Finite Group Databasebeta

← #36352853

Subgroups — #36352853

Order 22144.

262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012233C211yesmaximal, characteristic, complemented
2#26011881C211yesmaximal, characteristic, complemented
3#26011880C212yesmaximal, complemented
4#26011230C212yesmaximal, complemented
5#329×C173C211yesmaximal, characteristic, complemented
6C2×#14583815—22no—
7C2×#14583813—21no—
8C2×D1384—22no—
9C2×#14583811—21no—
10#19149676C2211yescharacteristic, complemented
11#19149668—21no—
12#19149667C2212yes—
13#19149665—22no—
14#19149649—22no—
15#19149647—21no—
16#19149590C2211yescharacteristic, complemented
17#124×C173C2211yescharacteristic
18#97×C173C2212yescomplemented
19C23×D173—21no—
20C2×Dic346D412yes—
21C2×D692—21no—
22C2×D692—21no—
23C2×D692—22no—
24C2×D692D412yes—
25C2×C22⋊D173—22no—
26#14583815—44no—
27#14583813—42no—
28D1384—44no—
29#14583811—42no—
30C2×#13063436—21no—
31C22⋊Dic173—21no—
32#14583803—22no—
33#14583803—22no—
34#14583803—22no—
35C4×Dic173—21no—
36C2×Q8×C173D411yescharacteristic
37C2×D4×C173D411yescharacteristic
38C2×C8×C173—22no—
39C22⋊C4×C173—22no—
40C42×C173C2311yescharacteristic
41C22×D173—22no—
42C22×D173—41no—
43C22×D173—82no—
44C22×D173—81no—
45Dic346—42no—
46D692—42no—
47D692—41no—
48D692—42no—
49D692—41no—
50D692—42no—
51D692—81no—
52C22⋊D173—82no—
53C2×Dic173—22no—
54#13063436—22no—
55C23×C173—21no—
56Q8×C173—41no—
57D4×C173—41no—
58C2×C4×C173—21no—
59C2×C4×C173C2×D412yes—
60C2×C4×C173C2×D411yescharacteristic
61C8×C173—24no—
62C22×C173—41no—
63C22×C173—42no—
64C22×C173#12011yescharacteristic
65C2×D173—42no—
66C2×D173—42no—
67C2×D173—42no—
68C2×D173—84no—
69C2×D173—82no—
70C2×D173—81no—
71C4×C173—22no—
72C4×C173—21no—
73C4×C173—41no—
74Dic173—42no—
75C2×C173—81no—
76C2×C173#45312yes—
77C2×C173#45311yescharacteristic
78D173—82no—
79D173—161no—
80C173#165511yescharacteristic, complemented, Sylow
81#1655—1731nomaximal, Sylow
82H(3,𝔽4)—1731no—
83#496—1731no—
84#496—1732no—
85#329—1732no—
86#329D17311yescharacteristic, complemented
87C2×SD16—3462no—
88C2×SD16—3461no—
89C2×D8—3462no—
90C2×D8—3461no—
91#127—1731no—
92#124—3462no—
93#124—3461no—
94#124—1732no—
95#124C2×D17311yescharacteristic
96#120—3462no—
97#120—3461no—
98#97—1731no—
99#97C2×D17312yescomplemented
100C24—3461no—
101C2×Q8—1732no—
102C2×Q8D69211yescharacteristic
103C2×D4—3461no—
104C2×D4—3462no—
105C2×D4—3462no—
106C2×D4—3461no—
107C2×D4—1732no—
108C2×D4D69211yescharacteristic
109SD16—6924no—
110SD16—6922no—
111D8—6924no—
112D8—6922no—
113C2×C8—22no—
114C2×C8—3461no—
115C22⋊C4—22no—
116C22⋊C4—3462no—
117C22⋊C4—3461no—
118C22⋊C4—3462no—
119C22⋊C4—3462no—
120C42—3461no—
121C42C22×D17311yescharacteristic
122C23—21no—
123C23—3462no—
124C23—13842no—
125C23—13841no—
126C23—6921no—
127Q8—41no—
128Q8—6922no—
129D4—41no—
130D4—13841no—
131D4—13842no—
132D4—6922no—
133D4—6921no—
134D4—6922no—
135D4—6921no—
136D4—6922no—
137C2×C4—21no—
138C2×C4—3462no—
139C2×C4D4×D17312yes—
140C2×C4C2×D69211yescharacteristic
141C8—24no—
142C8—3462no—
143C22—42no—
144C22—41no—
145C22—13844no—
146C22—13842no—
147C22—13841no—
148C22—6922no—
149C22—6922no—
150C22—6922no—
151C22#1914964911yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—6922no—
156C2—81no—
157C2—13842no—
158C2—27681no—
159C2#2601166012yes—
160C2#2601166111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#329×C173C2
socleC22×C173#120
hypercenter#329D173
hypocenterC173#1655
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #26012233C2semidirect
2. #26011881C2semidirect
3. #26011880C2semidirect
4. #26011230C2semidirect
5. #329×C173C2semidirect
10. #19149676C22semidirect
16. #19149590C22semidirect
18. #97×C173C22semidirect
80. C173#1655semidirect
86. #329D173semidirect
99. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122332
#260118812
#260118802
#260112302
#329×C1732
#1655173