Finite Group Databasebeta

← #36355540

Subgroups — #36355540

Order 22144.

206 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 47 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#26012304C211yesmaximal, characteristic
2#26011932C211yesmaximal, characteristic, complemented
3#26011763C211yesmaximal, characteristic, complemented
4#26011307C211yesmaximal, characteristic, complemented
5#26011299C211yesmaximal, characteristic, complemented
6#26011274C211yesmaximal, characteristic, complemented
7C2×Q32×C173C211yesmaximal, characteristic, complemented
8#19149727—21no—
9#19149711—22no—
10#19149710—21no—
11#19149710—21no—
12#19149702—21no—
13#19149701C2211yescharacteristic
14Q16×D173—21no—
15#19149685C2211yescharacteristic
16#19149625—21no—
17#19149622—21no—
18#19149619C2211yescharacteristic
19#19149615C2211yescharacteristic
20#19149598—21no—
21C2×Q16×C173C2211yescharacteristic
22C2×Q16×C173C2211yescharacteristic
23Q32×C173—22no—
24C2×C16×C173C2211yescharacteristic
25#14583827—22no—
26#14583827—41no—
27#14583826—21no—
28#14583826—42no—
29#14583826D411yescharacteristic
30Q8×D173—22no—
31Q8×D173—41no—
32Dic692D411yescharacteristic
33#14583816—22no—
34#14583816—41no—
35#14583815—21no—
36#14583813—22no—
37#14583813—41no—
38D1384D411yescharacteristic
39#14583810—21no—
40#14583810D411yescharacteristic
41#14583809—21no—
42C8×D173—21no—
43#14583801—21no—
44C2×Q8×C173—21no—
45C2×Q8×C173—21no—
46Q16×C173—21no—
47Q16×C173—21no—
48Q16×C173—21no—
49Q16×C173D412yes—
50C2×C8×C173C2311yescharacteristic
51C16×C173—21no—
52Dic346—21no—
53Dic346—42no—
54Dic346D811yescharacteristic
55D692—21no—
56D692—42no—
57D692D811yescharacteristic
58C22⋊D173—21no—
59C22⋊D173—41no—
60C4×D173—21no—
61C4×D173—42no—
62C4×D173—42no—
63C4×D173—41no—
64#13063436—21no—
65#13063436—41no—
66Q8×C173—22no—
67Q8×C173—41no—
68Q8×C173—41no—
69Q8×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173—41no—
72C2×C4×C173C2×D411yescharacteristic
73C8×C173C2×D411yescharacteristic
74C8×C173C2×D411yescharacteristic
75C22×C173AGL(1,ℤ/8)11yescharacteristic
76C2×D173—21no—
77C2×D173—41no—
78C4×C173—42no—
79C4×C173—81no—
80C4×C173#12011yescharacteristic
81C4×C173C2×D811yescharacteristic
82Dic173—21no—
83Dic173—41no—
84C2×C173—21no—
85C2×C173#44711yescharacteristic
86D173—41no—
87D173—81no—
88C173#219511yescharacteristic, complemented, Sylow
89#2195—1731nomaximal, Sylow
90#578—1731no—
91#510—1731no—
92C2×Q32D17311yescharacteristic, complemented
93#473—1731no—
94#362—1731no—
95#359—1731no—
96#350—1731no—
972−1+4—3461no—
98#137—3462no—
99#137—3461no—
100#137—3461no—
101#135—3461no—
102#135—1731no—
103C2×Q16—3461no—
104C2×Q16C2×D17311yescharacteristic
105C2×Q16C2×D17311yescharacteristic
106#131—1731no—
107Q32—22no—
108Q32—3461no—
109SD32—3461no—
110M32—1731no—
111C2×C16C2×D17311yescharacteristic
112#108—1731no—
113#101—3461no—
114#41—3462no—
115#41—3461no—
116#41—1731no—
117#41—6922no—
118#41—6921no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—3462no—
122C2×Q8—6921no—
123Q16—21no—
124Q16—21no—
125Q16—21no—
126Q16—3462no—
127Q16—1731no—
128Q16—6921no—
129Q16C22⋊D17312yes—
130SD16—3462no—
131SD16—3461no—
132SD16—6921no—
133D8—1731no—
134M16—3461no—
135M16—3461no—
136M16—1731no—
137C2×C8—3461no—
138C2×C8C22×D17311yescharacteristic
139C16—21no—
140C16—3461no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144Q8—41no—
145Q8—3461no—
146Q8—1731no—
147Q8—6922no—
148D4—3461no—
149D4—3461no—
150D4—1731no—
151D4—6922no—
152D4—6921no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—3461no—
156C2×C4—6922no—
157C2×C4—6922no—
158C2×C4—6921no—
159C2×C4D4×D17311yescharacteristic
160C8—3461no—
161C8—6921no—
162C8D4×D17311yescharacteristic
163C8C2×C22⋊D17311yescharacteristic
164C22—3461no—
165C22—6921no—
166C22#1914970311yescharacteristic
167C4—42no—
168C4—81no—
169C4—3461no—
170C4—6921no—
171C4#1914964711yescharacteristic
172C4D8×D17311yescharacteristic
173C2—21no—
174C2—13841no—
175C2—6921no—
176C2#2601162311yescharacteristic
subgroupG/H
centerC2#26011623
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
FittingC2×Q32×C173C2
socleC2×C173#447
prime radical#26012304C2
hypercenterC2×Q32D173
hypocenterC173#2195
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q32

Complements

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

NH ≅ G/Ntype
2. #26011932C2semidirect
3. #26011763C2semidirect
4. #26011307C2semidirect
5. #26011299C2semidirect
6. #26011274C2semidirect
7. C2×Q32×C173C2semidirect
88. C173#2195semidirect
92. C2×Q32D173semidirect

Maximal subgroups

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

subgroupindex
#260123042
#260119322
#260117632
#260113072
#260112992
#260112742
C2×Q32×C1732
#2195173