Finite Group Databasebeta

← #36352764

Subgroups — #36352764

Order 22144.

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

normal subgroups only

This group has 250 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#26012161C211yesmaximal, characteristic
2#26011837C211yesmaximal, characteristic, complemented
3#26011777C211yesmaximal, characteristic, complemented
4#26011697C211yesmaximal, characteristic, complemented
5#26011221C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#331×C173C211yesmaximal, characteristic, complemented
8C2×#14583825—21no—
9C2×#14583815—21no—
10#19149679C2211yescharacteristic
11#19149675—21no—
12#19149651—22no—
13#19149650—21no—
14C4×Dic346C2211yescharacteristic
15C4×D692C2211yescharacteristic
16D4×Dic173—21no—
17#19149611—21no—
18#19149604—21no—
19#19149604—21no—
20#19149600—22no—
21C4×#13063436C2211yescharacteristic
22#127×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25#14583825—42no—
26#14583825—42no—
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C22⋊D173—22no—
31C2×C22⋊D173—22no—
32C22×Dic173—22no—
33C2×C4×D173—21no—
34#14583815—42no—
35C2×#13063436—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173D411yescharacteristic
39C4⋊Dic173D411yescharacteristic
40#14583805—21no—
41C22⋊Dic173—22no—
42#14583803—21no—
43C4×Dic173—21no—
44C2×D4×C173—21no—
45C2×D4×C173—21no—
46C2×D4×C173D412yes—
47C2×C8×C173—21no—
48C2×C8×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51Dic346—21no—
52Dic346—41no—
53Dic346—41no—
54Dic346SD1612yescomplemented
55D692—21no—
56D692—21no—
57C22⋊D173—44no—
58C22⋊D173—82no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—44no—
63C2×Dic173—41no—
64C4×D173—41no—
65C4×D173—41no—
66#13063436—41no—
67C23×C173—22no—
68D4×C173—42no—
69D4×C173—41no—
70D4×C173—41no—
71D4×C173—41no—
72D4×C173—41no—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C8×C173—41no—
77C8×C173—41no—
78C22×C173—42no—
79C22×C173—42no—
80C22×C173—42no—
81C22×C173#12011yescharacteristic
82C2×D173—41no—
83C2×D173—41no—
84C2×D173—41no—
85C4×C173—21no—
86C4×C173C2×SD1611yescharacteristic
87C4×C173#13511yescharacteristic
88C4×C173AGL(1,ℤ/8)11yescharacteristic
89C4×C173AGL(1,ℤ/8)11yescharacteristic
90Dic173—22no—
91Dic173—41no—
92Dic173—41no—
93Dic173—81no—
94C2×C173—82no—
95C2×C173#44911yescharacteristic
96C2×C173#45011yescharacteristic
97C2×C173#45511yescharacteristic
98D173—81no—
99C173#164211yescharacteristic, complemented, Sylow
100#1642—1731nomaximal, Sylow
101#550—1731no—
102#486—1731no—
103#475—1731no—
104#461—1731no—
105#331D17311yescharacteristic, complemented
106Q8⋊C8—1731no—
107D4⋊C8—1731no—
108C2×#41—3461no—
109C2×SD16—3461no—
110C4⋊Q8—1731no—
111#127—3461no—
112#127C2×D17311yescharacteristic
113#121—3462no—
114#121—3461no—
115C4×Q8—1731no—
116C4×D4—3461no—
117C4×D4—1731no—
118#106—3461no—
119C4⋊C8C2×D17311yescharacteristic
120C4⋊C8C2×D17311yescharacteristic
121#103—3461no—
122#103—3461no—
123#102—3462no—
124C4×C8—1731no—
125#41—6922no—
126#41—6922no—
127C2×Q8—3461no—
128C2×Q8—1731no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3462no—
132C2×D4—3462no—
133C2×D4—1731no—
134C2×D4D69212yes—
135C22×C4—3462no—
136C22×C4—3461no—
137SD16—6922no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3461no—
141C4⋊C4—3461no—
142C4⋊C4—3461no—
143C4⋊C4—3461no—
144C4⋊C4—1731no—
145C4⋊C4—1731no—
146C22⋊C4—3462no—
147C22⋊C4—3461no—
148C42—3461no—
149C42C22×D17311yescharacteristic
150C23—22no—
151C23—3461no—
152Q8—3461no—
153Q8—1732no—
154Q8—6921no—
155Q8—6921no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—3461no—
162D4—3461no—
163D4—13842no—
164D4—6924no—
165C2×C4—3461no—
166C2×C4—3461no—
167C2×C4—3461no—
168C2×C4—6924no—
169C2×C4—6921no—
170C2×C4—6921no—
171C2×C4—6921no—
172C2×C4D4×D17311yescharacteristic
173C2×C4D4×D17311yescharacteristic
174C2×C4C2×D69211yescharacteristic
175C8—41no—
176C8—41no—
177C8—6921no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—6921no—
182C22—6921no—
183C22—6921no—
184C22#1914964911yescharacteristic
185C4—21no—
186C4—3462no—
187C4—13841no—
188C4—6921no—
189C4—6921no—
190C4SD16×D17311yescharacteristic
191C4#1914970011yescharacteristic
192C4#1914970311yescharacteristic
193C4#1914970311yescharacteristic
194C2—82no—
195C2—13841no—
196C2#2601163711yescharacteristic
197C2#2601164311yescharacteristic
198C2#2601167111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#331×C173C2
socleC22×C173#120
prime radical#26012161C2
hypercenter#331D173
hypocenterC173#1642
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #26011837C2semidirect
3. #26011777C2semidirect
4. #26011697C2semidirect
5. #26011221C2semidirect
6. #26011218C2semidirect
7. #331×C173C2semidirect
54. Dic346SD16semidirect
99. C173#1642semidirect
105. #331D173semidirect

Maximal subgroups

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

subgroupindex
#260121612
#260118372
#260117772
#260116972
#260112212
#260112182
#331×C1732
#1642173