Finite Group Databasebeta

← #34187652

Subgroups — #34187652

Order 20224.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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#23773723C211yesmaximal, characteristic
2#23763531C211yesmaximal, characteristic, complemented
3#23761965C212yesmaximal, complemented
4#23761961C212yesmaximal
5C79×#1450C211yesmaximal, characteristic, complemented
6#19099120C2211yescharacteristic
7#19099113C2211yescharacteristic
8#19098918—21no—
9#19098918—21no—
10#19098914C412yescomplemented
11#19098910C412yescomplemented
12#19098560—22no—
13#19098429C2212yes—
14#19098326—22no—
15#19098301—24no—
16#19098298—24no—
17#431×C79C2212yes—
18C2×C4×C8×C79C2211yescharacteristic
19#322×C79—21no—
20C82×C79—21no—
21C2×#11929624—21no—
22#14579161C2×C412yes—
23#14579161C2×C411yescharacteristic
24#14579158C2×C411yescharacteristic
25#14579153—21no—
26#14579149—21no—
27#14579140—22no—
28#14579135—22no—
29C4×Dic158—22no—
30C4×D316—22no—
31#14579120—21no—
32C2×C4⋊Dic79—22no—
33#14579086—22no—
34#14579086—22no—
35#14579083—22no—
36#14579083—22no—
37C79⋊#95—22no—
38C79⋊#95—22no—
39C22×C8×C79D412yes—
40C2×C42×C79C2311yescharacteristic
41C4⋊C8×C79—22no—
42C22⋊C8×C79D414yes—
43C4×C8×C79—22no—
44C4×C8×C79C2×C412yes—
45C4×C8×C79D411yescharacteristic
46C4×C8×C79D411yescharacteristic
47#11929624—44no—
48C2×Dic158—21no—
49C2×Dic158—22no—
50C2×D316—21no—
51C2×D316—22no—
52C2×C22⋊D79—41no—
53C22×Dic79—42no—
54C2×C4×D79—41no—
55C4⋊Dic79—22no—
56C4⋊Dic79—22no—
57C4⋊Dic79—22no—
58C4⋊Dic79—44no—
59Dic79⋊C4—42no—
60C22⋊Dic79—41no—
61(C2×D79)⋊C4—42no—
62C4×Dic79—41no—
63C22×C4×C79C2×D412yes—
64C22×C4×C79C2×D411yescharacteristic
65C2×C8×C79—24no—
66C2×C8×C79—24no—
67C2×C8×C79C22⋊C412yes—
68C2×C8×C79C22⋊C412yes—
69C42×C79C22×C411yescharacteristic
70C42×C79C2×D411yescharacteristic
71C42×C79#4112yes—
72C23×C79#12711yescharacteristic
73C22×D79—41no—
74C2×C4×C79—42no—
75C2×C4×C79C2×C22⋊C411yescharacteristic
76C2×C4×C79C4×D412yes—
77C2×C4×C79#12112yes—
78C2×C4×C79#12411yescharacteristic
79C2×C4×C79#13514yes—
80C22⋊D79—82no—
81C2×Dic79—42no—
82C2×Dic79—41no—
83C2×Dic79—84no—
84C2×Dic79—82no—
85D316—42no—
86D316—82no—
87C4×D79—82no—
88Dic158—42no—
89Dic158—82no—
90C8×C79—22no—
91C8×C79—22no—
92C8×C79—44no—
93C22×C79—22no—
94C22×C79#39011yescharacteristic
95C22×C79#49512yes—
96C2×D79—81no—
97C2×D79—82no—
98C4×C79—22no—
99C4×C79—22no—
100C4×C79#41612yes—
101C4×C79#48812yes—
102Dic79—81no—
103Dic79—82no—
104Dic79—162no—
105#9757—791nomaximal, Sylow
106C2×C79—22no—
107C2×C79#195811yescharacteristic
108C2×C79#197612yes—
109D79—161no—
110#3131—791no—
111#1835—791no—
112#1518—792no—
113#1517—792no—
114#1450D7911yescharacteristic, complemented
115C79#975711yescharacteristic, complemented, Sylow
116#533—791no—
117#532—791no—
118#488—1581no—
119#488—1581no—
120#487—792no—
121#486—792no—
122#431C2×D7912yes—
123#416—1582no—
124C2×C4×C8C2×D7911yescharacteristic
125#382—792no—
126#340—1582no—
127#328—1584no—
128#327—1584no—
129#322—21no—
130C82—21no—
131C2×#41—1581no—
132C22×C8D31612yes—
133C4⋊Q8—791no—
134C4⋊Q8—792no—
135#127—791no—
136#125—1581no—
137#124—1581no—
138C22⋊Q8—1582no—
139#121—1582no—
140C4×Q8—1582no—
141C4×D4—1582no—
142#117—1581no—
143C2×C4⋊C4—1582no—
144C2×C42C22×D7911yescharacteristic
145C4⋊C8—22no—
146#103—1582no—
147#103—1582no—
148#102—1582no—
149#102—1582no—
150C22⋊C8D31614yes—
151C4×C8—22no—
152C4×C8C4×D7912yes—
153C4×C8C22⋊D7911yescharacteristic
154C4×C8C22⋊D7911yescharacteristic
155#95—1582no—
156#95—1582no—
157#41—3164no—
158C2×Q8—1581no—
159C2×Q8—1582no—
160C2×D4—3161no—
161C2×D4—1581no—
162C2×D4—1582no—
163C22×C4—3161no—
164C22×C4—3162no—
165C22×C4C2×D31612yes—
166C22×C4C2×D31611yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8(C2×D79)⋊C412yes—
170C2×C8(C2×D79)⋊C412yes—
171C4⋊C4—3162no—
172C4⋊C4—3164no—
173C4⋊C4—1582no—
174C4⋊C4—1582no—
175C4⋊C4—1582no—
176C22⋊C4—3161no—
177C22⋊C4—3162no—
178C42—3161no—
179C42C2×C4×D7911yescharacteristic
180C42C2×C22⋊D7911yescharacteristic
181C42#1192962412yes—
182C23—3161no—
183C23#1457915811yescharacteristic
184Q8—3162no—
185Q8—6322no—
186D4—3162no—
187D4—6322no—
188D4—6322no—
189C2×C4—42no—
190C2×C4—3162no—
191C2×C4—3161no—
192C2×C4—6322no—
193C2×C4—6324no—
194C2×C4—6322no—
195C2×C4C2×(C2×D79)⋊C411yescharacteristic
196C2×C4C4×D31612yes—
197C2×C4#1457913512yes—
198C2×C4#1457914911yescharacteristic
199C2×C4#1457918314yes—
200C8—22no—
201C8—22no—
202C8—44no—
203C22—22no—
204C22—6322no—
205C22—6321no—
206C22#1909846511yescharacteristic
207C22#1909894712yes—
208C4—22no—
209C4—22no—
210C4—6322no—
211C4—6321no—
212C4—12642no—
213C4#1909856012yes—
214C4#1909891812yes—
215C2—22no—
216C2—12641no—
217C2#2376421111yescharacteristic
218C2#2376429812yes—
subgroupG/H
centerC22#19098465
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1450C2
socleC22×C79#390
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1450D79
hypocenterC79#9757
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #23763531C2semidirect
3. #23761965C2semidirect
5. C79×#1450C2semidirect
10. #19098914C4semidirect
11. #19098910C4semidirect
114. #1450D79semidirect
115. C79#9757semidirect

Maximal subgroups

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

subgroupindex
#237737232
#237635312
#237619652
#237619612
C79×#14502
#975779