	—	158	1	no	—
203	#117	—	158	1	no	—
204	C₂×C₄⋊C₄	—	79	1	no	—
205	C₂×C₄⋊C₄	—	79	1	no	—
206	C₂×C₂²⋊C₄	—	79	1	no	—
207	C₂×C₂²⋊C₄	—	79	1	no	—
208	C₂×C₄²	—	79	1	no	—
209	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	79	2	no	—
215	C₄⋊C₈	—	158	4	no	—
216	C₄⋊C₈	—	158	1	no	—
217	C₄⋊C₈	—	158	1	no	—
218	C₄⋊C₈	—	158	1	no	—
219	C₄⋊C₈	—	158	1	no	—
220	C₄⋊C₈	C₂²⋊D₇₉	1	2	yes	—
221	C₂²⋊C₈	—	316	2	no	—
222	C₂²⋊C₈	—	79	2	no	—
223	C₂²⋊C₈	C₄×D₇₉	1	2	yes	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	158	1	no	—
226	C₂⁴	—	79	1	no	—
227	C₂²×C₄	—	79	1	no	—
228	C₂²×C₄	—	79	1	no	—
229	C₂²×C₄	—	79	1	no	—
230	C₂²×C₄	—	79	2	no	—
231	C₂²×C₄	—	79	2	no	—
232	C₂²×C₄	—	158	1	no	—
233	C₂²×C₄	—	158	1	no	—
234	C₂²×C₄	—	158	1	no	—
235	C₂²×C₄	—	158	1	no	—
236	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	316	4	no	—
242	M₁₆	—	316	1	no	—
243	M₁₆	—	316	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	316	4	no	—
250	C₂×C₈	—	316	1	no	—
251	C₂×C₈	—	316	1	no	—
252	C₂×C₈	—	316	1	no	—
253	C₂×C₈	—	158	2	no	—
254	C₂×C₈	—	158	2	no	—
255	C₄⋊C₄	—	158	1	no	—
256	C₄⋊C₄	—	158	1	no	—
257	C₄⋊C₄	—	158	1	no	—
258	C₄⋊C₄	—	158	1	no	—
259	C₂²⋊C₄	—	158	1	no	—
260	C₂²⋊C₄	—	158	1	no	—
261	C₂²⋊C₄	—	158	1	no	—
262	C₂²⋊C₄	—	158	1	no	—
263	C₄²	—	158	1	no	—
264	C₄²	—	158	1	no	—
265	C₄²	D₄×D₇₉	1	1	yes	characteristic
266	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
267	C₄²	#11929624	1	1	yes	characteristic
268	C₄²	#11929625	1	1	yes	characteristic
269	C₂³	—	316	1	no	—
270	C₂³	—	316	1	no	—
271	C₂³	—	79	2	no	—
272	C₂³	—	158	1	no	—
273	C₂³	—	158	1	no	—
274	C₂³	C₄⋊C₄×D₇₉	1	1	yes	characteristic
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	316	1	no	—
278	C₂×C₄	—	316	1	no	—
279	C₂×C₄	—	79	2	no	—
280	C₂×C₄	—	158	1	no	—
281	C₂×C₄	—	158	1	no	—
282	C₂×C₄	—	158	1	no	—
283	C₂×C₄	—	158	1	no	—
284	C₂×C₄	—	158	2	no	—
285	C₂×C₄	—	158	2	no	—
286	C₂×C₄	—	158	2	no	—
287	C₂×C₄	—	158	2	no	—
288	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
289	C₂×C₄	#14579123	1	1	yes	characteristic
290	C₂×C₄	#14579129	1	1	yes	characteristic
291	C₂×C₄	#14579137	1	1	yes	characteristic
292	C₂×C₄	#14579139	1	1	yes	characteristic
293	C₂×C₄	#14579146	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
295	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
296	C₂×C₄	#14579188	1	1	yes	characteristic
297	C₂×C₄	#14579190	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	316	2	no	—
302	C₈	—	316	2	no	—
303	C₈	—	632	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	316	2	no	—
307	C₂²	—	316	1	no	—
308	C₂²	—	316	1	no	—
309	C₂²	—	632	1	no	—
310	C₂²	—	158	2	no	—
311	C₂²	#19098434	1	1	yes	characteristic
312	C₂²	#106×D₇₉	1	1	yes	characteristic
313	C₂²	#19098602	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	316	1	no	—
319	C₄	—	316	1	no	—
320	C₄	—	158	2	no	—
321	C₄	#19098700	1	1	yes	characteristic
322	C₄	#19098712	1	1	yes	characteristic
323	C₄	#19098883	1	1	yes	characteristic
324	C₄	#19098895	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	316	2	no	—
328	C₂	#23763591	1	1	yes	characteristic
329	C₂	#23763618	1	1	yes	characteristic
330	C₂	#23763684	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1570	C₂
socle	C₂²×C₇₉	#385
prime radical	C₂³×D₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1570	D₇₉
hypocenter	C₇₉	#8862
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

Complements

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

N	H ≅ G/N	type
3. #23762170	C₂	semidirect
5. #23761296	C₂	semidirect
6. #23761295	C₂	semidirect
7. C₇₉×#1570	C₂	semidirect
169. #1570	D₇₉	semidirect
173. C₇₉	#8862	semidirect

Maximal subgroups

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

subgroup	index
#23771962	2
#23763381	2
#23762170	2
#23761712	2
#23761296	2
#23761295	2
C₇₉×#1570	2
#8862	79