Finite Group Databasebeta

← #37510224

Subgroups — #37510224

Order 22912.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#26039020C211yesmaximal, characteristic
2C2×#19153041C212yesmaximal, complemented
3#26038438C212yesmaximal, complemented
4#26038208C211yesmaximal, characteristic, complemented
5#384×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2212yes—
7C2×#15668243C2212yes—
8#19153093—22no—
9#19153088—22no—
10#19153087—22no—
11#19153041—24no—
12#19153036—22no—
13#19153033—22no—
14C2×C4⋊C4×C179—22no—
15C2×C4⋊C4×C179C2211yescharacteristic
16C2×C4⋊C4×C179C2211yescharacteristic
17C2×C42×C179C2211yescharacteristic
18#15668259—22no—
19#15668259—22no—
20#15668259—22no—
21#15668259—22no—
22C2×Dic358D412yes—
23C2×D716—22no—
24C2×D716D412yes—
25C2×C22⋊D179—22no—
26C2×C4×D179—22no—
27#15668243—22no—
28#15668243—22no—
29C2×C179⋊C8—22no—
30C4⋊Dic179—22no—
31(C2×D179)⋊C4—24no—
32C22×C4×C179—21no—
33C22×C4×C179—21no—
34C22×C4×C179D412yes—
35C22×C4×C179C2311yescharacteristic
36C4⋊C4×C179—22no—
37C4⋊C4×C179—22no—
38C4⋊C4×C179—44no—
39C42×C179—21no—
40C42×C179D412yescomplemented
41C22×D179—22no—
42Dic358—22no—
43Dic358—22no—
44D716—22no—
45D716—22no—
46D716—82no—
47C22⋊D179—42no—
48C22⋊D179—42no—
49C2×Dic179—22no—
50C4×D179—42no—
51C4×D179—42no—
52C179⋊C8—42no—
53C23×C179C2×D411yescharacteristic
54C2×C4×C179—22no—
55C2×C4×C179—22no—
56C2×C4×C179—22no—
57C2×C4×C179—22no—
58C2×C4×C179—42no—
59C2×C4×C179—42no—
60C2×C4×C179—41no—
61C2×C4×C179C2×D412yes—
62C2×C4×C179C2×D411yescharacteristic
63C2×C4×C179#4111yescharacteristic
64C2×C4×C179#4112yes—
65C22×C179—21no—
66C22×C179—21no—
67C22×C179#12011yescharacteristic
68C22×C179#12111yescharacteristic
69C22×C179#12411yescharacteristic
70C2×D179—42no—
71C2×D179—42no—
72C2×D179—42no—
73C4×C179—24no—
74C4×C179—42no—
75C4×C179—81no—
76C4×C179#12112yes—
77C4×C179#12312yes—
78Dic179—42no—
79C2×C179—21no—
80C2×C179—21no—
81C2×C179#39411yescharacteristic
82C2×C179#45311yescharacteristic
83C2×C179#45611yescharacteristic
84D179—82no—
85C179#203911yescharacteristic, complemented, Sylow
86#2039—1791nomaximal, Sylow
87#537—1791no—
88C2×#104—1792no—
89#417—1792no—
90#384D17911yescharacteristic, complemented
91#343—1791no—
92C2×#41—1792no—
93C2×M16—1792no—
94C22⋊Q8—3582no—
95#121—3582no—
96#121—3582no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D17911yescharacteristic
99C2×C4⋊C4C2×D17911yescharacteristic
100C2×C42C2×D17911yescharacteristic
101#104—3584no—
102#103—3582no—
103#102—3582no—
104#41—3582no—
105#41—3582no—
106#41—3582no—
107#41—3582no—
108C2×Q8—1792no—
109C2×D4—3582no—
110C2×D4—3582no—
111C2×D4—1792no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—3582no—
115C22×C4C22⋊D17912yes—
116C22×C4C22×D17911yescharacteristic
117M16—3582no—
118M16—3582no—
119C2×C8—3582no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—3582no—
124C22⋊C4—3584no—
125C42—21no—
126C42D71612yescomplemented
127C23—3582no—
128C23C2×C22⋊D17911yescharacteristic
129Q8—3582no—
130Q8—3582no—
131D4—3582no—
132D4—3582no—
133D4—14322no—
134D4—7162no—
135D4—7162no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—3582no—
144C2×C4—7162no—
145C2×C4—7162no—
146C2×C4D4×D17912yes—
147C2×C4C2×D71611yescharacteristic
148C2×C4#1566825911yescharacteristic
149C2×C4#1566826012yes—
150C8—7162no—
151C22—21no—
152C22—21no—
153C22—7162no—
154C22—7162no—
155C22—7162no—
156C22#1915308111yescharacteristic
157C22#1915308711yescharacteristic
158C22#1915310211yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—7162no—
163C4#1915308812yes—
164C4#1915309812yes—
165C2—21no—
166C2—21no—
167C2—14322no—
168C2#2603835611yescharacteristic
169C2#2603860611yescharacteristic
170C2#2603862111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#384×C179C2
socleC22×C179#120
prime radical#26039020C2
hypercenter#384D179
hypocenterC179#2039
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#19153041C2semidirect
3. #26038438C2semidirect
4. #26038208C2semidirect
5. #384×C179C2semidirect
40. C42×C179D4semidirect
85. C179#2039semidirect
90. #384D179semidirect
126. C42D716semidirect

Maximal subgroups

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

subgroupindex
#260390202
C2×#191530412
#260384382
#260382082
#384×C1792
#2039179