Subgroups — #34179467

Order 20224.

446 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#19098315	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098313	C₂	1	1	yes	maximal, characteristic
5	#23761234	C₂	1	1	yes	maximal, characteristic
6	#23761234	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14579072	C₄	1	2	yes	—
11	C₂×#14579072	C₄	1	2	yes	—
12	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	#19098318	—	2	1	no	—
16	#19098315	C₄	1	4	yes	complemented
17	#19098313	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₇₉	—	2	1	no	—
21	C₄³×C₇₉	C₂²	1	1	yes	characteristic
22	#336×C₇₉	—	2	1	no	—
23	C₂²×C₇₉⋊C₈	—	2	1	no	—
24	C₂²×C₇₉⋊C₈	—	2	1	no	—
25	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
29	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
30	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
31	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
32	#14579072	—	2	4	no	—
33	#14579072	C₈	1	8	yes	complemented
34	C₄×C₇₉⋊C₈	—	2	1	no	—
35	C₄×C₇₉⋊C₈	—	2	1	no	—
36	C₄×C₇₉⋊C₈	—	4	2	no	—
37	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₉	—	2	1	no	—
40	C₂²×C₈×C₇₉	—	2	1	no	—
41	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
45	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₇₉	—	2	1	no	—
47	C₄⋊C₈×C₇₉	—	2	1	no	—
48	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
49	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₇₉	—	4	2	no	—
51	C₂×C₇₉⋊C₈	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	2	2	no	—
53	C₂×C₇₉⋊C₈	—	2	2	no	—
54	C₂×C₇₉⋊C₈	—	2	2	no	—
55	C₂×C₇₉⋊C₈	—	2	2	no	—
56	C₂×C₇₉⋊C₈	—	4	1	no	—
57	C₂×C₇₉⋊C₈	—	4	1	no	—
58	C₂×C₇₉⋊C₈	—	4	1	no	—
59	C₂×C₇₉⋊C₈	—	4	1	no	—
60	C₂×C₇₉⋊C₈	C₄²	1	2	yes	—
61	C₂×C₇₉⋊C₈	C₄²	1	2	yes	—
62	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
63	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₇₉	—	2	2	no	—
70	C₂×C₈×C₇₉	—	2	2	no	—
71	C₂×C₈×C₇₉	—	4	1	no	—
72	C₂×C₈×C₇₉	—	4	1	no	—
73	C₂×C₈×C₇₉	—	4	1	no	—
74	C₂×C₈×C₇₉	—	4	1	no	—
75	C₄²×C₇₉	—	2	1	no	—
76	C₄²×C₇₉	—	2	1	no	—
77	C₄²×C₇₉	—	2	1	no	—
78	C₄²×C₇₉	—	2	1	no	—
79	C₄²×C₇₉	—	4	2	no	—
80	C₄²×C₇₉	C₂×C₈	1	4	yes	—
81	C₄²×C₇₉	M₁₆	1	4	yes	—
82	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₇₉	#41	1	1	yes	characteristic
85	C₄²×C₇₉	#41	1	1	yes	characteristic
86	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₇₉	—	2	2	no	—
88	C₂×C₄×C₇₉	—	2	2	no	—
89	C₂×C₄×C₇₉	—	2	2	no	—
90	C₂×C₄×C₇₉	—	2	2	no	—
91	C₂×C₄×C₇₉	—	2	2	no	—
92	C₂×C₄×C₇₉	C₄⋊C₈	1	4	yes	—
93	C₂×C₄×C₇₉	C₄⋊C₈	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	1	yes	characteristic
97	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
105	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
106	C₈×C₇₉	—	4	2	no	—
107	C₈×C₇₉	—	4	2	no	—
108	C₇₉⋊C₈	—	2	2	no	—
109	C₇₉⋊C₈	—	2	4	no	—
110	C₇₉⋊C₈	—	4	2	no	—
111	C₇₉⋊C₈	—	4	2	no	—
112	C₇₉⋊C₈	C₄×C₈	1	4	yes	complemented
113	C₂²×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₇₉	C₂×C₄⋊C₈	1	2	yes	—
115	C₂²×C₇₉	#428	1	1	yes	characteristic
116	C₂²×C₇₉	#428	1	1	yes	characteristic
117	C₂²×C₇₉	#430	1	2	yes	—
118	C₄×C₇₉	—	2	4	no	—
119	C₄×C₇₉	—	2	4	no	—
120	C₄×C₇₉	—	2	1	no	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
123	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
124	C₄×C₇₉	#431	1	2	yes	—
125	C₄×C₇₉	#440	1	1	yes	characteristic
126	C₄×C₇₉	#440	1	1	yes	characteristic
127	C₄×C₇₉	#441	1	1	yes	characteristic
128	C₄×C₇₉	#442	1	1	yes	characteristic
129	#8275	—	79	1	no	maximal, Sylow
130	C₂×C₇₉	#1565	1	2	yes	—
131	C₂×C₇₉	#1565	1	2	yes	—
132	C₂×C₇₉	C₄×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₇₉	#1776	1	1	yes	characteristic
134	C₂×C₇₉	#1778	1	1	yes	characteristic
135	C₄×C₄⋊C₈	—	79	1	no	—
136	C₄×C₄⋊C₈	D₇₉	1	1	yes	characteristic
137	C₄×#97	—	79	1	no	—
138	C₂×#335	—	79	1	no	—
139	C₂×#334	—	79	1	no	—
140	#1277	—	79	1	no	—
141	#1277	—	79	1	no	—
142	C₇₉	#8275	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊C₈	—	79	1	no	—
144	C₂×C₄⋊C₈	—	79	1	no	—
145	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
146	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
147	C₂×#97	—	79	2	no	—
148	C₂×#97	—	79	2	no	—
149	C₂×C₄×C₈	—	2	1	no	—
150	C₂×C₄×C₈	—	79	1	no	—
151	C₂×C₄×C₈	—	79	1	no	—
152	C₂×C₄×C₈	—	158	1	no	—
153	C₄³	C₂×D₇₉	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#336	—	158	1	no	—
156	#335	—	79	4	no	—
157	#334	—	79	4	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	79	1	no	—
161	C₂²×C₈	—	79	1	no	—
162	C₂²×C₈	—	79	2	no	—
163	C₂²×C₈	—	158	1	no	—
164	C₂²×C₈	—	158	1	no	—
165	C₂×C₄²	Dic₁₅₈	1	2	yes	—
166	C₂×C₄²	C₄×D₇₉	1	2	yes	—
167	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	79	2	no	—
173	C₄⋊C₈	—	79	2	no	—
174	C₄⋊C₈	—	158	1	no	—
175	C₄⋊C₈	—	158	1	no	—
176	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
177	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
178	#97	—	79	8	no	—
179	#97	—	158	4	no	—
180	C₄×C₈	—	4	2	no	—
181	C₄×C₈	—	316	2	no	—
182	C₄×C₈	—	79	2	no	—
183	C₄×C₈	—	79	2	no	—
184	C₄×C₈	—	158	1	no	—
185	C₄×C₈	—	158	1	no	—
186	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
187	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
188	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	316	1	no	—
198	C₂×C₈	—	316	1	no	—
199	C₂×C₈	—	316	1	no	—
200	C₂×C₈	—	316	1	no	—
201	C₂×C₈	—	79	2	no	—
202	C₂×C₈	—	79	4	no	—
203	C₂×C₈	—	79	2	no	—
204	C₂×C₈	—	79	4	no	—
205	C₂×C₈	—	158	2	no	—
206	C₂×C₈	—	158	2	no	—
207	C₂×C₈	—	158	2	no	—
208	C₂×C₈	—	158	2	no	—
209	C₂×C₈	—	158	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	2	no	—
215	C₄²	C₈×D₇₉	1	4	yes	—
216	C₄²	#11929607	1	4	yes	—
217	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
218	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
219	C₄²	#11929624	1	1	yes	characteristic
220	C₄²	#11929624	1	1	yes	characteristic
221	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	Dic₇₉⋊C₈	1	4	yes	—
228	C₂×C₄	Dic₇₉⋊C₈	1	4	yes	—
229	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
230	C₂×C₄	#14579118	1	1	yes	characteristic
231	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
232	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
233	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
234	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
236	C₂×C₄	C₂×#11929607	1	2	yes	—
237	C₂×C₄	#14579186	1	1	yes	characteristic
238	C₂×C₄	#14579187	1	1	yes	characteristic
239	C₂×C₄	#14579191	1	1	yes	characteristic
240	C₂×C₄	#14579192	1	1	yes	characteristic
241	C₈	—	4	2	no	—
242	C₈	—	4	2	no	—
243	C₈	—	316	2	no	—
244	C₈	—	316	2	no	—
245	C₈	—	79	4	no	—
246	C₈	—	158	2	no	—
247	C₈	—	158	4	no	—
248	C₂²	C₄×Dic₇₉⋊C₄	1	1	yes	characteristic
249	C₂²	C₂×Dic₇₉⋊C₈	1	2	yes	—
250	C₂²	#19098603	1	1	yes	characteristic
251	C₂²	#19098606	1	1	yes	characteristic
252	C₂²	#19098611	1	2	yes	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	4	no	—
257	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
258	C₄	C₄×#11929607	1	1	yes	characteristic
259	C₄	#19098614	1	2	yes	—
260	C₄	#19098661	1	1	yes	characteristic
261	C₄	#19098663	1	1	yes	characteristic
262	C₄	#19098668	1	1	yes	characteristic
263	C₄	#19098673	1	1	yes	characteristic
264	C₂	#23762150	1	2	yes	—
265	C₂	#23762153	1	2	yes	—
266	C₂	C₄×Dic₇₉⋊C₈	1	1	yes	characteristic
267	C₂	#23763298	1	1	yes	characteristic
268	C₂	#23763307	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#8275
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. #19098315	C₄	semidirect
17. #19098313	C₄	semidirect
33. #14579072	C₈	semidirect
112. C₇₉⋊C₈	C₄×C₈	semidirect
142. C₇₉	#8275	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×#19098315	2
C₂×#19098313	2
#23761234	2
C₄×C₄⋊C₈×C₇₉	2
#8275	79