Subgroups — #34296444

Order 20224.

746 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773722	C₂	1	1	yes	maximal, characteristic
2	#23772334	C₂	1	1	yes	maximal, characteristic, complemented
3	#23764376	C₂	1	1	yes	maximal, characteristic, complemented
4	#23764229	C₂	1	1	yes	maximal, characteristic
5	#23764114	C₂	1	4	yes	maximal, complemented
6	#23763537	C₂	1	1	yes	maximal, characteristic, complemented
7	#23763242	C₂	1	4	yes	maximal, complemented
8	C₄×#14579071	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2931	C₂	1	1	yes	maximal, characteristic
10	#19099120	C₂²	1	1	yes	characteristic
11	#19098917	C₄	1	4	yes	complemented
12	#19098913	C₄	1	2	yes	complemented
13	#19098909	C₄	1	2	yes	complemented
14	#19098693	—	2	2	no	—
15	C₈×Dic₁₅₈	—	2	2	no	—
16	#19098589	C₂²	1	2	yes	—
17	#19098587	C₂²	1	1	yes	characteristic
18	C₂×#14579088	C₂²	1	4	yes	—
19	#19098559	C₂²	1	4	yes	—
20	#19098518	C₂²	1	2	yes	—
21	C₂×#14579071	C₂²	1	2	yes	—
22	C₂×#14579071	C₂²	1	2	yes	—
23	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
24	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
25	#19098327	C₂²	1	4	yes	—
26	#19098324	C₂²	1	4	yes	—
27	C₂×#131×C₇₉	C₂²	1	2	yes	—
28	#532×C₇₉	C₂²	1	1	yes	characteristic
29	#436×C₇₉	—	2	2	no	—
30	C₈×D₄×C₇₉	—	2	2	no	—
31	#423×C₇₉	C₂²	1	1	yes	characteristic
32	#408×C₇₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×#11929608	C₂³	1	2	yes	—
35	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
36	#14579161	C₂×C₄	1	2	yes	—
37	#14579153	C₂×C₄	1	2	yes	—
38	#14579140	—	2	2	no	—
39	C₄×Dic₁₅₈	—	2	2	no	—
40	#14579120	C₂³	1	2	yes	—
41	C₂×C₄×Dic₇₉	—	2	2	no	—
42	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
43	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
44	Dic₇₉⋊C₈	—	2	4	no	—
45	#14579088	—	2	8	no	—
46	#14579085	C₂×C₄	1	8	yes	—
47	#14579082	C₂×C₄	1	8	yes	—
48	#14579071	—	2	2	no	—
49	#14579071	—	2	4	no	—
50	#14579071	—	2	4	no	—
51	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
52	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
53	C₈×Dic₇₉	—	2	2	no	—
54	C₂×#41×C₇₉	C₂³	1	2	yes	—
55	#131×C₇₉	—	2	8	no	—
56	C₂×M₁₆×C₇₉	—	2	2	no	—
57	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
58	C₂²×C₈×C₇₉	—	2	2	no	—
59	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
62	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
63	#124×C₇₉	C₂×C₄	1	2	yes	—
64	#121×C₇₉	—	2	2	no	—
65	C₄×D₄×C₇₉	—	2	2	no	—
66	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₇₉	—	2	2	no	—
68	C₂²⋊C₈×C₇₉	—	2	4	no	—
69	C₄×C₈×C₇₉	D₄	1	2	yes	—
70	C₄×C₈×C₇₉	D₄	1	2	yes	—
71	C₂×Dic₁₅₈	—	2	2	no	—
72	C₂²×Dic₇₉	—	2	2	no	—
73	#11929608	—	2	4	no	—
74	C₂×C₇₉⋊C₈	—	2	2	no	—
75	C₂×C₇₉⋊C₈	—	2	2	no	—
76	C₂×C₇₉⋊C₈	—	2	2	no	—
77	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
78	C₄⋊Dic₇₉	—	2	2	no	—
79	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
80	Dic₇₉⋊C₄	—	2	4	no	—
81	C₂²⋊Dic₇₉	—	2	2	no	—
82	C₄×Dic₇₉	—	2	2	no	—
83	C₄×Dic₇₉	—	2	2	no	—
84	C₄×Dic₇₉	—	4	4	no	—
85	C₄×Dic₇₉	#41	1	4	yes	—
86	#41×C₇₉	—	2	4	no	—
87	#41×C₇₉	—	2	4	no	—
88	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₇₉	—	2	2	no	—
90	C₂×D₄×C₇₉	—	2	2	no	—
91	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₇₉	—	2	2	no	—
93	C₂²×C₄×C₇₉	#41	1	2	yes	—
94	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₇₉	—	2	4	no	—
96	M₁₆×C₇₉	—	4	4	no	—
97	C₂×C₈×C₇₉	—	2	2	no	—
98	C₂×C₈×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	—	4	4	no	—
100	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	#41	1	4	yes	—
104	C₄⋊C₄×C₇₉	—	2	2	no	—
105	C₂²⋊C₄×C₇₉	—	2	4	no	—
106	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₇₉	C₂×D₄	1	2	yes	—
109	C₂³×C₇₉	—	2	2	no	—
110	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
111	Q₈×C₇₉	—	2	4	no	—
112	D₄×C₇₉	—	2	4	no	—
113	D₄×C₇₉	—	4	4	no	—
114	D₄×C₇₉	—	4	4	no	—
115	C₂×C₄×C₇₉	—	2	4	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	4	2	no	—
118	C₂×C₄×C₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
124	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
125	C₂×Dic₇₉	—	2	4	no	—
126	C₂×Dic₇₉	—	2	4	no	—
127	C₂×Dic₇₉	—	4	4	no	—
128	Dic₁₅₈	—	4	4	no	—
129	C₈×C₇₉	—	2	2	no	—
130	C₈×C₇₉	—	4	2	no	—
131	C₈×C₇₉	#124	1	2	yes	—
132	C₈×C₇₉	#124	1	2	yes	—
133	C₇₉⋊C₈	—	2	4	no	—
134	C₇₉⋊C₈	—	4	2	no	—
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	—	4	4	no	—
137	C₂²×C₇₉	—	4	2	no	—
138	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₇₉	#529	1	2	yes	—
140	C₄×C₇₉	—	2	2	no	—
141	C₄×C₇₉	—	4	2	no	—
142	C₄×C₇₉	#444	1	4	yes	—
143	C₄×C₇₉	C₄×#41	1	2	yes	—
144	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
145	C₄×C₇₉	#532	1	1	yes	characteristic
146	Dic₇₉	—	4	4	no	—
147	Dic₇₉	—	4	4	no	—
148	#21660	—	79	1	no	maximal, Sylow
149	C₂×C₇₉	—	2	2	no	—
150	C₂×C₇₉	—	8	2	no	—
151	C₂×C₇₉	C₄×#124	1	1	yes	characteristic
152	C₂×C₇₉	C₂×#444	1	2	yes	—
153	#3131	—	79	1	no	—
154	#2960	—	79	1	no	—
155	#2931	D₇₉	1	1	yes	characteristic
156	#1994	—	79	1	no	—
157	#1962	—	79	1	no	—
158	#1938	—	79	4	no	—
159	#1837	—	79	1	no	—
160	#1765	—	79	4	no	—
161	C₄×#97	—	79	1	no	—
162	C₇₉	#21660	1	1	yes	characteristic, complemented, Sylow
163	C₂×#131	C₂×D₇₉	1	2	yes	—
164	#533	—	79	1	no	—
165	#532	C₂×D₇₉	1	1	yes	characteristic
166	#488	—	79	4	no	—
167	#487	—	79	2	no	—
168	#486	—	79	2	no	—
169	#446	—	158	2	no	—
170	C₈×Q₈	—	158	2	no	—
171	#436	—	2	2	no	—
172	C₈×D₄	—	2	2	no	—
173	#424	—	79	2	no	—
174	#423	—	79	1	no	—
175	#423	C₂×D₇₉	1	1	yes	characteristic
176	C₂×#104	—	79	4	no	—
177	#416	—	79	4	no	—
178	#408	C₂×D₇₉	1	2	yes	—
179	#405	—	79	2	no	—
180	C₂×#97	—	79	2	no	—
181	C₂×#97	—	79	2	no	—
182	C₂×C₄×C₈	—	79	1	no	—
183	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
184	C₄³	—	79	1	no	—
185	#341	—	79	4	no	—
186	#339	—	79	4	no	—
187	C₂×#41	C₂²×D₇₉	1	2	yes	—
188	#131	—	2	8	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	79	2	no	—
191	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	79	2	no	—
194	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
196	C₄⋊Q₈	—	79	2	no	—
197	C₄⋊Q₈	C₄×D₇₉	1	1	yes	characteristic
198	#127	C₄×D₇₉	1	1	yes	characteristic
199	#125	—	79	2	no	—
200	#124	C₄×D₇₉	1	2	yes	—
201	C₂²⋊Q₈	—	158	2	no	—
202	#121	—	2	2	no	—
203	C₄×Q₈	—	158	2	no	—
204	C₄×D₄	—	2	2	no	—
205	#117	—	79	2	no	—
206	C₂×C₄²	—	79	2	no	—
207	C₂×C₄²	—	158	2	no	—
208	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	158	4	no	—
211	C₄⋊C₈	—	158	2	no	—
212	#104	—	158	8	no	—
213	#103	—	79	8	no	—
214	#102	—	79	8	no	—
215	C₂²⋊C₈	—	2	4	no	—
216	#97	—	158	4	no	—
217	#97	—	158	2	no	—
218	#97	—	158	4	no	—
219	C₄×C₈	—	79	2	no	—
220	C₄×C₈	—	79	2	no	—
221	C₄×C₈	—	158	2	no	—
222	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
223	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
224	#41	—	2	4	no	—
225	#41	—	2	4	no	—
226	C₂×Q₈	—	158	2	no	—
227	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	C₂×C₄×D₇₉	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	158	2	no	—
233	C₂²×C₄	#11929624	1	2	yes	—
234	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	4	4	no	—
237	M₁₆	—	158	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	—	79	4	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	—	158	2	no	—
244	C₂×C₈	—	158	2	no	—
245	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
246	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
248	C₂×C₈	#11929623	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	79	4	no	—
251	C₄⋊C₄	—	158	4	no	—
252	C₄⋊C₄	—	158	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	158	2	no	—
255	C₄²	—	316	4	no	—
256	C₄²	—	79	4	no	—
257	C₄²	—	158	2	no	—
258	C₄²	—	158	2	no	—
259	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
260	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
261	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#11929624	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	316	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	316	4	no	—
274	C₂×C₄	—	158	4	no	—
275	C₂×C₄	—	158	4	no	—
276	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
277	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
280	C₂×C₄	#41×D₇₉	1	2	yes	—
281	C₂×C₄	C₂×#11929623	1	2	yes	—
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	316	2	no	—
285	C₈	—	158	4	no	—
286	C₈	#14579148	1	2	yes	—
287	C₈	#14579148	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	C₂×C₄×C₂²⋊D₇₉	1	1	yes	characteristic
292	C₂²	#19099095	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	316	4	no	—
296	C₄	—	316	4	no	—
297	C₄	#19098683	1	4	yes	—
298	C₄	C₄×#11929623	1	2	yes	—
299	C₄	C₂×#14579148	1	1	yes	characteristic
300	C₄	#19099109	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	8	2	no	—
303	C₂	C₄×#14579148	1	1	yes	characteristic
304	C₂	C₂×#19098683	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#2931	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#532×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2931	D₇₉
hypocenter	C₇₉	#21660
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #23772334	C₂	semidirect
3. #23764376	C₂	semidirect
5. #23764114	C₂	semidirect
6. #23763537	C₂	semidirect
7. #23763242	C₂	semidirect
8. C₄×#14579071	C₂	semidirect
11. #19098917	C₄	semidirect
12. #19098913	C₄	semidirect
13. #19098909	C₄	semidirect
162. C₇₉	#21660	semidirect

Maximal subgroups

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

subgroup	index
#23773722	2
#23772334	2
#23764376	2
#23764229	2
#23764114	2
#23763537	2
#23763242	2
C₄×#14579071	2
C₇₉×#2931	2
#21660	79