Finite Group Databasebeta

← #34177569

Subgroups — #34177569

Order 20224.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764963C212yesmaximal
2#23764938C211yesmaximal, characteristic
3#23761226C211yesmaximal, characteristic
4#23761223C212yesmaximal
5C79×#2734C211yesmaximal, characteristic
6C2×#14579094—21no—
7C2×#14579085—21no—
8#19098433C2211yescharacteristic
9#19098325—22no—
10#19098325—22no—
11#19098318C2211yescharacteristic
12#19098318C2212yes—
13C2×C4⋊Q8×C79—21no—
14#398×C79—21no—
15#395×C79—22no—
16#391×C79C2211yescharacteristic
17#389×C79C2212yes—
18#387×C79—22no—
19C22×C79⋊C8—21no—
20C22×C79⋊C8—22no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79—22no—
23#14579094—42no—
24#14579085—42no—
25C22×Q8×C79—21no—
26C22×Q8×C79D411yescharacteristic
27C4⋊Q8×C79—42no—
28C4⋊Q8×C79—42no—
29C2×C4⋊C4×C79—22no—
30C2×C4⋊C4×C79—22no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79D412yes—
33C2×C4⋊C4×C79D412yes—
34C2×C4⋊C4×C79D411yescharacteristic
35C2×C42×C79—21no—
36C2×C42×C79C2311yescharacteristic
37#95×C79—21no—
38#95×C79—22no—
39#95×C79—22no—
40#95×C79—22no—
41C22×Dic79—41no—
42C2×C79⋊C8—22no—
43C2×C79⋊C8—41no—
44C2×C79⋊C8—42no—
45C2×C79⋊C8—42no—
46C2×C79⋊C8—41no—
47C2×C79⋊C8—42no—
48C4⋊Dic79—41no—
49C4⋊Dic79—41no—
50C4⋊Dic79—44no—
51C2×Q8×C79—21no—
52C2×Q8×C79—42no—
53C2×Q8×C79—41no—
54C2×Q8×C79—41no—
55C2×Q8×C79—81no—
56C2×Q8×C79—81no—
57C2×Q8×C79SD1612yes—
58C22×C4×C79—22no—
59C22×C4×C79—21no—
60C22×C4×C79—21no—
61C22×C4×C79—22no—
62C22×C4×C79C2×D411yescharacteristic
63C22×C4×C79C2×D412yes—
64C4⋊C4×C79—22no—
65C4⋊C4×C79—44no—
66C4⋊C4×C79—42no—
67C4⋊C4×C79—41no—
68C4⋊C4×C79—42no—
69C4⋊C4×C79—42no—
70C4⋊C4×C79—81no—
71C4⋊C4×C79Q1614yes—
72C42×C79—41no—
73C42×C79—42no—
74C23×C79#12011yescharacteristic
75Q8×C79—42no—
76Q8×C79—81no—
77Q8×C79—84no—
78C2×C4×C79—21no—
79C2×C4×C79—22no—
80C2×C4×C79—22no—
81C2×C4×C79—21no—
82C2×C4×C79—22no—
83C2×C4×C79—24no—
84C2×C4×C79—42no—
85C2×C4×C79—41no—
86C2×C4×C79—42no—
87C2×C4×C79—41no—
88C2×C4×C79—42no—
89C2×C4×C79—41no—
90C2×C4×C79—42no—
91C2×C4×C79—42no—
92C2×C4×C79—42no—
93C2×C4×C79C2×SD1611yescharacteristic
94C2×C4×C79C2×Q1612yes—
95C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C79#13712yes—
97C2×Dic79—81no—
98C2×Dic79—82no—
99C79⋊C8—42no—
100C79⋊C8—82no—
101C22×C79#45011yescharacteristic
102C22×C79#45112yes—
103C22×C79#45311yescharacteristic
104C22×C79#45412yes—
105C22×C79#45811yescharacteristic
106C4×C79—21no—
107C4×C79—21no—
108C4×C79—22no—
109C4×C79—22no—
110C4×C79—42no—
111C4×C79—44no—
112C4×C79—82no—
113C4×C79—81no—
114Dic79—161no—
115#7829—791nomaximal, Sylow
116C2×C79#161311yescharacteristic
117C2×C79#161712yes—
118C2×C79#163311yescharacteristic
119C2×C79#163712yes—
120C2×C79#165711yescharacteristic
121#2734D7911yescharacteristic
122#2100—792no—
123#2095—791no—
124#1272—791no—
125#1271—792no—
126C79#782911yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—21no—
128C2×#107—1581no—
129C2×#103—1581no—
130#398—21no—
131#395—22no—
132#391C2×D7911yescharacteristic
133#389C2×D7912yes—
134#387—22no—
135#384—791no—
136#340—1582no—
137#340—1582no—
138#336—791no—
139#336—792no—
140C22×Q8—21no—
141C22×Q8C22⋊D7911yescharacteristic
142C22×C8—1582no—
143C22×C8—1581no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—1582no—
150C2×C4⋊C4—1581no—
151C2×C4⋊C4C22⋊D7912yes—
152C2×C4⋊C4C22⋊D7912yes—
153C2×C4⋊C4C22⋊D7911yescharacteristic
154C2×C42—21no—
155C2×C42C22×D7911yescharacteristic
156#107—3162no—
157#103—3162no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—22no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8#1192961212yes—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—3161no—
174C22×C4C2×C22⋊D7911yescharacteristic
175C22×C4C2×C22⋊D7912yes—
176C2×C8—3161no—
177C2×C8—3161no—
178C2×C8—3162no—
179C2×C8—3162no—
180C2×C8—3162no—
181C2×C8—1582no—
182C4⋊C4—22no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4—3161no—
190C4⋊C4—3161no—
191C4⋊C4—3164no—
192C4⋊C4#1192961414yes—
193C42—41no—
194C42—42no—
195C23#1457913011yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—84no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—6321no—
215C2×C4—6322no—
216C2×C4C2×#1192961211yescharacteristic
217C2×C4C2×#1192961412yes—
218C2×C4#1457918711yescharacteristic
219C2×C4#1457919212yes—
220C8—3162no—
221C8—6322no—
222C22#1909871411yescharacteristic
223C22#1909871912yes—
224C22#1909873111yescharacteristic
225C22#1909873612yes—
226C22#1909875311yescharacteristic
227C4—22no—
228C4—22no—
229C4—21no—
230C4—21no—
231C4—44no—
232C4—42no—
233C4—81no—
234C4—82no—
235C4—12641no—
236C2#2376242011yescharacteristic
237C2#2376244212yes—
238C2#2376254011yescharacteristic
239C2#2376256512yes—
240C2#2376269811yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2734C2
socle, prime radicalC23×C79#120
hypercenter#2734D79
hypocenterC79#7829
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
126. C79#7829semidirect

Maximal subgroups

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

subgroupindex
#237649632
#237649382
#237612262
#237612232
C79×#27342
#782979