Subgroups — #34179009

Order 20224.

460 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772361	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579089	C₂	1	1	yes	maximal, characteristic
3	#23763146	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761780	C₂	1	1	yes	maximal, characteristic
5	#23761690	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761236	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11929624	C₂²	1	1	yes	characteristic
9	#19098690	C₄	1	2	yes	complemented
10	#19098640	C₄	1	2	yes	complemented
11	#19098631	—	2	2	no	—
12	#19098614	—	2	1	no	—
13	C₂×#14579089	C₄	1	2	yes	—
14	C₄×#11929608	C₂²	1	1	yes	characteristic
15	C₄×#11929607	—	2	1	no	—
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	#19098323	C₄	1	2	yes	—
19	#19098318	—	2	2	no	—
20	#19098293	C₄	1	2	yes	complemented
21	#19098290	C₄	1	2	yes	complemented
22	#19098285	—	2	1	no	—
23	#19098284	—	2	1	no	—
24	#431×C₇₉	C₂²	1	1	yes	characteristic
25	#431×C₇₉	C₂²	1	1	yes	characteristic
26	C₄³×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
28	C₂×#11929608	C₂×C₄	1	2	yes	—
29	C₂×#11929607	—	2	2	no	—
30	C₂²×C₇₉⋊C₈	—	2	1	no	—
31	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
32	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
34	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
35	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₇₉	—	2	1	no	—
37	C₄×C₂²⋊D₇₉	—	2	1	no	—
38	#14579120	C₂×C₄	1	1	yes	characteristic
39	#14579118	—	2	1	no	—
40	C₄²×D₇₉	—	2	1	no	—
41	Dic₇₉⋊C₈	—	2	2	no	—
42	#14579089	—	2	4	no	—
43	(C₂×D₇₉)⋊C₈	—	2	2	no	—
44	#14579072	—	2	1	no	—
45	#14579071	—	2	2	no	—
46	C₄×C₇₉⋊C₈	—	2	1	no	—
47	C₄×C₇₉⋊C₈	—	2	1	no	—
48	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
50	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	—	2	1	no	—
53	C₂×C₄²×C₇₉	—	2	1	no	—
54	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₈×C₇₉	—	2	1	no	—
59	C₂²⋊C₈×C₇₉	—	2	1	no	—
60	C₄×C₈×C₇₉	—	2	1	no	—
61	C₄×C₈×C₇₉	—	2	1	no	—
62	#11929624	—	2	2	no	—
63	#11929624	—	2	2	no	—
64	C₂×Dic₁₅₈	C₄²	1	1	yes	characteristic
65	C₂×D₃₁₆	C₄²	1	1	yes	characteristic
66	C₂×C₂²⋊D₇₉	—	2	1	no	—
67	C₂×C₄×D₇₉	—	2	1	no	—
68	C₂×C₄×D₇₉	—	2	1	no	—
69	C₂×C₄×D₇₉	—	2	1	no	—
70	#11929608	—	2	2	no	—
71	#11929608	—	2	2	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	2	no	—
76	C₂×C₇₉⋊C₈	—	4	2	no	—
77	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
79	C₄⋊Dic₇₉	C₄²	1	1	yes	characteristic
80	C₄⋊Dic₇₉	C₄²	1	1	yes	characteristic
81	Dic₇₉⋊C₄	—	2	1	no	—
82	Dic₇₉⋊C₄	—	2	1	no	—
83	C₂²⋊Dic₇₉	—	2	1	no	—
84	(C₂×D₇₉)⋊C₄	—	2	1	no	—
85	(C₂×D₇₉)⋊C₄	—	2	1	no	—
86	C₄×Dic₇₉	—	2	1	no	—
87	C₄×Dic₇₉	—	2	1	no	—
88	C₄×Dic₇₉	—	2	1	no	—
89	C₂²×C₄×C₇₉	—	2	2	no	—
90	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	2	2	no	—
95	C₄²×C₇₉	—	2	2	no	—
96	C₄²×C₇₉	—	4	2	no	—
97	C₄²×C₇₉	—	4	2	no	—
98	C₄²×C₇₉	M₁₆	1	4	yes	complemented
99	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₉	#41	1	1	yes	characteristic
103	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	2	4	no	—
108	C₂×C₄×C₇₉	—	4	2	no	—
109	C₂×C₄×C₇₉	—	4	2	no	—
110	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₇₉	#131	1	2	yes	—
118	C₂²⋊D₇₉	—	4	2	no	—
119	C₂×Dic₇₉	—	2	1	no	—
120	C₂×Dic₇₉	—	2	1	no	—
121	C₂×Dic₇₉	—	2	1	no	—
122	D₃₁₆	—	2	1	no	—
123	D₃₁₆	—	2	1	no	—
124	C₄×D₇₉	—	4	2	no	—
125	C₄×D₇₉	—	4	2	no	—
126	C₄×D₇₉	—	4	2	no	—
127	Dic₁₅₈	—	2	1	no	—
128	Dic₁₅₈	—	2	1	no	—
129	C₈×C₇₉	—	4	2	no	—
130	C₈×C₇₉	—	4	2	no	—
131	C₇₉⋊C₈	—	2	2	no	—
132	C₇₉⋊C₈	—	2	2	no	—
133	C₇₉⋊C₈	—	4	2	no	—
134	C₇₉⋊C₈	—	4	2	no	—
135	C₂²×C₇₉	—	2	1	no	—
136	C₂²×C₇₉	—	2	1	no	—
137	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₇₉	#435	1	2	yes	—
139	C₂×D₇₉	—	4	1	no	—
140	C₂×D₇₉	—	4	1	no	—
141	C₂×D₇₉	—	4	1	no	—
142	C₄×C₇₉	—	2	1	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	4	4	no	—
145	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
146	C₄×C₇₉	#405	1	1	yes	characteristic
147	C₄×C₇₉	#421	1	2	yes	—
148	C₄×C₇₉	#436	1	2	yes	—
149	C₄×C₇₉	#444	1	2	yes	—
150	Dic₇₉	—	4	1	no	—
151	Dic₇₉	—	4	1	no	—
152	Dic₇₉	—	4	1	no	—
153	#8195	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	—	2	2	no	—
155	C₂×C₇₉	#1753	1	1	yes	characteristic
156	C₂×C₇₉	#1759	1	1	yes	characteristic
157	C₂×C₇₉	#1765	1	1	yes	characteristic
158	D₇₉	—	8	1	no	—
159	#2965	—	79	1	no	—
160	C₄×#104	—	79	1	no	—
161	#1744	—	79	1	no	—
162	#1476	—	79	1	no	—
163	#1451	—	79	1	no	—
164	#1278	—	79	1	no	—
165	#1276	D₇₉	1	1	yes	characteristic, complemented
166	C₇₉	#8195	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	—	79	1	no	—
168	#446	—	79	2	no	—
169	#436	—	79	2	no	—
170	#435	—	158	2	no	—
171	#431	—	158	1	no	—
172	#431	C₂×D₇₉	1	1	yes	characteristic
173	#431	C₂×D₇₉	1	1	yes	characteristic
174	C₂×#104	—	79	2	no	—
175	C₄×M₁₆	—	79	1	no	—
176	C₄×M₁₆	—	158	1	no	—
177	C₂×C₄×C₈	—	79	1	no	—
178	C₂×C₄×C₈	—	79	1	no	—
179	C₄³	C₂×D₇₉	1	1	yes	characteristic
180	#339	—	79	2	no	—
181	#336	—	158	2	no	—
182	Q₈⋊C₈	—	79	2	no	—
183	D₄⋊C₈	—	79	2	no	—
184	#322	—	158	1	no	—
185	#322	—	158	1	no	—
186	C₂×#41	—	79	1	no	—
187	C₂×M₁₆	—	79	2	no	—
188	C₂×M₁₆	—	158	2	no	—
189	C₂²×C₈	—	79	2	no	—
190	C₂²×C₈	—	158	1	no	—
191	C₄×Q₈	—	79	1	no	—
192	C₄×Q₈	—	79	1	no	—
193	C₄×D₄	—	79	1	no	—
194	C₄×D₄	—	79	1	no	—
195	C₄×D₄	—	158	1	no	—
196	C₄×D₄	—	158	1	no	—
197	#117	—	79	1	no	—
198	#117	—	158	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	158	1	no	—
202	C₂×C₄²	C₄×D₇₉	1	2	yes	—
203	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
204	C₄⋊C₈	—	158	2	no	—
205	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
206	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
207	#104	—	158	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	158	2	no	—
211	#97	—	158	2	no	—
212	#97	—	158	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	79	2	no	—
216	C₄×C₈	—	79	2	no	—
217	C₄×C₈	—	79	1	no	—
218	C₄×C₈	—	79	1	no	—
219	C₄×C₈	—	158	1	no	—
220	C₄×C₈	—	158	1	no	—
221	#41	—	158	2	no	—
222	#41	—	158	2	no	—
223	C₂×Q₈	—	79	1	no	—
224	C₂×D₄	—	79	1	no	—
225	C₂×D₄	—	158	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	158	1	no	—
228	C₂²×C₄	—	158	1	no	—
229	C₂²×C₄	—	158	1	no	—
230	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
232	C₂²×C₄	#11929623	1	1	yes	characteristic
233	M₁₆	—	316	4	no	—
234	M₁₆	—	158	2	no	—
235	M₁₆	—	158	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	316	2	no	—
239	C₂×C₈	—	316	2	no	—
240	C₂×C₈	—	79	2	no	—
241	C₂×C₈	—	79	2	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	—	158	2	no	—
244	C₄⋊C₄	—	79	1	no	—
245	C₄⋊C₄	—	79	1	no	—
246	C₄⋊C₄	—	158	1	no	—
247	C₄⋊C₄	—	158	1	no	—
248	C₂²⋊C₄	—	158	1	no	—
249	C₂²⋊C₄	—	158	1	no	—
250	C₂²⋊C₄	—	158	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	158	1	no	—
255	C₄²	—	158	1	no	—
256	C₄²	—	158	1	no	—
257	C₄²	#11929607	1	4	yes	complemented
258	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
260	C₄²	D₄×D₇₉	1	1	yes	characteristic
261	C₄²	#11929625	1	1	yes	characteristic
262	C₂³	—	158	1	no	—
263	C₂³	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
264	Q₈	—	158	1	no	—
265	Q₈	—	158	1	no	—
266	D₄	—	316	2	no	—
267	D₄	—	158	1	no	—
268	D₄	—	158	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	316	2	no	—
275	C₂×C₄	—	316	2	no	—
276	C₂×C₄	—	316	2	no	—
277	C₂×C₄	—	158	1	no	—
278	C₂×C₄	—	158	1	no	—
279	C₂×C₄	—	158	1	no	—
280	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
282	C₂×C₄	#14579119	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#14579123	1	1	yes	characteristic
285	C₂×C₄	#14579123	1	1	yes	characteristic
286	C₂×C₄	C₂×#11929607	1	2	yes	—
287	C₂×C₄	#14579169	1	2	yes	—
288	C₈	—	4	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	316	2	no	—
291	C₈	—	316	2	no	—
292	C₈	—	158	2	no	—
293	C₈	—	158	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	316	1	no	—
297	C₂²	—	316	1	no	—
298	C₂²	—	316	1	no	—
299	C₂²	#19098410	1	1	yes	characteristic
300	C₂²	#19098631	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	4	no	—
304	C₄	—	316	1	no	—
305	C₄	—	316	1	no	—
306	C₄	—	316	1	no	—
307	C₄	C₄×#11929607	1	1	yes	characteristic
308	C₄	#19098520	1	1	yes	characteristic
309	C₄	#19098578	1	2	yes	—
310	C₄	#19098640	1	2	yes	—
311	C₄	#19098682	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	632	1	no	—
314	C₂	#23763180	1	1	yes	characteristic
315	C₂	#23763209	1	1	yes	characteristic
316	C₂	#23763241	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1276	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1276	D₇₉
hypocenter	C₇₉	#8195
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
3. #23763146	C₂	semidirect
5. #23761690	C₂	semidirect
6. #23761236	C₂	semidirect
7. C₇₉×#1276	C₂	semidirect
9. #19098690	C₄	semidirect
10. #19098640	C₄	semidirect
20. #19098293	C₄	semidirect
21. #19098290	C₄	semidirect
98. C₄²×C₇₉	M₁₆	semidirect
165. #1276	D₇₉	semidirect
166. C₇₉	#8195	semidirect
257. C₄²	#11929607	semidirect

Maximal subgroups

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

subgroup	index
#23772361	2
C₄×#14579089	2
#23763146	2
#23761780	2
#23761690	2
#23761236	2
C₇₉×#1276	2
#8195	79