Finite Group Databasebeta

← #37508005

Subgroups — #37508005

Order 22912.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#26038753C212yesmaximal, complemented
3#26038369C211yesmaximal, characteristic, complemented
4#26038174C212yesmaximal, complemented
5#324×C179C211yesmaximal, characteristic, complemented
6C2×#15668259—21no—
7#19153093—22no—
8#19153088—22no—
9#19153087—21no—
10#19153087C2212yes—
11C2×C4⋊Dic179C2211yescharacteristic
12C2×Dic179⋊C4—21no—
13#19153047—22no—
14#19153035—22no—
15C179⋊#95—21no—
16C179⋊#95C2211yescharacteristic
17C2×C4⋊C4×C179C2211yescharacteristic
18C22⋊C8×C179C2212yes—
19#15668259—42no—
20#15668259—42no—
21C2×Dic358—21no—
22C2×D716—21no—
23C2×D716—21no—
24C2×D716D412yes—
25C2×C22⋊D179—22no—
26C22×Dic179—21no—
27C22×Dic179—21no—
28C2×C4×D179—22no—
29C4⋊Dic179—21no—
30C4⋊Dic179—22no—
31C4⋊Dic179D412yes—
32Dic179⋊C4—42no—
33(C2×D179)⋊C4—22no—
34(C2×D179)⋊C4—22no—
35C22×C4×C179D411yescharacteristic
36C22×C4×C179D411yescharacteristic
37C22×C4×C179C2311yescharacteristic
38C2×C8×C179—22no—
39C4⋊C4×C179—22no—
40C22×D179—22no—
41Dic358—42no—
42D716—42no—
43D716—42no—
44D716—81no—
45C22⋊D179—42no—
46C22⋊D179—42no—
47C2×Dic179—22no—
48C2×Dic179—22no—
49C2×Dic179—41no—
50C2×Dic179—41no—
51C2×Dic179—41no—
52C2×Dic179—41no—
53C2×Dic179—41no—
54C4×D179—44no—
55C23×C179C2×D411yescharacteristic
56C2×C4×C179—21no—
57C2×C4×C179—21no—
58C2×C4×C179—21no—
59C2×C4×C179—41no—
60C2×C4×C179—41no—
61C2×C4×C179D812yes—
62C2×C4×C179C2×D412yes—
63C8×C179—42no—
64C22×C179—21no—
65C22×C179—21no—
66C22×C179#12011yescharacteristic
67C22×C179C2×D811yescharacteristic
68C22×C179AGL(1,ℤ/8)11yescharacteristic
69C2×D179—42no—
70C2×D179—42no—
71C2×D179—42no—
72C4×C179—22no—
73C4×C179—22no—
74C4×C179—41no—
75Dic179—42no—
76Dic179—42no—
77Dic179—81no—
78C2×C179—21no—
79C2×C179—21no—
80C2×C179#44711yescharacteristic
81C2×C179#45611yescharacteristic
82C2×C179#45811yescharacteristic
83D179—82no—
84C179#161011yescharacteristic, complemented, Sylow
85#1610—1791nomaximal, Sylow
86#537—1791no—
87#480—1792no—
88#398—1791no—
89#327—1792no—
90#324D17911yescharacteristic, complemented
91C2×#41—3581no—
92C22⋊Q8—3582no—
93#121—3582no—
94#121—3581no—
95#121—1792no—
96C2×C4⋊C4—3581no—
97C2×C4⋊C4—1791no—
98C2×C4⋊C4C2×D17911yescharacteristic
99#107—3582no—
100#102—3582no—
101C22⋊C8C2×D17912yes—
102#95—3581no—
103#95—1791no—
104#41—7162no—
105#41—7162no—
106C2×Q8—3581no—
107C2×D4—3581no—
108C2×D4—3581no—
109C2×D4—3582no—
110C2×D4—1792no—
111C22×C4—3582no—
112C22×C4—3581no—
113C22×C4—3581no—
114C22×C4D71611yescharacteristic
115C22×C4D71611yescharacteristic
116C22×C4C22×D17911yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—3581no—
120C4⋊C4—3582no—
121C4⋊C4—1792no—
122C4⋊C4—7162no—
123C22⋊C4—3582no—
124C22⋊C4—3582no—
125C23—3582no—
126C23C2×D71611yescharacteristic
127Q8—7162no—
128D4—14321no—
129D4—7162no—
130D4—7162no—
131D4—7162no—
132D4—7162no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—3582no—
139C2×C4—3582no—
140C2×C4—7164no—
141C2×C4—7161no—
142C2×C4—7161no—
143C2×C4—7161no—
144C2×C4—7161no—
145C2×C4—7161no—
146C2×C4D143212yes—
147C2×C4D4×D17912yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—7162no—
152C22—7162no—
153C22—7162no—
154C22#1915308311yescharacteristic
155C22C2×D143211yescharacteristic
156C22#1915314111yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—14321no—
161C4—7162no—
162C4—7162no—
163C2—21no—
164C2—21no—
165C2—14322no—
166C2#2603857511yescharacteristic
167C2#2603862411yescharacteristic
168C2#2603863011yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#324×C179C2
socleC22×C179#120
prime radical#26039020C2
hypercenter#324D179
hypocenterC179#1610
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #26038753C2semidirect
3. #26038369C2semidirect
4. #26038174C2semidirect
5. #324×C179C2semidirect
84. C179#1610semidirect
90. #324D179semidirect

Maximal subgroups

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

subgroupindex
#260390202
#260387532
#260383692
#260381742
#324×C1792
#1610179