Subgroups — #34186956

Order 20224.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764163	C₂	1	2	yes	maximal
2	#23763560	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
4	#23761284	C₂	1	2	yes	maximal
5	C₇₉×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579095	—	2	1	no	—
7	C₂×#14579095	—	2	1	no	—
8	C₂×#14579092	—	2	2	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
10	#19098433	C₂²	1	2	yes	—
11	#19098325	—	2	2	no	—
12	#19098314	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×#97×C₇₉	—	2	1	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
17	#336×C₇₉	—	2	1	no	—
18	C₂²×C₇₉⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	—	2	2	no	—
22	#14579095	—	4	2	no	—
23	#14579095	—	4	2	no	—
24	#14579092	—	4	4	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₇₉	—	2	1	no	—
28	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	—	2	1	no	—
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₉	—	2	1	no	—
35	C₄⋊C₈×C₇₉	Q₈	1	2	yes	—
36	#97×C₇₉	—	4	2	no	—
37	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₉	Q₈	1	2	yes	—
39	#95×C₇₉	—	2	1	no	—
40	C₂²×Dic₇₉	—	4	2	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	C₄⋊Dic₇₉	—	4	2	no	—
46	C₄⋊Dic₇₉	—	4	2	no	—
47	C₄⋊Dic₇₉	—	4	2	no	—
48	C₄⋊Dic₇₉	—	8	2	no	—
49	C₂²×C₄×C₇₉	—	2	1	no	—
50	C₂²×C₄×C₇₉	—	2	1	no	—
51	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₉	—	2	2	no	—
55	C₂×C₈×C₇₉	—	2	1	no	—
56	C₂×C₈×C₇₉	—	2	1	no	—
57	C₂×C₈×C₇₉	—	4	1	no	—
58	C₂×C₈×C₇₉	—	4	1	no	—
59	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
62	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
63	C₄⋊C₄×C₇₉	—	2	2	no	—
64	C₄⋊C₄×C₇₉	—	2	2	no	—
65	C₄²×C₇₉	—	4	1	no	—
66	C₄²×C₇₉	—	4	1	no	—
67	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₇₉	#41	1	1	yes	characteristic
71	C₂³×C₇₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	—	4	1	no	—
75	C₂×C₄×C₇₉	—	4	1	no	—
76	C₂×C₄×C₇₉	—	4	1	no	—
77	C₂×C₄×C₇₉	—	4	1	no	—
78	C₂×C₄×C₇₉	—	4	1	no	—
79	C₂×C₄×C₇₉	—	4	1	no	—
80	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#137	1	2	yes	—
92	C₂×Dic₇₉	—	8	2	no	—
93	C₂×Dic₇₉	—	8	2	no	—
94	C₂×Dic₇₉	—	8	2	no	—
95	C₈×C₇₉	—	2	2	no	—
96	C₈×C₇₉	—	4	2	no	—
97	C₈×C₇₉	#106	1	4	yes	complemented
98	C₇₉⋊C₈	—	4	4	no	—
99	C₂²×C₇₉	#389	1	1	yes	characteristic
100	C₂²×C₇₉	#460	1	1	yes	characteristic
101	C₂²×C₇₉	#461	1	1	yes	characteristic
102	C₂²×C₇₉	#469	1	1	yes	characteristic
103	C₂²×C₇₉	#470	1	1	yes	characteristic
104	C₂²×C₇₉	#472	1	1	yes	characteristic
105	C₂²×C₇₉	#473	1	1	yes	characteristic
106	C₄×C₇₉	—	2	1	no	—
107	C₄×C₇₉	—	2	1	no	—
108	C₄×C₇₉	—	8	1	no	—
109	C₄×C₇₉	—	8	1	no	—
110	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
111	C₄×C₇₉	#428	1	1	yes	characteristic
112	C₄×C₇₉	#440	1	2	yes	—
113	C₄×C₇₉	#475	1	1	yes	characteristic
114	C₄×C₇₉	#476	1	1	yes	characteristic
115	C₄×C₇₉	#501	1	2	yes	—
116	Dic₇₉	—	16	2	no	—
117	#9621	—	79	1	no	maximal, Sylow
118	C₂×C₇₉	#1688	1	1	yes	characteristic
119	C₂×C₇₉	#1690	1	1	yes	characteristic
120	C₂×C₇₉	#1693	1	1	yes	characteristic
121	C₂×C₇₉	#1695	1	1	yes	characteristic
122	C₂×C₇₉	#1944	1	1	yes	characteristic
123	C₂×C₇₉	#1948	1	1	yes	characteristic
124	C₂×C₇₉	#1952	1	1	yes	characteristic
125	#1948	—	79	2	no	—
126	#1845	—	79	1	no	—
127	#1770	D₇₉	1	1	yes	characteristic
128	C₂×#334	—	79	1	no	—
129	#1296	—	79	2	no	—
130	C₇₉	#9621	1	1	yes	characteristic, complemented, Sylow
131	C₂×#107	—	158	1	no	—
132	C₂×#107	—	158	1	no	—
133	C₂×#106	—	158	2	no	—
134	C₂×C₄⋊C₈	—	79	2	no	—
135	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
136	C₂×#97	—	2	1	no	—
137	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
138	#384	—	79	2	no	—
139	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
140	#340	—	158	2	no	—
141	#336	—	2	1	no	—
142	#334	—	79	4	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	158	2	no	—
145	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	158	2	no	—
148	C₂×C₄⋊C₄	—	158	2	no	—
149	C₂×C₄⋊C₄	—	158	2	no	—
150	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
151	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
154	#107	—	316	2	no	—
155	#107	—	316	2	no	—
156	#106	—	316	4	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	79	4	no	—
159	C₄⋊C₈	—	158	2	no	—
160	C₄⋊C₈	Dic₁₅₈	1	2	yes	—
161	#97	—	4	2	no	—
162	C₄×C₈	Dic₁₅₈	1	2	yes	—
163	C₄×C₈	C₄×D₇₉	1	2	yes	—
164	#95	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	316	2	no	—
168	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
170	C₂²×C₄	#11929624	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	316	2	no	—
177	C₂×C₈	—	316	2	no	—
178	C₂×C₈	—	158	4	no	—
179	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
180	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
181	C₂×C₈	#11929611	1	2	yes	—
182	C₂×C₈	#11929612	1	2	yes	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	316	2	no	—
186	C₄⋊C₄	—	316	2	no	—
187	C₄⋊C₄	—	316	2	no	—
188	C₄⋊C₄	—	316	2	no	—
189	C₄⋊C₄	—	632	2	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	4	1	no	—
192	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
193	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
194	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
195	C₄²	#11929624	1	1	yes	characteristic
196	C₂³	#14579135	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	632	2	no	—
206	C₂×C₄	—	632	2	no	—
207	C₂×C₄	—	632	2	no	—
208	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
210	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
211	C₂×C₄	#14579140	1	1	yes	characteristic
212	C₂×C₄	#14579153	1	1	yes	characteristic
213	C₂×C₄	#14579161	1	1	yes	characteristic
214	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
215	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
216	C₂×C₄	#14579186	1	1	yes	characteristic
217	C₂×C₄	#14579189	1	2	yes	—
218	C₂×C₄	#14579191	1	1	yes	characteristic
219	C₂×C₄	#14579194	1	2	yes	—
220	C₈	—	2	2	no	—
221	C₈	—	4	2	no	—
222	C₈	—	316	4	no	—
223	C₈	#14579092	1	4	yes	complemented
224	C₂²	#19098461	1	1	yes	characteristic
225	C₂²	#19098763	1	1	yes	characteristic
226	C₂²	#19098768	1	1	yes	characteristic
227	C₂²	#19098816	1	1	yes	characteristic
228	C₂²	#19098821	1	1	yes	characteristic
229	C₂²	#19098833	1	1	yes	characteristic
230	C₂²	#19098838	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	8	1	no	—
234	C₄	—	8	1	no	—
235	C₄	—	1264	2	no	—
236	C₄	C₂×#14579092	1	1	yes	characteristic
237	C₄	#19098606	1	1	yes	characteristic
238	C₄	#19098665	1	2	yes	—
239	C₄	#19098848	1	1	yes	characteristic
240	C₄	#19098853	1	1	yes	characteristic
241	C₄	#19098972	1	2	yes	—
242	C₂	#23762884	1	1	yes	characteristic
243	C₂	#23762894	1	1	yes	characteristic
244	C₂	#23762913	1	1	yes	characteristic
245	C₂	#23762923	1	1	yes	characteristic
246	C₂	#23764143	1	1	yes	characteristic
247	C₂	#23764163	1	1	yes	characteristic
248	C₂	#23764185	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579135
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1770	C₂
socle, prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1770	D₇₉
hypocenter	C₇₉	#9621
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
12. #19098314	C₄	semidirect
97. C₈×C₇₉	#106	semidirect
130. C₇₉	#9621	semidirect
223. C₈	#14579092	semidirect

Maximal subgroups

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

subgroup	index
#23764163	2
#23763560	2
C₂×#19098314	2
#23761284	2
C₇₉×#1770	2
#9621	79