Finite Group Databasebeta

← #36355589

Subgroups — #36355589

Order 22144.

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

normal subgroups only

This group has 236 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#26011673C211yesmaximal, characteristic, complemented
2#26011663C212yesmaximal, complemented
3#26011290C211yesmaximal, characteristic, complemented
4#26011288C212yesmaximal, complemented
5#555×C173C211yesmaximal, characteristic, complemented
6#19149727C2211yescharacteristic
7#19149710—21no—
8#19149705—22no—
9#19149607—22no—
10#19149607—21no—
11#19149598C2212yes—
12#19149598C2211yescharacteristic
13C4⋊Q8×C173C2211yescharacteristic
14#124×C173—21no—
15#124×C173C2212yes—
16#121×C173—22no—
17C4×D4×C173—21no—
18#117×C173—22no—
19#14583827—41no—
20#14583826—22no—
21#14583826—21no—
22Q8×D173—41no—
23#14583816—41no—
24#14583814—42no—
25#14583813—41no—
26#14583811—42no—
27#14583810—22no—
28#14583810—21no—
29C2×Q8×C173D411yescharacteristic
30C2×Q8×C173C2311yescharacteristic
31C2×D4×C173—21no—
32C2×D4×C173—21no—
33C2×D4×C173D412yes—
34C22×C4×C173—22no—
35C4⋊C4×C173—21no—
36C4⋊C4×C173—22no—
37C22⋊C4×C173—22no—
38C22⋊C4×C173—22no—
39C22⋊C4×C173—22no—
40C42×C173—21no—
41C42×C173D411yescharacteristic
42C42×C173D412yes—
43Dic346—22no—
44Dic346—21no—
45D692—22no—
46D692—21no—
47C22⋊D173—41no—
48C4×D173—42no—
49C4×D173—41no—
50#13063436—42no—
51#13063436—41no—
52C23×C173—22no—
53Q8×C173—41no—
54Q8×C173—41no—
55D4×C173—42no—
56D4×C173—42no—
57D4×C173—81no—
58C2×C4×C173—21no—
59C2×C4×C173—22no—
60C2×C4×C173—21no—
61C2×C4×C173—44no—
62C2×C4×C173C2×D411yescharacteristic
63C2×C4×C173C2×D412yes—
64C22×C173—42no—
65C22×C173—42no—
66C22×C173—42no—
67C22×C173#12011yescharacteristic
68C2×D173—41no—
69C4×C173—22no—
70C4×C173—22no—
71C4×C173—21no—
72C4×C173—42no—
73C4×C173—41no—
74Dic173—41no—
75C2×C173—21no—
76C2×C173—82no—
77C2×C173B(4,𝔽2)11yescharacteristic
78D173—81no—
79C173#220411yescharacteristic, complemented, Sylow
80#2204—1731nomaximal, Sylow
81#555D17311yescharacteristic, complemented
82#456—1731no—
83#454—1732no—
84#356—1731no—
85#355—1732no—
862−1+4—1731no—
87#137—3461no—
88AGL(1,ℤ/8)—3462no—
89C4⋊Q8C2×D17311yescharacteristic
90#124—21no—
91#124C2×D17312yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—3462no—
96#104—3461no—
97#101—1731no—
98#101—1732no—
99#41—3462no—
100#41—3461no—
101#41—6921no—
102C2×Q8—6921no—
103C2×Q8C22⋊D17311yescharacteristic
104C2×Q8C22×D17311yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D17312yes—
108C22×C4—22no—
109Q16—6921no—
110SD16—6922no—
111SD16—6921no—
112D8—6922no—
113M16—3462no—
114M16—3461no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D17312yes—
122C42C22⋊D17311yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—3462no—
127Q8—3461no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—3462no—
132D4—3461no—
133D4—6921no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—6922no—
139C2×C4—6921no—
140C2×C4C2×C22⋊D17311yescharacteristic
141C2×C4C2×C22⋊D17312yes—
142C8—6922no—
143C8—6921no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—6921no—
148C22#1914964811yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—6921no—
155C2—21no—
156C2—82no—
157C2—13841no—
158C2#2601167811yescharacteristic
subgroupG/H
centerC2#26011678
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#555×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#555D173
hypocenterC173#2204
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #26011673C2semidirect
2. #26011663C2semidirect
3. #26011290C2semidirect
4. #26011288C2semidirect
5. #555×C173C2semidirect
79. C173#2204semidirect
81. #555D173semidirect

Maximal subgroups

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

subgroupindex
#260116732
#260116632
#260112902
#260112882
#555×C1732
#2204173