Finite Group Databasebeta

← #37507940

Subgroups — #37507940

Order 22912.

266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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#26038589C212yesmaximal
2#26038356C211yesmaximal, characteristic
3#26038176C212yesmaximal, complemented
4#26038165C211yesmaximal, characteristic, complemented
5#544×C179C211yesmaximal, characteristic, complemented
6C22×D716C2211yescharacteristic
7C2×#15668246—22no—
8C2×(C2×D179)⋊C4—21no—
9C2×(C2×D179)⋊C4—21no—
10#19153033—22no—
11C22⋊(C179⋊C8)C2212yes—
12C179⋊#95C2211yescharacteristic
13C4⋊Q8×C179—22no—
14#125×C179—22no—
15C22⋊Q8×C179—21no—
16C22⋊Q8×C179C2212yes—
17#117×C179—21no—
18C2×C4⋊C4×C179C2211yescharacteristic
19C23×D179—21no—
20C2×D716—22no—
21C2×D716—42no—
22C22×Dic179—21no—
23#15668246—44no—
24C2×C179⋊C8—22no—
25(C2×D179)⋊C4—42no—
26(C2×D179)⋊C4—42no—
27C2×Q8×C179—21no—
28C2×Q8×C179D412yes—
29C22×C4×C179D411yescharacteristic
30C22×C4×C179D411yescharacteristic
31C22×C4×C179C2311yescharacteristic
32C4⋊C4×C179—21no—
33C4⋊C4×C179—21no—
34C4⋊C4×C179—21no—
35C4⋊C4×C179—22no—
36C4⋊C4×C179—22no—
37C4⋊C4×C179—22no—
38C4⋊C4×C179D412yes—
39C22⋊C4×C179—22no—
40C42×C179—22no—
41C22×D179—22no—
42C22×D179—41no—
43C22×D179—41no—
44C22×D179—41no—
45C22×D179—41no—
46C22×D179—41no—
47C22×D179—41no—
48D716—24no—
49D716—42no—
50C2×Dic179—41no—
51C2×Dic179—41no—
52C2×Dic179—41no—
53C179⋊C8—42no—
54C23×C179C2×D411yescharacteristic
55Q8×C179—42no—
56Q8×C179—81no—
57C2×C4×C179—21no—
58C2×C4×C179—21no—
59C2×C4×C179—21no—
60C2×C4×C179—22no—
61C2×C4×C179—22no—
62C2×C4×C179—41no—
63C2×C4×C179—41no—
64C2×C4×C179SD1612yescomplemented
65C2×C4×C179C2×D412yes—
66C22×C179—21no—
67C22×C179—21no—
68C22×C179#12011yescharacteristic
69C22×C179C2×SD1611yescharacteristic
70C22×C179#13711yescharacteristic
71C2×D179—42no—
72C2×D179—42no—
73C2×D179—42no—
74C2×D179—81no—
75C2×D179—81no—
76C2×D179—81no—
77C2×D179—81no—
78C4×C179—22no—
79C4×C179—22no—
80C4×C179—42no—
81C4×C179—41no—
82C4×C179—42no—
83Dic179—81no—
84C2×C179—21no—
85C2×C179—21no—
86C2×C179#44811yescharacteristic
87C2×C179#45411yescharacteristic
88C2×C179B(4,𝔽2)11yescharacteristic
89D179—82no—
90C179#159911yescharacteristic, complemented, Sylow
91#1599—1791nomaximal, Sylow
92#544D17911yescharacteristic, complemented
93#450—1792no—
94#394—1791no—
95#328—1792no—
96#324—1791no—
97C22×D4—1791no—
98C2×SD16—3582no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C2×D17912yes—
103#117—21no—
104C2×C4⋊C4C2×D17911yescharacteristic
105C2×C22⋊C4—3581no—
106C2×C22⋊C4—3581no—
107#102—3582no—
108C22⋊C8—1792no—
109#95—1791no—
110C24—3581no—
111C2×Q8—21no—
112C2×Q8C22⋊D17912yes—
113C2×D4—3582no—
114C2×D4—7162no—
115C22×C4—3581no—
116C22×C4C22⋊D17911yescharacteristic
117C22×C4C22⋊D17911yescharacteristic
118C22×C4C22×D17911yescharacteristic
119SD16—7164no—
120C2×C8—3582no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4C22⋊D17912yes—
128C22⋊C4—22no—
129C22⋊C4—7162no—
130C22⋊C4—7162no—
131C42—22no—
132C23—3582no—
133C23—7161no—
134C23—7161no—
135C23—7161no—
136C23—7161no—
137C23—7161no—
138C23—7161no—
139C23C2×C22⋊D17911yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—3584no—
143D4—7162no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—7161no—
152C2×C4—7161no—
153C2×C4—7161no—
154C2×C4#1566824612yescomplemented
155C2×C4C2×C22⋊D17912yes—
156C8—7162no—
157C22—21no—
158C22—21no—
159C22—14321no—
160C22—14321no—
161C22—14321no—
162C22—14321no—
163C22—7162no—
164C22—7162no—
165C22—7162no—
166C22#1915308211yescharacteristic
167C22C2×#1566824611yescharacteristic
168C22#1915314411yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—14321no—
175C2—21no—
176C2—21no—
177C2—14322no—
178C2#2603857811yescharacteristic
179C2#2603861211yescharacteristic
180C2#2603862611yescharacteristic
subgroupG/H
centerC22#19153082
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#544×C179C2
socleC22×C179#120
prime radicalC22×D716C22
hypercenter#544D179
hypocenterC179#1599
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #26038176C2semidirect
4. #26038165C2semidirect
5. #544×C179C2semidirect
64. C2×C4×C179SD16semidirect
90. C179#1599semidirect
92. #544D179semidirect
154. C2×C4#15668246semidirect

Maximal subgroups

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

subgroupindex
#260385892
#260383562
#260381762
#260381652
#544×C1792
#1599179