Finite Group Databasebeta

← #36352845

Subgroups — #36352845

Order 22144.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012146C211yesmaximal, characteristic
2#26011887C211yesmaximal, characteristic, complemented
3#26011768C211yesmaximal, characteristic, complemented
4#26011236C211yesmaximal, characteristic, complemented
5D4⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#461×C173C211yesmaximal, characteristic, complemented
8C2×#14583826—21no—
9#19149679C2211yescharacteristic
10#19149661—22no—
11#19149658—21no—
12#19149650—21no—
13C4×D692C2211yescharacteristic
14C4⋊C4×D173—21no—
15#19149612—21no—
16#19149610—22no—
17C4⋊#13063436C2211yescharacteristic
18#19149599—21no—
19C4×#13063436C2211yescharacteristic
20C2×SD16×C173—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24#103×C173—21no—
25#14583826—42no—
26#14583826—42no—
27C2×Dic346—21no—
28C2×D692D411yescharacteristic
29C2×C22⋊D173—22no—
30C2×C4×D173—22no—
31C2×C4×D173—21no—
32C2×#13063436—21no—
33C2×#13063436—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173D412yes—
36C4⋊Dic173D411yescharacteristic
37#14583805—22no—
38C22⋊Dic173—22no—
39#14583803—21no—
40C2×Q8×C173—21no—
41C2×D4×C173D411yescharacteristic
42C22×C4×C173—21no—
43SD16×C173—42no—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51Dic346—41no—
52Dic346—41no—
53D692—21no—
54D692SD1612yescomplemented
55C22⋊D173—44no—
56C2×Dic173—22no—
57C2×Dic173—21no—
58C4×D173—44no—
59C4×D173—41no—
60C4×D173—41no—
61#13063436—22no—
62#13063436—41no—
63C23×C173—21no—
64Q8×C173—41no—
65Q8×C173—41no—
66D4×C173—21no—
67D4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173—41no—
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C8×C173—41no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173—41no—
79C22×C173#12011yescharacteristic
80C2×D173—22no—
81C2×D173—41no—
82C2×D173—41no—
83C4×C173—21no—
84C4×C173—41no—
85C4×C173—81no—
86C4×C173C2×SD1611yescharacteristic
87C4×C173#13511yescharacteristic
88C4×C173AGL(1,ℤ/8)11yescharacteristic
89C4×C173AGL(1,ℤ/8)11yescharacteristic
90Dic173—41no—
91Dic173—42no—
92C2×C173—81no—
93C2×C173#44911yescharacteristic
94C2×C173#45011yescharacteristic
95C2×C173#45611yescharacteristic
96D173—42no—
97C173#165411yescharacteristic, complemented, Sylow
98#1654—1731nomaximal, Sylow
99#547—1731no—
100#498—1731no—
101#474—1731no—
102#461D17311yescharacteristic, complemented
103#332—1731no—
104D4⋊C8—1731no—
105D4⋊C8—1731no—
106C2×#41—3461no—
107C2×SD16—21no—
108C4⋊Q8—1731no—
109C4⋊Q8C2×D17311yescharacteristic
110#123—3462no—
111C22⋊Q8—3461no—
112#121—3461no—
113C4×D4—1731no—
114C4×D4C2×D17311yescharacteristic
115C2×C4⋊C4—3461no—
116#107—3461no—
117#106—3462no—
118C4⋊C8—1731no—
119C4⋊C8C2×D17311yescharacteristic
120#103—21no—
121#102—3461no—
122C4×C8—1731no—
123#41—6922no—
124#41—6922no—
125C2×Q8—21no—
126C2×Q8—3461no—
127C2×D4—3462no—
128C2×D4—1731no—
129C2×D4C22⋊D17311yescharacteristic
130C22×C4—21no—
131C22×C4—3462no—
132C22×C4—3461no—
133SD16—42no—
134C2×C8—21no—
135C2×C8—3461no—
136C2×C8—3461no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—3462no—
140C4⋊C4—3461no—
141C4⋊C4—1732no—
142C4⋊C4—1731no—
143C4⋊C4C22⋊D17311yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—3462no—
146C22⋊C4—3461no—
147C42C22×D17311yescharacteristic
148C23—21no—
149C23—3461no—
150Q8—41no—
151Q8—41no—
152Q8—6921no—
153Q8—6921no—
154D4—21no—
155D4—21no—
156D4—3461no—
157D4—1732no—
158D4—6924no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—3461no—
164C2×C4—3462no—
165C2×C4—6924no—
166C2×C4—6921no—
167C2×C4—6921no—
168C2×C4D4×D17311yescharacteristic
169C2×C4D4×D17311yescharacteristic
170C2×C4C2×C22⋊D17311yescharacteristic
171C8—41no—
172C8—3462no—
173C8—6921no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—3462no—
178C22—6921no—
179C22—6921no—
180C22#1914964711yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—6921no—
185C4—6922no—
186C4SD16×D17311yescharacteristic
187C4#1914969911yescharacteristic
188C4#1914970511yescharacteristic
189C4#1914970611yescharacteristic
190C2—81no—
191C2—6922no—
192C2#2601163411yescharacteristic
193C2#2601164011yescharacteristic
194C2#2601167211yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#461×C173C2
socleC22×C173#120
prime radical#26012146C2
hypercenter#461D173
hypocenterC173#1654
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #26011887C2semidirect
3. #26011768C2semidirect
4. #26011236C2semidirect
5. D4⋊#13063436C2semidirect
6. #26011218C2semidirect
7. #461×C173C2semidirect
54. D692SD16semidirect
97. C173#1654semidirect
102. #461D173semidirect

Maximal subgroups

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

subgroupindex
#260121462
#260118872
#260117682
#260112362
D4⋊#130634362
#260112182
#461×C1732
#1654173