Finite Group Databasebeta

← #36368924

Subgroups — #36368924

Order 22144.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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#26012513C211yesmaximal, characteristic, complemented
2#26012509C211yesmaximal, characteristic, complemented
3#26012339C214yesmaximal, complemented
4#26012042C211yesmaximal, characteristic
5#19149746—21no—
6C4⋊#13063443C2212yes—
7#19149736—21no—
8#19149736—21no—
9#19149734—21no—
10C4×#13063443C2212yes—
11C2×#14583827—21no—
12#19149676C411yescharacteristic
13#19149668—21no—
14#19149654—21no—
15#19149654C412yes—
16#19149641—21no—
17#19149641C2212yes—
18C42×D173C2211yescharacteristic
19C4⋊Q8×C173C411yescharacteristic
20C2×#13063443—22no—
21C2×#13063443—22no—
22#14583827—44no—
23C2×D692—22no—
24C2×D692—21no—
25C2×D692—21no—
26C2×D692C2×C412yes—
27C2×C4×D173—22no—
28C2×C4×D173C2×C411yescharacteristic
29C2×C4×D173D411yescharacteristic
30C2×C4×D173D411yescharacteristic
31#14583803—22no—
32#14583803—22no—
33C4×Dic173D411yescharacteristic
34C4×Dic173D411yescharacteristic
35C4×Dic173C2311yescharacteristic
36C2×Q8×C173—21no—
37C4⋊C4×C173—21no—
38C4⋊C4×C173C2×C412yes—
39C42×C173C2×C411yescharacteristic
40#13063443—42no—
41#13063443—42no—
42C22×D173—22no—
43C22×D173C22⋊C411yescharacteristic
44D692—24no—
45D692—42no—
46D692—42no—
47D692—44no—
48C2×Dic173C22⋊C411yescharacteristic
49C2×Dic173C2×D411yescharacteristic
50C2×Dic173C2×D411yescharacteristic
51C4×D173—22no—
52C4×D173—22no—
53C4×D173—41no—
54C4×D173—44no—
55Q8×C173—42no—
56C2×C4×C173—22no—
57C2×C4×C173C22⋊C411yescharacteristic
58C2×C4×C173C22⋊C411yescharacteristic
59C2×C4×C173C22×C411yescharacteristic
60C22×C173C2×C22⋊C411yescharacteristic
61C2×D173—21no—
62C2×D173—21no—
63C2×D173—21no—
64C2×D173—42no—
65C2×D173—42no—
66C2×D173—42no—
67C4×C173—21no—
68C4×C173—21no—
69C4×C173—42no—
70C4×C173#13112yes—
71Dic173—21no—
72Dic173—21no—
73Dic173AGL(1,ℤ/8)12yes—
74C2×C173#40811yescharacteristic
75C2×C173#41811yescharacteristic
76C2×C173#42111yescharacteristic
77D173—41no—
78D173—82no—
79C173#149611yescharacteristic, complemented, Sylow
80#1496—1731nomaximal, Sylow
81#532—1731no—
82#432—1731no—
83#431—1731no—
84D4⋊C8—1734no—
85C2×#41—3461no—
86C4⋊Q8#1062192411yescharacteristic
87#127—1731no—
88#124—3461no—
89#121—3461no—
90#121—1732no—
91C4×D4—3461no—
92C4×D4—1732no—
93C2×C42—1731no—
94C4⋊C8—3461no—
95C4⋊C8—1732no—
96C22⋊C8—3461no—
97C22⋊C8—3461no—
98#97—3461no—
99C4×C8—1732no—
100#41—6924no—
101C2×Q8—21no—
102C2×D4—3462no—
103C2×D4—3461no—
104C2×D4—3461no—
105C2×D4—1732no—
106C22×C4—3462no—
107C22×C4—1731no—
108C22×C4—1731no—
109C22×C4—1731no—
110C2×C8—3462no—
111C2×C8—3462no—
112C4⋊C4—21no—
113C4⋊C4C2×#1062192412yes—
114C22⋊C4—3462no—
115C22⋊C4—3462no—
116C42—1731no—
117C42—1731no—
118C42—1731no—
119C42C2×#1062192411yescharacteristic
120C23—3462no—
121C23—1731no—
122Q8—42no—
123D4—3464no—
124D4—6922no—
125D4—6922no—
126D4—6924no—
127C2×C4—22no—
128C2×C4—3462no—
129C2×C4—3462no—
130C2×C4—1731no—
131C2×C4—1731no—
132C2×C4—1731no—
133C2×C4—6921no—
134C2×C4—6924no—
135C2×C4C22⋊#1062192411yescharacteristic
136C2×C4C22⋊#1062192411yescharacteristic
137C2×C4C22×#1062192411yescharacteristic
138C8—6922no—
139C8—6922no—
140C22—3461no—
141C22—3461no—
142C22—3461no—
143C22—6922no—
144C22—6922no—
145C22—6922no—
146C22C2×C22⋊#1062192411yescharacteristic
147C4—21no—
148C4—21no—
149C4—42no—
150C4—3461no—
151C4—3461no—
152C4—1732no—
153C4#1914976712yes—
154C2—13842no—
155C2—6921no—
156C2#2601246611yescharacteristic
157C2#2601248311yescharacteristic
158C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012042C2
cosocleC4×Dic173C23
hypercenterC4⋊Q8#10621924
hypocenterC173#1496
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26012513C2semidirect
2. #26012509C2semidirect
3. #26012339C2semidirect
79. C173#1496semidirect

Maximal subgroups

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

subgroupindex
#260125132
#260125092
#260123392
#260120422
#1496173