Subgroups — #34179486

Order 20224.

394 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763253	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
5	#23761235	C₂	1	1	yes	maximal, characteristic
6	#23761235	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1744	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×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14579072	—	2	1	no	—
13	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
14	#19098318	—	2	1	no	—
15	#19098318	—	2	1	no	—
16	#19098316	C₄	1	4	yes	complemented
17	#19098314	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	C₄³×C₇₉	C₂²	1	1	yes	characteristic
21	#336×C₇₉	C₄	1	4	yes	—
22	C₂²×C₇₉⋊C₈	—	2	1	no	—
23	C₂²×C₇₉⋊C₈	—	2	1	no	—
24	C₂²×C₇₉⋊C₈	—	2	1	no	—
25	C₂²×C₇₉⋊C₈	—	2	1	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
27	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
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	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
33	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
34	#14579072	—	4	2	no	—
35	C₄×C₇₉⋊C₈	—	4	2	no	—
36	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
42	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₇₉	—	2	1	no	—
44	C₄×C₈×C₇₉	—	2	1	no	—
45	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	2	2	no	—
49	C₂×C₇₉⋊C₈	—	2	2	no	—
50	C₂×C₇₉⋊C₈	—	2	2	no	—
51	C₂×C₇₉⋊C₈	—	4	1	no	—
52	C₂×C₇₉⋊C₈	—	4	1	no	—
53	C₂×C₇₉⋊C₈	—	4	1	no	—
54	C₂×C₇₉⋊C₈	—	4	1	no	—
55	C₂×C₇₉⋊C₈	—	4	1	no	—
56	C₂×C₇₉⋊C₈	—	4	1	no	—
57	C₂×C₇₉⋊C₈	—	4	1	no	—
58	C₂×C₇₉⋊C₈	—	4	1	no	—
59	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₉	—	2	2	no	—
65	C₂×C₈×C₇₉	—	2	4	no	—
66	C₂×C₈×C₇₉	—	2	4	no	—
67	C₂×C₈×C₇₉	C₄²	1	4	yes	—
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	1	no	—
73	C₄²×C₇₉	—	2	1	no	—
74	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇₉	#41	1	1	yes	characteristic
77	C₄²×C₇₉	#41	1	1	yes	characteristic
78	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	—	2	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	#108	1	4	yes	—
86	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₇₉	#131	1	4	yes	—
93	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
97	C₈×C₇₉	—	2	4	no	—
98	C₈×C₇₉	—	4	2	no	—
99	C₇₉⋊C₈	—	4	2	no	—
100	C₇₉⋊C₈	—	4	2	no	—
101	C₇₉⋊C₈	—	4	2	no	—
102	C₇₉⋊C₈	—	4	2	no	—
103	C₂²×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₇₉	#424	1	2	yes	—
105	C₂²×C₇₉	#428	1	1	yes	characteristic
106	C₂²×C₇₉	#428	1	1	yes	characteristic
107	C₂²×C₇₉	C₂×#108	1	2	yes	—
108	C₄×C₇₉	—	2	4	no	—
109	C₄×C₇₉	—	2	1	no	—
110	C₄×C₇₉	—	2	1	no	—
111	C₄×C₇₉	—	4	2	no	—
112	C₄×C₇₉	#405	1	2	yes	—
113	C₄×C₇₉	#433	1	2	yes	—
114	C₄×C₇₉	#440	1	1	yes	characteristic
115	C₄×C₇₉	#440	1	1	yes	characteristic
116	C₄×C₇₉	#441	1	1	yes	characteristic
117	C₄×C₇₉	#442	1	1	yes	characteristic
118	#8279	—	79	1	no	maximal, Sylow
119	C₂×C₇₉	#1573	1	2	yes	—
120	C₂×C₇₉	#1574	1	2	yes	—
121	C₂×C₇₉	#1768	1	1	yes	characteristic
122	C₂×C₇₉	#1776	1	1	yes	characteristic
123	C₂×C₇₉	C₄×#108	1	1	yes	characteristic
124	#1767	—	79	1	no	—
125	C₄×C₄⋊C₈	—	79	1	no	—
126	#1744	D₇₉	1	1	yes	characteristic
127	C₂×#335	—	79	1	no	—
128	C₂×#334	—	79	1	no	—
129	#1277	—	79	1	no	—
130	#1277	—	79	1	no	—
131	C₇₉	#8279	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊C₈	—	79	1	no	—
133	C₂×C₄⋊C₈	—	79	1	no	—
134	C₂×C₄⋊C₈	—	79	1	no	—
135	C₂×C₄⋊C₈	—	79	1	no	—
136	C₂×#97	—	158	1	no	—
137	C₂×C₄×C₈	—	158	1	no	—
138	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
140	C₄³	C₂×D₇₉	1	1	yes	characteristic
141	#336	—	158	1	no	—
142	#336	—	158	1	no	—
143	#336	Dic₇₉	1	4	yes	—
144	#335	—	79	4	no	—
145	#334	—	79	4	no	—
146	C₂²×C₈	—	158	1	no	—
147	C₂²×C₈	—	158	1	no	—
148	C₂²×C₈	—	158	1	no	—
149	C₂²×C₈	—	158	1	no	—
150	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
151	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
152	C₂×C₄²	Dic₁₅₈	1	2	yes	—
153	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
154	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
156	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
157	C₄⋊C₈	—	79	2	no	—
158	C₄⋊C₈	—	79	2	no	—
159	C₄⋊C₈	—	79	2	no	—
160	C₄⋊C₈	—	79	2	no	—
161	C₄⋊C₈	—	158	1	no	—
162	C₄⋊C₈	—	158	1	no	—
163	C₄⋊C₈	—	158	1	no	—
164	C₄⋊C₈	—	158	1	no	—
165	#97	—	316	2	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	316	2	no	—
169	C₄×C₈	C₄×D₇₉	1	2	yes	—
170	C₄×C₈	C₄×D₇₉	1	2	yes	—
171	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
172	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	316	1	no	—
180	C₂×C₈	—	316	1	no	—
181	C₂×C₈	—	316	1	no	—
182	C₂×C₈	—	316	1	no	—
183	C₂×C₈	—	316	1	no	—
184	C₂×C₈	—	316	1	no	—
185	C₂×C₈	—	316	1	no	—
186	C₂×C₈	—	316	1	no	—
187	C₂×C₈	—	158	2	no	—
188	C₂×C₈	—	158	2	no	—
189	C₂×C₈	—	158	2	no	—
190	C₂×C₈	—	158	2	no	—
191	C₂×C₈	C₄×Dic₇₉	1	4	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
199	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
200	C₄²	#11929624	1	1	yes	characteristic
201	C₄²	#11929624	1	1	yes	characteristic
202	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	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	#14579097	1	4	yes	—
210	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
211	C₂×C₄	#14579120	1	1	yes	characteristic
212	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
213	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
214	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
215	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
216	C₂×C₄	#14579168	1	4	yes	—
217	C₂×C₄	#14579189	1	1	yes	characteristic
218	C₂×C₄	#14579189	1	1	yes	characteristic
219	C₂×C₄	#14579194	1	1	yes	characteristic
220	C₂×C₄	#14579194	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₈	—	316	2	no	—
224	C₈	—	316	2	no	—
225	C₈	—	316	2	no	—
226	C₈	—	316	2	no	—
227	C₂²	C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
228	C₂²	#19098590	1	2	yes	—
229	C₂²	#19098605	1	1	yes	characteristic
230	C₂²	#19098605	1	1	yes	characteristic
231	C₂²	C₂×#14579097	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	2	no	—
236	C₄	#19098519	1	2	yes	—
237	C₄	#19098624	1	2	yes	—
238	C₄	#19098665	1	1	yes	characteristic
239	C₄	#19098665	1	1	yes	characteristic
240	C₄	#19098671	1	1	yes	characteristic
241	C₄	#19098676	1	1	yes	characteristic
242	C₂	#23762185	1	2	yes	—
243	C₂	#23762189	1	2	yes	—
244	C₂	#23763257	1	1	yes	characteristic
245	C₂	#23763297	1	1	yes	characteristic
246	C₂	C₄×#14579097	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#8279
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
16. #19098316	C₄	semidirect
17. #19098314	C₄	semidirect
131. C₇₉	#8279	semidirect

Maximal subgroups

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

subgroup	index
#23763253	2
C₄×C₄⋊(C₇₉⋊C₈)	2
C₂×#19098316	2
C₂×#19098314	2
#23761235	2
C₇₉×#1744	2
#8279	79