Finite Group Databasebeta

← #34186111

Subgroups — #34186111

Order 20224.

474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 127 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#23772481C211yesmaximal, characteristic
2#23768762C211yesmaximal, characteristic
3#23761828C212yesmaximal
4#23761260C212yesmaximal
5C79×#1801C211yesmaximal, characteristic
6C2×C4×Dic158C2211yescharacteristic
7Q8×C79⋊C8—24no—
8#19098616—22no—
9#19098616C2212yes—
10#19098502C412yes—
11#19098483C412yes—
12#19098470—21no—
13#19098459—21no—
14#19098447—22no—
15#19098429C2211yescharacteristic
16#19098293—24no—
17C2×C4×Q8×C79C2211yescharacteristic
18#423×C79C2212yes—
19C2×#101×C79—21no—
20#343×C79—21no—
21C22×Dic158C2×C411yescharacteristic
22C4×Dic158—22no—
23C4×Dic158—22no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×Dic79⋊C4—22no—
27C2×Dic79⋊C4—21no—
28C2×C4×Dic79—21no—
29C4⋊(C79⋊C8)—24no—
30C4⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32C4×C79⋊C8—24no—
33C4×C79⋊C8D414yes—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—21no—
38C79⋊#95C2×C412yes—
39C22×Q8×C79D411yescharacteristic
40C2×M16×C79—22no—
41C4×Q8×C79—22no—
42C4×Q8×C79—22no—
43C2×C4⋊C4×C79C2×C412yes—
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C42×C79D412yes—
46C2×C42×C79C2311yescharacteristic
47C4⋊C8×C79—22no—
48C4⋊C8×C79—22no—
49#101×C79—42no—
50C2×Dic158—22no—
51C2×Dic158—42no—
52C22×Dic79—21no—
53C22×Dic79—22no—
54C22×Dic79—21no—
55C2×C79⋊C8—24no—
56C4⋊Dic79—22no—
57C4⋊Dic79—42no—
58Dic79⋊C4—44no—
59Dic79⋊C4—41no—
60Dic79⋊C4—41no—
61C4×Dic79—42no—
62C2×Q8×C79—22no—
63C2×Q8×C79—22no—
64C2×Q8×C79SD1612yes—
65C2×Q8×C79Q1612yes—
66C22×C4×C79C22⋊C412yes—
67C22×C4×C79C22⋊C412yes—
68C22×C4×C79C22×C411yescharacteristic
69C22×C4×C79C2×D411yescharacteristic
70C22×C4×C79C2×D411yescharacteristic
71M16×C79—42no—
72M16×C79—42no—
73C2×C8×C79—42no—
74C4⋊C4×C79—24no—
75C4⋊C4×C79—22no—
76C42×C79—22no—
77C42×C79—22no—
78C42×C79C2×D412yes—
79C42×C79#4112yes—
80C23×C79C2×C22⋊C411yescharacteristic
81Q8×C79—24no—
82Q8×C79—24no—
83C2×C4×C79—22no—
84C2×C4×C79—22no—
85C2×C4×C79—22no—
86C2×C4×C79—41no—
87C2×C4×C79—41no—
88C2×C4×C79—42no—
89C2×C4×C79#10314yes—
90C2×C4×C79C4×D412yes—
91C2×C4×C79#12011yescharacteristic
92C2×C4×C79#12112yes—
93C2×C4×C79#12311yescharacteristic
94C2×C4×C79C2×SD1611yescharacteristic
95C2×C4×C79C2×Q1611yescharacteristic
96C2×C4×C79#13512yes—
97C2×Dic79—22no—
98C2×Dic79—41no—
99C2×Dic79—41no—
100C2×Dic79—42no—
101C2×Dic79—42no—
102C2×Dic79—42no—
103C2×Dic79—41no—
104C2×Dic79—42no—
105Dic158—24no—
106Dic158—42no—
107C8×C79—82no—
108C79⋊C8—44no—
109C22×C79—21no—
110C22×C79—21no—
111C22×C79#38611yescharacteristic
112C22×C79C2×#10311yescharacteristic
113C22×C79#41611yescharacteristic
114C4×C79—22no—
115C4×C79—24no—
116C4×C79—21no—
117C4×C79—21no—
118C4×C79—42no—
119C4×C79#46512yes—
120C4×C79#46712yes—
121Dic79—42no—
122Dic79—82no—
123Dic79—81no—
124#9484—791nomaximal, Sylow
125C2×C79—21no—
126C2×C79—21no—
127C2×C79#189111yescharacteristic
128C2×C79#189211yescharacteristic
129C2×C79#191211yescharacteristic
130#2982—791no—
131#2581—791no—
132#1801D7911yescharacteristic
133#1487—792no—
134#1286—792no—
135C79#948411yescharacteristic, complemented, Sylow
136C2×C4×Q8—791no—
137C2×C4×Q8C2×D7911yescharacteristic
138C8×Q8—1584no—
139#431—792no—
140#431—1582no—
141#423C2×D7912yes—
142C2×#101—21no—
143#400—792no—
144#395—792no—
145#391—1581no—
146#389—1581no—
147#387—1582no—
148#382—791no—
149#343—21no—
150Q8⋊C8—1584no—
151C22×Q8—791no—
152C22×Q8C22⋊D7911yescharacteristic
153C2×M16—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1582no—
157C4×Q8—1582no—
158C2×C4⋊C4—791no—
159C2×C4⋊C4—1582no—
160C2×C4⋊C4—1581no—
161C2×C4⋊C4—1581no—
162C2×C4⋊C4C4×D7912yes—
163C2×C4⋊C4C22⋊D7911yescharacteristic
164C2×C42—1581no—
165C2×C42D31612yes—
166C2×C42C22×D7911yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—1584no—
170C4⋊C8—1582no—
171#101—42no—
172C22⋊C8—1582no—
173C4×C8—794no—
174C4×C8—1584no—
175#95—792no—
176#95—1582no—
177#95—1582no—
178#95—1582no—
179#95—1581no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—3162no—
183C2×Q8—1582no—
184C2×Q8#1192961212yes—
185C2×Q8#1192961412yes—
186C22×C4—1582no—
187C22×C4—1581no—
188C22×C4—1581no—
189C22×C4(C2×D79)⋊C412yes—
190C22×C4(C2×D79)⋊C412yes—
191C22×C4C2×C4×D7911yescharacteristic
192C22×C4C2×C22⋊D7911yescharacteristic
193C22×C4C2×D31611yescharacteristic
194M16—42no—
195M16—42no—
196C2×C8—42no—
197C2×C8—1584no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—3164no—
201C4⋊C4—3161no—
202C4⋊C4—3161no—
203C4⋊C4—3162no—
204C4⋊C4—1582no—
205C42—22no—
206C42—22no—
207C42—3162no—
208C42D4×D7912yes—
209C42#1192962312yes—
210C23C2×(C2×D79)⋊C411yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—3162no—
214Q8—1584no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—3161no—
222C2×C4—3161no—
223C2×C4—3162no—
224C2×C4—3162no—
225C2×C4—3162no—
226C2×C4—3161no—
227C2×C4—3162no—
228C2×C4—1582no—
229C2×C4#1457908414yes—
230C2×C4(C22⋊D79)⋊C412yes—
231C2×C4#1457913111yescharacteristic
232C2×C4#1457913312yes—
233C2×C4#1457914511yescharacteristic
234C2×C4C2×#1192961211yescharacteristic
235C2×C4C2×#1192961411yescharacteristic
236C2×C4#1457918112yes—
237C8—82no—
238C8—3164no—
239C22—21no—
240C22—21no—
241C22#1909844011yescharacteristic
242C22C2×#1457908411yescharacteristic
243C22#1909855711yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—3162no—
250C4—6322no—
251C4—6321no—
252C4#1909879012yes—
253C4#1909880212yes—
254C2—21no—
255C2—21no—
256C2#2376381611yescharacteristic
257C2#2376382111yescharacteristic
258C2#2376393511yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1801C2
socleC22×C79#386
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1801D79
hypocenterC79#9484
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
135. C79#9484semidirect

Maximal subgroups

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

subgroupindex
#237724812
#237687622
#237618282
#237612602
C79×#18012
#948479