Finite Group Databasebeta

← #34197705

Subgroups — #34197705

Order 20224.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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#23765212C212yesmaximal, complemented
2#23761351C212yesmaximal, complemented
3#23761336C212yesmaximal
4C79×#2921C211yesmaximal, characteristic
5#19098384—22no—
6#19098345C2214yes—
7#19098341—22no—
8C2×#117×C79C411yescharacteristic
9#433×C79—22no—
10#433×C79—22no—
11#423×C79C411yescharacteristic
12#423×C79C2211yescharacteristic
13#407×C79C411yescharacteristic
14#407×C79C2211yescharacteristic
15C4×M16×C79C411yescharacteristic
16C4×M16×C79C2211yescharacteristic
17#14579101—24no—
18C23×C4×C79C2×C411yescharacteristic
19C2×M16×C79C2×C411yescharacteristic
20C2×M16×C79C2×C411yescharacteristic
21C2×M16×C79C2×C411yescharacteristic
22C2×M16×C79C2311yescharacteristic
23#117×C79—21no—
24#117×C79—21no—
25#117×C79—22no—
26C2×C4⋊C4×C79—21no—
27C2×C22⋊C4×C79—21no—
28C2×C42×C79C2×C411yescharacteristic
29C2×C42×C79C2×C411yescharacteristic
30C4⋊C8×C79—22no—
31C4⋊C8×C79—22no—
32C22⋊C8×C79—22no—
33C22⋊C8×C79—22no—
34#97×C79—21no—
35#97×C79D412yes—
36C4×C8×C79—21no—
37C4×C8×C79D412yes—
38C79⋊C16—44no—
39C24×C79M1611yescharacteristic
40C22×C4×C79—21no—
41C22×C4×C79—21no—
42C22×C4×C79—21no—
43C22×C4×C79—21no—
44C22×C4×C79—41no—
45C22×C4×C79—41no—
46C22×C4×C79M1611yescharacteristic
47C22×C4×C79C22×C411yescharacteristic
48M16×C79—41no—
49M16×C79—41no—
50M16×C79—41no—
51M16×C79—41no—
52C2×C8×C79—21no—
53C2×C8×C79—21no—
54C2×C8×C79C22⋊C412yes—
55C2×C8×C79C2×D412yes—
56C4⋊C4×C79—22no—
57C4⋊C4×C79—22no—
58C22⋊C4×C79—22no—
59C22⋊C4×C79—22no—
60C42×C79—21no—
61C42×C79—21no—
62C42×C79C22⋊C411yescharacteristic
63C42×C79C22⋊C411yescharacteristic
64C42×C79M1612yes—
65C23×C79—21no—
66C23×C79—41no—
67C23×C79—81no—
68C23×C79C2×M1611yescharacteristic
69C2×C4×C79—21no—
70C2×C4×C79—21no—
71C2×C4×C79—21no—
72C2×C4×C79—21no—
73C2×C4×C79—22no—
74C2×C4×C79—21no—
75C2×C4×C79—42no—
76C2×C4×C79—41no—
77C2×C4×C79—41no—
78C2×C4×C79—41no—
79C2×C4×C79—81no—
80C2×C4×C79—81no—
81C2×C4×C79C2×C22⋊C411yescharacteristic
82C2×C4×C79C2×M1611yescharacteristic
83C8×C79—22no—
84C8×C79—22no—
85C8×C79—41no—
86C8×C79—41no—
87C22×C79—21no—
88C22×C79—41no—
89C22×C79—42no—
90C22×C79—41no—
91C22×C79—82no—
92C22×C79#40711yescharacteristic
93C4×C79—21no—
94C4×C79—41no—
95C4×C79—41no—
96C4×C79—42no—
97C4×C79—41no—
98C4×C79#41011yescharacteristic
99C4×C79#41311yescharacteristic
100#11522—791nomaximal, Sylow
101C2×C79—21no—
102C2×C79—41no—
103C2×C79—81no—
104C2×C79#146411yescharacteristic
105#2921D7911yescharacteristic
106#2165—792no—
107#1327—792no—
108#1320—792no—
109C79#1152211yescharacteristic, complemented, Sylow
110C2×#117Dic7911yescharacteristic
111#433—22no—
112#433—22no—
113#423Dic7911yescharacteristic
114#423C2×D7911yescharacteristic
115#407Dic7911yescharacteristic
116#407C2×D7911yescharacteristic
117C4×M16Dic7911yescharacteristic
118C4×M16C2×D7911yescharacteristic
119#364—1582no—
120#349—794no—
121#347—1582no—
122C23×C4C2×Dic7911yescharacteristic
123C2×M16C2×Dic7911yescharacteristic
124C2×M16C2×Dic7911yescharacteristic
125C2×M16C2×Dic7911yescharacteristic
126C2×M16C22×D7911yescharacteristic
127#117—21no—
128#117—21no—
129#117—22no—
130C2×C4⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C42C2×Dic7911yescharacteristic
133C2×C42C2×Dic7911yescharacteristic
134M32—1584no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139#97—21no—
140#97C22⋊D7912yes—
141C4×C8—21no—
142C4×C8C22⋊D7912yes—
143C24#1192960811yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4#1192960811yescharacteristic
151C22×C4C22×Dic7911yescharacteristic
152M16—41no—
153M16—41no—
154M16—41no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8C22⋊Dic7912yes—
159C2×C8C2×C22⋊D7912yes—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C42—21no—
165C42—21no—
166C42C22⋊Dic7911yescharacteristic
167C42C22⋊Dic7911yescharacteristic
168C42#1192960812yes—
169C16—3164no—
170C23—21no—
171C23—41no—
172C23—81no—
173C23C2×#1192960811yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4C2×C22⋊Dic7911yescharacteristic
187C2×C4C2×#1192960811yescharacteristic
188C8—22no—
189C8—22no—
190C8—41no—
191C8—41no—
192C22—21no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22—82no—
197C22#1909852711yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—42no—
202C4—41no—
203C4#1909853711yescharacteristic
204C4#1909854711yescharacteristic
205C2—21no—
206C2—41no—
207C2—81no—
208C2#2376173411yescharacteristic
subgroupG/H
centerC4#19098537
commutatorC22×C4×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#2921C2
socleC2×C79#1464
prime radicalC24×C79M16
cosocleC2×M16×C79C23
hypercenter#2921D79
hypocenterC79#11522
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. #23765212C2semidirect
2. #23761351C2semidirect
109. C79#11522semidirect

Maximal subgroups

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

subgroupindex
#237652122
#237613512
#237613362
C79×#29212
#1152279