Finite Group Databasebeta

← #36352566

Subgroups — #36352566

Order 22144.

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

normal subgroups only

This group has 254 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#26011802C212yesmaximal, complemented
2#26011421C211yesmaximal, characteristic, complemented
3#26011225C212yesmaximal, complemented
4#26011217C211yesmaximal, characteristic, complemented
5#537×C173C211yesmaximal, characteristic
6C2×C4⋊Dic173C2211yescharacteristic
7C2×#14583805—21no—
8#19149612—22no—
9#19149600—22no—
10C22⋊#13063436C2212yes—
11#19149586—21no—
12#19149586C2211yescharacteristic
13C2×#41×C173—21no—
14C22⋊Q8×C173—22no—
15#121×C173—22no—
16#121×C173—21no—
17#121×C173C2212yes—
18C2×C4⋊C4×C173C2211yescharacteristic
19C22×Dic173—21no—
20C22×Dic173—21no—
21C2×#13063436—22no—
22C4⋊Dic173—22no—
23#14583805—42no—
24#41×C173—42no—
25#41×C173—42no—
26C2×Q8×C173—21no—
27C2×D4×C173—21no—
28C2×D4×C173—22no—
29C2×D4×C173—21no—
30C2×D4×C173D412yes—
31C22×C4×C173—22no—
32C22×C4×C173D411yescharacteristic
33C22×C4×C173D411yescharacteristic
34C22×C4×C173C2311yescharacteristic
35C4⋊C4×C173—21no—
36C4⋊C4×C173—22no—
37C4⋊C4×C173D412yes—
38C22⋊C4×C173—22no—
39C22⋊C4×C173—22no—
40C2×Dic173—22no—
41C2×Dic173—41no—
42C2×Dic173—41no—
43C2×Dic173—41no—
44C2×Dic173—41no—
45C2×Dic173—41no—
46#13063436—42no—
47C23×C173—22no—
48C23×C173C2×D411yescharacteristic
49Q8×C173—42no—
50D4×C173—42no—
51D4×C173—42no—
52D4×C173—42no—
53D4×C173—42no—
54D4×C173—81no—
55C2×C4×C173—21no—
56C2×C4×C173—21no—
57C2×C4×C173—21no—
58C2×C4×C173—22no—
59C2×C4×C173—41no—
60C2×C4×C173—44no—
61C2×C4×C173—41no—
62C2×C4×C173D812yes—
63C2×C4×C173C2×D412yes—
64C22×C173—21no—
65C22×C173—21no—
66C22×C173—42no—
67C22×C173—42no—
68C22×C173—42no—
69C22×C173#12011yescharacteristic
70C22×C173C2×D811yescharacteristic
71C22×C173AGL(1,ℤ/8)11yescharacteristic
72C4×C173—22no—
73C4×C173—22no—
74C4×C173—42no—
75C4×C173—41no—
76Dic173—42no—
77Dic173—81no—
78C2×C173—21no—
79C2×C173—21no—
80C2×C173—82no—
81C2×C173#44711yescharacteristic
82C2×C173#45611yescharacteristic
83C2×C173#45811yescharacteristic
84C173#161011yescharacteristic, complemented, Sylow
85#1610—1731nomaximal, Sylow
86#537D17311yescharacteristic
87#480—1732no—
88#398—1731no—
89#327—1732no—
90#324—1731no—
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C2×D17312yes—
96C2×C4⋊C4—3461no—
97C2×C4⋊C4—1731no—
98C2×C4⋊C4C2×D17311yescharacteristic
99#107—3462no—
100#102—3462no—
101C22⋊C8—1732no—
102#95—3461no—
103#95—1731no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C22⋊D17312yes—
111C22×C4—22no—
112C22×C4—3461no—
113C22×C4—3461no—
114C22×C4C22⋊D17311yescharacteristic
115C22×C4C22⋊D17311yescharacteristic
116C22×C4C22×D17311yescharacteristic
117C2×C8—3462no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—3462no—
121C4⋊C4—6922no—
122C4⋊C4C22⋊D17312yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×C22⋊D17311yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—44no—
139C2×C4—41no—
140C2×C4—3462no—
141C2×C4—6921no—
142C2×C4—6921no—
143C2×C4—6921no—
144C2×C4—6921no—
145C2×C4—6921no—
146C2×C4#1458381112yes—
147C2×C4C2×C22⋊D17312yes—
148C8—6922no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#1914964811yescharacteristic
155C22C2×#1458381111yescharacteristic
156C22#1914970511yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—42no—
161C4—13841no—
162C4—6922no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2601162511yescharacteristic
167C2#2601167411yescharacteristic
168C2#2601168111yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting, prime radical#537×C173C2
socleC22×C173#120
hypercenter#537D173
hypocenterC173#1610
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #26011802C2semidirect
2. #26011421C2semidirect
3. #26011225C2semidirect
4. #26011217C2semidirect
84. C173#1610semidirect

Maximal subgroups

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

subgroupindex
#260118022
#260114212
#260112252
#260112172
#537×C1732
#1610173