Finite Group Databasebeta

← #36352762

Subgroups — #36352762

Order 22144.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#26012163C211yesmaximal, characteristic
2#26011839C211yesmaximal, characteristic, complemented
3#26011693C211yesmaximal, characteristic, complemented
4#26011234C211yesmaximal, characteristic, complemented
5Q8⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#475×C173C211yesmaximal, characteristic, complemented
8C2×#14583827—21no—
9C2×#14583813—21no—
10#19149676—21no—
11#19149676C2211yescharacteristic
12#19149654—21no—
13#19149654—22no—
14C4×D692—21no—
15C4×D692C2211yescharacteristic
16C4⋊#13063436C2211yescharacteristic
17#19149603—21no—
18#19149599—22no—
19C4×#13063436C2211yescharacteristic
20C4⋊Q8×C173C2211yescharacteristic
21C4×Q8×C173C2211yescharacteristic
22#106×C173—21no—
23C4⋊C8×C173C2211yescharacteristic
24#103×C173—21no—
25#14583827—42no—
26#14583827—42no—
27C2×D692—22no—
28C2×D692—22no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692D411yescharacteristic
32C2×D692D412yes—
33C2×C4×D173—22no—
34C2×C4×D173—21no—
35#14583813—42no—
36C2×#13063436—21no—
37C2×#13063436—21no—
38C4⋊Dic173D411yescharacteristic
39#14583803—22no—
40#14583803—21no—
41C2×Q8×C173—21no—
42C2×Q8×C173D411yescharacteristic
43C2×C8×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D411yescharacteristic
48C42×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51C22×D173—22no—
52D692—21no—
53D692—21no—
54D692—42no—
55D692—44no—
56D692—41no—
57D692—41no—
58D692—41no—
59D692—41no—
60D692—82no—
61C2×Dic173—21no—
62C4×D173—44no—
63C4×D173—41no—
64C4×D173—41no—
65#13063436—41no—
66#13063436—41no—
67Q8×C173—21no—
68Q8×C173—41no—
69Q8×C173—41no—
70Q8×C173SD1612yescomplemented
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173—41no—
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C2×C4×C173C2×D411yescharacteristic
77C8×C173—41no—
78C22×C173#12011yescharacteristic
79C2×D173—41no—
80C2×D173—42no—
81C2×D173—42no—
82C2×D173—41no—
83C2×D173—42no—
84C2×D173—41no—
85C4×C173—22no—
86C4×C173—21no—
87C4×C173—41no—
88C4×C173—81no—
89C4×C173C2×SD1611yescharacteristic
90C4×C173#13511yescharacteristic
91C4×C173AGL(1,ℤ/8)11yescharacteristic
92C4×C173AGL(1,ℤ/8)11yescharacteristic
93Dic173—41no—
94C2×C173#44911yescharacteristic
95C2×C173#45011yescharacteristic
96C2×C173#45511yescharacteristic
97D173—81no—
98D173—82no—
99C173#164211yescharacteristic, complemented, Sylow
100#1642—1731nomaximal, Sylow
101#550—1731no—
102#486—1731no—
103#475D17311yescharacteristic, complemented
104#461—1731no—
105#331—1731no—
106Q8⋊C8—1731no—
107D4⋊C8—1731no—
108C2×#41—3461no—
109C2×SD16—3461no—
110C4⋊Q8C2×D17311yescharacteristic
111#127—3461no—
112#127—1731no—
113#121—3462no—
114#121—3461no—
115C4×Q8C2×D17311yescharacteristic
116C4×D4—3461no—
117C4×D4—1731no—
118#106—21no—
119C4⋊C8—1731no—
120C4⋊C8C2×D17311yescharacteristic
121#103—21no—
122#103—3461no—
123#102—3462no—
124C4×C8—1731no—
125#41—6922no—
126#41—6922no—
127C2×Q8—21no—
128C2×Q8C22⋊D17311yescharacteristic
129C2×D4—3462no—
130C2×D4—3462no—
131C2×D4—3461no—
132C2×D4—3461no—
133C2×D4—1731no—
134C2×D4—1732no—
135C22×C4—3462no—
136C22×C4—3461no—
137SD16—6922no—
138C2×C8—21no—
139C2×C8—3461no—
140C2×C8—3461no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1731no—
145C4⋊C4C22⋊D17311yescharacteristic
146C22⋊C4—3462no—
147C22⋊C4—3461no—
148C42—21no—
149C42C22×D17311yescharacteristic
150C23—3461no—
151C23—3462no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1458381312yescomplemented
156D4—3461no—
157D4—3461no—
158D4—13842no—
159D4—6922no—
160D4—6924no—
161D4—6921no—
162D4—6921no—
163D4—6921no—
164D4—6921no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—3461no—
169C2×C4—6924no—
170C2×C4—6921no—
171C2×C4—6921no—
172C2×C4D4×D17311yescharacteristic
173C2×C4D4×D17311yescharacteristic
174C2×C4C2×C22⋊D17311yescharacteristic
175C8—41no—
176C8—6921no—
177C8—6921no—
178C22—6921no—
179C22—6922no—
180C22—6921no—
181C22—6922no—
182C22—6922no—
183C22—6921no—
184C22#1914964711yescharacteristic
185C4—21no—
186C4—22no—
187C4—41no—
188C4—81no—
189C4—6921no—
190C4C2×#1458381311yescharacteristic
191C4#1914970011yescharacteristic
192C4#1914970411yescharacteristic
193C4#1914970611yescharacteristic
194C2—13841no—
195C2—13842no—
196C2#2601163511yescharacteristic
197C2#2601164111yescharacteristic
198C2#2601166711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#475×C173C2
socleC22×C173#120
prime radical#26012163C2
hypercenter#475D173
hypocenterC173#1642
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #26011839C2semidirect
3. #26011693C2semidirect
4. #26011234C2semidirect
5. Q8⋊#13063436C2semidirect
6. #26011218C2semidirect
7. #475×C173C2semidirect
70. Q8×C173SD16semidirect
99. C173#1642semidirect
103. #475D173semidirect
155. Q8#14583813semidirect

Maximal subgroups

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

subgroupindex
#260121632
#260118392
#260116932
#260112342
Q8⋊#130634362
#260112182
#475×C1732
#1642173