Finite Group Databasebeta

← #36352267

Subgroups — #36352267

Order 22144.

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

normal subgroups only

This group has 242 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#26012042C211yesmaximal, characteristic
2#26011547C211yesmaximal, characteristic, complemented
3#26011543C211yesmaximal, characteristic, complemented
4#26011237C211yesmaximal, characteristic, complemented
5#26011234C211yesmaximal, characteristic, complemented
6#26011234C211yesmaximal, characteristic, complemented
7#332×C173C211yesmaximal, characteristic, complemented
8C2×#14583827—21no—
9#19149676C2211yescharacteristic
10#19149668—21no—
11#19149654—21no—
12#19149654C412yes—
13#19149641—21no—
14#19149641C412yes—
15C42×D173C2211yescharacteristic
16C4⋊#13063436C2211yescharacteristic
17C4⋊#13063436C2211yescharacteristic
18#19149591—21no—
19#19149591—21no—
20#19149589—21no—
21C8×Dic173—21no—
22C4⋊Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25#14583827—42no—
26#14583827—42no—
27C2×D692—22no—
28C2×D692—21no—
29C2×D692—21no—
30C2×D692C2×C412yes—
31C2×C4×D173—22no—
32C2×C4×D173C2×C411yescharacteristic
33C2×C4×D173D411yescharacteristic
34C2×C4×D173D411yescharacteristic
35C2×#13063436—21no—
36C2×#13063436—21no—
37#14583803—22no—
38#14583803—22no—
39C4×Dic173C2×C411yescharacteristic
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C2×Q8×C173—21no—
43C2×C8×C173—21no—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173C2×C412yes—
47C42×C173C2311yescharacteristic
48C22×D173—22no—
49C22×D173C22⋊C411yescharacteristic
50D692—42no—
51D692—44no—
52D692—41no—
53D692—41no—
54D692—41no—
55D692—41no—
56C2×Dic173C22⋊C411yescharacteristic
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C22⋊C411yescharacteristic
59C4×D173—21no—
60C4×D173—21no—
61C4×D173—21no—
62C4×D173—21no—
63C4×D173—41no—
64C4×D173—44no—
65#13063436—41no—
66#13063436—41no—
67Q8×C173—41no—
68Q8×C173—41no—
69C2×C4×C173—22no—
70C2×C4×C173C22×C411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C8×C173—41no—
74C8×C173—41no—
75C22×C173C2×C22⋊C411yescharacteristic
76C2×D173—21no—
77C2×D173—21no—
78C2×D173—21no—
79C2×D173—42no—
80C2×D173—42no—
81C2×D173—42no—
82C4×C173—21no—
83C4×C173—42no—
84C4×C173#13511yescharacteristic
85C4×C173#13511yescharacteristic
86C4×C173AGL(1,ℤ/8)11yescharacteristic
87C4×C173AGL(1,ℤ/8)11yescharacteristic
88Dic173—21no—
89Dic173—21no—
90Dic173—21no—
91C2×C173#41311yescharacteristic
92C2×C173#41611yescharacteristic
93C2×C173#41811yescharacteristic
94D173—41no—
95D173—82no—
96C173#155311yescharacteristic, complemented, Sylow
97#1553—1731nomaximal, Sylow
98#532—1731no—
99#432—1731no—
100#431—1731no—
101#332—1731no—
102#332D17311yescharacteristic, complemented
103#331—1731no—
104#331—1731no—
105C2×#41—3461no—
106C4⋊Q8C2×D17311yescharacteristic
107#127—1731no—
108#124—3461no—
109#121—3461no—
110#121—1732no—
111C4×D4—3461no—
112C4×D4—1732no—
113C2×C42—1731no—
114C4⋊C8—1731no—
115C4⋊C8—1731no—
116C4⋊C8C2×D17311yescharacteristic
117C4⋊C8C2×D17311yescharacteristic
118C22⋊C8—3461no—
119C22⋊C8—3461no—
120#97—3461no—
121C4×C8—3461no—
122#41—6922no—
123#41—6922no—
124C2×Q8—21no—
125C2×D4—3462no—
126C2×D4—3461no—
127C2×D4—3461no—
128C2×D4—1732no—
129C22×C4—3462no—
130C22×C4—1731no—
131C22×C4—1731no—
132C22×C4—1731no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3461no—
136C2×C8—3461no—
137C4⋊C4—21no—
138C4⋊C4C4×D17312yes—
139C22⋊C4—3462no—
140C22⋊C4—3462no—
141C42—1731no—
142C42—1731no—
143C42—1731no—
144C42C22×D17311yescharacteristic
145C23—3462no—
146C23—1731no—
147Q8—41no—
148Q8—41no—
149D4—6922no—
150D4—6924no—
151D4—6921no—
152D4—6921no—
153D4—6921no—
154D4—6921no—
155C2×C4—22no—
156C2×C4—3461no—
157C2×C4—3461no—
158C2×C4—3461no—
159C2×C4—3461no—
160C2×C4—1731no—
161C2×C4—1731no—
162C2×C4—1731no—
163C2×C4—6921no—
164C2×C4—6924no—
165C2×C4C2×C4×D17311yescharacteristic
166C2×C4D4×D17311yescharacteristic
167C2×C4D4×D17311yescharacteristic
168C8—41no—
169C8—41no—
170C8—6921no—
171C8—6921no—
172C22—3461no—
173C22—3461no—
174C22—3461no—
175C22—6922no—
176C22—6922no—
177C22—6922no—
178C22C22⋊C4×D17311yescharacteristic
179C4—21no—
180C4—42no—
181C4—3461no—
182C4—3461no—
183C4—3461no—
184C4#1914970011yescharacteristic
185C4#1914970211yescharacteristic
186C4#1914970311yescharacteristic
187C4#1914970611yescharacteristic
188C2—13842no—
189C2—6921no—
190C2#2601147411yescharacteristic
191C2#2601148611yescharacteristic
192C2#2601149411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC42C22×D173
Fitting#332×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012042C2
cosocleC42×C173C23
hypercenter#332D173
hypocenterC173#1553
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #26011547C2semidirect
3. #26011543C2semidirect
4. #26011237C2semidirect
5. #26011234C2semidirect
6. #26011234C2semidirect
7. #332×C173C2semidirect
96. C173#1553semidirect
102. #332D173semidirect

Maximal subgroups

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

subgroupindex
#260120422
#260115472
#260115432
#260112372
#260112342
#332×C1732
#1553173