Subgroups — #34292543

Order 20224.

766 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23766051	C₂	1	2	yes	maximal
2	#23764262	C₂	1	2	yes	maximal
3	#23764134	C₂	1	2	yes	maximal
4	#23764092	C₂	1	4	yes	maximal
5	C₄×#14579095	C₂	1	1	yes	maximal, characteristic
6	#23763230	C₂	1	2	yes	maximal
7	#23763086	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579095	C₂²	1	2	yes	—
10	C₂×#14579085	C₂²	1	4	yes	—
11	C₂×#14579072	C₂²	1	1	yes	characteristic
12	C₂×#14579071	C₂²	1	2	yes	—
13	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
14	#19098470	C₂²	1	4	yes	—
15	#19098462	C₂²	1	4	yes	—
16	#19098449	—	2	4	no	—
17	#19098431	—	2	2	no	—
18	#19098429	—	2	2	no	—
19	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
20	#19098409	C₂²	1	1	yes	characteristic
21	#19098326	C₂²	1	2	yes	—
22	#19098325	C₂²	1	4	yes	—
23	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₇₉	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₇₉	C₄	1	8	yes	complemented
26	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#103×C₇₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₇₉	—	2	4	no	—
31	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
32	C₂×C₄×Dic₇₉	—	2	2	no	—
33	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
34	#14579095	—	2	4	no	—
35	#14579085	—	2	8	no	—
36	#14579072	—	2	2	no	—
37	#14579071	—	2	4	no	—
38	C₈×Dic₇₉	—	2	4	no	—
39	C₇₉⋊#95	—	2	2	no	—
40	C₇₉⋊#95	—	2	4	no	—
41	C₇₉⋊#95	—	2	4	no	—
42	C₇₉⋊#95	—	2	4	no	—
43	C₇₉⋊#95	D₄	1	1	yes	characteristic
44	C₇₉⋊#95	D₄	1	1	yes	characteristic
45	C₇₉⋊#95	D₄	1	2	yes	—
46	C₂²×Q₈×C₇₉	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₇₉	—	2	4	no	—
48	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₇₉	—	2	4	no	—
52	C₄×Q₈×C₇₉	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₉	—	2	2	no	—
56	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
57	#107×C₇₉	C₂×C₄	1	4	yes	—
58	#103×C₇₉	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
60	C₂²×Dic₇₉	—	2	4	no	—
61	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
62	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
63	C₂×C₇₉⋊C₈	—	2	2	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	#41	1	4	yes	—
66	C₄⋊Dic₇₉	—	2	4	no	—
67	C₄⋊Dic₇₉	—	2	4	no	—
68	C₄⋊Dic₇₉	—	8	4	no	—
69	C₄×Dic₇₉	—	2	2	no	—
70	C₄×Dic₇₉	—	2	2	no	—
71	C₄×Dic₇₉	—	8	2	no	—
72	C₂×Q₈×C₇₉	—	2	2	no	—
73	C₂×Q₈×C₇₉	—	2	4	no	—
74	C₂×Q₈×C₇₉	—	2	4	no	—
75	C₂×Q₈×C₇₉	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₇₉	—	2	2	no	—
77	C₂²×C₄×C₇₉	—	2	2	no	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₇₉	—	2	8	no	—
82	C₂×C₈×C₇₉	—	2	2	no	—
83	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₇₉	#41	1	4	yes	—
86	C₄⋊C₄×C₇₉	—	2	8	no	—
87	C₄⋊C₄×C₇₉	—	2	2	no	—
88	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
89	C₄²×C₇₉	—	2	8	no	—
90	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
91	C₄²×C₇₉	#41	1	2	yes	—
92	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₇₉	—	2	4	no	—
94	Q₈×C₇₉	—	2	4	no	—
95	Q₈×C₇₉	—	4	4	no	—
96	C₂×C₄×C₇₉	—	2	1	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	2	1	no	—
99	C₂×C₄×C₇₉	—	2	4	no	—
100	C₂×C₄×C₇₉	—	2	4	no	—
101	C₂×C₄×C₇₉	—	4	4	no	—
102	C₂×C₄×C₇₉	—	4	2	no	—
103	C₂×C₄×C₇₉	—	4	2	no	—
104	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
108	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
110	C₂×Dic₇₉	—	2	2	no	—
111	C₂×Dic₇₉	—	2	2	no	—
112	C₂×Dic₇₉	—	2	2	no	—
113	C₂×Dic₇₉	—	2	2	no	—
114	C₂×Dic₇₉	—	2	2	no	—
115	C₂×Dic₇₉	—	2	2	no	—
116	C₂×Dic₇₉	—	4	4	no	—
117	C₂×Dic₇₉	—	4	4	no	—
118	C₂×Dic₇₉	—	4	4	no	—
119	C₈×C₇₉	—	2	2	no	—
120	C₈×C₇₉	#117	1	4	yes	—
121	C₇₉⋊C₈	—	2	4	no	—
122	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₇₉	#532	1	2	yes	—
124	C₂²×C₇₉	#576	1	1	yes	characteristic
125	C₂²×C₇₉	#578	1	2	yes	—
126	C₂²×C₇₉	#578	1	1	yes	characteristic
127	C₄×C₇₉	—	2	2	no	—
128	C₄×C₇₉	—	2	2	no	—
129	C₄×C₇₉	—	4	4	no	—
130	C₄×C₇₉	—	4	4	no	—
131	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
132	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
133	C₄×C₇₉	#529	1	2	yes	—
134	Dic₇₉	—	4	2	no	—
135	Dic₇₉	—	4	2	no	—
136	Dic₇₉	—	8	4	no	—
137	#21281	—	79	1	no	maximal, Sylow
138	C₂×C₇₉	#2293	1	1	yes	characteristic
139	C₂×C₇₉	#2950	1	1	yes	characteristic
140	C₂×C₇₉	#2951	1	1	yes	characteristic
141	C₂×C₇₉	#3175	1	2	yes	—
142	C₂×C₇₉	#3177	1	2	yes	—
143	C₂×C₄×Q₁₆	D₇₉	1	1	yes	characteristic
144	#2320	—	79	2	no	—
145	#1969	—	79	2	no	—
146	#1942	—	79	2	no	—
147	#1935	—	79	4	no	—
148	C₄×#107	—	79	1	no	—
149	#1763	—	79	2	no	—
150	#1726	—	79	1	no	—
151	C₇₉	#21281	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
153	C₂×C₄×Q₈	C₂×D₇₉	1	2	yes	—
154	C₄×Q₁₆	Dic₇₉	1	8	yes	complemented
155	C₂×#107	—	79	2	no	—
156	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
157	C₂×#103	—	79	4	no	—
158	C₂×#103	C₂×D₇₉	1	2	yes	—
159	C₂×#97	—	79	2	no	—
160	C₂×#97	—	79	1	no	—
161	C₂×C₄×C₈	—	79	2	no	—
162	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
163	#391	—	79	4	no	—
164	#389	—	79	4	no	—
165	#387	—	158	4	no	—
166	#383	—	158	2	no	—
167	#382	—	158	2	no	—
168	C₄×C₄⋊C₄	—	79	2	no	—
169	#376	—	79	1	no	—
170	#340	—	79	2	no	—
171	#340	—	79	4	no	—
172	C₂²×Q₈	C₂²×D₇₉	1	2	yes	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	C₂×Dic₇₉	1	4	yes	—
175	C₂²×C₈	—	79	2	no	—
176	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	C₂×Dic₇₉	1	8	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	79	4	no	—
182	C₂×C₄⋊C₄	—	158	4	no	—
183	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	79	2	no	—
186	C₂×C₄²	—	158	2	no	—
187	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
188	#107	—	158	4	no	—
189	#107	C₂×Dic₇₉	1	4	yes	—
190	#103	—	158	8	no	—
191	#103	C₂×Dic₇₉	1	8	yes	—
192	#97	—	158	4	no	—
193	#97	—	158	2	no	—
194	C₄×C₈	—	158	4	no	—
195	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
196	#95	—	79	1	no	—
197	#95	—	79	1	no	—
198	#95	—	79	2	no	—
199	#95	—	158	4	no	—
200	#95	—	158	4	no	—
201	#95	—	158	4	no	—
202	#95	—	158	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	C₂²×Dic₇₉	1	4	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	79	2	no	—
210	C₂²×C₄	—	79	2	no	—
211	C₂²×C₄	—	158	4	no	—
212	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
214	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
215	Q₁₆	—	2	8	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	79	4	no	—
218	C₂×C₈	—	158	2	no	—
219	C₂×C₈	—	158	2	no	—
220	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
221	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
222	C₂×C₈	#11929624	1	4	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	632	4	no	—
226	C₄⋊C₄	—	158	4	no	—
227	C₄⋊C₄	—	158	4	no	—
228	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
229	C₄²	—	2	8	no	—
230	C₄²	—	632	2	no	—
231	C₄²	—	158	2	no	—
232	C₄²	—	158	2	no	—
233	C₄²	C₂²×Dic₇₉	1	2	yes	—
234	C₄²	#11929623	1	2	yes	—
235	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	2	4	no	—
238	Q₈	—	4	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	316	4	no	—
248	C₂×C₄	—	316	4	no	—
249	C₂×C₄	—	316	4	no	—
250	C₂×C₄	—	158	2	no	—
251	C₂×C₄	—	158	2	no	—
252	C₂×C₄	—	158	2	no	—
253	C₂×C₄	—	158	2	no	—
254	C₂×C₄	—	158	2	no	—
255	C₂×C₄	—	158	2	no	—
256	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
257	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
258	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
259	C₂×C₄	#41×D₇₉	1	2	yes	—
260	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
261	C₂×C₄	C₂×#11929624	1	2	yes	—
262	C₈	—	2	2	no	—
263	C₈	—	158	4	no	—
264	C₈	#14579120	1	4	yes	—
265	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
266	C₂²	#19099110	1	2	yes	—
267	C₂²	#19099365	1	1	yes	characteristic
268	C₂²	#19099376	1	2	yes	—
269	C₂²	#19099377	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	4	no	—
273	C₄	—	4	4	no	—
274	C₄	—	316	2	no	—
275	C₄	—	316	2	no	—
276	C₄	—	632	4	no	—
277	C₄	C₂×#14579120	1	1	yes	characteristic
278	C₄	#41×Dic₇₉	1	1	yes	characteristic
279	C₄	#19099095	1	2	yes	—
280	C₂	#23765865	1	1	yes	characteristic
281	C₂	#23772268	1	1	yes	characteristic
282	C₂	#23772276	1	1	yes	characteristic
283	C₂	#23774041	1	2	yes	—
284	C₂	#23774055	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₄×Q₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇₉
hypocenter	C₇₉	#21281
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₇₉	C₄	semidirect
151. C₇₉	#21281	semidirect
154. C₄×Q₁₆	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23766051	2
#23764262	2
#23764134	2
#23764092	2
C₄×#14579095	2
#23763230	2
#23763086	2
C₂×C₄×Q₁₆×C₇₉	2
#21281	79