Finite Group Databasebeta

← #36354704

Subgroups — #36354704

Order 22144.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#19149679C211yesmaximal, characteristic
2C2×#19149603C212yesmaximal
3#26011253C212yesmaximal
4#26011246C211yesmaximal, characteristic
5#391×C173C211yesmaximal, characteristic
6C22×Dic346—21no—
7C22×#13063436C2212yes—
8#19149679—24no—
9C2×C4⋊Dic173—21no—
10C2×C4⋊Dic173C2212yes—
11#19149603—24no—
12C22×Q8×C173C2211yescharacteristic
13C2×C4⋊C4×C173C2211yescharacteristic
14C2×C42×C173C2211yescharacteristic
15#95×C173—21no—
16#95×C173—21no—
17C2×Dic346—24no—
18C2×Dic346—42no—
19C2×Dic346—42no—
20C22×Dic173—22no—
21C2×#13063436—22no—
22C2×#13063436—22no—
23C2×#13063436—22no—
24C4⋊Dic173—24no—
25C4⋊Dic173—22no—
26C4⋊Dic173D414yes—
27C2×Q8×C173—21no—
28C2×Q8×C173—22no—
29C2×Q8×C173—22no—
30C2×Q8×C173Q812yescomplemented
31C22×C4×C173—21no—
32C22×C4×C173—21no—
33C22×C4×C173D411yescharacteristic
34C22×C4×C173D411yescharacteristic
35C22×C4×C173C2311yescharacteristic
36C4⋊C4×C173—22no—
37C42×C173—22no—
38Dic346—48no—
39C2×Dic173—24no—
40C2×Dic173—42no—
41C2×Dic173—42no—
42#13063436—42no—
43C23×C173C2×D411yescharacteristic
44Q8×C173—24no—
45Q8×C173—42no—
46C2×C4×C173—21no—
47C2×C4×C173—21no—
48C2×C4×C173—21no—
49C2×C4×C173—21no—
50C2×C4×C173—22no—
51C2×C4×C173—41no—
52C2×C4×C173—41no—
53C2×C4×C173—41no—
54C2×C4×C173—41no—
55C2×C4×C173—41no—
56C2×C4×C173SD1612yes—
57C2×C4×C173Q1612yes—
58C2×C4×C173C2×D412yes—
59C2×C4×C173C2×Q811yescharacteristic
60C2×C4×C173#4112yes—
61C2×C4×C173#4111yescharacteristic
62C22×C173#12011yescharacteristic
63C22×C173C22⋊Q811yescharacteristic
64C22×C173#12411yescharacteristic
65C22×C173C2×SD1611yescharacteristic
66C22×C173C2×Q1611yescharacteristic
67C22×C173AGL(1,ℤ/8)11yescharacteristic
68C22×C173#13711yescharacteristic
69C4×C173—22no—
70C4×C173—22no—
71C4×C173—42no—
72C4×C173—81no—
73C4×C173C22⋊Q812yes—
74C4×C173#12412yes—
75Dic173—44no—
76C2×C173#39311yescharacteristic
77C2×C173#45011yescharacteristic
78C2×C173#45111yescharacteristic
79C2×C173#47511yescharacteristic
80C2×C173#47711yescharacteristic
81C2×C173#48711yescharacteristic
82C2×C173#48911yescharacteristic
83C173#202411yescharacteristic, complemented, Sylow
84#2024—1731nomaximal, Sylow
85C2×C4⋊Q8—1731no—
86C2×#103—1732no—
87#391D17311yescharacteristic
88#340—1732no—
89#336—1731no—
90C22×Q8—3461no—
91C22×Q8C2×D17311yescharacteristic
92C22×C8—1732no—
93C4⋊Q8—3464no—
94C2×C4⋊C4—3461no—
95C2×C4⋊C4—1732no—
96C2×C4⋊C4C2×D17311yescharacteristic
97C2×C42C2×D17311yescharacteristic
98#103—3464no—
99#95—21no—
100#95—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—3464no—
105C2×Q8—6922no—
106C2×Q8—6922no—
107C2×Q8Dic34612yescomplemented
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—3462no—
111C22×C4C22⋊D17311yescharacteristic
112C22×C4C22⋊D17311yescharacteristic
113C22×C4C22×D17311yescharacteristic
114C2×C8—3462no—
115C2×C8—3462no—
116C2×C8—3462no—
117C4⋊C4—22no—
118C4⋊C4—3464no—
119C4⋊C4—3462no—
120C4⋊C4—1734no—
121C42—22no—
122C23C2×C22⋊D17311yescharacteristic
123Q8—24no—
124Q8—42no—
125Q8—6928no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—3464no—
137C2×C4—6922no—
138C2×C4—6922no—
139C2×C4#1458381312yes—
140C2×C4#1458381612yes—
141C2×C4D4×D17312yes—
142C2×C4C2×Dic34611yescharacteristic
143C2×C4#1458382512yes—
144C2×C4#1458382611yescharacteristic
145C8—6922no—
146C22#1914964711yescharacteristic
147C22#1914965811yescharacteristic
148C22#1914966611yescharacteristic
149C22C2×#1458381311yescharacteristic
150C22C2×#1458381611yescharacteristic
151C22#1914970411yescharacteristic
152C22#1914970911yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—81no—
157C4—6924no—
158C4#1914965612yes—
159C4#1914966512yes—
160C2#2601140311yescharacteristic
161C2#2601164111yescharacteristic
162C2#2601164611yescharacteristic
163C2#2601177611yescharacteristic
164C2#2601178611yescharacteristic
165C2#2601184111yescharacteristic
166C2#2601185011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D173
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#391×C173C2
socle, prime radicalC23×C173C2×D4
hypercenter#391D173
hypocenterC173#2024
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C173Q8semidirect
83. C173#2024semidirect
107. C2×Q8Dic346semidirect

Maximal subgroups

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

subgroupindex
C2×#191496792
C2×#191496032
#260112532
#260112462
#391×C1732
#2024173