Finite Group Databasebeta

← #37509402

Subgroups — #37509402

Order 22912.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#19153101C211yesmaximal, characteristic
2#26038916C211yesmaximal, characteristic, complemented
3#26038445C211yesmaximal, characteristic
4#26038443C211yesmaximal, characteristic
5C2×#19153031C211yesmaximal, characteristic, complemented
6#26038199C211yesmaximal, characteristic, complemented
7#339×C179C211yesmaximal, characteristic, complemented
8C22×Dic358C2211yescharacteristic
9C22×D716C2211yescharacteristic
10C2×#15668243C2211yescharacteristic
11#19153101—22no—
12#19153101C414yescomplemented
13C4×Dic358—22no—
14#19153078—22no—
15#19153072C2211yescharacteristic
16#19153069—22no—
17C2×(C2×D179)⋊C4—22no—
18#19153035—22no—
19#19153033—22no—
20#19153031—22no—
21C2×M16×C179C2211yescharacteristic
22#117×C179C2211yescharacteristic
23C2×C42×C179C2211yescharacteristic
24C23×D179—21no—
25C2×Dic358—21no—
26C2×Dic358—21no—
27C2×Dic358—21no—
28C2×Dic358—21no—
29C2×Dic358C2×C412yes—
30C2×Dic358D412yes—
31C2×D716—21no—
32C2×D716—21no—
33C2×D716—21no—
34C2×D716—21no—
35C2×D716—21no—
36C2×D716C2×C412yes—
37C22×Dic179D412yes—
38#15668243—21no—
39#15668243—21no—
40C2×C179⋊C8—21no—
41C4⋊Dic179D412yes—
42Dic179⋊C4—24no—
43C22⋊Dic179—21no—
44(C2×D179)⋊C4—28no—
45C4×Dic179—24no—
46C4×Dic179—21no—
47C22×C4×C179—21no—
48C22×C4×C179C2311yescharacteristic
49M16×C179—21no—
50M16×C179—21no—
51C2×C8×C179—21no—
52C4⋊C4×C179D412yes—
53C22⋊C4×C179—21no—
54C42×C179—21no—
55C42×C179—21no—
56C42×C179C2×C412yes—
57C22×D179—22no—
58C22×D179—44no—
59C22×D179—41no—
60C22×D179—41no—
61Dic358—22no—
62Dic358—22no—
63Dic358—22no—
64Dic358—22no—
65D716—22no—
66D716—22no—
67D716—41no—
68D716—41no—
69C2×Dic179—22no—
70C2×Dic179—22no—
71C2×Dic179—22no—
72C2×Dic179C22⋊C414yes—
73C179⋊C8—41no—
74C23×C179C2×D411yescharacteristic
75C2×C4×C179—22no—
76C2×C4×C179—22no—
77C2×C4×C179—42no—
78C2×C4×C179C22×C411yescharacteristic
79C2×C4×C179C2×D411yescharacteristic
80C2×C4×C179C2×D411yescharacteristic
81C2×C4×C179C2×D411yescharacteristic
82C2×C4×C179#4111yescharacteristic
83C2×C4×C179#4111yescharacteristic
84C8×C179—41no—
85C22×C179—22no—
86C22×C179C2×C22⋊C411yescharacteristic
87C22×C179#12011yescharacteristic
88C22×C179#12311yescharacteristic
89C2×D179—44no—
90C2×D179—42no—
91C2×D179—82no—
92C2×D179—82no—
93C4×C179—42no—
94C4×C179—42no—
95C4×C179C4×D411yescharacteristic
96C4×C179C4×D411yescharacteristic
97C4×C179#12111yescharacteristic
98C4×C179#12111yescharacteristic
99Dic179—24no—
100Dic179—42no—
101C2×C179—21no—
102C2×C179—21no—
103C2×C179#38611yescharacteristic
104C2×C179#45412yes—
105D179—82no—
106C179#188811yescharacteristic, complemented, Sylow
107#1888—1791nomaximal, Sylow
108C2×#124—1791no—
109#519—1791no—
110#418—1791no—
111#418—1791no—
112C2×#101—1791no—
113#339—1791no—
114#339D17911yescharacteristic, complemented
115C22×Q8—1791no—
116C22×D4—1791no—
117C2×M16—1791no—
118C2×M16C2×D17911yescharacteristic
119#124—3582no—
120#124—1794no—
121C4×Q8—3582no—
122C4×Q8—3582no—
123#117—3582no—
124#117—1791no—
125#117C2×D17911yescharacteristic
126C2×C22⋊C4—3582no—
127C2×C42C2×D17911yescharacteristic
128#102—3582no—
129#102—3582no—
130#101—3582no—
131C24—3581no—
132C2×Q8—3581no—
133C2×Q8—3581no—
134C2×Q8—3581no—
135C2×Q8—3581no—
136C2×Q8—1792no—
137C2×Q8—1792no—
138C2×D4—3581no—
139C2×D4—3581no—
140C2×D4—3581no—
141C2×D4—3581no—
142C2×D4—3581no—
143C2×D4—1792no—
144C22×C4—21no—
145C22×C4—1792no—
146C22×C4C22×D17911yescharacteristic
147M16—21no—
148M16—21no—
149M16—3581no—
150M16—3581no—
151C2×C8—21no—
152C2×C8—3581no—
153C4⋊C4—3584no—
154C4⋊C4—1792no—
155C4⋊C4D71612yes—
156C22⋊C4—21no—
157C22⋊C4—3588no—
158C22⋊C4—3581no—
159C42—21no—
160C42—21no—
161C42—3584no—
162C42—3581no—
163C42C4×D17912yes—
164C23—3582no—
165C23—7164no—
166C23—7161no—
167C23—7161no—
168C23D4×D17911yescharacteristic
169Q8—3582no—
170Q8—3582no—
171Q8—3582no—
172Q8—3582no—
173D4—3582no—
174D4—3582no—
175D4—7161no—
176D4—7161no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—3582no—
181C2×C4—3582no—
182C2×C4—3582no—
183C2×C4—1794no—
184C2×C4C2×C4×D17911yescharacteristic
185C2×C4D4×D17911yescharacteristic
186C2×C4D4×D17911yescharacteristic
187C2×C4C2×D71611yescharacteristic
188C2×C4#1566825911yescharacteristic
189C2×C4#1566826011yescharacteristic
190C8—41no—
191C8—7161no—
192C22—22no—
193C22—14322no—
194C22—14322no—
195C22—7164no—
196C22—7162no—
197C22C22⋊C4×D17911yescharacteristic
198C22#1915308311yescharacteristic
199C22#1915309811yescharacteristic
200C4—42no—
201C4—42no—
202C4—3584no—
203C4—7162no—
204C4#1915307511yescharacteristic
205C4C4×D71611yescharacteristic
206C4#1915308611yescharacteristic
207C4#1915308811yescharacteristic
208C2—21no—
209C2—21no—
210C2—14322no—
211C2#2603831711yescharacteristic
212C2#2603861312yes—
subgroupG/H
centerC22#19153083
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#339×C179C2
socleC22×C179#120
prime radicalC22×D716C22
cosocleC22×C4×C179C23
hypercenter#339D179
hypocenterC179#1888
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #26038916C2semidirect
5. C2×#19153031C2semidirect
6. #26038199C2semidirect
7. #339×C179C2semidirect
12. #19153101C4semidirect
106. C179#1888semidirect
114. #339D179semidirect

Maximal subgroups

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

subgroupindex
C2×#191531012
#260389162
#260384452
#260384432
C2×#191530312
#260381992
#339×C1792
#1888179