Subgroups — #34288891

Order 20224.

718 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772037	C₂	1	2	yes	maximal
2	C₂×#19098517	C₂	1	1	yes	maximal, characteristic
3	#23763572	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763540	C₂	1	2	yes	maximal, complemented
5	#23763439	C₂	1	2	yes	maximal
6	#23763435	C₂	1	2	yes	maximal
7	#23763308	C₂	1	2	yes	maximal, complemented
8	#108×Dic₇₉	C₂	1	2	yes	maximal, complemented
9	C₇₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11929607	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
12	C₂×#14579118	C₂²	1	1	yes	characteristic
13	#19098623	—	2	8	no	—
14	C₂×#14579097	C₂²	1	2	yes	—
15	C₂×#14579097	C₂²	1	2	yes	—
16	#19098587	C₂²	1	2	yes	—
17	#19098526	C₂²	1	4	yes	—
18	#19098517	C₄	1	8	yes	—
19	M₁₆×Dic₇₉	C₂²	1	4	yes	—
20	C₂×#14579072	C₂²	1	1	yes	characteristic
21	C₂×#14579071	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
23	#19098327	C₂²	1	4	yes	—
24	#19098327	C₂²	1	4	yes	—
25	C₂×#108×C₇₉	C₂²	1	2	yes	—
26	C₂×#108×C₇₉	C₂²	1	2	yes	—
27	#423×C₇₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₇₉	C₂³	1	1	yes	characteristic
30	C₂×#11929608	C₂³	1	4	yes	—
31	C₂×#11929607	—	2	2	no	—
32	C₂×#11929607	—	2	2	no	—
33	C₂×#11929607	C₂×C₄	1	4	yes	—
34	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₇₉	—	2	2	no	—
37	C₂×C₈×D₇₉	—	2	2	no	—
38	C₂×C₈×D₇₉	C₂×C₄	1	4	yes	—
39	#14579118	C₂×C₄	1	4	yes	—
40	#14579118	D₄	1	4	yes	—
41	C₂×Dic₇₉⋊C₄	Q₈	1	2	yes	—
42	C₂×(C₂×D₇₉)⋊C₄	Q₈	1	2	yes	—
43	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
44	#14579097	—	2	4	no	—
45	#14579097	—	2	4	no	—
46	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
47	(C₂×D₇₉)⋊C₈	—	2	8	no	—
48	#14579072	C₂×C₄	1	4	yes	—
49	#14579071	—	2	4	no	—
50	#14579071	C₂×C₄	1	4	yes	—
51	C₈×Dic₇₉	—	2	4	no	—
52	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
54	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
57	#108×C₇₉	—	2	4	no	—
58	#108×C₇₉	—	2	4	no	—
59	C₄⋊C₈×C₇₉	—	2	4	no	—
60	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
61	C₂³×D₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×Dic₇₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×Dic₇₉	C₂×Q₈	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
70	#11929608	—	2	4	no	—
71	#11929608	—	2	4	no	—
72	#11929607	—	2	8	no	—
73	C₂×C₇₉⋊C₈	—	2	4	no	—
74	C₂×C₇₉⋊C₈	—	2	1	no	—
75	C₂×C₇₉⋊C₈	—	2	1	no	—
76	C₂×C₇₉⋊C₈	—	2	1	no	—
77	C₂×C₇₉⋊C₈	—	2	1	no	—
78	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
80	C₈×D₇₉	—	2	8	no	—
81	Dic₇₉⋊C₄	C₄⋊C₄	1	8	yes	—
82	(C₂×D₇₉)⋊C₄	C₄⋊C₄	1	8	yes	—
83	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
84	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₇₉	—	2	4	no	—
88	M₁₆×C₇₉	—	2	4	no	—
89	C₂×C₈×C₇₉	—	2	4	no	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₇₉	#41	1	2	yes	—
93	C₂×C₈×C₇₉	#41	1	2	yes	—
94	C₂×C₈×C₇₉	#41	1	2	yes	—
95	C₂×C₈×C₇₉	#41	1	2	yes	—
96	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
97	C₄²×C₇₉	C₂×D₄	1	2	yes	—
98	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×D₇₉	—	2	1	no	—
100	C₂²×D₇₉	—	2	1	no	—
101	C₂²×D₇₉	—	4	2	no	—
102	C₂²×D₇₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₇₉	—	4	2	no	—
104	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₇₉	—	2	1	no	—
112	C₂×Dic₇₉	—	2	1	no	—
113	C₂×Dic₇₉	—	4	2	no	—
114	C₂×Dic₇₉	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×Dic₇₉	C₂×C₄⋊C₄	1	2	yes	—
116	C₄×D₇₉	—	2	2	no	—
117	C₄×D₇₉	—	2	2	no	—
118	C₄×D₇₉	—	2	2	no	—
119	C₄×D₇₉	—	2	2	no	—
120	C₈×C₇₉	—	4	4	no	—
121	C₈×C₇₉	#117	1	4	yes	—
122	C₈×C₇₉	#117	1	4	yes	—
123	C₇₉⋊C₈	—	2	2	no	—
124	C₇₉⋊C₈	—	2	2	no	—
125	C₇₉⋊C₈	—	4	4	no	—
126	C₂²×C₇₉	—	2	2	no	—
127	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₇₉	#533	1	1	yes	characteristic
129	C₂²×C₇₉	#533	1	1	yes	characteristic
130	C₂×D₇₉	—	2	4	no	—
131	C₂×D₇₉	—	2	2	no	—
132	C₂×D₇₉	—	4	2	no	—
133	C₂×D₇₉	—	4	2	no	—
134	C₄×C₇₉	—	2	4	no	—
135	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
136	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
137	C₄×C₇₉	#532	1	1	yes	characteristic
138	C₄×C₇₉	#532	1	1	yes	characteristic
139	Dic₇₉	—	2	4	no	—
140	Dic₇₉	—	2	2	no	—
141	#20828	—	79	1	no	maximal, Sylow
142	C₂×C₇₉	—	2	1	no	—
143	C₂×C₇₉	—	2	1	no	—
144	C₂×C₇₉	#2285	1	1	yes	characteristic
145	C₂×C₇₉	#2916	1	2	yes	—
146	D₇₉	—	4	2	no	—
147	#2921	—	79	2	no	—
148	C₂×#405	—	79	1	no	—
149	#1848	—	79	1	no	—
150	#1848	D₇₉	1	1	yes	characteristic, complemented
151	#1838	—	79	2	no	—
152	#1811	—	79	2	no	—
153	#1810	—	79	2	no	—
154	#1778	—	79	2	no	—
155	C₄×#108	—	79	2	no	—
156	C₇₉	#20828	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	79	1	no	—
158	C₂³×C₈	—	79	1	no	—
159	C₂×#117	—	79	1	no	—
160	#433	—	158	8	no	—
161	C₂×#108	—	79	2	no	—
162	C₂×#108	—	79	2	no	—
163	C₂×#108	C₂×D₇₉	1	2	yes	—
164	C₂×#108	C₂×D₇₉	1	2	yes	—
165	#423	—	79	2	no	—
166	#423	C₂×D₇₉	1	2	yes	—
167	#407	—	79	4	no	—
168	#405	—	79	8	no	—
169	C₄×M₁₆	—	79	4	no	—
170	C₂×#97	—	79	1	no	—
171	C₂×#97	—	79	1	no	—
172	C₂×C₄×C₈	—	79	1	no	—
173	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
174	#341	—	79	4	no	—
175	#341	—	79	4	no	—
176	C₂³×C₄	—	79	1	no	—
177	C₂×M₁₆	—	79	4	no	—
178	C₂×M₁₆	—	79	4	no	—
179	C₂×M₁₆	—	158	2	no	—
180	C₂×M₁₆	—	158	2	no	—
181	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
182	C₂²×C₈	—	79	1	no	—
183	C₂²×C₈	—	79	1	no	—
184	C₂²×C₈	—	79	4	no	—
185	C₂²×C₈	—	158	2	no	—
186	C₂²×C₈	—	158	2	no	—
187	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
189	#117	—	79	4	no	—
190	#117	—	79	4	no	—
191	C₂×C₄⋊C₄	—	79	2	no	—
192	C₂×C₂²⋊C₄	—	79	2	no	—
193	C₂×C₄²	—	79	1	no	—
194	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
195	#108	—	2	4	no	—
196	#108	—	2	4	no	—
197	#108	—	158	4	no	—
198	#108	—	158	4	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	158	4	no	—
201	C₂²⋊C₈	—	158	8	no	—
202	#97	—	79	4	no	—
203	#97	—	79	4	no	—
204	#97	—	158	4	no	—
205	C₄×C₈	—	79	4	no	—
206	C₄×C₈	—	158	4	no	—
207	C₄×C₈	C₄×D₇₉	1	4	yes	—
208	C₂⁴	—	79	1	no	—
209	C₂²×C₄	—	79	2	no	—
210	C₂²×C₄	—	79	1	no	—
211	C₂²×C₄	—	79	2	no	—
212	C₂²×C₄	—	79	2	no	—
213	C₂²×C₄	—	158	1	no	—
214	C₂²×C₄	—	158	1	no	—
215	C₂²×C₄	—	158	1	no	—
216	C₂²×C₄	—	158	1	no	—
217	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
218	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	158	4	no	—
222	M₁₆	—	158	4	no	—
223	M₁₆	—	158	8	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	79	2	no	—
226	C₂×C₈	—	79	2	no	—
227	C₂×C₈	—	158	4	no	—
228	C₂×C₈	—	158	1	no	—
229	C₂×C₈	—	158	1	no	—
230	C₂×C₈	—	158	1	no	—
231	C₂×C₈	—	158	1	no	—
232	C₂×C₈	—	158	8	no	—
233	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
235	C₂×C₈	#11929623	1	2	yes	—
236	C₂×C₈	#11929623	1	2	yes	—
237	C₂×C₈	#11929625	1	2	yes	—
238	C₂×C₈	#11929625	1	2	yes	—
239	C₄⋊C₄	—	79	8	no	—
240	C₂²⋊C₄	—	79	8	no	—
241	C₄²	—	79	2	no	—
242	C₄²	—	79	2	no	—
243	C₄²	C₂×C₄×D₇₉	1	2	yes	—
244	C₄²	D₄×D₇₉	1	2	yes	—
245	C₂³	—	316	2	no	—
246	C₂³	—	79	2	no	—
247	C₂³	—	158	1	no	—
248	C₂³	—	158	1	no	—
249	C₂³	C₂×Q₈×D₇₉	1	1	yes	characteristic
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	316	2	no	—
252	C₂×C₄	—	79	4	no	—
253	C₂×C₄	—	79	2	no	—
254	C₂×C₄	—	158	1	no	—
255	C₂×C₄	—	158	1	no	—
256	C₂×C₄	—	158	2	no	—
257	C₂×C₄	—	158	2	no	—
258	C₂×C₄	—	158	2	no	—
259	C₂×C₄	—	158	2	no	—
260	C₂×C₄	C₄⋊C₄×D₇₉	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
263	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
264	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
265	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
266	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
267	C₈	—	4	4	no	—
268	C₈	—	316	4	no	—
269	C₈	—	158	2	no	—
270	C₈	—	158	2	no	—
271	C₈	#14579119	1	4	yes	—
272	C₈	#14579119	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	316	2	no	—
275	C₂²	—	316	2	no	—
276	C₂²	—	158	2	no	—
277	C₂²	—	158	4	no	—
278	C₂²	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
279	C₂²	#19099118	1	1	yes	characteristic
280	C₂²	#19099118	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	158	2	no	—
283	C₄	—	158	4	no	—
284	C₄	C₂×#14579119	1	1	yes	characteristic
285	C₄	C₂×#14579119	1	1	yes	characteristic
286	C₄	#19099111	1	1	yes	characteristic
287	C₄	#19099114	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	316	2	no	—
291	C₂	#23765805	1	1	yes	characteristic
292	C₂	#23772008	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1848	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1848	D₇₉
hypocenter	C₇₉	#20828
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
3. #23763572	C₂	semidirect
4. #23763540	C₂	semidirect
7. #23763308	C₂	semidirect
8. #108×Dic₇₉	C₂	semidirect
9. C₇₉×#1848	C₂	semidirect
150. #1848	D₇₉	semidirect
156. C₇₉	#20828	semidirect

Maximal subgroups

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

subgroup	index
#23772037	2
C₂×#19098517	2
#23763572	2
#23763540	2
#23763439	2
#23763435	2
#23763308	2
#108×Dic₇₉	2
C₇₉×#1848	2
#20828	79