Finite Group Databasebeta

← #36355205

Subgroups — #36355205

Order 22144.

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

normal subgroups only

This group has 264 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#26011998C211yesmaximal, characteristic
2#26011465C211yesmaximal, characteristic, complemented
3#26011282C212yesmaximal, complemented
4#26011279C212yesmaximal, complemented
5#413×C173C211yesmaximal, characteristic, complemented
6C22×C4×D173C411yescharacteristic, complemented
7#19149660C411yescharacteristic, complemented
8#19149660C2211yescharacteristic
9#19149655—21no—
10#19149650—21no—
11#19149647C411yescharacteristic, complemented
12#19149647C2211yescharacteristic
13C4×C22⋊D173—21no—
14C4×C22⋊D173—21no—
15#19149638—21no—
16#19149638C411yescharacteristic, complemented
17#19149594—21no—
18#19149594C2212yes—
19C2×#41×C173C2211yescharacteristic
20C2×M16×C173—21no—
21#101×C173—21no—
22#100×C173C2212yes—
23C23×D173C2×C411yescharacteristic
24C2×C22⋊D173—21no—
25C2×C22⋊D173—21no—
26C22×Dic173C2×C411yescharacteristic
27C2×C4×D173—21no—
28C2×C4×D173—21no—
29C2×C4×D173—41no—
30C2×C4×D173—41no—
31C4⋊Dic173—21no—
32C4⋊Dic173—41no—
33#14583805—21no—
34#14583805—21no—
35C22⋊Dic173—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—22no—
38C22⋊Dic173C2×C411yescharacteristic
39C22⋊Dic173C2×C411yescharacteristic
40#14583803—21no—
41#14583803—21no—
42C4×Dic173—21no—
43C4×Dic173—41no—
44#41×C173—41no—
45#41×C173—41no—
46C2×Q8×C173D411yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C2×D4×C173D411yescharacteristic
49C2×D4×C173C2311yescharacteristic
50C22×C4×C173C2×C411yescharacteristic
51C22×C4×C173D411yescharacteristic
52C22×C4×C173D411yescharacteristic
53M16×C173—22no—
54M16×C173—41no—
55C2×C8×C173—41no—
56C22×D173—21no—
57C22×D173—42no—
58C22×D173—41no—
59C22⋊D173—42no—
60C22⋊D173—42no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173—41no—
65C2×Dic173—42no—
66C4×D173—42no—
67C4×D173—41no—
68C4×D173—41no—
69C23×C173C22⋊C411yescharacteristic
70C23×C173C22×C411yescharacteristic
71C23×C173C2×D411yescharacteristic
72Q8×C173—41no—
73D4×C173—41no—
74D4×C173—41no—
75D4×C173—41no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—21no—
81C2×C4×C173—21no—
82C2×C4×C173C22⋊C411yescharacteristic
83C2×C4×C173C22⋊C411yescharacteristic
84C2×C4×C173C22⋊C411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C8×C173—42no—
87C22×C173—21no—
88C22×C173—21no—
89C22×C173—21no—
90C22×C173—41no—
91C22×C173—41no—
92C22×C173—41no—
93C22×C173C2×C22⋊C411yescharacteristic
94C2×D173—22no—
95C2×D173—41no—
96C2×D173—42no—
97C2×D173—41no—
98C2×D173—42no—
99C4×C173—21no—
100C4×C173—21no—
101C4×C173—21no—
102C4×C173#9911yescharacteristic
103C4×C173#9911yescharacteristic
104Dic173—41no—
105Dic173—41no—
106Dic173—41no—
107Dic173—82no—
108C2×C173—21no—
109C2×C173—41no—
110C2×C173—41no—
111C2×C173—41no—
112C2×C173C2×#9911yescharacteristic
113D173—42no—
114C173#212011yescharacteristic, complemented, Sylow
115#2120—1731nomaximal, Sylow
116#525—1731no—
117#413D17311yescharacteristic, complemented
118#410—1731no—
119#352—1732no—
120AΣL(1,𝔽16)—1732no—
121C2×#41C2×D17311yescharacteristic
122C23×C4—1731no—
123C2×M16—21no—
124#123—1731no—
125#123—1731no—
126C22⋊Q8—3461no—
127#121—3461no—
128#120—1731no—
129#120—1731no—
130C4×D4—3461no—
131C4×D4—3461no—
132#117—3461no—
133#117—1731no—
134#101—21no—
135#100C2×D17312yes—
136#99—3461no—
137#99—1732no—
138C24—1731no—
139#41—41no—
140#41—41no—
141C2×Q8C22⋊D17311yescharacteristic
142C2×D4—3461no—
143C2×D4—3461no—
144C2×D4C4×D17311yescharacteristic
145C2×D4D69211yescharacteristic
146C2×D4C22×D17311yescharacteristic
147C22×C4—3461no—
148C22×C4—3461no—
149C22×C4—1731no—
150C22×C4—6921no—
151C22×C4—6921no—
152C22×C4C4×D17311yescharacteristic
153C22×C4C22⋊D17311yescharacteristic
154C22×C4D69211yescharacteristic
155M16—22no—
156M16—41no—
157C2×C8—41no—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—6921no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—3462no—
167C22⋊C4—1731no—
168C22⋊C4—1731no—
169C42—3461no—
170C42—6921no—
171C23—3461no—
172C23—6921no—
173C23—6922no—
174C23#1458380311yescharacteristic
175C23C2×C4×D17311yescharacteristic
176C23C2×D69211yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—6922no—
182D4—6922no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—3461no—
190C2×C4—3461no—
191C2×C4—3461no—
192C2×C4—6922no—
193C2×C4—6922no—
194C2×C4—6921no—
195C2×C4—6921no—
196C2×C4—6921no—
197C2×C4#1458380311yescharacteristic
198C2×C4#1458380311yescharacteristic
199C2×C4#1458380311yescharacteristic
200C2×C4C2×C22⋊D17311yescharacteristic
201C8—42no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—3462no—
209C22—6921no—
210C22—6922no—
211C22—6922no—
212C22—6921no—
213C22C2×#1458380311yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—13842no—
218C4—6921no—
219C4—6921no—
220C4—6921no—
221C4#1914959311yescharacteristic
222C4#1914959311yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—6922no—
228C2C2×#1914959311yescharacteristic
subgroupG/H
centerC4#19149593
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#413×C173C2
socleC2×C173C2×#99
prime radical#26011998C2
cosocleC2×D4×C173C23
hypercenter#413D173
hypocenterC173#2120
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #26011465C2semidirect
3. #26011282C2semidirect
4. #26011279C2semidirect
5. #413×C173C2semidirect
6. C22×C4×D173C4semidirect
7. #19149660C4semidirect
11. #19149647C4semidirect
16. #19149638C4semidirect
114. C173#2120semidirect
117. #413D173semidirect

Maximal subgroups

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

subgroupindex
#260119982
#260114652
#260112822
#260112792
#413×C1732
#2120173