Subgroups — #34178921

Order 20224.

530 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	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579072	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
4	#23761241	C₂	1	2	yes	maximal
5	C₇₉×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14579072	—	2	2	no	—
8	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
10	#19098318	—	2	1	no	—
11	Q₈⋊(C₇₉⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
13	#391×C₇₉	C₄	1	2	yes	—
14	#389×C₇₉	C₄	1	2	yes	—
15	#387×C₇₉	—	2	2	no	—
16	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
17	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
18	#376×C₇₉	—	2	1	no	—
19	C₄³×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
24	#14579072	—	2	8	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	4	2	no	—
27	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₇₉	—	2	2	no	—
30	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	—	2	1	no	—
36	C₂×C₄²×C₇₉	—	2	1	no	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	D₄	1	2	yes	—
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	#95×C₇₉	—	2	2	no	—
43	#95×C₇₉	—	2	2	no	—
44	#95×C₇₉	—	2	2	no	—
45	#95×C₇₉	—	2	1	no	—
46	C₂×C₇₉⋊C₈	—	2	8	no	—
47	C₂×C₇₉⋊C₈	—	2	4	no	—
48	C₂×C₇₉⋊C₈	—	2	4	no	—
49	C₂×C₇₉⋊C₈	—	4	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×Q₈×C₇₉	—	2	1	no	—
52	C₂×Q₈×C₇₉	—	2	4	no	—
53	C₂×Q₈×C₇₉	C₄²	1	2	yes	—
54	C₂²×C₄×C₇₉	—	2	1	no	—
55	C₂²×C₄×C₇₉	—	2	1	no	—
56	C₂²×C₄×C₇₉	—	2	2	no	—
57	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₇₉	—	2	1	no	—
63	C₄⋊C₄×C₇₉	—	2	4	no	—
64	C₄⋊C₄×C₇₉	—	2	4	no	—
65	C₄⋊C₄×C₇₉	—	4	4	no	—
66	C₄⋊C₄×C₇₉	C₄²	1	2	yes	—
67	C₄²×C₇₉	—	2	4	no	—
68	C₄²×C₇₉	—	2	4	no	—
69	C₄²×C₇₉	—	2	4	no	—
70	C₄²×C₇₉	—	2	1	no	—
71	C₄²×C₇₉	—	2	1	no	—
72	C₄²×C₇₉	—	2	2	no	—
73	C₄²×C₇₉	—	4	1	no	—
74	C₄²×C₇₉	—	4	1	no	—
75	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₇₉	#41	1	2	yes	—
77	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₇₉	—	2	4	no	—
79	Q₈×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	2	no	—
83	C₂×C₄×C₇₉	—	2	2	no	—
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	2	2	no	—
86	C₂×C₄×C₇₉	—	2	2	no	—
87	C₂×C₄×C₇₉	—	2	2	no	—
88	C₂×C₄×C₇₉	—	4	2	no	—
89	C₂×C₄×C₇₉	—	4	2	no	—
90	C₂×C₄×C₇₉	—	4	2	no	—
91	C₂×C₄×C₇₉	—	4	2	no	—
92	C₂×C₄×C₇₉	—	4	2	no	—
93	C₂×C₄×C₇₉	#104	1	4	yes	—
94	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₇₉	#131	1	4	yes	—
99	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#137	1	2	yes	—
102	C₇₉⋊C₈	—	2	8	no	—
103	C₇₉⋊C₈	—	4	4	no	—
104	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₇₉	#408	1	2	yes	—
106	C₂²×C₇₉	#417	1	1	yes	characteristic
107	C₂²×C₇₉	#419	1	1	yes	characteristic
108	C₂²×C₇₉	C₂×#104	1	2	yes	—
109	C₄×C₇₉	—	2	4	no	—
110	C₄×C₇₉	—	2	2	no	—
111	C₄×C₇₉	—	4	2	no	—
112	C₄×C₇₉	—	4	2	no	—
113	C₄×C₇₉	—	4	2	no	—
114	C₄×C₇₉	—	8	2	no	—
115	C₄×C₇₉	#405	1	2	yes	—
116	C₄×C₇₉	#433	1	2	yes	—
117	C₄×C₇₉	#440	1	2	yes	—
118	C₄×C₇₉	#441	1	2	yes	—
119	#8180	—	79	1	no	maximal, Sylow
120	C₂×C₇₉	#1483	1	2	yes	—
121	C₂×C₇₉	#1485	1	2	yes	—
122	C₂×C₇₉	#1750	1	1	yes	characteristic
123	C₂×C₇₉	C₄×#104	1	1	yes	characteristic
124	C₂×C₇₉	#1763	1	1	yes	characteristic
125	#2295	D₇₉	1	1	yes	characteristic
126	C₄×C₄⋊C₈	—	79	1	no	—
127	C₄×#97	—	79	1	no	—
128	C₂×Q₈⋊C₈	—	79	2	no	—
129	#1279	—	79	2	no	—
130	C₇₉	#8180	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	79	2	no	—
133	C₂×#97	—	158	2	no	—
134	C₂×C₄×C₈	—	79	2	no	—
135	C₂×C₄×C₈	—	158	1	no	—
136	#391	Dic₇₉	1	2	yes	—
137	#389	Dic₇₉	1	2	yes	—
138	#387	—	2	2	no	—
139	C₄×C₄⋊C₄	—	2	1	no	—
140	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
141	#376	—	2	1	no	—
142	C₄³	C₂×D₇₉	1	1	yes	characteristic
143	#336	—	158	1	no	—
144	Q₈⋊C₈	—	79	8	no	—
145	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
146	C₂²×C₈	—	158	2	no	—
147	C₂²×C₈	—	158	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	C₄×D₇₉	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
161	C₄⋊C₈	—	79	4	no	—
162	C₄⋊C₈	—	158	2	no	—
163	#97	—	158	8	no	—
164	C₄×C₈	—	316	2	no	—
165	C₄×C₈	—	79	4	no	—
166	C₄×C₈	—	158	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	#95	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	C₄×Dic₇₉	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
182	C₂×C₈	—	316	2	no	—
183	C₂×C₈	—	316	2	no	—
184	C₂×C₈	—	158	4	no	—
185	C₂×C₈	—	158	8	no	—
186	C₂×C₈	—	158	4	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	C₄×Dic₇₉	1	2	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×C₄×D₇₉	1	2	yes	—
201	C₄²	#11929624	1	2	yes	—
202	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	#14579088	1	4	yes	—
219	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
220	C₂×C₄	#14579120	1	1	yes	characteristic
221	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
222	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
223	C₂×C₄	#14579168	1	4	yes	—
224	C₂×C₄	#14579186	1	1	yes	characteristic
225	C₂×C₄	#14579191	1	1	yes	characteristic
226	C₂×C₄	#14579192	1	2	yes	—
227	C₈	—	316	4	no	—
228	C₈	—	158	8	no	—
229	C₂²	C₄×C₂²⋊Dic₇₉	1	1	yes	characteristic
230	C₂²	#19098530	1	2	yes	—
231	C₂²	#19098564	1	1	yes	characteristic
232	C₂²	#19098572	1	1	yes	characteristic
233	C₂²	C₂×#14579088	1	2	yes	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	8	2	no	—
240	C₄	#19098519	1	2	yes	—
241	C₄	#19098624	1	2	yes	—
242	C₄	#19098661	1	2	yes	—
243	C₄	#19098668	1	2	yes	—
244	C₂	#23761813	1	2	yes	—
245	C₂	#23761821	1	2	yes	—
246	C₂	#23763166	1	1	yes	characteristic
247	C₂	C₄×#14579088	1	1	yes	characteristic
248	C₂	#23763230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2295	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2295	D₇₉
hypocenter	C₇₉	#8180
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
11. Q₈⋊(C₇₉⋊C₈)	C₄	semidirect
130. C₇₉	#8180	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₇₉⋊C₈)	2
C₄×#14579072	2
C₂×Q₈⋊(C₇₉⋊C₈)	2
#23761241	2
C₇₉×#2295	2
#8180	79