Subgroups — #34175455

Order 20224.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763560	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579095	—	2	1	no	—
5	C₂×#14579095	C₄	1	2	yes	—
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	#19098433	—	2	1	no	—
8	#19098316	—	2	1	no	—
9	#19098316	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₉	—	2	1	no	—
13	C₄⋊C₁₆×C₇₉	C₄	1	2	yes	complemented
14	C₂²×C₇₉⋊C₈	—	4	1	no	—
15	C₂×C₄⋊Dic₇₉	—	2	2	no	—
16	C₂×C₄⋊Dic₇₉	—	4	1	no	—
17	C₂×C₄⋊Dic₇₉	—	4	1	no	—
18	#14579095	—	4	1	no	—
19	#14579095	—	4	1	no	—
20	#14579095	—	4	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	—	2	2	no	—
27	C₂×C₁₆×C₇₉	—	4	1	no	—
28	C₂×C₁₆×C₇₉	—	4	1	no	—
29	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₉	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	8	1	no	—
36	C₂×C₇₉⋊C₈	—	8	1	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₄⋊Dic₇₉	—	8	2	no	—
40	C₄⋊Dic₇₉	—	8	1	no	—
41	C₄⋊Dic₇₉	—	8	1	no	—
42	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₉	—	2	1	no	—
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
52	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
55	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
58	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
59	C₁₆×C₇₉	—	4	2	no	—
60	C₂³×C₇₉	#102	1	1	yes	characteristic
61	C₂×C₄×C₇₉	—	2	1	no	—
62	C₂×C₄×C₇₉	—	2	1	no	—
63	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
65	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
73	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
75	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
76	C₂×C₄×C₇₉	Q₃₂	1	1	yes	characteristic
77	C₂×Dic₇₉	—	8	2	no	—
78	C₂×Dic₇₉	—	8	2	no	—
79	C₂×Dic₇₉	—	8	2	no	—
80	C₈×C₇₉	—	2	1	no	—
81	C₈×C₇₉	—	2	1	no	—
82	C₈×C₇₉	#102	1	1	yes	characteristic
83	C₈×C₇₉	#103	1	1	yes	characteristic
84	C₈×C₇₉	#106	1	1	yes	characteristic
85	C₈×C₇₉	#107	1	1	yes	characteristic
86	C₇₉⋊C₈	—	8	2	no	—
87	C₂²×C₇₉	#331	1	1	yes	characteristic
88	C₂²×C₇₉	#332	1	1	yes	characteristic
89	C₂²×C₇₉	#340	1	1	yes	characteristic
90	C₂²×C₇₉	#357	1	1	yes	characteristic
91	C₂²×C₇₉	#358	1	1	yes	characteristic
92	C₂²×C₇₉	#361	1	1	yes	characteristic
93	C₂²×C₇₉	#362	1	1	yes	characteristic
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	#332	1	1	yes	characteristic
97	C₄×C₇₉	#333	1	1	yes	characteristic
98	C₄×C₇₉	#340	1	1	yes	characteristic
99	C₄×C₇₉	#343	1	1	yes	characteristic
100	C₄×C₇₉	#357	1	1	yes	characteristic
101	C₄×C₇₉	#358	1	1	yes	characteristic
102	C₄×C₇₉	#366	1	1	yes	characteristic
103	C₄×C₇₉	#367	1	1	yes	characteristic
104	Dic₇₉	—	16	2	no	—
105	#7274	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	#1296	1	1	yes	characteristic
107	C₂×C₇₉	#1361	1	1	yes	characteristic
108	C₂×C₇₉	#1362	1	1	yes	characteristic
109	C₂×C₇₉	#1364	1	1	yes	characteristic
110	C₂×C₇₉	#1365	1	1	yes	characteristic
111	C₂×C₇₉	#1380	1	1	yes	characteristic
112	C₂×C₇₉	#1387	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
114	#1845	—	79	1	no	—
115	C₂×#335	—	79	1	no	—
116	C₇₉	#7274	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₁₆	—	2	1	no	—
118	C₂×#107	—	79	2	no	—
119	C₂×#107	—	158	1	no	—
120	C₂×C₄⋊C₈	—	158	1	no	—
121	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
122	#384	—	158	1	no	—
123	C₄⋊C₁₆	—	2	1	no	—
124	C₄⋊C₁₆	Dic₇₉	1	2	yes	complemented
125	#335	—	79	2	no	—
126	#335	—	158	1	no	—
127	C₂²×C₈	—	316	1	no	—
128	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	316	1	no	—
131	C₂×C₄⋊C₄	—	316	1	no	—
132	C₂×C₄⋊C₄	—	158	2	no	—
133	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	4	1	no	—
136	C₂×C₁₆	—	4	1	no	—
137	#107	—	316	1	no	—
138	#107	—	316	1	no	—
139	#107	—	316	2	no	—
140	C₄⋊C₈	—	316	1	no	—
141	C₄⋊C₈	—	158	2	no	—
142	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
143	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂²×C₄	—	316	2	no	—
147	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
149	C₂²×C₄	#11929613	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	—	632	1	no	—
154	C₂×C₈	—	632	1	no	—
155	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
157	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
158	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
159	C₂×C₈	#11929609	1	1	yes	characteristic
160	C₂×C₈	#11929611	1	1	yes	characteristic
161	C₂×C₈	#11929612	1	1	yes	characteristic
162	C₂×C₈	#11929614	1	1	yes	characteristic
163	C₄⋊C₄	—	316	2	no	—
164	C₄⋊C₄	—	316	2	no	—
165	C₄⋊C₄	—	632	2	no	—
166	C₄⋊C₄	—	632	1	no	—
167	C₄⋊C₄	—	632	1	no	—
168	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
169	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
170	C₄²	#11929613	1	1	yes	characteristic
171	C₄²	Dic₃₁₆	1	1	yes	characteristic
172	C₁₆	—	4	2	no	—
173	C₂³	#14579083	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	632	2	no	—
177	C₂×C₄	—	632	2	no	—
178	C₂×C₄	—	632	2	no	—
179	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
180	C₂×C₄	#14579077	1	1	yes	characteristic
181	C₂×C₄	#14579079	1	1	yes	characteristic
182	C₂×C₄	#14579081	1	1	yes	characteristic
183	C₂×C₄	#14579083	1	1	yes	characteristic
184	C₂×C₄	#14579084	1	1	yes	characteristic
185	C₂×C₄	#14579086	1	1	yes	characteristic
186	C₂×C₄	#14579086	1	1	yes	characteristic
187	C₂×C₄	#14579093	1	1	yes	characteristic
188	C₂×C₄	#14579095	1	1	yes	characteristic
189	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
190	C₂×C₄	#14579106	1	1	yes	characteristic
191	C₂×C₄	#14579106	1	1	yes	characteristic
192	C₂×C₄	Dic₆₃₂	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	632	2	no	—
196	C₈	#14579082	1	1	yes	characteristic
197	C₈	#14579085	1	1	yes	characteristic
198	C₈	#14579092	1	1	yes	characteristic
199	C₈	#14579094	1	1	yes	characteristic
200	C₂²	#19098306	1	1	yes	characteristic
201	C₂²	#19098308	1	1	yes	characteristic
202	C₂²	#19098326	1	1	yes	characteristic
203	C₂²	#19098365	1	1	yes	characteristic
204	C₂²	#19098368	1	1	yes	characteristic
205	C₂²	#19098377	1	1	yes	characteristic
206	C₂²	#19098380	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	1264	2	no	—
210	C₄	#19098310	1	1	yes	characteristic
211	C₄	#19098311	1	1	yes	characteristic
212	C₄	#19098325	1	1	yes	characteristic
213	C₄	#19098331	1	1	yes	characteristic
214	C₄	#19098365	1	1	yes	characteristic
215	C₄	#19098368	1	1	yes	characteristic
216	C₄	#19098389	1	1	yes	characteristic
217	C₄	#19098391	1	1	yes	characteristic
218	C₂	#23761284	1	1	yes	characteristic
219	C₂	#23761447	1	1	yes	characteristic
220	C₂	#23761450	1	1	yes	characteristic
221	C₂	#23761454	1	1	yes	characteristic
222	C₂	#23761457	1	1	yes	characteristic
223	C₂	#23761492	1	1	yes	characteristic
224	C₂	#23761509	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579083
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#102
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#7274
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. #19098316	C₄	semidirect
13. C₄⋊C₁₆×C₇₉	C₄	semidirect
116. C₇₉	#7274	semidirect
124. C₄⋊C₁₆	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23763560	2
C₂×#19098316	2
C₂×C₄⋊C₁₆×C₇₉	2
#7274	79