Finite Group Databasebeta

← #36355588

Subgroups — #36355588

Order 22144.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012207C211yesmaximal, characteristic, complemented
2#26011664C212yesmaximal, complemented
3#26011290C211yesmaximal, characteristic, complemented
4#26011287C212yesmaximal, complemented
5#456×C173C211yesmaximal, characteristic, complemented
6#19149704—22no—
7#19149668—21no—
8#19149668C2212yes—
9#19149654—22no—
10C4×D692—21no—
11#19149637—22no—
12#19149606—22no—
13#19149598C2212yes—
142−1+4×C173C2211yescharacteristic
15#137×C173—21no—
16C4⋊Q8×C173C2211yescharacteristic
17#104×C173—21no—
18#101×C173C2211yescharacteristic
19C2×D692—21no—
20C2×D692—21no—
21C2×D692D412yes—
22C2×C4×D173—22no—
23#14583813—42no—
24#14583811—42no—
25#14583810—22no—
26C4⋊Dic173—21no—
27#14583803—22no—
28#14583803—22no—
29#14583803—22no—
30C4×Dic173—21no—
31C4×Dic173D412yes—
32#41×C173—21no—
33#41×C173—22no—
34#41×C173—41no—
35C2×Q8×C173—41no—
36C2×Q8×C173D411yescharacteristic
37C2×Q8×C173C2311yescharacteristic
38Q16×C173—41no—
39SD16×C173—41no—
40M16×C173—21no—
41C4⋊C4×C173—22no—
42C42×C173D411yescharacteristic
43C22×D173—22no—
44D692—42no—
45D692—42no—
46D692—81no—
47C2×Dic173—22no—
48C4×D173—44no—
49#13063436—42no—
50Q8×C173—22no—
51Q8×C173—21no—
52Q8×C173—41no—
53Q8×C173—41no—
54D4×C173—22no—
55D4×C173—21no—
56D4×C173—41no—
57C2×C4×C173—21no—
58C2×C4×C173—21no—
59C2×C4×C173—41no—
60C2×C4×C173—42no—
61C2×C4×C173C2×D412yes—
62C2×C4×C173C2×D411yescharacteristic
63C8×C173—41no—
64C22×C173—41no—
65C22×C173#12011yescharacteristic
66C2×D173—42no—
67C2×D173—42no—
68C2×D173—42no—
69C4×C173—22no—
70C4×C173—21no—
71C4×C173—22no—
72C4×C173—41no—
73C4×C173—41no—
74Dic173—42no—
75C2×C173—21no—
76C2×C173—81no—
77C2×C173B(4,𝔽2)11yescharacteristic
78D173—82no—
79C173#220411yescharacteristic, complemented, Sylow
80#2204—1731nomaximal, Sylow
81#555—1731no—
82#456D17311yescharacteristic, complemented
83#454—1732no—
84#356—1731no—
85#355—1732no—
862−1+4C2×D17311yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—3462no—
89C4⋊Q8C2×D17311yescharacteristic
90#124—3461no—
91#124—1732no—
92#121—3462no—
93C4×D4—3461no—
94#117—3462no—
95#104—21no—
96#104—3462no—
97#101—1732no—
98#101C2×D17311yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D17311yescharacteristic
104C2×Q8C22×D17311yescharacteristic
105C2×D4—3461no—
106C2×D4—3461no—
107C2×D4—1732no—
108C22×C4—3462no—
109Q16—41no—
110SD16—41no—
111SD16—6922no—
112D8—6922no—
113M16—21no—
114M16—3462no—
115C4⋊C4—22no—
116C4⋊C4—3461no—
117C22⋊C4—3462no—
118C22⋊C4—3462no—
119C22⋊C4—3462no—
120C42—3461no—
121C42—1732no—
122C42C22⋊D17311yescharacteristic
123C23—3462no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—13841no—
132D4—6922no—
133D4—6922no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—3462no—
139C2×C4—6924no—
140C2×C4D4×D17312yes—
141C2×C4C2×C22⋊D17311yescharacteristic
142C8—41no—
143C8—6922no—
144C22—41no—
145C22—6922no—
146C22—6922no—
147C22—6922no—
148C22#1914964711yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—6922no—
155C2—21no—
156C2—81no—
157C2—13842no—
158C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#456×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#456D173
hypocenterC173#2204
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26012207C2semidirect
2. #26011664C2semidirect
3. #26011290C2semidirect
4. #26011287C2semidirect
5. #456×C173C2semidirect
79. C173#2204semidirect
82. #456D173semidirect

Maximal subgroups

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

subgroupindex
#260122072
#260116642
#260112902
#260112872
#456×C1732
#2204173