Finite Group Databasebeta

← #37510230

Subgroups — #37510230

Order 22912.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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×#19153039C212yesmaximal, complemented
3#26038441C212yesmaximal, complemented
4#26038304C211yesmaximal, characteristic, complemented
5#343×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2212yes—
7#19153093—22no—
8#19153088—22no—
9#19153087—22no—
10C2×C4⋊Dic179C2211yescharacteristic
11C2×C4×Dic179C2211yescharacteristic
12#19153039—24no—
13#19153038—22no—
14#19153035—22no—
15C179⋊#95—22no—
16C2×M16×C179C2212yes—
17C2×C4⋊C4×C179C2211yescharacteristic
18#15668259—22no—
19#15668259—22no—
20#15668259—22no—
21#15668259—22no—
22C2×Dic358D412yes—
23C2×D716—22no—
24C2×D716D412yes—
25C2×C22⋊D179—22no—
26C22×Dic179—21no—
27C22×Dic179—21no—
28C2×C4×D179—22no—
29C4⋊Dic179—22no—
30C4⋊Dic179—22no—
31(C2×D179)⋊C4—24no—
32C4×Dic179—21no—
33C4×Dic179D412yescomplemented
34C22×C4×C179D412yes—
35C22×C4×C179C2311yescharacteristic
36M16×C179—22no—
37M16×C179—22no—
38C2×C8×C179—22no—
39C4⋊C4×C179—22no—
40C22×D179—22no—
41Dic358—22no—
42Dic358—22no—
43D716—22no—
44D716—22no—
45D716—82no—
46C22⋊D179—42no—
47C22⋊D179—42no—
48C2×Dic179—22no—
49C2×Dic179—22no—
50C2×Dic179—41no—
51C2×Dic179—41no—
52C2×Dic179—42no—
53C2×Dic179—41no—
54C4×D179—42no—
55C4×D179—42no—
56C23×C179C2×D411yescharacteristic
57C2×C4×C179—22no—
58C2×C4×C179—42no—
59C2×C4×C179C2×D411yescharacteristic
60C2×C4×C179C2×D412yes—
61C2×C4×C179#4112yes—
62C2×C4×C179#4111yescharacteristic
63C8×C179—42no—
64C22×C179—21no—
65C22×C179—21no—
66C22×C179#12011yescharacteristic
67C22×C179#12111yescharacteristic
68C22×C179#12411yescharacteristic
69C2×D179—42no—
70C2×D179—42no—
71C2×D179—42no—
72C4×C179—42no—
73C4×C179#12112yes—
74C4×C179#12312yes—
75Dic179—42no—
76Dic179—42no—
77Dic179—81no—
78C2×C179—21no—
79C2×C179—21no—
80C2×C179#39411yescharacteristic
81C2×C179#45411yescharacteristic
82C2×C179#45511yescharacteristic
83D179—82no—
84C179#204011yescharacteristic, complemented, Sylow
85#2040—1791nomaximal, Sylow
86#537—1791no—
87C2×#104—1792no—
88#417—1792no—
89#382—1791no—
90#343D17911yescharacteristic, complemented
91C2×#41—1792no—
92C2×M16C2×D17912yes—
93C22⋊Q8—3582no—
94#121—3582no—
95#121—3582no—
96C2×C4⋊C4—1791no—
97C2×C4⋊C4C2×D17911yescharacteristic
98C2×C42—1791no—
99#104—3584no—
100#103—3582no—
101#102—3582no—
102#95—3582no—
103#41—3582no—
104#41—3582no—
105#41—3582no—
106#41—3582no—
107C2×Q8—1792no—
108C2×D4—3582no—
109C2×D4—3582no—
110C2×D4—1792no—
111C22×C4—3581no—
112C22×C4—3582no—
113C22×C4—3581no—
114C22×C4D71612yes—
115C22×C4C22×D17911yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—3582no—
121C4⋊C4—3582no—
122C22⋊C4—3584no—
123C42—3581no—
124C42—1792no—
125C23—3582no—
126C23C2×D71611yescharacteristic
127Q8—3582no—
128Q8—3582no—
129D4—3582no—
130D4—3582no—
131D4—14322no—
132D4—7162no—
133D4—7162no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3582no—
137C2×C4—3582no—
138C2×C4—7162no—
139C2×C4—7161no—
140C2×C4—7161no—
141C2×C4—7161no—
142C2×C4—7162no—
143C2×C4—7162no—
144C2×C4D4×D17911yescharacteristic
145C2×C4D4×D17912yes—
146C2×C4#1566825912yes—
147C2×C4#1566826011yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—7162no—
152C22—7162no—
153C22—7162no—
154C22#1915308311yescharacteristic
155C22#1915308811yescharacteristic
156C22#1915310111yescharacteristic
157C4—42no—
158C4—14321no—
159C4—7162no—
160C4—7162no—
161C4#1915308612yes—
162C4#1915309812yes—
163C2—21no—
164C2—21no—
165C2—14322no—
166C2#2603835711yescharacteristic
167C2#2603861411yescharacteristic
168C2#2603861911yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#343×C179C2
socleC22×C179#120
prime radical#26039020C2
hypercenter#343D179
hypocenterC179#2040
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#19153039C2semidirect
3. #26038441C2semidirect
4. #26038304C2semidirect
5. #343×C179C2semidirect
33. C4×Dic179D4semidirect
84. C179#2040semidirect
90. #343D179semidirect

Maximal subgroups

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

subgroupindex
#260390202
C2×#191530392
#260384412
#260383042
#343×C1792
#2040179