Finite Group Databasebeta

← #36355254

Subgroups — #36355254

Order 22144.

262 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 65 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#26012055C211yesmaximal, characteristic
2#26011475C211yesmaximal, characteristic, complemented
3#26011287C214yesmaximal, complemented
4#413×C173C211yesmaximal, characteristic, complemented
5C22×D692C411yescharacteristic
6C2×#14583810—21no—
7#19149675C412yes—
8#19149668C2212yes—
9#19149653—21no—
10#19149653—21no—
11#19149647—21no—
12#19149647—21no—
13#19149638C411yescharacteristic
14#19149598C2212yescomplemented
15#19149596—21no—
16C2×#41×C173C2211yescharacteristic
17C2×M16×C173—21no—
18#101×C173C2212yescomplemented
19#100×C173—21no—
20C23×D173—21no—
21C2×D692—21no—
22C2×D692—41no—
23C2×D692—41no—
24C2×D692C2×C412yes—
25C2×C22⋊D173—22no—
26C2×C22⋊D173—22no—
27#14583810—22no—
28#14583810—41no—
29C2×#13063436—41no—
30C4⋊Dic173—21no—
31C22⋊Dic173—21no—
32#14583803—22no—
33#14583803—22no—
34C4×Dic173C2×C412yes—
35#41×C173—41no—
36#41×C173—41no—
37C2×Q8×C173C2311yescharacteristic
38C2×D4×C173C2×C411yescharacteristic
39C2×D4×C173D411yescharacteristic
40C2×D4×C173D411yescharacteristic
41C22×C4×C173C2×C411yescharacteristic
42C22×C4×C173D411yescharacteristic
43C22×C4×C173D411yescharacteristic
44M16×C173—22no—
45M16×C173—41no—
46C2×C8×C173—41no—
47C22×D173—22no—
48C22×D173—42no—
49C22×D173—82no—
50D692—42no—
51D692—41no—
52D692—41no—
53C22⋊D173—82no—
54C22⋊D173—82no—
55C2×Dic173—22no—
56#13063436—42no—
57C23×C173C22⋊C411yescharacteristic
58C23×C173C22⋊C411yescharacteristic
59C23×C173C22⋊C411yescharacteristic
60Q8×C173—41no—
61D4×C173—41no—
62D4×C173—41no—
63D4×C173—41no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C22×C411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C8×C173—42no—
75C22×C173—21no—
76C22×C173—21no—
77C22×C173—21no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173—41no—
81C22×C173C2×C22⋊C411yescharacteristic
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—84no—
86C4×C173—21no—
87C4×C173—21no—
88C4×C173—21no—
89C4×C173#9911yescharacteristic
90C4×C173#9911yescharacteristic
91Dic173—42no—
92C2×C173—21no—
93C2×C173—41no—
94C2×C173—41no—
95C2×C173—41no—
96C2×C173C2×#9911yescharacteristic
97D173—82no—
98C173#213411yescharacteristic, complemented, Sylow
99#2134—1731nomaximal, Sylow
100#535—1731no—
101#413—1731no—
102#413D17311yescharacteristic, complemented
103#355—1734no—
104C2×#41C2×D17311yescharacteristic
105C22×D4—1731no—
106C2×M16—21no—
107C2×M16—3461no—
108#127—1732no—
109#124—1732no—
110#121—3461no—
111#121—3461no—
112#120—3461no—
113#120—3461no—
114#117—1731no—
115#101—1732no—
116#101C2×D17312yescomplemented
117#100—21no—
118#100—3461no—
119C24—3461no—
120#41—41no—
121#41—41no—
122C2×Q8C22×D17311yescharacteristic
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—3461no—
126C2×D4—1732no—
127C2×D4—6921no—
128C2×D4—6921no—
129C2×D4C4×D17311yescharacteristic
130C2×D4C22⋊D17311yescharacteristic
131C2×D4D69211yescharacteristic
132C22×C4C4×D17311yescharacteristic
133C22×C4C22⋊D17311yescharacteristic
134C22×C4D69211yescharacteristic
135M16—22no—
136M16—41no—
137M16—3462no—
138M16—6921no—
139C2×C8—41no—
140C2×C8—6921no—
141C4⋊C4—3461no—
142C22⋊C4—3461no—
143C22⋊C4—3462no—
144C22⋊C4—3462no—
145C42—1732no—
146C23—3462no—
147C23—13842no—
148C23—6922no—
149C23#1458380311yescharacteristic
150C23#1458380311yescharacteristic
151C23#1458380311yescharacteristic
152Q8—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—13842no—
157D4—13842no—
158D4—6922no—
159D4—6921no—
160D4—6921no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—3462no—
168C2×C4#1458380311yescharacteristic
169C2×C4C2×C4×D17311yescharacteristic
170C2×C4C2×C22⋊D17311yescharacteristic
171C2×C4C2×D69211yescharacteristic
172C8—42no—
173C8—6922no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—13844no—
181C22—6922no—
182C22—6922no—
183C22—6922no—
184C22C2×#1458380311yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—6922no—
189C4#1914959311yescharacteristic
190C4#1914959311yescharacteristic
191C2—21no—
192C2—41no—
193C2—41no—
194C2—41no—
195C2—13842no—
196C2C2×#1914959311yescharacteristic
subgroupG/H
centerC2C2×#19149593
commutatorC2×C4×C173C22×C4
FrattiniC2×Q8C22×D173
Fitting#413×C173C2
socleC2×C173C2×#99
prime radical#26012055C2
cosocleC2×Q8×C173C23
hypercenter#413D173
hypocenterC173#2134
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #26011475C2semidirect
3. #26011287C2semidirect
4. #413×C173C2semidirect
14. #19149598C22semidirect
18. #101×C173C22semidirect
98. C173#2134semidirect
102. #413D173semidirect
116. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120552
#260114752
#260112872
#413×C1732
#2134173