Finite Group Databasebeta

← #34201093

Subgroups — #34201093

Order 20224.

392 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764556C211yesmaximal, characteristic
2#23762441C212yesmaximal, complemented
3#23761427C212yesmaximal, complemented
4#23761332C211yesmaximal, characteristic
5C79×#2804C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098898—22no—
8#19098568—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13#400×C79C2212yes—
14#393×C79—22no—
15#391×C79—21no—
16#390×C79C2211yescharacteristic
17#386×C79—21no—
18#385×C79—22no—
19C2×#11929608—21no—
20#14579160—41no—
21#14579152—41no—
22#14579140—22no—
23#14579120—21no—
24#14579094—42no—
25#14579084—42no—
26#14579082—41no—
27#14579082—41no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79—21no—
30C22×D4×C79D411yescharacteristic
31C23×C4×C79—21no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79D412yes—
34C2×C4⋊C4×C79C2311yescharacteristic
35C2×C22⋊C4×C79—22no—
36C2×C22⋊C4×C79—22no—
37C2×C22⋊C4×C79—21no—
38C2×C42×C79D411yescharacteristic
39#95×C79—22no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44#11929608—41no—
45#11929608—41no—
46C2×C79⋊C8—42no—
47C2×C79⋊C8—41no—
48C4⋊Dic79—21no—
49C4⋊Dic79—21no—
50C4⋊Dic79—22no—
51Dic79⋊C4—42no—
52C22⋊Dic79—41no—
53C4×Dic79—41no—
54C24×C79—21no—
55C2×Q8×C79—42no—
56C2×Q8×C79—81no—
57C2×Q8×C79—81no—
58C2×D4×C79—21no—
59C2×D4×C79—41no—
60C2×D4×C79—41no—
61C2×D4×C79Q1612yes—
62C22×C4×C79—21no—
63C22×C4×C79—22no—
64C22×C4×C79—22no—
65C22×C4×C79—82no—
66C22×C4×C79—81no—
67C22×C4×C79C2×D411yescharacteristic
68C22×C4×C79C2×D412yes—
69C4⋊C4×C79—41no—
70C4⋊C4×C79—42no—
71C4⋊C4×C79—81no—
72C22⋊C4×C79—44no—
73C22⋊C4×C79—82no—
74C22⋊C4×C79—81no—
75C42×C79—41no—
76C23×C79—22no—
77C23×C79—42no—
78C23×C79—42no—
79C23×C79—41no—
80C23×C79—41no—
81C23×C79#12011yescharacteristic
82Q8×C79—84no—
83D4×C79—42no—
84D4×C79—81no—
85C2×C4×C79—21no—
86C2×C4×C79—21no—
87C2×C4×C79—22no—
88C2×C4×C79—22no—
89C2×C4×C79—41no—
90C2×C4×C79—41no—
91C2×C4×C79—42no—
92C2×C4×C79—42no—
93C2×C4×C79—42no—
94C2×C4×C79—42no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—84no—
99C2×C4×C79C2×Q1611yescharacteristic
100C2×C4×C79#13711yescharacteristic
101C2×Dic79—41no—
102C2×Dic79—41no—
103Dic158—82no—
104C79⋊C8—81no—
105C79⋊C8—82no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79—44no—
109C22×C79—42no—
110C22×C79—82no—
111C22×C79—81no—
112C22×C79—81no—
113C22×C79#45111yescharacteristic
114C22×C79#45511yescharacteristic
115C22×C79B(4,𝔽2)11yescharacteristic
116C4×C79—21no—
117C4×C79—21no—
118C4×C79—42no—
119C4×C79—42no—
120C4×C79—82no—
121C4×C79—82no—
122Dic79—81no—
123Dic79—81no—
124#12221—791nomaximal, Sylow
125C2×C79—21no—
126C2×C79—21no—
127C2×C79—82no—
128C2×C79#160211yescharacteristic
129C2×C79#219811yescharacteristic
130C2×C79#220211yescharacteristic
131#2804D7911yescharacteristic
132#2027—791no—
133#1617—792no—
134#1353—792no—
135#1318—791no—
136C79#1222111yescharacteristic, complemented, Sylow
137#544—791no—
138#484—1582no—
139#418—1581no—
140#400C2×D7912yes—
141#393—22no—
142#391—21no—
143#390C2×D7911yescharacteristic
144#386—21no—
145#385—22no—
146#343—791no—
147#339—1581no—
148#328—1582no—
149#324—792no—
150C22×Q8—21no—
151C22×D4C22⋊D7911yescharacteristic
152C23×C4—21no—
153C2×M16—1581no—
154C4⋊Q8—3161no—
155#125—3161no—
156C22⋊Q8—1582no—
157#117—1581no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C22⋊D7912yes—
160C2×C4⋊C4C22×D7911yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42C22⋊D7911yescharacteristic
165#107—3162no—
166#103—3162no—
167#102—3161no—
168#102—3161no—
169C22⋊C8—1582no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95C22⋊D7912yes—
174C24—21no—
175C2×Q8—42no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×Q8—1582no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1192961412yes—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—82no—
187C22×C4—81no—
188C22×C4C2×C22⋊D7911yescharacteristic
189C22×C4C2×C22⋊D7912yes—
190M16—3161no—
191M16—3161no—
192C2×C8—3162no—
193C2×C8—3161no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—3162no—
198C4⋊C4—1581no—
199C4⋊C4—1581no—
200C4⋊C4—1582no—
201C22⋊C4—44no—
202C22⋊C4—82no—
203C22⋊C4—81no—
204C22⋊C4—3161no—
205C42—41no—
206C42—3161no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#1457913011yescharacteristic
213Q8—84no—
214Q8—6322no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—3161no—
232C2×C4—3161no—
233C2×C4C2×#1192961411yescharacteristic
234C2×C4#1457919211yescharacteristic
235C8—6321no—
236C8—6322no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1909871911yescharacteristic
245C22#1909874111yescharacteristic
246C22#1909875011yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—82no—
252C4—82no—
253C4—6321no—
254C4—6321no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2#2376235711yescharacteristic
259C2#2376539611yescharacteristic
260C2#2376541711yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2804C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
hypercenter#2804D79
hypocenterC79#12221
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #23762441C2semidirect
3. #23761427C2semidirect
136. C79#12221semidirect

Maximal subgroups

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

subgroupindex
#237645562
#237624412
#237614272
#237613322
C79×#28042
#1222179