Finite Group Databasebeta

← #37509548

Subgroups — #37509548

Order 22912.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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#26039206C211yesmaximal, characteristic, complemented
2#26038423C211yesmaximal, characteristic, complemented
3C2×#19153030C212yesmaximal, complemented
4#26038209C212yesmaximal, complemented
5#535×C179C211yesmaximal, characteristic
6#19153120—24no—
7C2×#15668243—22no—
8C2×#15668243C2212yes—
9C2×#15668243C2212yes—
10#19153032—21no—
11#19153030—24no—
12#19153030—21no—
13C2×#41×C179C2211yescharacteristic
14C22×D4×C179C2211yescharacteristic
15#127×C179—21no—
16#124×C179—21no—
17#121×C179—22no—
18#120×C179C414yes—
19#117×C179C2211yescharacteristic
20#15668243—24no—
21#15668243—22no—
22#15668243—22no—
23#15668243—22no—
24#15668243D414yescomplemented
25C2×C179⋊C8—24no—
26C2×C179⋊C8—22no—
27C2×C179⋊C8—22no—
28C24×C179C2×C412yes—
29#41×C179—22no—
30#41×C179—22no—
31C2×Q8×C179D411yescharacteristic
32C2×D4×C179—24no—
33C2×D4×C179—21no—
34C2×D4×C179—21no—
35C2×D4×C179—22no—
36C2×D4×C179—22no—
37C2×D4×C179C2×C412yes—
38C2×D4×C179D411yescharacteristic
39C22×C4×C179D412yes—
40C22×C4×C179C2311yescharacteristic
41C4⋊C4×C179—21no—
42C22⋊C4×C179—24no—
43C22⋊C4×C179C2×C412yes—
44C42×C179—21no—
45C179⋊C8—24no—
46C179⋊C8—42no—
47C23×C179—22no—
48C23×C179—24no—
49C23×C179—82no—
50C23×C179C22⋊C412yes—
51C23×C179C22×C411yescharacteristic
52Q8×C179—22no—
53D4×C179—24no—
54D4×C179—22no—
55D4×C179—41no—
56D4×C179—42no—
57D4×C179—41no—
58D4×C179—84no—
59C2×C4×C179—22no—
60C2×C4×C179—22no—
61C2×C4×C179—22no—
62C2×C4×C179C22⋊C412yes—
63C2×C4×C179C2×D412yes—
64C2×C4×C179C2×D411yescharacteristic
65C2×C4×C179C2×D411yescharacteristic
66C2×C4×C179#4112yes—
67C22×C179—22no—
68C22×C179—41no—
69C22×C179—42no—
70C22×C179—44no—
71C22×C179—42no—
72C22×C179—84no—
73C22×C179C2×C22⋊C411yescharacteristic
74C22×C179C4×D412yes—
75C4×C179—22no—
76C4×C179—42no—
77C4×C179#12011yescharacteristic
78C4×C179#12112yes—
79C4×C179#12311yescharacteristic
80C2×C179—21no—
81C2×C179—22no—
82C2×C179—42no—
83C2×C179—82no—
84C2×C179#38611yescharacteristic
85C179#191411yescharacteristic, complemented, Sylow
86#1914—1791nomaximal, Sylow
87#568—1791no—
88#535D17911yescharacteristic
89#413—1791no—
90C2×#100—1792no—
91#344—1792no—
92C2×#41C2×D17911yescharacteristic
93C22×D4C2×D17911yescharacteristic
94#131—3584no—
95C2×M16—3582no—
96C2×M16—1792no—
97C2×M16—1792no—
98#127—21no—
99#124—21no—
100#121—22no—
101#120Dic17914yes—
102#117C2×D17911yescharacteristic
103#101—3581no—
104#100—3581no—
105#100—3584no—
106C24C2×Dic17912yes—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D17911yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×Dic17912yes—
116C2×D4C22⋊D17911yescharacteristic
117C22×C4C22⋊D17912yes—
118C22×C4C22×D17911yescharacteristic
119M16—3584no—
120M16—3582no—
121M16—3582no—
122M16—3582no—
123M16—1794no—
124C2×C8—3584no—
125C2×C8—3582no—
126C2×C8—3582no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×Dic17912yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊Dic17912yes—
135C23C22×Dic17911yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—42no—
141D4—41no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊Dic17912yes—
147C2×C4D4×D17912yes—
148C2×C4C2×C22⋊D17911yescharacteristic
149C2×C4C2×C22⋊D17911yescharacteristic
150C2×C4#1566825812yes—
151C8—3584no—
152C8—7162no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊Dic17911yescharacteristic
160C22D4×Dic17912yes—
161C4—22no—
162C4—42no—
163C4#1915308111yescharacteristic
164C4#1915308512yes—
165C4#1915309411yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2603831911yescharacteristic
subgroupG/H
centerC2#26038319
commutatorC23×C179C22×C4
FrattiniC22×C4C22×D179
Fitting, prime radical#535×C179C2
socleC2×C179#386
cosocleC22×C4×C179C23
hypercenter#535D179
hypocenterC179#1914
LCSG ▷ C23×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26039206C2semidirect
2. #26038423C2semidirect
3. C2×#19153030C2semidirect
4. #26038209C2semidirect
24. #15668243D4semidirect
85. C179#1914semidirect

Maximal subgroups

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

subgroupindex
#260392062
#260384232
C2×#191530302
#260382092
#535×C1792
#1914179