Subgroups — #34196895

Order 20224.

730 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579096	C₂	1	2	yes	maximal
2	C₂×#19098587	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098327	C₂	1	2	yes	maximal
4	C₂×#19098318	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂³×C₇₉⋊C₈	C₂²	1	2	yes	—
8	C₂×#14579096	—	2	4	no	—
9	C₂×#14579096	—	2	4	no	—
10	C₂×#14579096	—	2	4	no	—
11	#19098587	—	2	4	no	—
12	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
13	#19098327	—	2	4	no	—
14	#19098318	C₄	1	2	yes	—
15	#19098318	C₄	1	2	yes	—
16	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	2	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	—	2	2	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	—	2	2	no	—
25	C₂×#11929608	D₄	1	4	yes	—
26	C₂²×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₇₉⋊C₈	—	2	2	no	—
28	C₂²×C₇₉⋊C₈	—	2	2	no	—
29	C₂²×C₇₉⋊C₈	—	2	2	no	—
30	C₂²×C₇₉⋊C₈	—	2	2	no	—
31	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
33	C₂²×C₇₉⋊C₈	Q₈	1	4	yes	—
34	#14579096	—	2	16	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
36	C₂³×C₄×C₇₉	Q₈	1	2	yes	—
37	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₉	—	2	1	no	—
39	C₂×M₁₆×C₇₉	—	2	1	no	—
40	C₂×M₁₆×C₇₉	—	2	2	no	—
41	C₂×M₁₆×C₇₉	—	2	2	no	—
42	C₂×M₁₆×C₇₉	—	2	2	no	—
43	C₂×M₁₆×C₇₉	—	2	2	no	—
44	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
45	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
46	C₂²×C₈×C₇₉	—	2	1	no	—
47	C₂²×C₈×C₇₉	—	2	1	no	—
48	C₂×C₄²×C₇₉	—	2	2	no	—
49	C₂×C₄²×C₇₉	—	2	2	no	—
50	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	D₄	1	2	yes	—
53	#11929608	—	2	4	no	—
54	#11929608	—	2	4	no	—
55	#11929608	—	4	2	no	—
56	#11929608	—	4	2	no	—
57	C₂×C₇₉⋊C₈	—	2	4	no	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	2	4	no	—
61	C₂×C₇₉⋊C₈	—	2	4	no	—
62	C₂×C₇₉⋊C₈	—	2	4	no	—
63	C₂×C₇₉⋊C₈	—	2	4	no	—
64	C₂×C₇₉⋊C₈	—	4	2	no	—
65	C₂×C₇₉⋊C₈	—	4	2	no	—
66	C₂×C₇₉⋊C₈	C₄⋊C₄	1	8	yes	—
67	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	—	2	4	no	—
69	C₂²×C₄×C₇₉	—	2	4	no	—
70	C₂²×C₄×C₇₉	—	4	2	no	—
71	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	#41	1	2	yes	—
82	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
83	M₁₆×C₇₉	—	2	4	no	—
84	M₁₆×C₇₉	—	2	4	no	—
85	M₁₆×C₇₉	—	4	2	no	—
86	M₁₆×C₇₉	—	4	2	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	4	1	no	—
90	C₂×C₈×C₇₉	—	4	1	no	—
91	C₂×C₈×C₇₉	—	4	1	no	—
92	C₂×C₈×C₇₉	—	4	1	no	—
93	C₄²×C₇₉	—	2	2	no	—
94	C₄²×C₇₉	—	2	2	no	—
95	C₄²×C₇₉	—	2	2	no	—
96	C₄²×C₇₉	—	2	2	no	—
97	C₂³×C₇₉	—	2	2	no	—
98	C₂³×C₇₉	—	2	2	no	—
99	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₇₉	#125	1	1	yes	characteristic
102	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	2	2	no	—
108	C₂×C₄×C₇₉	—	2	2	no	—
109	C₂×C₄×C₇₉	—	4	8	no	—
110	C₂×C₄×C₇₉	#108	1	4	yes	complemented
111	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#120	1	2	yes	—
117	C₂×C₄×C₇₉	#121	1	2	yes	—
118	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
124	C₂×C₄×C₇₉	#123	1	2	yes	—
125	C₂×C₄×C₇₉	#125	1	2	yes	—
126	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
129	C₈×C₇₉	—	4	2	no	—
130	C₈×C₇₉	—	4	2	no	—
131	C₇₉⋊C₈	—	2	8	no	—
132	C₇₉⋊C₈	—	4	4	no	—
133	C₂²×C₇₉	—	2	2	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	—	2	4	no	—
136	C₂²×C₇₉	—	2	2	no	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	#384	1	1	yes	characteristic
139	C₂²×C₇₉	#385	1	1	yes	characteristic
140	C₂²×C₇₉	#389	1	1	yes	characteristic
141	C₂²×C₇₉	#395	1	2	yes	—
142	C₂²×C₇₉	#398	1	2	yes	—
143	C₂²×C₇₉	C₂×#108	1	2	yes	—
144	C₂²×C₇₉	#430	1	2	yes	—
145	C₄×C₇₉	—	2	4	no	—
146	C₄×C₇₉	—	4	2	no	—
147	C₄×C₇₉	#385	1	2	yes	—
148	C₄×C₇₉	#389	1	2	yes	—
149	C₄×C₇₉	#392	1	1	yes	characteristic
150	C₄×C₇₉	#395	1	1	yes	characteristic
151	C₄×C₇₉	#398	1	1	yes	characteristic
152	C₄×C₇₉	#399	1	1	yes	characteristic
153	#11326	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	—	2	2	no	—
155	C₂×C₇₉	—	2	2	no	—
156	C₂×C₇₉	#1444	1	1	yes	characteristic
157	C₂×C₇₉	#1848	1	1	yes	characteristic
158	C₂×C₇₉	#1849	1	1	yes	characteristic
159	C₂×C₇₉	#1855	1	2	yes	—
160	C₂×C₇₉	#1950	1	2	yes	—
161	C₂²×#108	—	79	2	no	—
162	C₂×#423	—	79	1	no	—
163	C₂×#341	—	79	2	no	—
164	#1728	D₇₉	1	1	yes	characteristic
165	C₂×#336	—	79	1	no	—
166	C₇₉	#11326	1	1	yes	characteristic, complemented, Sylow
167	C₂²×M₁₆	—	79	2	no	—
168	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
169	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
170	C₂³×C₈	—	79	2	no	—
171	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
172	C₂×#108	—	158	4	no	—
173	C₂×#108	—	158	4	no	—
174	C₂×#108	—	158	4	no	—
175	#423	—	158	4	no	—
176	C₂×C₄⋊C₈	—	158	2	no	—
177	#341	—	158	4	no	—
178	#336	—	2	2	no	—
179	#336	—	79	2	no	—
180	#336	—	79	2	no	—
181	C₂³×C₄	Dic₁₅₈	1	2	yes	—
182	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	79	4	no	—
190	C₂×M₁₆	—	158	2	no	—
191	C₂×M₁₆	—	158	2	no	—
192	C₂×M₁₆	—	158	2	no	—
193	C₂×M₁₆	—	158	2	no	—
194	C₂×M₁₆	—	158	2	no	—
195	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
196	C₂×M₁₆	D₃₁₆	1	2	yes	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	79	4	no	—
200	C₂²×C₈	—	79	4	no	—
201	C₂²×C₈	—	79	4	no	—
202	C₂²×C₈	—	158	2	no	—
203	C₂²×C₈	—	158	2	no	—
204	C₂²×C₈	—	158	2	no	—
205	C₂²×C₈	—	158	2	no	—
206	C₂²×C₈	—	158	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
212	#108	—	158	16	no	—
213	C₄⋊C₈	—	158	8	no	—
214	C₂⁴	C₂×Dic₁₅₈	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
219	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₇₉	1	2	yes	—
222	C₂²×C₄	D₄×D₇₉	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
227	C₂²×C₄	#11929623	1	2	yes	—
228	C₂²×C₄	#11929624	1	1	yes	characteristic
229	C₂²×C₄	#11929624	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	316	2	no	—
235	M₁₆	—	316	2	no	—
236	M₁₆	—	158	4	no	—
237	M₁₆	—	158	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	316	2	no	—
245	C₂×C₈	—	316	2	no	—
246	C₂×C₈	—	79	8	no	—
247	C₂×C₈	—	158	4	no	—
248	C₂×C₈	—	158	4	no	—
249	C₂×C₈	—	158	4	no	—
250	C₂×C₈	—	158	4	no	—
251	C₂×C₈	—	158	4	no	—
252	C₂×C₈	—	158	4	no	—
253	C₂×C₈	—	158	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
261	C₂³	#14579138	1	2	yes	—
262	C₂³	#14579153	1	1	yes	characteristic
263	C₂³	#14579160	1	2	yes	—
264	C₂³	#14579161	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	#14579096	1	4	yes	complemented
272	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
273	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
274	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
275	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
276	C₂×C₄	#14579129	1	1	yes	characteristic
277	C₂×C₄	#14579131	1	2	yes	—
278	C₂×C₄	#14579134	1	2	yes	—
279	C₂×C₄	#14579135	1	1	yes	characteristic
280	C₂×C₄	#14579138	1	2	yes	—
281	C₂×C₄	#14579139	1	2	yes	—
282	C₂×C₄	#14579140	1	1	yes	characteristic
283	C₂×C₄	#14579141	1	2	yes	—
284	C₂×C₄	#14579142	1	1	yes	characteristic
285	C₂×C₄	#14579143	1	2	yes	—
286	C₂×C₄	#14579151	1	2	yes	—
287	C₂×C₄	#14579157	1	1	yes	characteristic
288	C₂×C₄	#14579159	1	1	yes	characteristic
289	C₂×C₄	#14579160	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	316	4	no	—
293	C₈	—	158	8	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	#19098432	1	1	yes	characteristic
300	C₂²	#19098435	1	1	yes	characteristic
301	C₂²	#19098461	1	1	yes	characteristic
302	C₂²	#19098484	1	2	yes	—
303	C₂²	#19098497	1	2	yes	—
304	C₂²	C₂×#14579096	1	2	yes	—
305	C₂²	#19098611	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	#19098437	1	2	yes	—
309	C₄	#19098460	1	2	yes	—
310	C₄	#19098472	1	1	yes	characteristic
311	C₄	#19098483	1	1	yes	characteristic
312	C₄	#19098493	1	1	yes	characteristic
313	C₄	#19098499	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#23761664	1	1	yes	characteristic
317	C₂	#23763571	1	1	yes	characteristic
318	C₂	#23763574	1	1	yes	characteristic
319	C₂	#23763604	1	2	yes	—
320	C₂	#23764175	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1728	C₂
socle	C₂³×C₇₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1728	D₇₉
hypocenter	C₇₉	#11326
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
110. C₂×C₄×C₇₉	#108	semidirect
166. C₇₉	#11326	semidirect
271. C₂×C₄	#14579096	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579096	2
C₂×#19098587	2
C₂×#19098327	2
C₂×#19098318	2
C₇₉×#1728	2
#11326	79