Finite Group Databasebeta

← #36355591

Subgroups — #36355591

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#26011673C211yesmaximal, characteristic, complemented
3#26011666C212yesmaximal, complemented
4#26011287C212yesmaximal, complemented
5#356×C173C211yesmaximal, characteristic, complemented
6#19149727C2211yescharacteristic
7#19149710—21no—
8#19149707—22no—
9#19149668—21no—
10#19149668C2212yes—
11#19149654—22no—
12C4×D692—21no—
13#19149637—22no—
14#19149607—21no—
15#19149605—22no—
16#19149598C2211yescharacteristic
17C4⋊Q8×C173C2211yescharacteristic
18#101×C173C2212yes—
19#14583827—41no—
20#14583826—22no—
21#14583826—21no—
22Q8×D173—41no—
23C2×D692—21no—
24C2×D692—21no—
25C2×D692D412yes—
26C2×C4×D173—22no—
27#14583816—41no—
28#14583815—42no—
29#14583813—41no—
30D1384—42no—
31#14583810—21no—
32C4⋊Dic173—21no—
33#14583803—22no—
34#14583803—22no—
35#14583803—22no—
36C4×Dic173—21no—
37C4×Dic173D412yes—
38C2×Q8×C173D411yescharacteristic
39C2×Q8×C173C2311yescharacteristic
40M16×C173—22no—
41C4⋊C4×C173—22no—
42C42×C173D411yescharacteristic
43C22×D173—22no—
44Dic346—22no—
45Dic346—21no—
46D692—22no—
47D692—21no—
48D692—42no—
49D692—42no—
50D692—81no—
51C22⋊D173—41no—
52C2×Dic173—22no—
53C4×D173—44no—
54C4×D173—42no—
55C4×D173—41no—
56#13063436—41no—
57Q8×C173—41no—
58Q8×C173—41no—
59C2×C4×C173—21no—
60C2×C4×C173—21no—
61C2×C4×C173C2×D412yes—
62C2×C4×C173C2×D411yescharacteristic
63C8×C173—42no—
64C22×C173#12011yescharacteristic
65C2×D173—42no—
66C2×D173—42no—
67C2×D173—41no—
68C2×D173—42no—
69C4×C173—22no—
70C4×C173—22no—
71C4×C173—21no—
72C4×C173—41no—
73Dic173—42no—
74Dic173—41no—
75C2×C173—21no—
76C2×C173B(4,𝔽2)11yescharacteristic
77D173—81no—
78D173—82no—
79C173#220411yescharacteristic, complemented, Sylow
80#2204—1731nomaximal, Sylow
81#555—1731no—
82#456—1731no—
83#454—1732no—
84#356D17311yescharacteristic, complemented
85#355—1732no—
862−1+4—1731no—
87#137—3461no—
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—3462no—
96#104—3461no—
97#101—1731no—
98#101C2×D17312yes—
99#41—3462no—
100#41—3461no—
101#41—6921no—
102C2×Q8—6921no—
103C2×Q8D69211yescharacteristic
104C2×Q8C22×D17311yescharacteristic
105C2×D4—3461no—
106C2×D4—3461no—
107C2×D4—1732no—
108C22×C4—3462no—
109Q16—6921no—
110SD16—6922no—
111SD16—6921no—
112D8—6922no—
113M16—22no—
114M16—3461no—
115C4⋊C4—22no—
116C4⋊C4—3461no—
117C22⋊C4—3462no—
118C22⋊C4—3462no—
119C22⋊C4—3462no—
120C42—3461no—
121C42—1732no—
122C42D69211yescharacteristic
123C23—3462no—
124Q8—41no—
125Q8—41no—
126Q8—3462no—
127Q8—3461no—
128D4—3462no—
129D4—3461no—
130D4—13841no—
131D4—6922no—
132D4—6922no—
133D4—6921no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—3462no—
137C2×C4—6924no—
138C2×C4—6922no—
139C2×C4—6921no—
140C2×C4D4×D17312yes—
141C2×C4C2×D69211yescharacteristic
142C8—42no—
143C8—6921no—
144C22—6922no—
145C22—6922no—
146C22—6921no—
147C22—6922no—
148C22#1914964911yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—6922no—
154C4—6921no—
155C2—21no—
156C2—13841no—
157C2—13842no—
158C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#356×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#356D173
hypocenterC173#2204
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #26012207C2semidirect
2. #26011673C2semidirect
3. #26011666C2semidirect
4. #26011287C2semidirect
5. #356×C173C2semidirect
79. C173#2204semidirect
84. #356D173semidirect

Maximal subgroups

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

subgroupindex
#260122072
#260116732
#260116662
#260112872
#356×C1732
#2204173