Finite Group Databasebeta

← #36352085

Subgroups — #36352085

Order 22144.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#26012022C211yesmaximal, characteristic
2#26011545C211yesmaximal, characteristic, complemented
3#26011342C211yesmaximal, characteristic, complemented
4#26011229C211yesmaximal, characteristic, complemented
5#26011227C211yesmaximal, characteristic, complemented
6#26011225C211yesmaximal, characteristic
7#327×C173C211yesmaximal, characteristic
8#19149672—21no—
9#19149670C411yescharacteristic, complemented
10#19149666C411yescharacteristic, complemented
11#19149660—21no—
12#19149658C2211yescharacteristic
13#19149644C411yescharacteristic, complemented
14D4×Dic173—21no—
15D4×Dic173C411yescharacteristic, complemented
16#19149635—21no—
17C2×#14583805—21no—
18C2×C4×Dic173—21no—
19C2×C4×Dic173C2211yescharacteristic
20#19149608—21no—
21C22⋊#13063436C2211yescharacteristic
22#19149589—21no—
23#19149586C2211yescharacteristic
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C2×Dic346C2×C411yescharacteristic
28C22×Dic173—21no—
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173C2×C411yescharacteristic
35#14583805—24no—
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39C22⋊Dic173—21no—
40C22⋊Dic173—21no—
41C22⋊Dic173—21no—
42C22⋊Dic173—21no—
43C4×Dic173—21no—
44C4×Dic173—42no—
45C4×Dic173C2×C411yescharacteristic
46C4×Dic173C2×C411yescharacteristic
47C4×Dic173D411yescharacteristic
48C4×Dic173D411yescharacteristic
49C2×D4×C173—21no—
50C2×D4×C173C2×C411yescharacteristic
51C22×C4×C173—21no—
52C22×C4×C173C2311yescharacteristic
53C2×C8×C173—21no—
54C4⋊C4×C173C2×C411yescharacteristic
55C22⋊C4×C173—21no—
56Dic346—41no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173—22no—
60C2×Dic173—22no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—41no—
64C2×Dic173—41no—
65C2×Dic173—42no—
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C22⋊C411yescharacteristic
68C2×Dic173C22⋊C411yescharacteristic
69C2×Dic173C22⋊C411yescharacteristic
70#13063436—41no—
71C23×C173—21no—
72C23×C173C2×D411yescharacteristic
73D4×C173—41no—
74D4×C173—41no—
75D4×C173—41no—
76C2×C4×C173—22no—
77C2×C4×C173—21no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173C22×C411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C8×C173—41no—
84C22×C173—21no—
85C22×C173—21no—
86C22×C173—41no—
87C22×C173—41no—
88C22×C173—41no—
89C22×C173C2×C22⋊C411yescharacteristic
90C22×C173AGL(1,ℤ/8)11yescharacteristic
91C22×C173#13711yescharacteristic
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—42no—
95C4×C173—41no—
96Dic173—21no—
97Dic173—21no—
98Dic173—21no—
99Dic173—42no—
100Dic173—41no—
101Dic173—41no—
102Dic173#10412yes—
103C2×C173—21no—
104C2×C173—21no—
105C2×C173—81no—
106C2×C173#41011yescharacteristic
107C2×C173#41711yescharacteristic
108C2×C173C2×#10411yescharacteristic
109C173#150911yescharacteristic, complemented, Sylow
110#1509—1731nomaximal, Sylow
111#529—1731no—
112#432—1731no—
113C4×C4⋊C4—1731no—
114#328—1731no—
115#328—1731no—
116#327—1731no—
117#327D17311yescharacteristic
118#126—3461no—
119#125—1731no—
120#124—1731no—
121#123—3461no—
122C22⋊Q8—1731no—
123#121C2×D17311yescharacteristic
124C4×Q8—1731no—
125C4×D4—3461no—
126C4×D4—1731no—
127#117—3461no—
128C2×C4⋊C4—3461no—
129C2×C42—3461no—
130C2×C42—1731no—
131C4⋊C8—3461no—
132C22⋊C8—1731no—
133C22⋊C8C2×D17311yescharacteristic
134#97—3461no—
135#95—1731no—
136#95C2×D17311yescharacteristic
137C2×Q8—1731no—
138C2×D4—21no—
139C2×D4C4×D17311yescharacteristic
140C22×C4—21no—
141C22×C4—3461no—
142C22×C4—3461no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4C22×D17311yescharacteristic
146C2×C8—21no—
147C2×C8—3461no—
148C4⋊C4—3464no—
149C4⋊C4—3461no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—1731no—
154C4⋊C4C4×D17311yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3461no—
157C22⋊C4—3461no—
158C22⋊C4—3461no—
159C22⋊C4—3461no—
160C42—3461no—
161C42—1731no—
162C42—1731no—
163C42—1731no—
164C42—1731no—
165C42—6922no—
166C23—21no—
167C23D4×D17311yescharacteristic
168Q8—6921no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—3461no—
178C2×C4—3461no—
179C2×C4—3462no—
180C2×C4—3462no—
181C2×C4—3461no—
182C2×C4—3461no—
183C2×C4—1731no—
184C2×C4—1731no—
185C2×C4—1731no—
186C2×C4—1731no—
187C2×C4—6922no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4C2×C4×D17311yescharacteristic
191C2×C4D4×D17311yescharacteristic
192C8—41no—
193C8—6921no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D17311yescharacteristic
200C22#1914970311yescharacteristic
201C22#1914970811yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—3461no—
207C4—3461no—
208C4—3461no—
209C4—1732no—
210C4—6922no—
211C4—6921no—
212C4—6921no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2601146311yescharacteristic
217C2#2601148911yescharacteristic
218C2#104×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173C2×C22⋊C4
prime radical#121×C173C22
cosocleC22×C4×C173C23
hypercenter#327D173
hypocenterC173#1509
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26011545C2semidirect
3. #26011342C2semidirect
4. #26011229C2semidirect
5. #26011227C2semidirect
9. #19149670C4semidirect
10. #19149666C4semidirect
13. #19149644C4semidirect
15. D4×Dic173C4semidirect
109. C173#1509semidirect

Maximal subgroups

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

subgroupindex
#260120222
#260115452
#260113422
#260112292
#260112272
#260112252
#327×C1732
#1509173