Subgroups — #34179347

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772111	C₂	1	1	yes	maximal, characteristic
2	#23763270	C₂	1	1	yes	maximal, characteristic, complemented
3	#23762050	C₂	1	1	yes	maximal, characteristic
4	#23761896	C₂	1	1	yes	maximal, characteristic
5	#23761693	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579169	—	2	1	no	—
9	#19099113	C₂²	1	1	yes	characteristic
10	#19098640	—	2	1	no	—
11	#19098640	C₄	1	2	yes	complemented
12	#19098628	—	2	1	no	—
13	#19098628	C₄	1	2	yes	complemented
14	#19098616	C₂²	1	1	yes	characteristic
15	#19098531	—	2	1	no	—
16	C₂×#14579071	—	2	1	no	—
17	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
19	#19098317	—	2	1	no	—
20	#19098310	C₄	1	2	yes	complemented
21	#19098306	C₄	1	2	yes	complemented
22	#19098298	C₄	1	4	yes	—
23	#19098285	—	2	1	no	—
24	#19098284	—	2	1	no	—
25	#431×C₇₉	C₂²	1	1	yes	characteristic
26	#423×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#11929624	—	2	1	no	—
29	#14579169	—	4	2	no	—
30	#14579169	—	4	2	no	—
31	C₂×#11929607	—	2	2	no	—
32	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₈×D₇₉	—	2	2	no	—
35	#14579161	C₂×C₄	1	1	yes	characteristic
36	#14579158	C₂×C₄	1	1	yes	characteristic
37	#14579149	—	2	1	no	—
38	#14579135	—	2	1	no	—
39	#14579135	C₂×C₄	1	2	yes	—
40	C₄×D₃₁₆	—	2	1	no	—
41	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₇₉	—	2	1	no	—
43	C₂×C₄×Dic₇₉	—	2	1	no	—
44	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
45	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
46	(C₂×D₇₉)⋊C₈	—	2	2	no	—
47	(C₂×D₇₉)⋊C₈	—	2	2	no	—
48	#14579071	—	4	2	no	—
49	C₄×C₇₉⋊C₈	—	2	1	no	—
50	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
53	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
54	C₇₉⋊#95	C₂×C₄	1	2	yes	—
55	C₂×M₁₆×C₇₉	—	2	1	no	—
56	C₂²×C₈×C₇₉	—	2	1	no	—
57	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₇₉	—	2	1	no	—
59	C₄⋊C₈×C₇₉	—	2	1	no	—
60	C₄⋊C₈×C₇₉	—	2	1	no	—
61	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₇₉	—	2	1	no	—
65	#11929624	—	4	1	no	—
66	#11929624	—	4	1	no	—
67	#11929624	—	4	1	no	—
68	#11929624	—	4	1	no	—
69	C₂×Dic₁₅₈	—	2	1	no	—
70	C₂×D₃₁₆	—	2	1	no	—
71	C₂×D₃₁₆	—	2	1	no	—
72	C₂×D₃₁₆	C₄²	1	2	yes	—
73	C₂×C₂²⋊D₇₉	—	2	2	no	—
74	C₂²×Dic₇₉	—	2	2	no	—
75	C₂×C₄×D₇₉	—	2	2	no	—
76	#11929607	—	4	4	no	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	2	1	no	—
79	C₂×C₇₉⋊C₈	—	2	1	no	—
80	C₂×C₇₉⋊C₈	—	2	1	no	—
81	C₂×C₇₉⋊C₈	—	2	1	no	—
82	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₈×D₇₉	—	4	4	no	—
87	C₄⋊Dic₇₉	—	2	1	no	—
88	C₄⋊Dic₇₉	—	4	2	no	—
89	C₄⋊Dic₇₉	C₄²	1	2	yes	—
90	(C₂×D₇₉)⋊C₄	—	2	2	no	—
91	(C₂×D₇₉)⋊C₄	—	2	2	no	—
92	C₄×Dic₇₉	—	4	1	no	—
93	C₄×Dic₇₉	—	4	1	no	—
94	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
97	M₁₆×C₇₉	—	4	1	no	—
98	M₁₆×C₇₉	—	4	1	no	—
99	C₂×C₈×C₇₉	—	2	2	no	—
100	C₂×C₈×C₇₉	—	2	2	no	—
101	C₂×C₈×C₇₉	—	2	2	no	—
102	C₂×C₈×C₇₉	—	4	1	no	—
103	C₂×C₈×C₇₉	—	4	1	no	—
104	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇₉	#41	1	1	yes	characteristic
108	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
109	C₂²×D₇₉	—	2	2	no	—
110	C₂×C₄×C₇₉	—	2	2	no	—
111	C₂×C₄×C₇₉	—	4	1	no	—
112	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
122	C₂²⋊D₇₉	—	4	2	no	—
123	C₂²⋊D₇₉	—	4	2	no	—
124	C₂×Dic₇₉	—	2	2	no	—
125	C₂×Dic₇₉	—	4	2	no	—
126	C₂×Dic₇₉	—	4	2	no	—
127	C₂×Dic₇₉	—	4	2	no	—
128	D₃₁₆	—	4	2	no	—
129	D₃₁₆	—	4	1	no	—
130	D₃₁₆	—	4	1	no	—
131	D₃₁₆	—	4	2	no	—
132	C₄×D₇₉	—	4	2	no	—
133	C₄×D₇₉	—	4	2	no	—
134	Dic₁₅₈	—	4	1	no	—
135	Dic₁₅₈	—	4	1	no	—
136	C₈×C₇₉	—	4	2	no	—
137	C₈×C₇₉	—	4	2	no	—
138	C₈×C₇₉	—	4	2	no	—
139	C₇₉⋊C₈	—	2	1	no	—
140	C₇₉⋊C₈	—	2	1	no	—
141	C₇₉⋊C₈	—	2	1	no	—
142	C₇₉⋊C₈	—	2	1	no	—
143	C₇₉⋊C₈	—	4	2	no	—
144	C₂²×C₇₉	—	2	1	no	—
145	C₂²×C₇₉	—	2	1	no	—
146	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₇₉	#440	1	1	yes	characteristic
148	C₂²×C₇₉	#442	1	1	yes	characteristic
149	C₂×D₇₉	—	4	2	no	—
150	C₂×D₇₉	—	4	2	no	—
151	C₂×D₇₉	—	4	2	no	—
152	C₄×C₇₉	—	2	1	no	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	#417	1	1	yes	characteristic
156	C₄×C₇₉	#418	1	1	yes	characteristic
157	C₄×C₇₉	#440	1	1	yes	characteristic
158	C₄×C₇₉	#442	1	1	yes	characteristic
159	C₄×C₇₉	#444	1	2	yes	—
160	Dic₇₉	—	4	2	no	—
161	Dic₇₉	—	8	2	no	—
162	#8253	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	—	2	1	no	—
164	C₂×C₇₉	—	2	1	no	—
165	C₂×C₇₉	#1757	1	1	yes	characteristic
166	C₂×C₇₉	#1762	1	1	yes	characteristic
167	C₂×C₇₉	#1765	1	1	yes	characteristic
168	D₇₉	—	8	2	no	—
169	#2931	—	79	1	no	—
170	#1770	—	79	1	no	—
171	#1540	—	79	1	no	—
172	#1502	—	79	1	no	—
173	#1452	—	79	1	no	—
174	#1303	—	79	1	no	—
175	#1301	D₇₉	1	1	yes	characteristic, complemented
176	C₇₉	#8253	1	1	yes	characteristic, complemented, Sylow
177	C₂×#131	—	158	1	no	—
178	#532	—	79	1	no	—
179	#436	—	79	2	no	—
180	#436	—	158	1	no	—
181	C₈×D₄	—	79	2	no	—
182	C₈×D₄	—	158	1	no	—
183	#431	—	79	1	no	—
184	#431	C₂×D₇₉	1	1	yes	characteristic
185	#423	C₂×D₇₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
187	#408	—	158	1	no	—
188	C₂×#97	—	158	1	no	—
189	C₂×C₄×C₈	—	79	1	no	—
190	C₄×C₄⋊C₄	—	79	1	no	—
191	#336	—	158	1	no	—
192	#332	—	79	2	no	—
193	#331	—	79	2	no	—
194	#327	—	79	4	no	—
195	#322	—	158	1	no	—
196	#322	—	158	1	no	—
197	C₂×#41	—	158	1	no	—
198	#131	—	316	2	no	—
199	#131	—	316	2	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	158	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	79	1	no	—
204	C₂²×C₈	—	79	1	no	—
205	C₂²×C₈	—	158	2	no	—
206	C₄⋊Q₈	—	79	1	no	—
207	#127	—	79	1	no	—
208	#124	—	158	1	no	—
209	#121	—	79	2	no	—
210	#121	—	158	1	no	—
211	C₄×D₄	—	79	2	no	—
212	C₄×D₄	—	158	1	no	—
213	C₂×C₄⋊C₄	—	158	1	no	—
214	C₂×C₄²	—	158	1	no	—
215	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	79	2	no	—
220	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
221	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
222	C₂²⋊C₈	—	158	2	no	—
223	C₂²⋊C₈	—	158	2	no	—
224	C₂²⋊C₈	—	158	1	no	—
225	C₂²⋊C₈	C₄×D₇₉	1	2	yes	—
226	#97	—	316	2	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	79	1	no	—
229	C₄×C₈	—	79	1	no	—
230	C₄×C₈	—	79	1	no	—
231	C₄×C₈	—	79	1	no	—
232	C₄×C₈	—	158	1	no	—
233	#95	—	79	2	no	—
234	#41	—	316	1	no	—
235	#41	—	316	1	no	—
236	#41	—	316	1	no	—
237	#41	—	316	1	no	—
238	C₂×Q₈	—	158	1	no	—
239	C₂×D₄	—	79	2	no	—
240	C₂×D₄	—	158	2	no	—
241	C₂×D₄	—	158	1	no	—
242	C₂×D₄	—	158	1	no	—
243	C₂²×C₄	—	158	2	no	—
244	C₂²×C₄	—	158	2	no	—
245	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
247	C₂²×C₄	#11929625	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	316	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	316	4	no	—
257	C₂×C₈	—	79	1	no	—
258	C₂×C₈	—	79	1	no	—
259	C₂×C₈	—	79	1	no	—
260	C₂×C₈	—	79	1	no	—
261	C₂×C₈	—	158	2	no	—
262	C₂×C₈	—	158	1	no	—
263	C₂×C₈	—	158	1	no	—
264	C₂×C₈	—	158	1	no	—
265	C₂×C₈	—	158	1	no	—
266	C₄⋊C₄	—	316	2	no	—
267	C₄⋊C₄	—	79	2	no	—
268	C₄⋊C₄	—	158	1	no	—
269	C₂²⋊C₄	—	158	2	no	—
270	C₂²⋊C₄	—	158	2	no	—
271	C₄²	—	316	1	no	—
272	C₄²	—	316	1	no	—
273	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
275	C₄²	D₄×D₇₉	1	1	yes	characteristic
276	C₄²	#11929623	1	1	yes	characteristic
277	C₂³	—	158	2	no	—
278	C₂³	#14579123	1	1	yes	characteristic
279	Q₈	—	316	1	no	—
280	Q₈	—	316	1	no	—
281	D₄	—	316	2	no	—
282	D₄	—	316	2	no	—
283	D₄	—	316	2	no	—
284	D₄	—	316	1	no	—
285	D₄	—	316	1	no	—
286	D₄	—	316	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	316	2	no	—
290	C₂×C₄	—	316	2	no	—
291	C₂×C₄	—	316	2	no	—
292	C₂×C₄	—	316	2	no	—
293	C₂×C₄	—	316	2	no	—
294	C₂×C₄	—	158	2	no	—
295	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
297	C₂×C₄	#14579119	1	1	yes	characteristic
298	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
300	C₂×C₄	#14579123	1	1	yes	characteristic
301	C₂×C₄	#14579185	1	1	yes	characteristic
302	C₂×C₄	#14579188	1	1	yes	characteristic
303	C₂×C₄	#14579188	1	1	yes	characteristic
304	C₂×C₄	#14579193	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	316	2	no	—
309	C₈	—	158	1	no	—
310	C₈	—	158	1	no	—
311	C₈	—	158	1	no	—
312	C₈	—	158	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	316	2	no	—
316	C₂²	—	316	2	no	—
317	C₂²	—	316	2	no	—
318	C₂²	#19098410	1	1	yes	characteristic
319	C₂²	#19098662	1	1	yes	characteristic
320	C₂²	#19098674	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	316	2	no	—
325	C₄	—	632	2	no	—
326	C₄	#19098563	1	1	yes	characteristic
327	C₄	#19098566	1	1	yes	characteristic
328	C₄	#19098666	1	1	yes	characteristic
329	C₄	#19098672	1	1	yes	characteristic
330	C₄	#19098684	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	632	2	no	—
334	C₂	#23763199	1	1	yes	characteristic
335	C₂	#23763224	1	1	yes	characteristic
336	C₂	#23763243	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1301	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	#19099113	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1301	D₇₉
hypocenter	C₇₉	#8253
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #23763270	C₂	semidirect
5. #23761693	C₂	semidirect
6. #23761301	C₂	semidirect
7. C₇₉×#1301	C₂	semidirect
11. #19098640	C₄	semidirect
13. #19098628	C₄	semidirect
20. #19098310	C₄	semidirect
21. #19098306	C₄	semidirect
175. #1301	D₇₉	semidirect
176. C₇₉	#8253	semidirect

Maximal subgroups

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

subgroup	index
#23772111	2
#23763270	2
#23762050	2
#23761896	2
#23761693	2
#23761301	2
C₇₉×#1301	2
#8253	79