Finite Group Databasebeta

← #36352104

Subgroups — #36352104

Order 22144.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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#26012002C211yesmaximal, characteristic
2#26011455C211yesmaximal, characteristic, complemented
3#26011227C212yesmaximal, complemented
4#26011224C212yesmaximal, complemented
5#380×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C411yescharacteristic, complemented
7C2×#14583810—21no—
8#19149662—21no—
9#19149658C411yescharacteristic, complemented
10#19149658C2211yescharacteristic
11#19149653C411yescharacteristic, complemented
12#19149653C2211yescharacteristic
13#19149648—21no—
14C4×C22⋊D173—21no—
15C4×C22⋊D173—21no—
16#19149638C411yescharacteristic, complemented
17C22⋊#13063436—21no—
18C22⋊#13063436C2212yes—
19C23×C4×C173C2211yescharacteristic
20C2×C4⋊C4×C173—21no—
21C2×C22⋊C4×C173—21no—
22#95×C173C2212yes—
23#14583826—42no—
24C2×Dic346C2×C411yescharacteristic
25C2×D692C2×C411yescharacteristic
26C2×C22⋊D173—21no—
27C2×C22⋊D173—21no—
28C2×C22⋊D173—21no—
29C2×C4×D173—21no—
30#14583810—42no—
31C2×#13063436—22no—
32C4⋊Dic173C2×C411yescharacteristic
33C4⋊Dic173C2×C411yescharacteristic
34#14583805—21no—
35#14583805—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39#14583803—21no—
40#14583803—21no—
41C4×Dic173—21no—
42C24×C173D411yescharacteristic
43C22×C4×C173—22no—
44C22×C4×C173—42no—
45C22×C4×C173C2×C411yescharacteristic
46C22×C4×C173C2×C411yescharacteristic
47C22×C4×C173D411yescharacteristic
48C22×C4×C173D411yescharacteristic
49C22×C4×C173D411yescharacteristic
50C22×C4×C173C2311yescharacteristic
51C4⋊C4×C173—42no—
52C22⋊C4×C173—42no—
53C22×D173—21no—
54Dic346—41no—
55D692—41no—
56C22⋊D173—42no—
57C22⋊D173—42no—
58C22⋊D173—41no—
59C22⋊D173—41no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C4×D173—42no—
64#13063436—42no—
65C23×C173—21no—
66C23×C173—21no—
67C23×C173—41no—
68C23×C173—41no—
69C23×C173C22⋊C411yescharacteristic
70C23×C173C22⋊C411yescharacteristic
71C23×C173C2×D411yescharacteristic
72C2×C4×C173—22no—
73C2×C4×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173—42no—
79C2×C4×C173—42no—
80C2×C4×C173—42no—
81C2×C4×C173C22⋊C411yescharacteristic
82C2×C4×C173C22⋊C411yescharacteristic
83C2×C4×C173C22×C411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C22×C173—21no—
86C22×C173—21no—
87C22×C173—21no—
88C22×C173—21no—
89C22×C173—21no—
90C22×C173—21no—
91C22×C173—21no—
92C22×C173—21no—
93C22×C173—42no—
94C22×C173—42no—
95C22×C173C2×C22⋊C411yescharacteristic
96C22×C173AGL(1,ℤ/8)11yescharacteristic
97C22×C173#13711yescharacteristic
98C2×D173—41no—
99C2×D173—41no—
100C2×D173—41no—
101C4×C173—21no—
102C4×C173—21no—
103C4×C173—21no—
104C4×C173—82no—
105C4×C173#9912yes—
106Dic173—41no—
107Dic173—41no—
108Dic173—41no—
109C2×C173—21no—
110C2×C173—21no—
111C2×C173—41no—
112C2×C173—41no—
113C2×C173C2×#9911yescharacteristic
114C2×C173#41711yescharacteristic
115C2×C173#42111yescharacteristic
116D173—81no—
117C173#151311yescharacteristic, complemented, Sylow
118#1513—1731nomaximal, Sylow
119#525—1731no—
120#407—1731no—
121#380D17311yescharacteristic, complemented
122#328—1732no—
123#327—1732no—
124C2×#41—1731no—
125C23×C4C2×D17311yescharacteristic
126C2×M16—3461no—
127#123—3461no—
128C22⋊Q8—1731no—
129C22⋊Q8—1731no—
130#121—1731no—
131#121—1731no—
132#120—3461no—
133C4×D4—3461no—
134C4×D4—3461no—
135#117—1731no—
136C2×C4⋊C4—21no—
137C2×C22⋊C4—21no—
138C22⋊C8—3461no—
139C22⋊C8—1732no—
140#95C2×D17312yes—
141C24C22⋊D17311yescharacteristic
142#41—6922no—
143C2×Q8—1731no—
144C2×D4—3461no—
145C2×D4—3461no—
146C2×D4—3461no—
147C2×D4—1731no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—3461no—
151C22×C4C4×D17311yescharacteristic
152C22×C4C4×D17311yescharacteristic
153C22×C4C22⋊D17311yescharacteristic
154C22×C4D69211yescharacteristic
155C22×C4D69211yescharacteristic
156C22×C4C22×D17311yescharacteristic
157M16—6922no—
158C2×C8—3462no—
159C4⋊C4—42no—
160C4⋊C4—3461no—
161C4⋊C4—3461no—
162C4⋊C4—1731no—
163C4⋊C4—1731no—
164C22⋊C4—42no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C42—3461no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—3461no—
176C23#1458380311yescharacteristic
177C23#1458380311yescharacteristic
178C23C2×C22⋊D17311yescharacteristic
179Q8—6921no—
180D4—6921no—
181D4—6922no—
182D4—6922no—
183D4—6921no—
184D4—6921no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—3461no—
197C2×C4—6922no—
198C2×C4#1458380311yescharacteristic
199C2×C4#1458380311yescharacteristic
200C2×C4C2×C4×D17311yescharacteristic
201C2×C4C2×D69211yescharacteristic
202C8—6922no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—6921no—
214C22—6921no—
215C22—6921no—
216C22C2×#1458380311yescharacteristic
217C22#1914970511yescharacteristic
218C22#1914971011yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—82no—
223C4—6921no—
224C4—6921no—
225C4—6921no—
226C4#1914959312yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—13841no—
232C2C2×#1914959311yescharacteristic
233C2#2601149011yescharacteristic
234C2#2601150911yescharacteristic
subgroupG/H
centerC22C2×#14583803
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#380×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012002C2
cosocleC22×C4×C173C23
hypercenter#380D173
hypocenterC173#1513
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #26011455C2semidirect
3. #26011227C2semidirect
4. #26011224C2semidirect
5. #380×C173C2semidirect
6. C2×#14583826C4semidirect
9. #19149658C4semidirect
11. #19149653C4semidirect
16. #19149638C4semidirect
117. C173#1513semidirect
121. #380D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
#260114552
#260112272
#260112242
#380×C1732
#1513173