Subgroups — #34181186

Order 20224.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
2	#23763389	C₂	1	2	yes	maximal
3	#23762156	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761274	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₉	C₄	1	2	yes	—
7	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
8	C₈×D₃₁₆	—	2	1	no	—
9	C₈×D₃₁₆	—	2	1	no	—
10	C₈×D₃₁₆	—	2	1	no	—
11	C₈×D₃₁₆	—	2	1	no	—
12	#19098587	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14579077	—	2	2	no	—
15	C₂×(C₂×D₇₉)⋊C₈	C₄	1	2	yes	—
16	#19098331	—	2	2	no	—
17	#19098314	—	2	1	no	—
18	#19098314	—	2	1	no	—
19	#423×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	#334×C₇₉	—	2	1	no	—
22	#334×C₇₉	—	2	1	no	—
23	C₂²×D₃₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₇₉	C₂×C₄	1	2	yes	—
25	C₂×#11929608	—	2	2	no	—
26	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₇₉	—	2	2	no	—
28	C₂×C₈×D₇₉	—	2	2	no	—
29	C₂×C₈×D₇₉	—	4	4	no	—
30	C₄×D₃₁₆	—	2	1	no	—
31	C₄×D₃₁₆	—	2	1	no	—
32	C₄×D₃₁₆	—	2	1	no	—
33	C₄×D₃₁₆	—	2	1	no	—
34	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
35	C₂×(C₂×D₇₉)⋊C₄	Q₈	1	2	yes	—
36	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
38	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
40	#14579077	—	4	4	no	—
41	(C₂×D₇₉)⋊C₈	—	2	2	no	—
42	(C₂×D₇₉)⋊C₈	—	2	2	no	—
43	C₂×M₁₆×C₇₉	—	2	2	no	—
44	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₉	—	2	2	no	—
48	C₄⋊C₈×C₇₉	—	2	2	no	—
49	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
53	C₂³×D₇₉	C₄⋊C₄	1	2	yes	—
54	C₂×D₃₁₆	—	2	1	no	—
55	C₂×D₃₁₆	—	2	1	no	—
56	C₂×D₃₁₆	—	2	2	no	—
57	C₂×D₃₁₆	SD₁₆	1	4	yes	—
58	C₂²×Dic₇₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×D₇₉	—	2	2	no	—
60	C₂×C₄×D₇₉	—	2	2	no	—
61	C₂×C₄×D₇₉	—	4	2	no	—
62	C₂×C₄×D₇₉	—	4	2	no	—
63	#11929608	—	4	2	no	—
64	#11929608	—	4	2	no	—
65	C₂×C₇₉⋊C₈	—	2	2	no	—
66	C₂×C₇₉⋊C₈	—	4	2	no	—
67	C₂×C₇₉⋊C₈	—	4	2	no	—
68	C₈×D₇₉	—	4	4	no	—
69	C₈×D₇₉	—	4	4	no	—
70	C₄⋊Dic₇₉	—	2	1	no	—
71	C₄⋊Dic₇₉	—	2	1	no	—
72	(C₂×D₇₉)⋊C₄	—	2	2	no	—
73	(C₂×D₇₉)⋊C₄	—	2	2	no	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
77	M₁₆×C₇₉	—	4	2	no	—
78	M₁₆×C₇₉	—	4	2	no	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₂×C₈×C₇₉	—	4	1	no	—
81	C₂×C₈×C₇₉	—	4	2	no	—
82	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₇₉	#41	1	1	yes	characteristic
89	C₄²×C₇₉	#41	1	1	yes	characteristic
90	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×D₇₉	—	2	2	no	—
92	C₂²×D₇₉	—	2	2	no	—
93	C₂²×D₇₉	—	4	2	no	—
94	C₂²×D₇₉	—	4	2	no	—
95	C₂²×D₇₉	#106	1	4	yes	—
96	C₂×C₄×C₇₉	—	2	1	no	—
97	C₂×C₄×C₇₉	—	2	1	no	—
98	C₂×C₄×C₇₉	—	4	1	no	—
99	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₇₉	#135	1	2	yes	—
107	C₂×Dic₇₉	—	2	2	no	—
108	C₂×Dic₇₉	—	4	2	no	—
109	D₃₁₆	—	2	4	no	—
110	D₃₁₆	—	2	4	no	—
111	C₄×D₇₉	—	4	4	no	—
112	C₄×D₇₉	—	4	2	no	—
113	C₄×D₇₉	—	4	2	no	—
114	C₈×C₇₉	—	2	1	no	—
115	C₈×C₇₉	—	2	1	no	—
116	C₈×C₇₉	—	2	1	no	—
117	C₈×C₇₉	—	8	2	no	—
118	C₈×C₇₉	#100	1	2	yes	complemented
119	C₇₉⋊C₈	—	4	2	no	—
120	C₇₉⋊C₈	—	8	2	no	—
121	C₂²×C₇₉	—	2	1	no	—
122	C₂²×C₇₉	—	2	1	no	—
123	C₂²×C₇₉	#380	1	1	yes	characteristic
124	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
125	C₂²×C₇₉	#427	1	1	yes	characteristic
126	C₂×D₇₉	—	2	4	no	—
127	C₂×D₇₉	—	4	4	no	—
128	C₂×D₇₉	—	4	2	no	—
129	C₂×D₇₉	—	4	2	no	—
130	C₂×D₇₉	—	4	4	no	—
131	C₄×C₇₉	—	2	1	no	—
132	C₄×C₇₉	—	2	1	no	—
133	C₄×C₇₉	—	2	1	no	—
134	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
135	C₄×C₇₉	#413	1	1	yes	characteristic
136	C₄×C₇₉	#465	1	2	yes	—
137	C₄×C₇₉	#465	1	2	yes	—
138	Dic₇₉	—	4	2	no	—
139	#8614	—	79	1	no	maximal, Sylow
140	C₂×C₇₉	—	2	1	no	—
141	C₂×C₇₉	—	2	1	no	—
142	C₂×C₇₉	#1800	1	1	yes	characteristic
143	C₂×C₇₉	#1807	1	1	yes	characteristic
144	C₂×C₇₉	#1813	1	1	yes	characteristic
145	D₇₉	—	4	4	no	—
146	C₂×C₈×D₄	—	79	1	no	—
147	#1800	—	79	2	no	—
148	#1566	—	79	1	no	—
149	#1566	D₇₉	1	1	yes	characteristic, complemented
150	#1292	—	79	2	no	—
151	C₇₉	#8614	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₈	—	79	2	no	—
153	C₂×C₄×D₄	—	79	1	no	—
154	C₈×D₄	—	158	1	no	—
155	C₈×D₄	—	158	1	no	—
156	C₈×D₄	—	158	1	no	—
157	C₈×D₄	—	158	1	no	—
158	#423	—	79	2	no	—
159	#423	C₂×D₇₉	1	2	yes	—
160	C₂×C₄⋊C₈	—	79	1	no	—
161	C₂×#100	—	158	2	no	—
162	C₂×C₂²⋊C₈	—	79	2	no	—
163	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
164	#343	—	158	2	no	—
165	#334	—	2	1	no	—
166	#334	—	2	1	no	—
167	#334	—	158	1	no	—
168	#334	—	158	1	no	—
169	C₂²×D₄	—	79	1	no	—
170	C₂³×C₄	—	79	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	158	2	no	—
173	C₂²×C₈	—	316	4	no	—
174	C₂²×C₈	—	79	2	no	—
175	C₂²×C₈	—	158	2	no	—
176	C₂²×C₈	—	158	2	no	—
177	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
179	C₄×D₄	—	158	1	no	—
180	C₄×D₄	—	158	1	no	—
181	C₄×D₄	—	158	1	no	—
182	C₄×D₄	—	158	1	no	—
183	C₂×C₄⋊C₄	—	79	1	no	—
184	C₂×C₂²⋊C₄	—	79	2	no	—
185	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	158	1	no	—
189	C₄⋊C₈	—	158	2	no	—
190	C₄⋊C₈	—	158	1	no	—
191	C₄⋊C₈	—	158	2	no	—
192	#100	—	316	4	no	—
193	C₂²⋊C₈	—	158	2	no	—
194	C₂²⋊C₈	—	158	2	no	—
195	C₄×C₈	D₃₁₆	1	1	yes	characteristic
196	C₄×C₈	D₃₁₆	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
198	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
199	C₂⁴	—	79	2	no	—
200	C₂×D₄	—	79	4	no	—
201	C₂×D₄	—	158	1	no	—
202	C₂×D₄	—	158	1	no	—
203	C₂×D₄	—	158	2	no	—
204	C₂²×C₄	—	316	2	no	—
205	C₂²×C₄	—	316	2	no	—
206	C₂²×C₄	—	79	2	no	—
207	C₂²×C₄	—	158	2	no	—
208	C₂²×C₄	—	158	2	no	—
209	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
211	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	316	2	no	—
215	M₁₆	—	316	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	316	4	no	—
220	C₂×C₈	—	316	4	no	—
221	C₂×C₈	—	316	2	no	—
222	C₂×C₈	—	316	2	no	—
223	C₂×C₈	—	158	2	no	—
224	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
225	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
226	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
227	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
228	C₄⋊C₄	—	158	1	no	—
229	C₄⋊C₄	—	158	1	no	—
230	C₂²⋊C₄	—	158	2	no	—
231	C₂²⋊C₄	—	158	2	no	—
232	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
233	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
234	C₄²	#11929623	1	1	yes	characteristic
235	C₄²	#11929625	1	1	yes	characteristic
236	C₂³	—	316	2	no	—
237	C₂³	—	316	2	no	—
238	C₂³	—	79	4	no	—
239	C₂³	—	158	2	no	—
240	C₂³	—	158	2	no	—
241	C₂³	C₄⋊C₄×D₇₉	1	1	yes	characteristic
242	D₄	—	158	4	no	—
243	D₄	—	158	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	316	4	no	—
248	C₂×C₄	—	316	2	no	—
249	C₂×C₄	—	316	2	no	—
250	C₂×C₄	—	316	2	no	—
251	C₂×C₄	—	158	2	no	—
252	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	#14579119	1	1	yes	characteristic
254	C₂×C₄	#14579132	1	1	yes	characteristic
255	C₂×C₄	#14579136	1	1	yes	characteristic
256	C₂×C₄	#14579137	1	1	yes	characteristic
257	C₂×C₄	#14579141	1	1	yes	characteristic
258	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
259	C₂×C₄	#14579182	1	2	yes	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	8	2	no	—
264	C₈	—	316	2	no	—
265	C₈	—	632	2	no	—
266	C₈	#14579077	1	2	yes	complemented
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	316	4	no	—
270	C₂²	—	316	4	no	—
271	C₂²	—	316	2	no	—
272	C₂²	—	316	2	no	—
273	C₂²	—	158	4	no	—
274	C₂²	#19098420	1	1	yes	characteristic
275	C₂²	#106×D₇₉	1	1	yes	characteristic
276	C₂²	#19098598	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	316	2	no	—
281	C₄	C₂×#14579077	1	1	yes	characteristic
282	C₄	#19098545	1	1	yes	characteristic
283	C₄	#19098789	1	2	yes	—
284	C₄	#19098793	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	316	4	no	—
288	C₂	#23763389	1	1	yes	characteristic
289	C₂	#23763422	1	1	yes	characteristic
290	C₂	#23763447	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098420
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1566	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂²×D₃₁₆	D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1566	D₇₉
hypocenter	C₇₉	#8614
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #23762156	C₂	semidirect
4. #23761274	C₂	semidirect
5. C₇₉×#1566	C₂	semidirect
118. C₈×C₇₉	#100	semidirect
149. #1566	D₇₉	semidirect
151. C₇₉	#8614	semidirect
266. C₈	#14579077	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₁₆	2
#23763389	2
#23762156	2
#23761274	2
C₇₉×#1566	2
#8614	79