Finite Group Databasebeta

← #36369101

Subgroups — #36369101

Order 22144.

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

normal subgroups only

This group has 254 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#26012496C212yesmaximal, complemented
2#26012351C212yesmaximal, complemented
3#26012349C212yesmaximal, complemented
4#26012040C211yesmaximal, characteristic
5C2×#14583836—22no—
6#19149746—22no—
7#19149746C2214yes—
8C2×#14583827C411yescharacteristic
9C2×#14583825C411yescharacteristic
10#19149677C2211yescharacteristic
11#19149675C2211yescharacteristic
12#19149667C411yescharacteristic
13#19149652—22no—
14#19149639—22no—
15C42×D173C2211yescharacteristic
16#124×C173C411yescharacteristic
17#14583836—44no—
18C2×#13063443—24no—
19#14583827—42no—
20#14583825—42no—
21C2×Dic346C2×C411yescharacteristic
22C2×D692—21no—
23C2×D692C2×C411yescharacteristic
24C2×C22⋊D173—21no—
25C2×C22⋊D173—22no—
26C22×Dic173—21no—
27C2×C4×D173—21no—
28C2×C4×D173C2×C411yescharacteristic
29C2×C4×D173D412yes—
30#14583805—22no—
31#14583803—22no—
32C4×Dic173D412yes—
33C4×Dic173C2311yescharacteristic
34C2×Q8×C173C2×C411yescharacteristic
35C2×D4×C173C2×C411yescharacteristic
36C22⋊C4×C173—22no—
37C42×C173C2×C411yescharacteristic
38#13063443—44no—
39C22×D173—21no—
40C22×D173C22⋊C411yescharacteristic
41Dic346—41no—
42D692—41no—
43D692—42no—
44C22⋊D173—42no—
45C22⋊D173—42no—
46C22⋊D173—42no—
47C2×Dic173—21no—
48C2×Dic173—42no—
49C2×Dic173C22⋊C411yescharacteristic
50C2×Dic173C2×D412yes—
51C4×D173—22no—
52C4×D173—22no—
53C4×D173—22no—
54C4×D173—42no—
55C23×C173—21no—
56Q8×C173—41no—
57D4×C173—41no—
58C2×C4×C173—21no—
59C2×C4×C173C22⋊C412yes—
60C2×C4×C173C22×C411yescharacteristic
61C22×C173—41no—
62C22×C173—42no—
63C22×C173C2×C22⋊C411yescharacteristic
64C2×D173—21no—
65C2×D173—22no—
66C2×D173—41no—
67C2×D173—42no—
68C4×C173—21no—
69C4×C173—22no—
70C4×C173—41no—
71Dic173—41no—
72Dic173#10012yes—
73Dic173AGL(1,ℤ/8)12yes—
74Dic173#13712yes—
75C2×C173—81no—
76C2×C173C2×#10011yescharacteristic
77C2×C173#41712yes—
78D173—41no—
79D173—81no—
80C173#156011yescharacteristic, complemented, Sylow
81#1560—1731nomaximal, Sylow
82#532—1731no—
83#423—1732no—
84#333—1732no—
85#331—1732no—
86C2×#41—1731no—
87C2×#41—1731no—
88C2×M16—3462no—
89C4⋊Q8—1731no—
90#127—1731no—
91#124—1731no—
92#124#1062192411yescharacteristic
93#121—3462no—
94C4×D4—3462no—
95C2×C42—1731no—
96C4⋊C8—3462no—
97C4⋊C8—1734no—
98#41—6922no—
99#41—6922no—
100C2×Q8—1731no—
101C2×Q8C2×#1062192411yescharacteristic
102C2×D4—3461no—
103C2×D4—3461no—
104C2×D4—3462no—
105C2×D4—1731no—
106C2×D4C2×#1062192411yescharacteristic
107C22×C4—3461no—
108C22×C4—3461no—
109C22×C4—1732no—
110C22×C4—1731no—
111M16—6924no—
112C2×C8—3464no—
113C4⋊C4—3462no—
114C22⋊C4—22no—
115C22⋊C4—3462no—
116C42—1732no—
117C42—1731no—
118C42C2×#1062192411yescharacteristic
119C23—21no—
120C23—3461no—
121C23—1731no—
122Q8—41no—
123Q8—6921no—
124D4—41no—
125D4—6921no—
126D4—6922no—
127D4—6922no—
128D4—6922no—
129D4—6922no—
130C2×C4—21no—
131C2×C4—3462no—
132C2×C4—3462no—
133C2×C4—3462no—
134C2×C4—3461no—
135C2×C4—1731no—
136C2×C4—1732no—
137C2×C4—6922no—
138C2×C4—6922no—
139C2×C4C22⋊#1062192412yes—
140C2×C4C22×#1062192411yescharacteristic
141C8—6924no—
142C22—41no—
143C22—42no—
144C22—3461no—
145C22—3462no—
146C22—6921no—
147C22—6922no—
148C22C2×C22⋊#1062192411yescharacteristic
149C4—21no—
150C4—22no—
151C4—41no—
152C4—1732no—
153C4—1732no—
154C4—1732no—
155C4—6921no—
156C2—81no—
157C2—13841no—
158C2—6921no—
159C2D173⋊#10011yescharacteristic
160C2#2601248212yes—
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012040C2
cosocleC4×Dic173C23
hypercenter#124#10621924
hypocenterC173#1560
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012496C2semidirect
2. #26012351C2semidirect
3. #26012349C2semidirect
80. C173#1560semidirect

Maximal subgroups

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

subgroupindex
#260124962
#260123512
#260123492
#260120402
#1560173