Finite Group Databasebeta

← #36352844

Subgroups — #36352844

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#26012144C211yesmaximal, characteristic
2#26011768C211yesmaximal, characteristic, complemented
3#26011693C211yesmaximal, characteristic, complemented
4#26011237C211yesmaximal, characteristic, complemented
5#26011220C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic, complemented
7#498×C173C211yesmaximal, characteristic, complemented
8C2×#14583827—21no—
9C2×#14583813—21no—
10#19149664—22no—
11#19149655—21no—
12#19149654—21no—
13C4×D692C2211yescharacteristic
14C4×D692C2211yescharacteristic
15C4⋊C4×D173—21no—
16#19149612—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149603—21no—
20#19149599—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#106×C173—22no—
24C4×C8×C173C2211yescharacteristic
25#14583827—42no—
26#14583827—42no—
27C2×D692—22no—
28C2×D692D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C4×D173—22no—
31C2×C4×D173—21no—
32C2×C4×D173—21no—
33#14583813—42no—
34C2×#13063436—21no—
35C2×#13063436—21no—
36C4⋊Dic173D411yescharacteristic
37C4⋊Dic173D411yescharacteristic
38#14583805—22no—
39#14583803—22no—
40#14583803—21no—
41#14583803—21no—
42C2×Q8×C173—21no—
43C2×Q8×C173—21no—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173—21no—
48C4⋊C4×C173D412yes—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51C22×D173—21no—
52D692—21no—
53D692—21no—
54D692—21no—
55D692—44no—
56D692SD1612yescomplemented
57C2×Dic173—21no—
58C2×Dic173—21no—
59C4×D173—44no—
60C4×D173—41no—
61C4×D173—41no—
62C4×D173—41no—
63C4×D173—41no—
64#13063436—41no—
65#13063436—41no—
66Q8×C173—41no—
67Q8×C173—41no—
68Q8×C173—41no—
69Q8×C173—41no—
70C2×C4×C173—22no—
71C2×C4×C173—41no—
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C8×C173—22no—
76C22×C173#12011yescharacteristic
77C2×D173—22no—
78C2×D173—41no—
79C2×D173—41no—
80C2×D173—41no—
81C2×D173—41no—
82C2×D173—41no—
83C4×C173—21no—
84C4×C173—42no—
85C4×C173—81no—
86C4×C173C2×SD1611yescharacteristic
87C4×C173#13511yescharacteristic
88C4×C173AGL(1,ℤ/8)11yescharacteristic
89C4×C173AGL(1,ℤ/8)11yescharacteristic
90Dic173—41no—
91Dic173—41no—
92C2×C173#44911yescharacteristic
93C2×C173#45011yescharacteristic
94C2×C173#45611yescharacteristic
95D173—42no—
96D173—81no—
97C173#165411yescharacteristic, complemented, Sylow
98#1654—1731nomaximal, Sylow
99#547—1731no—
100#498D17311yescharacteristic, complemented
101#474—1731no—
102#461—1731no—
103#332—1731no—
104D4⋊C8—1731no—
105D4⋊C8—1731no—
106C2×#41—3461no—
107C2×SD16—3461no—
108C4⋊Q8C2×D17311yescharacteristic
109C4⋊Q8C2×D17311yescharacteristic
110#123—3462no—
111C22⋊Q8—3461no—
112#121—3461no—
113C4×D4—1731no—
114C4×D4—1731no—
115C2×C4⋊C4—3461no—
116#107—3461no—
117#106—22no—
118C4⋊C8—1731no—
119C4⋊C8—1731no—
120#103—3461no—
121#102—3461no—
122C4×C8C2×D17311yescharacteristic
123#41—6922no—
124#41—6922no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—3462no—
128C2×D4—1731no—
129C2×D4—1731no—
130C22×C4—3462no—
131C22×C4—3461no—
132C22×C4—3461no—
133SD16—6922no—
134C2×C8—21no—
135C2×C8—3461no—
136C2×C8—3461no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—3462no—
141C4⋊C4—1731no—
142C4⋊C4—1731no—
143C4⋊C4C22⋊D17312yes—
144C22⋊C4—3462no—
145C22⋊C4—3461no—
146C22⋊C4—3461no—
147C42C22×D17311yescharacteristic
148C23—3461no—
149C23—3461no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—3461no—
155D4—3461no—
156D4—3461no—
157D4—1732no—
158D4—6924no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—3461no—
162C2×C4—3461no—
163C2×C4—6924no—
164C2×C4—6921no—
165C2×C4—6921no—
166C2×C4—6921no—
167C2×C4—6921no—
168C2×C4D4×D17311yescharacteristic
169C2×C4D4×D17311yescharacteristic
170C2×C4C2×C22⋊D17311yescharacteristic
171C8—22no—
172C8—6921no—
173C8—6921no—
174C22—3462no—
175C22—6921no—
176C22—6921no—
177C22—6921no—
178C22—6921no—
179C22—6921no—
180C22#1914964711yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—6921no—
185C4—6921no—
186C4SD16×D17311yescharacteristic
187C4#1914970011yescharacteristic
188C4#1914970611yescharacteristic
189C4#1914970611yescharacteristic
190C2—13841no—
191C2—6922no—
192C2#2601163511yescharacteristic
193C2#2601164011yescharacteristic
194C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#498×C173C2
socleC22×C173#120
prime radical#26012144C2
hypercenter#498D173
hypocenterC173#1654
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #26011768C2semidirect
3. #26011693C2semidirect
4. #26011237C2semidirect
5. #26011220C2semidirect
6. #26011220C2semidirect
7. #498×C173C2semidirect
56. D692SD16semidirect
97. C173#1654semidirect
100. #498D173semidirect

Maximal subgroups

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

subgroupindex
#260121442
#260117682
#260116932
#260112372
#260112202
#498×C1732
#1654173