Subgroups — #34181339

Order 20224.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	#23763528	C₂	1	1	yes	maximal, characteristic
3	#23763406	C₂	1	2	yes	maximal
4	#23761284	C₂	1	2	yes	maximal
5	C₂×#334×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₅₈	—	2	2	no	—
8	C₈×Dic₁₅₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₇₉⋊C₈	—	2	1	no	—
11	C₂×#14579086	—	2	2	no	—
12	C₂×C₈×Dic₇₉	—	2	1	no	—
13	#19098433	C₂²	1	2	yes	—
14	#19098326	—	2	2	no	—
15	#19098326	—	2	1	no	—
16	#19098326	—	2	1	no	—
17	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#334×C₇₉	—	2	2	no	—
20	C₂²×Dic₁₅₈	D₄	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	1	no	—
22	C₄×Dic₁₅₈	—	2	1	no	—
23	C₄×Dic₁₅₈	—	2	1	no	—
24	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₅₈	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	—	2	2	no	—
29	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
30	C₂×Dic₇₉⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₇₉	—	2	1	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
33	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₇₉⋊C₈	—	2	4	no	—
35	#14579086	—	4	4	no	—
36	C₈×Dic₇₉	—	2	4	no	—
37	C₂²×C₈×C₇₉	—	2	2	no	—
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₇₉	—	2	2	no	—
42	C₄⋊C₈×C₇₉	—	2	2	no	—
43	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₇₉	D₄	1	2	yes	—
45	C₂×Dic₁₅₈	—	2	1	no	—
46	C₂×Dic₁₅₈	—	2	2	no	—
47	C₂×Dic₁₅₈	C₄⋊C₄	1	2	yes	—
48	C₂×Dic₁₅₈	SD₁₆	1	4	yes	—
49	C₂²×Dic₇₉	—	2	1	no	—
50	C₂²×Dic₇₉	—	2	1	no	—
51	C₂²×Dic₇₉	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	2	2	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₄⋊Dic₇₉	—	2	1	no	—
55	C₄⋊Dic₇₉	—	4	2	no	—
56	C₄⋊Dic₇₉	—	4	2	no	—
57	C₄⋊Dic₇₉	—	4	2	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	C₄⋊Dic₇₉	—	8	2	no	—
60	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
61	Dic₇₉⋊C₄	—	2	2	no	—
62	Dic₇₉⋊C₄	—	2	2	no	—
63	C₄×Dic₇₉	—	2	2	no	—
64	C₄×Dic₇₉	—	2	2	no	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₇₉	—	2	2	no	—
69	C₂×C₈×C₇₉	—	2	2	no	—
70	C₂×C₈×C₇₉	—	4	2	no	—
71	C₂×C₈×C₇₉	—	4	2	no	—
72	C₂×C₈×C₇₉	—	4	2	no	—
73	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₇₉	#41	1	1	yes	characteristic
79	C₂³×C₇₉	#121	1	1	yes	characteristic
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₇₉	#135	1	2	yes	—
90	C₂×C₄×C₇₉	#135	1	2	yes	—
91	C₂×C₄×C₇₉	#137	1	2	yes	—
92	C₂×Dic₇₉	—	2	2	no	—
93	C₂×Dic₇₉	—	2	4	no	—
94	C₂×Dic₇₉	—	2	2	no	—
95	C₂×Dic₇₉	—	4	1	no	—
96	C₂×Dic₇₉	—	4	1	no	—
97	C₂×Dic₇₉	—	8	2	no	—
98	C₂×Dic₇₉	—	8	2	no	—
99	C₂×Dic₇₉	—	8	2	no	—
100	Dic₁₅₈	—	2	2	no	—
101	Dic₁₅₈	—	2	4	no	—
102	Dic₁₅₈	#106	1	4	yes	—
103	C₈×C₇₉	—	2	2	no	—
104	C₈×C₇₉	—	2	2	no	—
105	C₈×C₇₉	—	8	2	no	—
106	C₇₉⋊C₈	—	4	2	no	—
107	C₂²×C₇₉	#380	1	1	yes	characteristic
108	C₂²×C₇₉	#464	1	2	yes	—
109	C₂²×C₇₉	#465	1	2	yes	—
110	C₂²×C₇₉	#472	1	1	yes	characteristic
111	C₂²×C₇₉	#473	1	1	yes	characteristic
112	C₄×C₇₉	—	2	1	no	—
113	C₄×C₇₉	—	2	1	no	—
114	C₄×C₇₉	#416	1	1	yes	characteristic
115	C₄×C₇₉	#419	1	1	yes	characteristic
116	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
117	C₄×C₇₉	#427	1	1	yes	characteristic
118	C₄×C₇₉	#465	1	2	yes	—
119	C₄×C₇₉	#479	1	2	yes	—
120	Dic₇₉	—	2	4	no	—
121	Dic₇₉	—	4	2	no	—
122	Dic₇₉	—	16	2	no	—
123	#8647	—	79	1	no	maximal, Sylow
124	C₂×C₇₉	#1678	1	2	yes	—
125	C₂×C₇₉	#1680	1	2	yes	—
126	C₂×C₇₉	#1805	1	1	yes	characteristic
127	C₂×C₇₉	#1807	1	1	yes	characteristic
128	C₂×C₇₉	#1816	1	1	yes	characteristic
129	C₂×C₈×Q₈	—	79	1	no	—
130	#1834	—	79	1	no	—
131	#1803	—	79	2	no	—
132	C₂×#334	D₇₉	1	1	yes	characteristic
133	#1296	—	79	2	no	—
134	C₇₉	#8647	1	1	yes	characteristic, complemented, Sylow
135	C₂×C₄×Q₈	—	79	1	no	—
136	C₈×Q₈	—	79	4	no	—
137	C₈×Q₈	—	158	2	no	—
138	C₂×C₄⋊C₈	—	79	1	no	—
139	C₂×C₄⋊C₈	—	158	1	no	—
140	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
141	C₂×#103	—	158	2	no	—
142	C₂×C₄×C₈	—	158	1	no	—
143	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
144	#384	—	79	2	no	—
145	#340	—	158	2	no	—
146	#340	—	158	1	no	—
147	#340	—	158	1	no	—
148	#334	—	2	2	no	—
149	C₂²×Q₈	—	79	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	158	1	no	—
152	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
153	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
154	C₄×Q₈	—	79	2	no	—
155	C₄×Q₈	—	79	2	no	—
156	C₄×Q₈	—	158	1	no	—
157	C₄×Q₈	—	158	1	no	—
158	C₂×C₄⋊C₄	—	79	1	no	—
159	C₂×C₄⋊C₄	—	158	2	no	—
160	C₂×C₄⋊C₄	—	158	2	no	—
161	C₂×C₄⋊C₄	—	158	2	no	—
162	C₂×C₄⋊C₄	—	158	1	no	—
163	C₂×C₄²	—	158	1	no	—
164	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	79	2	no	—
168	C₄⋊C₈	—	158	4	no	—
169	C₄⋊C₈	—	158	1	no	—
170	#103	—	316	4	no	—
171	C₄×C₈	—	158	4	no	—
172	C₄×C₈	C₄×D₇₉	1	2	yes	—
173	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
174	C₂×Q₈	—	79	2	no	—
175	C₂×Q₈	—	79	4	no	—
176	C₂×Q₈	—	158	1	no	—
177	C₂×Q₈	—	158	2	no	—
178	C₂²×C₄	—	316	2	no	—
179	C₂²×C₄	—	158	1	no	—
180	C₂²×C₄	—	158	1	no	—
181	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
183	C₂²×C₄	#11929625	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	316	2	no	—
190	C₂×C₈	—	158	2	no	—
191	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
192	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
193	C₄⋊C₄	—	316	2	no	—
194	C₄⋊C₄	—	316	2	no	—
195	C₄⋊C₄	—	316	2	no	—
196	C₄⋊C₄	—	316	2	no	—
197	C₄⋊C₄	—	79	2	no	—
198	C₄⋊C₄	—	632	2	no	—
199	C₄⋊C₄	—	158	2	no	—
200	C₄⋊C₄	—	158	1	no	—
201	C₄⋊C₄	—	158	2	no	—
202	C₄²	—	158	2	no	—
203	C₄²	—	158	2	no	—
204	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
205	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
206	C₄²	Q₈×D₇₉	1	1	yes	characteristic
207	C₄²	#11929623	1	1	yes	characteristic
208	C₂³	#14579136	1	1	yes	characteristic
209	Q₈	—	79	4	no	—
210	Q₈	—	158	4	no	—
211	Q₈	—	158	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	316	1	no	—
215	C₂×C₄	—	316	1	no	—
216	C₂×C₄	—	632	2	no	—
217	C₂×C₄	—	632	2	no	—
218	C₂×C₄	—	632	2	no	—
219	C₂×C₄	—	158	2	no	—
220	C₂×C₄	—	158	2	no	—
221	C₂×C₄	—	158	4	no	—
222	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
224	C₂×C₄	#14579119	1	1	yes	characteristic
225	C₂×C₄	#14579132	1	1	yes	characteristic
226	C₂×C₄	#14579137	1	1	yes	characteristic
227	C₂×C₄	#14579141	1	1	yes	characteristic
228	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
229	C₂×C₄	#14579182	1	2	yes	—
230	C₂×C₄	#14579183	1	2	yes	—
231	C₂×C₄	#14579194	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	8	2	no	—
235	C₈	—	316	2	no	—
236	C₂²	#19098420	1	1	yes	characteristic
237	C₂²	#19098786	1	2	yes	—
238	C₂²	#19098793	1	2	yes	—
239	C₂²	#19098834	1	1	yes	characteristic
240	C₂²	#19098839	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	316	2	no	—
244	C₄	—	158	4	no	—
245	C₄	—	1264	2	no	—
246	C₄	#19098560	1	1	yes	characteristic
247	C₄	#19098573	1	1	yes	characteristic
248	C₄	#106×D₇₉	1	1	yes	characteristic
249	C₄	#19098598	1	1	yes	characteristic
250	C₄	#19098789	1	2	yes	—
251	C₄	#19098871	1	2	yes	—
252	C₂	#23762817	1	2	yes	—
253	C₂	#23762831	1	2	yes	—
254	C₂	#23763416	1	1	yes	characteristic
255	C₂	#23763422	1	1	yes	characteristic
256	C₂	#23763461	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579136
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#334×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#334	D₇₉
hypocenter	C₇₉	#8647
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₅₈	C₄	semidirect
134. C₇₉	#8647	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₅₈	2
#23763528	2
#23763406	2
#23761284	2
C₂×#334×C₇₉	2
#8647	79