Finite Group Databasebeta

← #36369072

Subgroups — #36369072

Order 22144.

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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#26012508C211yesmaximal, characteristic
2#26012494C211yesmaximal, characteristic
3#26012350C214yesmaximal, complemented
4C2×#19149678C211yesmaximal, characteristic
5C2×#14583835—21no—
6#19149746C2212yes—
7#19149746C2212yes—
8C4⋊#13063443—21no—
9C22⋊#13063443—21no—
10C4×#13063443—21no—
11C22×Dic346—21no—
12#19149678—21no—
13#19149678—21no—
14#19149678C412yes—
15#19149678C2212yes—
16C2×C4⋊Dic173C411yescharacteristic
17C2×#14583805—21no—
18C2×C4×Dic173C2211yescharacteristic
19C2×C4⋊C4×C173C411yescharacteristic
20#14583835—42no—
21C2×#13063443—22no—
22C2×#13063443—22no—
23C2×Dic346—22no—
24C2×Dic346—22no—
25C2×Dic346—42no—
26C2×Dic346—42no—
27C22×Dic173—21no—
28C22×Dic173D411yescharacteristic
29C22×Dic173D411yescharacteristic
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C412yes—
32#14583805—22no—
33#14583805—22no—
34C4×Dic173C2×C411yescharacteristic
35C4×Dic173D411yescharacteristic
36C4×Dic173D411yescharacteristic
37C4×Dic173C2311yescharacteristic
38C22×C4×C173—21no—
39C22×C4×C173C2×C411yescharacteristic
40C4⋊C4×C173—21no—
41C4⋊C4×C173C2×C412yes—
42#13063443—42no—
43#13063443—42no—
44Dic346—44no—
45Dic346—42no—
46Dic346—42no—
47C2×Dic173—22no—
48C2×Dic173—22no—
49C2×Dic173—42no—
50C2×Dic173C22⋊C411yescharacteristic
51C2×Dic173C22⋊C411yescharacteristic
52C2×Dic173D812yes—
53C2×Dic173SD1612yes—
54C2×Dic173C2×D411yescharacteristic
55C2×Dic173C2×D411yescharacteristic
56C23×C173C22⋊C411yescharacteristic
57C2×C4×C173—22no—
58C2×C4×C173—22no—
59C2×C4×C173—42no—
60C2×C4×C173C22⋊C411yescharacteristic
61C2×C4×C173C22×C411yescharacteristic
62C22×C173—21no—
63C22×C173—21no—
64C22×C173#10212yes—
65C22×C173C2×C22⋊C411yescharacteristic
66C4×C173—21no—
67C4×C173—42no—
68C4×C173#10112yes—
69Dic173—21no—
70Dic173—42no—
71Dic173#10212yes—
72Dic173C2×D811yescharacteristic
73Dic173C2×SD1611yescharacteristic
74Dic173#13712yes—
75C2×C173—22no—
76C2×C173C2×#10111yescharacteristic
77C2×C173C2×#10211yescharacteristic
78C2×C173#41911yescharacteristic
79C173#154611yescharacteristic, complemented, Sylow
80#1546—1731nomaximal, Sylow
81C2×C4⋊Q8—1731no—
82#431—1731no—
83#423—1731no—
84#332—1734no—
85C22×Q8—3461no—
86C2×M16—3461no—
87C4⋊Q8—3461no—
88C4⋊Q8—3461no—
89C4⋊Q8—1732no—
90C4⋊Q8—1732no—
91C2×C4⋊C4—3461no—
92C2×C4⋊C4—1731no—
93C2×C4⋊C4#1062192411yescharacteristic
94C2×C42—1731no—
95C4⋊C8—3461no—
96C4⋊C8—1732no—
97C4⋊C8—1732no—
98C22⋊C8—3461no—
99C4×C8—3461no—
100C2×Q8—3462no—
101C2×Q8—3462no—
102C2×Q8—6922no—
103C2×Q8—6922no—
104C22×C4—21no—
105C22×C4—3461no—
106C22×C4—1731no—
107C22×C4—1731no—
108C22×C4C2×#1062192411yescharacteristic
109M16—6922no—
110C2×C8—3462no—
111C2×C8—3462no—
112C4⋊C4—21no—
113C4⋊C4—3462no—
114C4⋊C4—3462no—
115C4⋊C4—3461no—
116C4⋊C4—1732no—
117C4⋊C4C2×#1062192412yes—
118C42—1731no—
119C42—1731no—
120C42—1731no—
121C42—1731no—
122C23C22⋊#1062192411yescharacteristic
123Q8—6922no—
124Q8—6924no—
125Q8—6922no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—42no—
129C2×C4—3462no—
130C2×C4—3462no—
131C2×C4—1731no—
132C2×C4—1732no—
133C2×C4—1732no—
134C2×C4—1731no—
135C2×C4—1731no—
136C2×C4—1731no—
137C2×C4—6922no—
138C2×C4C22⋊#1062192411yescharacteristic
139C2×C4C22×#1062192411yescharacteristic
140C8—6922no—
141C8—6922no—
142C22—21no—
143C22—21no—
144C22#1914974112yes—
145C22C2×C22⋊#1062192411yescharacteristic
146C4—21no—
147C4—42no—
148C4—3461no—
149C4—1731no—
150C4—1732no—
151C4—1731no—
152C4—1732no—
153C4—6922no—
154C4#1914974012yes—
155C2—22no—
156C2C2×#1914974011yescharacteristic
157C2C2×#1914974111yescharacteristic
158C2#2601248411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×C173C22⋊C4
cosocleC4×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1546
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012350C2semidirect
79. C173#1546semidirect

Maximal subgroups

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

subgroupindex
#260125082
#260124942
#260123502
C2×#191496782
#1546173