Finite Group Databasebeta

← #36352605

Subgroups — #36352605

Order 22144.

202 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 57 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#26011824C212yesmaximal
2#26011430C211yesmaximal, characteristic
3#26011228C212yesmaximal
4#26011217C211yesmaximal, characteristic
5#544×C173C211yesmaximal, characteristic
6C2×C4⋊Dic173C2211yescharacteristic
7#19149612—22no—
8#19149603—22no—
9C22⋊#13063436C2212yes—
10#19149586—21no—
11#19149586—21no—
12#19149586C2211yescharacteristic
13C4⋊Q8×C173—22no—
14#125×C173—22no—
15C22⋊Q8×C173—21no—
16C22⋊Q8×C173C2212yes—
17#117×C173—21no—
18C2×C4⋊C4×C173C2211yescharacteristic
19C22×Dic173—21no—
20C22×Dic173—21no—
21C2×#13063436—22no—
22C4⋊Dic173—22no—
23C2×Q8×C173—21no—
24C2×Q8×C173D412yes—
25C22×C4×C173D411yescharacteristic
26C22×C4×C173D411yescharacteristic
27C22×C4×C173C2311yescharacteristic
28C4⋊C4×C173—21no—
29C4⋊C4×C173—22no—
30C4⋊C4×C173—21no—
31C4⋊C4×C173—21no—
32C4⋊C4×C173—22no—
33C4⋊C4×C173—22no—
34C4⋊C4×C173D412yes—
35C22⋊C4×C173—22no—
36C42×C173—22no—
37C2×Dic173—41no—
38C2×Dic173—41no—
39C2×Dic173—42no—
40C2×Dic173—41no—
41C2×Dic173—41no—
42#13063436—42no—
43C23×C173C2×D411yescharacteristic
44Q8×C173—42no—
45Q8×C173—81no—
46C2×C4×C173—21no—
47C2×C4×C173—21no—
48C2×C4×C173—22no—
49C2×C4×C173—22no—
50C2×C4×C173—21no—
51C2×C4×C173—41no—
52C2×C4×C173—41no—
53C2×C4×C173Q1612yes—
54C2×C4×C173C2×D412yes—
55C22×C173—21no—
56C22×C173—21no—
57C22×C173#12011yescharacteristic
58C22×C173C2×Q1611yescharacteristic
59C22×C173#13711yescharacteristic
60C4×C173—22no—
61C4×C173—22no—
62C4×C173—42no—
63C4×C173—41no—
64C4×C173—42no—
65Dic173—81no—
66Dic173—81no—
67C2×C173—21no—
68C2×C173—21no—
69C2×C173#45111yescharacteristic
70C2×C173#45411yescharacteristic
71C2×C173#45811yescharacteristic
72C173#161711yescharacteristic, complemented, Sylow
73#1617—1731nomaximal, Sylow
74#544D17311yescharacteristic
75#484—1732no—
76#400—1731no—
77#328—1732no—
78#324—1731no—
79C4⋊Q8—22no—
80#125—22no—
81C22⋊Q8—21no—
82C22⋊Q8C2×D17312yes—
83#117—21no—
84C2×C4⋊C4—1731no—
85C2×C4⋊C4C2×D17311yescharacteristic
86#107—3462no—
87#103—3462no—
88C22⋊C8—1732no—
89#95—3461no—
90#95—3461no—
91#95—1731no—
92C2×Q8—21no—
93C2×Q8C22⋊D17312yes—
94C22×C4—3461no—
95C22×C4—3461no—
96C22×C4C22⋊D17311yescharacteristic
97C22×C4C22⋊D17311yescharacteristic
98C22×C4C22×D17311yescharacteristic
99C2×C8—3462no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—3462no—
107C4⋊C4C22⋊D17312yes—
108C22⋊C4—22no—
109C42—22no—
110C23C2×C22⋊D17311yescharacteristic
111Q8—42no—
112Q8—81no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—6921no—
121C2×C4—6921no—
122C2×C4—6922no—
123C2×C4—6921no—
124C2×C4—6921no—
125C2×C4#1458381612yes—
126C2×C4C2×C22⋊D17312yes—
127C8—6922no—
128C22—21no—
129C22—21no—
130C22#1914964811yescharacteristic
131C22C2×#1458381611yescharacteristic
132C22#1914971011yescharacteristic
133C4—22no—
134C4—22no—
135C4—42no—
136C4—41no—
137C4—42no—
138C4—13841no—
139C4—13841no—
140C2—21no—
141C2—21no—
142C2#2601164711yescharacteristic
143C2#2601166411yescharacteristic
144C2#2601168111yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#544×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#544D173
hypocenterC173#1617
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
72. C173#1617semidirect

Maximal subgroups

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

subgroupindex
#260118242
#260114302
#260112282
#260112172
#544×C1732
#1617173