Finite Group Databasebeta

← #36369751

Subgroups — #36369751

Order 22144.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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#26012496C211yesmaximal, characteristic
2#26012427C211yesmaximal, characteristic
3#26012361C214yesmaximal
4#26012046C211yesmaximal, characteristic
5C2×#14583836C2212yes—
6C2×C4×#10621924C2212yes—
7#19149746—21no—
8#19149746—21no—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4—21no—
11#19149679C411yescharacteristic, complemented
12#19149670C412yescomplemented
13#19149659—21no—
14#19149655—21no—
15Q8×Dic173—21no—
16#19149644—21no—
17#19149637C2212yes—
18C42×D173C2211yescharacteristic
19C4⋊Q8×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21#14583836—24no—
22C2×#13063443—22no—
23C4×#10621924—22no—
24C4×#10621924Q814yescomplemented
25C2×Dic346—21no—
26C2×C4×D173D411yescharacteristic
27C2×C4×D173D411yescharacteristic
28C2×C4×D173C2311yescharacteristic
29C4⋊Dic173—22no—
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C412yes—
32#14583805—22no—
33#14583803—22no—
34C4×Dic173—22no—
35C4×Dic173C2×C411yescharacteristic
36C4×Dic173D411yescharacteristic
37C4×Dic173D411yescharacteristic
38C2×Q8×C173—21no—
39C4⋊C4×C173—21no—
40C4⋊C4×C173C2×C412yes—
41C42×C173C2×C411yescharacteristic
42C2×#10621924—24no—
43C2×#10621924—42no—
44C2×#10621924—42no—
45#13063443—42no—
46C22×D173C2×D411yescharacteristic
47Dic346—42no—
48C2×Dic173—22no—
49C2×Dic173C22⋊C411yescharacteristic
50C2×Dic173C22⋊C411yescharacteristic
51C2×Dic173C2×D411yescharacteristic
52C4×D173—41no—
53C4×D173—41no—
54C4×D173C2×Q812yes—
55C4×D173#4112yes—
56Q8×C173—42no—
57C2×C4×C173—22no—
58C2×C4×C173C22⋊C411yescharacteristic
59C2×C4×C173C22⋊C411yescharacteristic
60C2×C4×C173C22×C411yescharacteristic
61C22×C173C2×C22⋊C411yescharacteristic
62C2×D173—21no—
63C2×D173—21no—
64C2×D173C22⋊Q812yes—
65#10621924—44no—
66C4×C173—21no—
67C4×C173—21no—
68C4×C173—42no—
69C4×C173C4×Q812yes—
70Dic173—21no—
71Dic173—21no—
72Dic173—42no—
73Dic173#12411yescharacteristic
74Dic173C4⋊Q811yescharacteristic
75C2×C173#39111yescharacteristic
76C2×C173#42111yescharacteristic
77C2×C173#42111yescharacteristic
78D173—22no—
79C173#199511yescharacteristic, complemented, Sylow
80#1995—1731nomaximal, Sylow
81#533—1731no—
82#423—1731no—
83#376—1731no—
84#339—1734no—
85C2×M16—1732no—
86C4⋊Q8—1731no—
87C4⋊Q8#1062192411yescharacteristic, complemented
88#125—1732no—
89C22⋊Q8—3461no—
90C22⋊Q8—3461no—
91C4×Q8—3461no—
92C4×Q8—3461no—
93#117—1732no—
94C2×C42—1732no—
95C2×C42—1731no—
96C4⋊C8—3461no—
97C4⋊C8—3461no—
98#95—3461no—
99#95—3461no—
100C2×Q8—21no—
101C2×Q8—3461no—
102C22×C4—3462no—
103C22×C4—1731no—
104C22×C4—1731no—
105C22×C4—1731no—
106M16—3464no—
107C2×C8—3462no—
108C4⋊C4—21no—
109C4⋊C4—3462no—
110C4⋊C4—3462no—
111C4⋊C4—3461no—
112C4⋊C4—1732no—
113C4⋊C4C2×#1062192412yes—
114C22⋊C4—3462no—
115C42—3462no—
116C42—3462no—
117C42—1734no—
118C42—1731no—
119C42—1731no—
120C42—1731no—
121C42C2×#1062192411yescharacteristic
122C23—1731no—
123Q8—42no—
124Q8—6922no—
125C2×C4—22no—
126C2×C4—3464no—
127C2×C4—3462no—
128C2×C4—1731no—
129C2×C4—1731no—
130C2×C4—1731no—
131C2×C4—1732no—
132C2×C4—1732no—
133C2×C4—6921no—
134C2×C4—6921no—
135C2×C4—6922no—
136C2×C4—6922no—
137C2×C4C22⋊#1062192411yescharacteristic
138C2×C4C22⋊#1062192411yescharacteristic
139C2×C4C22×#1062192411yescharacteristic
140C8—6922no—
141C22—3461no—
142C22—3461no—
143C22—1732no—
144C22C2×C22⋊#1062192411yescharacteristic
145C4—21no—
146C4—21no—
147C4—42no—
148C4—3461no—
149C4—3461no—
150C4—1731no—
151C4—1731no—
152C4—6924no—
153C4—6922no—
154C4Q8×#1062192412yes—
155C2—3462no—
156C2(C2×#10621924)⋊Q811yescharacteristic
157C2#2601248911yescharacteristic
158C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D173C2×D4
cosocleC2×C4×D173C23
hypercenterC4⋊Q8#10621924
hypocenterC173#1995
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
11. #19149679C4semidirect
12. #19149670C4semidirect
19. C4⋊Q8×C173C4semidirect
24. C4×#10621924Q8semidirect
79. C173#1995semidirect
87. C4⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124962
#260124272
#260123612
#260120462
#1995173