Finite Group Databasebeta

← #36355526

Subgroups — #36355526

Order 22144.

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

normal subgroups only

This group has 210 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, complemented
2#26011927C211yesmaximal, characteristic, complemented
3#26011758C211yesmaximal, characteristic, complemented
4#26011307C211yesmaximal, characteristic, complemented
5#26011299C211yesmaximal, characteristic, complemented
6#26011274C211yesmaximal, characteristic, complemented
7C2×SD32×C173C211yesmaximal, characteristic, complemented
8#19149727—21no—
9#19149711—22no—
10#19149710—21no—
11#19149705—21no—
12#19149702—21no—
13#19149701C2211yescharacteristic
14Q16×D173—21no—
15#19149685C2211yescharacteristic
16#19149623—21no—
17#19149620—21no—
18#19149619C2211yescharacteristic
19#19149615C2211yescharacteristic
20#19149596—21no—
21C2×Q16×C173C2211yescharacteristic
22C2×D8×C173C2211yescharacteristic
23SD32×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#14583815—21no—
35#14583814—41no—
36#14583813—22no—
37D1384D411yescharacteristic
38#14583811—41no—
39#14583810—21no—
40#14583810D411yescharacteristic
41#14583809—21no—
42C8×D173—21no—
43#14583801—21no—
44C2×Q8×C173—21no—
45C2×D4×C173—21no—
46Q16×C173—21no—
47Q16×C173D412yes—
48D8×C173—21no—
49D8×C173—21no—
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—
66C23×C173—41no—
67Q8×C173—22no—
68Q8×C173—41no—
69D4×C173—41no—
70D4×C173—41no—
71C2×C4×C173—41no—
72C2×C4×C173C2×D411yescharacteristic
73C8×C173C2×D411yescharacteristic
74C8×C173C2×D411yescharacteristic
75C22×C173—81no—
76C22×C173—161no—
77C22×C173AGL(1,ℤ/8)11yescharacteristic
78C2×D173—21no—
79C2×D173—41no—
80C4×C173—42no—
81C4×C173#12011yescharacteristic
82C4×C173C2×D811yescharacteristic
83Dic173—21no—
84Dic173—41no—
85C2×C173—21no—
86C2×C173—161no—
87C2×C173#44711yescharacteristic
88D173—41no—
89D173—81no—
90C173#219311yescharacteristic, complemented, Sylow
91#2193—1731nomaximal, Sylow
92#578—1731no—
93#509—1731no—
94C2×SD32D17311yescharacteristic, complemented
95#472—1731no—
96#362—1731no—
97#359—1731no—
98#350—1731no—
992−1+4—3461no—
100#137—3462no—
101#137—3461no—
102AGL(1,ℤ/8)—3461no—
103#135—3461no—
104#135—1731no—
105C2×Q16—3461no—
106C2×Q16C2×D17311yescharacteristic
107C2×D8C2×D17311yescharacteristic
108#131—1731no—
109SD32—22no—
110SD32—3461no—
111D16—3461no—
112M32—1731no—
113C2×C16C2×D17311yescharacteristic
114#108—1731no—
115#100—3461no—
116#41—3462no—
117#41—3461no—
118#41—1731no—
119#41—6922no—
120#41—6921no—
121C2×Q8—21no—
122C2×Q8—3462no—
123C2×Q8—6921no—
124C2×D4—21no—
125Q16—21no—
126Q16—3462no—
127Q16—1731no—
128Q16C22⋊D17312yes—
129SD16—3462no—
130SD16—3461no—
131SD16—6921no—
132D8—21no—
133D8—21no—
134D8—1731no—
135D8—6921no—
136M16—3461no—
137M16—3461no—
138M16—1731no—
139C2×C8—3461no—
140C2×C8C22×D17311yescharacteristic
141C16—21no—
142C16—3461no—
143C23—41no—
144Q8—22no—
145Q8—41no—
146Q8—3461no—
147Q8—1731no—
148Q8—6922no—
149D4—41no—
150D4—41no—
151D4—3461no—
152D4—3461no—
153D4—1731no—
154D4—6922no—
155D4—6921no—
156C2×C4—41no—
157C2×C4—3461no—
158C2×C4—6922no—
159C2×C4—6922no—
160C2×C4—6921no—
161C2×C4D4×D17311yescharacteristic
162C8—3461no—
163C8—6921no—
164C8D4×D17311yescharacteristic
165C8C2×C22⋊D17311yescharacteristic
166C22—81no—
167C22—161no—
168C22—3461no—
169C22—6921no—
170C22#1914970311yescharacteristic
171C4—42no—
172C4—3461no—
173C4—6921no—
174C4#1914964711yescharacteristic
175C4D8×D17311yescharacteristic
176C2—21no—
177C2—161no—
178C2—13841no—
179C2—6921no—
180C2#2601162311yescharacteristic
subgroupG/H
centerC2#26011623
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
FittingC2×SD32×C173C2
socleC2×C173#447
hypercenterC2×SD32D173
hypocenterC173#2193
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #26012304C2semidirect
2. #26011927C2semidirect
3. #26011758C2semidirect
4. #26011307C2semidirect
5. #26011299C2semidirect
6. #26011274C2semidirect
7. C2×SD32×C173C2semidirect
90. C173#2193semidirect
94. C2×SD32D173semidirect

Maximal subgroups

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

subgroupindex
#260123042
#260119272
#260117582
#260113072
#260112992
#260112742
C2×SD32×C1732
#2193173