Finite Group Databasebeta

← #36352255

Subgroups — #36352255

Order 22144.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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#26012039C211yesmaximal, characteristic
2#26011547C211yesmaximal, characteristic, complemented
3C4⋊C8×D173C211yesmaximal, characteristic, complemented
4#26011239C211yesmaximal, characteristic, complemented
5#26011239C211yesmaximal, characteristic, complemented
6#26011235C211yesmaximal, characteristic, complemented
7#331×C173C211yesmaximal, characteristic, complemented
8C2×#14583825C412yes—
9C2×C8×D173—21no—
10#19149679C2211yescharacteristic
11#19149666C412yes—
12#19149650—21no—
13#19149650—21no—
14D4×Dic173—21no—
15D4×Dic173—21no—
16C42×D173C2211yescharacteristic
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149608—21no—
20#19149591—21no—
21#19149589—21no—
22#127×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25#14583825—44no—
26C2×Dic346C2×C412yes—
27C2×C22⋊D173—22no—
28C22×Dic173—22no—
29C2×C4×D173C2×C411yescharacteristic
30C2×C4×D173D411yescharacteristic
31C2×C4×D173D411yescharacteristic
32C2×#13063436—21no—
33C2×#13063436—21no—
34C8×D173—42no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37C22⋊Dic173—22no—
38C22⋊Dic173—22no—
39C4×Dic173C2×C411yescharacteristic
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C2×D4×C173—21no—
43C2×D4×C173—21no—
44C2×D4×C173C2×C412yes—
45C2×C8×C173—21no—
46C2×C8×C173—21no—
47C42×C173C2311yescharacteristic
48C22×D173C22⋊C411yescharacteristic
49Dic346—42no—
50C22⋊D173—44no—
51C2×Dic173—22no—
52C2×Dic173—44no—
53C2×Dic173C22⋊C411yescharacteristic
54C2×Dic173C22⋊C411yescharacteristic
55C2×Dic173C22⋊C411yescharacteristic
56C4×D173—22no—
57C4×D173—21no—
58C4×D173—21no—
59C4×D173—21no—
60C4×D173—21no—
61#13063436—41no—
62#13063436—41no—
63C23×C173—22no—
64D4×C173—42no—
65D4×C173—41no—
66D4×C173—41no—
67D4×C173—41no—
68D4×C173—41no—
69C2×C4×C173C22×C411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C8×C173—41no—
73C8×C173—41no—
74C22×C173—42no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173C2×C22⋊C411yescharacteristic
78C2×D173—21no—
79C2×D173—21no—
80C2×D173#10012yes—
81C4×C173—21no—
82C4×C173#13511yescharacteristic
83C4×C173#13511yescharacteristic
84C4×C173AGL(1,ℤ/8)11yescharacteristic
85C4×C173#13711yescharacteristic
86Dic173—21no—
87Dic173—21no—
88Dic173—21no—
89Dic173—42no—
90C2×C173—82no—
91C2×C173C2×#10011yescharacteristic
92C2×C173#41611yescharacteristic
93C2×C173#41711yescharacteristic
94D173—22no—
95C173#155111yescharacteristic, complemented, Sylow
96#1551—1731nomaximal, Sylow
97#532—1731no—
98#432—1731no—
99C2×C4⋊C8—1731no—
100#333—1731no—
101#333—1731no—
102#331—1731no—
103#331D17311yescharacteristic, complemented
104C2×#41—1732no—
105C22×C8—3461no—
106C4⋊Q8—1731no—
107#127C2×D17311yescharacteristic
108#124—1732no—
109#121—3461no—
110#121—3461no—
111C4×D4—3461no—
112C4×D4—3461no—
113C2×C42—1731no—
114C4⋊C8—3461no—
115C4⋊C8—1731no—
116C4⋊C8—1731no—
117C4⋊C8C2×D17311yescharacteristic
118C4⋊C8C2×D17311yescharacteristic
119C22⋊C8—3461no—
120#97—3461no—
121#41—6924no—
122C2×Q8—1732no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—3462no—
126C2×D4C4×D17312yes—
127C22×C4—3462no—
128C22×C4—1731no—
129C22×C4—1731no—
130C22×C4—1731no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—3461no—
134C2×C8—3461no—
135C2×C8—6922no—
136C4⋊C4—3461no—
137C4⋊C4—3461no—
138C22⋊C4—3462no—
139C22⋊C4—3462no—
140C42—1731no—
141C42—1731no—
142C42—1731no—
143C42C22×D17311yescharacteristic
144C23—22no—
145C23—1731no—
146Q8—6922no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—6924no—
153C2×C4—3462no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4—3461no—
158C2×C4—3462no—
159C2×C4—1731no—
160C2×C4—1731no—
161C2×C4—1731no—
162C2×C4—6924no—
163C2×C4C2×C4×D17311yescharacteristic
164C2×C4D4×D17311yescharacteristic
165C2×C4D4×D17311yescharacteristic
166C8—41no—
167C8—41no—
168C8—6921no—
169C8—6921no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—3461no—
174C22—3461no—
175C22—1732no—
176C22C22⋊C4×D17311yescharacteristic
177C4—21no—
178C4—3461no—
179C4—3461no—
180C4—3461no—
181C4—6922no—
182C4#1914969811yescharacteristic
183C4#1914970011yescharacteristic
184C4#1914970311yescharacteristic
185C4#1914970811yescharacteristic
186C2—82no—
187C2—3462no—
188C2#100×D17311yescharacteristic
189C2#2601148411yescharacteristic
190C2#2601148911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC42C22×D173
Fitting#331×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012039C2
cosocleC42×C173C23
hypercenter#331D173
hypocenterC173#1551
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #26011547C2semidirect
3. C4⋊C8×D173C2semidirect
4. #26011239C2semidirect
5. #26011239C2semidirect
6. #26011235C2semidirect
7. #331×C173C2semidirect
95. C173#1551semidirect
103. #331D173semidirect

Maximal subgroups

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

subgroupindex
#260120392
#260115472
C4⋊C8×D1732
#260112392
#260112352
#331×C1732
#1551173