Finite Group Databasebeta

← #34199456

Subgroups — #34199456

Order 20224.

296 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 75 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#23774050C211yesmaximal, characteristic
2#23765005C211yesmaximal, characteristic, complemented
3#23761432C214yesmaximal, complemented
4C79×#2166C211yesmaximal, characteristic, complemented
5#19099168C412yes—
6C2×#14579101—21no—
7#19098965C411yescharacteristic
8#19098940C411yescharacteristic
9#19098928C2212yes—
10#19098816—21no—
11#19098799—21no—
12#19098676—21no—
13C4×D632—21no—
14#19098341—21no—
15#19098341C2212yes—
16#405×C79C2211yescharacteristic
17#364×C79—21no—
18#349×C79—21no—
19#347×C79C2212yes—
20C2×D632—21no—
21C2×D632—21no—
22#14579158C2×C412yes—
23#14579153C2×C412yes—
24#14579146—42no—
25#14579135—22no—
26#14579134—42no—
27C4×D316—22no—
28#14579101—22no—
29#14579101—22no—
30C2×C79⋊C16—22no—
31#14579095—21no—
32#14579093—21no—
33#14579083—22no—
34#14579083—22no—
35C2×M16×C79D411yescharacteristic
36C22×C8×C79D411yescharacteristic
37#117×C79C2×C411yescharacteristic
38M32×C79—22no—
39#97×C79D411yescharacteristic
40#97×C79C2311yescharacteristic
41C4×C8×C79C2×C411yescharacteristic
42C4×C8×C79D411yescharacteristic
43C2×D316—22no—
44C2×D316—42no—
45C2×C22⋊D79—42no—
46C2×C4×D79—42no—
47D632—42no—
48D632—41no—
49D632—41no—
50C4⋊Dic79—22no—
51Dic79⋊C4—42no—
52(C2×D79)⋊C4—42no—
53(C2×D79)⋊C4—42no—
54C79⋊C16—24no—
55C22×C4×C79C22⋊C411yescharacteristic
56M16×C79—41no—
57C2×C8×C79—21no—
58C2×C8×C79—21no—
59C2×C8×C79C22⋊C411yescharacteristic
60C2×C8×C79C22⋊C411yescharacteristic
61C2×C8×C79C2×D411yescharacteristic
62C2×C8×C79C2×D411yescharacteristic
63C4⋊C4×C79—21no—
64C22⋊C4×C79—21no—
65C42×C79C22⋊C411yescharacteristic
66C42×C79C22×C411yescharacteristic
67C16×C79—42no—
68C23×C79#10011yescharacteristic
69C22×D79—42no—
70C2×C4×C79—21no—
71C2×C4×C79—21no—
72C2×C4×C79—21no—
73C2×C4×C79—21no—
74C2×C4×C79#10011yescharacteristic
75C2×C4×C79C2×C22⋊C411yescharacteristic
76C22⋊D79—162no—
77C2×Dic79—42no—
78D316—42no—
79D316—42no—
80D316—162no—
81C4×D79—162no—
82C8×C79—21no—
83C8×C79—21no—
84C8×C79—21no—
85C8×C79#10411yescharacteristic
86C8×C79#10411yescharacteristic
87C22×C79—21no—
88C22×C79—41no—
89C22×C79C2×#10011yescharacteristic
90C2×D79—82no—
91C2×D79—82no—
92C2×D79—82no—
93C4×C79—21no—
94C4×C79—41no—
95C4×C79—41no—
96C4×C79C2×#10411yescharacteristic
97C4×C79#42111yescharacteristic
98Dic79—82no—
99#11913—791nomaximal, Sylow
100C2×C79—21no—
101C2×C79—41no—
102C2×C79#153011yescharacteristic
103D79—162no—
104#3176—791no—
105#2166D7911yescharacteristic, complemented
106#2110—791no—
107#1355—794no—
108C79#1191311yescharacteristic, complemented, Sylow
109#540—792no—
110C2×M32—1581no—
111#499—791no—
112#493—791no—
113#490—792no—
114#469—1581no—
115#466—1581no—
116#442—1581no—
117C4×D8—1581no—
118#405C2×D7911yescharacteristic
119#364—21no—
120#349—21no—
121#347—792no—
122#347—1581no—
123#347C2×D7912yes—
124C2×D8—1581no—
125C2×D8—1581no—
126C2×M16C22⋊D7911yescharacteristic
127C22×C8D31611yescharacteristic
128#127—792no—
129#125—792no—
130#123—3162no—
131#121—3162no—
132#121—1582no—
133C4×D4—1582no—
134#117C4×D7911yescharacteristic
135M32—22no—
136M32—1582no—
137M32—1582no—
138C2×C16—1582no—
139#107—1581no—
140#106—1581no—
141#102—1582no—
142#102—1582no—
143#97D31611yescharacteristic
144#97C22×D7911yescharacteristic
145C4×C8C4×D7911yescharacteristic
146C4×C8C22⋊D7911yescharacteristic
147C2×D4—3162no—
148C2×D4—3162no—
149C2×D4—1582no—
150C22×C4—3162no—
151C22×C4(C2×D79)⋊C411yescharacteristic
152D8—3162no—
153D8—3161no—
154D8—3161no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8(C2×D79)⋊C411yescharacteristic
159C2×C8(C2×D79)⋊C411yescharacteristic
160C2×C8C2×C22⋊D7911yescharacteristic
161C2×C8C2×D31611yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—3162no—
164C4⋊C4—1582no—
165C22⋊C4—21no—
166C22⋊C4—3162no—
167C22⋊C4—3162no—
168C42(C2×D79)⋊C411yescharacteristic
169C42C2×C4×D7911yescharacteristic
170C16—42no—
171C16—1584no—
172C23—3162no—
173C23#1457907711yescharacteristic
174D4—3162no—
175D4—3162no—
176D4—12642no—
177D4—12642no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—3162no—
183C2×C4—12642no—
184C2×C4#1457907711yescharacteristic
185C2×C4C2×(C2×D79)⋊C411yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8#1457908711yescharacteristic
190C8#1457908711yescharacteristic
191C22—21no—
192C22—41no—
193C22—6322no—
194C22—6322no—
195C22—6322no—
196C22C2×#1457907711yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—6322no—
201C4C2×#1457908711yescharacteristic
202C4#1909858111yescharacteristic
203C2—21no—
204C2—41no—
205C2—12642no—
206C2#2376201211yescharacteristic
subgroupG/H
centerC2#23762012
commutatorC42×C79C22×C4
Frattini#97C22×D79
FittingC79×#2166C2
socleC2×C79#1530
prime radical#23774050C2
cosocle#97×C79C23
hypercenter#2166D79
hypocenterC79#11913
LCSG ▷ C42×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #23765005C2semidirect
3. #23761432C2semidirect
4. C79×#2166C2semidirect
105. #2166D79semidirect
108. C79#11913semidirect

Maximal subgroups

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

subgroupindex
#237740502
#237650052
#237614322
C79×#21662
#1191379