Finite Group Databasebeta

← #36369817

Subgroups — #36369817

Order 22144.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#26012478C211yesmaximal, characteristic, complemented
2#26012471C211yesmaximal, characteristic, complemented
3#26012393C212yesmaximal, complemented
4#26012392C212yesmaximal, complemented
5#26011999C211yesmaximal, characteristic
6C2×#14583835—21no—
7#19149758—21no—
8#19149740—21no—
9#19149739C2212yes—
10#19149737—21no—
11#19149737C2212yes—
12C2×#14583825C2211yescharacteristic
13C22×C4×D173C411yescharacteristic, complemented
14#19149662C2211yescharacteristic
15#19149657—21no—
16#19149652—21no—
17#19149649C2211yescharacteristic
18#19149647C411yescharacteristic, complemented
19#19149641—21no—
20#19149639—21no—
21#19149635C411yescharacteristic, complemented
22#123×C173C411yescharacteristic, complemented
23#14583835—22no—
24#14583835—41no—
25C2×#13063443—41no—
26C4⋊#10621924—41no—
27C22⋊#10621924—22no—
28C4×#10621924—41no—
29C23×D173C2×C411yescharacteristic
30#14583825—41no—
31#14583825—41no—
32C2×Dic346D411yescharacteristic
33C2×D692—21no—
34C2×C22⋊D173—21no—
35C2×C22⋊D173D411yescharacteristic
36C2×C22⋊D173C2311yescharacteristic
37C22×Dic173C2×C411yescharacteristic
38C22×Dic173D411yescharacteristic
39C2×C4×D173—21no—
40C2×C4×D173—21no—
41C2×C4×D173—41no—
42C2×C4×D173—41no—
43C2×C4×D173D411yescharacteristic
44#14583805—21no—
45#14583805—21no—
46C22⋊Dic173C2×C411yescharacteristic
47#14583803—21no—
48#14583803—21no—
49#14583803—21no—
50C4×Dic173—21no—
51C2×D4×C173C2×C411yescharacteristic
52C22×C4×C173C2×C411yescharacteristic
53C4⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22⋊C4×C173C2×C411yescharacteristic
56C2×#10621924—42no—
57#13063443—42no—
58C22×D173—21no—
59C22×D173—42no—
60C22×D173—41no—
61C22×D173C2×D411yescharacteristic
62Dic346—41no—
63D692—42no—
64C22⋊D173—41no—
65C22⋊D173—41no—
66C22⋊D173—42no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173C22⋊C411yescharacteristic
73C2×Dic173C22⋊C411yescharacteristic
74C2×Dic173C2×D411yescharacteristic
75C4×D173—21no—
76C4×D173—21no—
77C4×D173—42no—
78C4×D173—41no—
79C4×D173—41no—
80C23×C173C22⋊C411yescharacteristic
81C23×C173C22×C411yescharacteristic
82D4×C173—41no—
83C2×C4×C173—21no—
84C2×C4×C173—21no—
85C2×C4×C173—41no—
86C2×C4×C173C22⋊C411yescharacteristic
87C22×C173—21no—
88C22×C173—21no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173C2×C22⋊C411yescharacteristic
92C2×D173—22no—
93C2×D173—21no—
94C2×D173—41no—
95C2×D173—42no—
96C2×D173—41no—
97C2×D173—41no—
98C2×D173—42no—
99#10621924—82no—
100C4×C173—21no—
101C4×C173—41no—
102C4×C173—41no—
103Dic173—21no—
104Dic173—21no—
105Dic173—41no—
106Dic173#9911yescharacteristic
107Dic173#9911yescharacteristic
108C2×C173—21no—
109C2×C173—41no—
110C2×C173—41no—
111C2×C173C2×#9911yescharacteristic
112D173—41no—
113D173—42no—
114C173#212011yescharacteristic, complemented, Sylow
115#2120—1731nomaximal, Sylow
116#525—1731no—
117#413—1731no—
118#410—1731no—
119#352—1732no—
120AΣL(1,𝔽16)—1732no—
121C2×#41—1731no—
122C23×C4—1731no—
123C2×M16—3461no—
124#123—1731no—
125#123#1062192411yescharacteristic, complemented
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—3461no—
135#100—1732no—
136#99—3461no—
137#99—1732no—
138C24—1731no—
139#41—6921no—
140#41—6921no—
141C2×Q8—1731no—
142C2×D4—3461no—
143C2×D4—3461no—
144C2×D4—1731no—
145C2×D4—1731no—
146C2×D4C2×#1062192411yescharacteristic
147C22×C4—3461no—
148C22×C4—3461no—
149C22×C4—1731no—
150C22×C4—1731no—
151C22×C4—1731no—
152C22×C4—6921no—
153C22×C4—6921no—
154C22×C4C2×#1062192411yescharacteristic
155M16—3462no—
156M16—6921no—
157C2×C8—6921no—
158C4⋊C4—21no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—6921no—
162C22⋊C4—21no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—3462no—
167C22⋊C4—1731no—
168C22⋊C4C2×#1062192411yescharacteristic
169C42—3461no—
170C42—6921no—
171C23—3461no—
172C23—1731no—
173C23—6921no—
174C23—6922no—
175C23C22⋊#1062192411yescharacteristic
176C23C22×#1062192411yescharacteristic
177Q8—6921no—
178D4—41no—
179D4—6922no—
180D4—6921no—
181D4—6921no—
182D4—6922no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—3461no—
187C2×C4—3461no—
188C2×C4—3461no—
189C2×C4—3461no—
190C2×C4—3461no—
191C2×C4—3461no—
192C2×C4—3461no—
193C2×C4—1731no—
194C2×C4—1731no—
195C2×C4—1731no—
196C2×C4—6922no—
197C2×C4—6921no—
198C2×C4—6921no—
199C2×C4—6922no—
200C2×C4C22⋊#1062192411yescharacteristic
201C8—6922no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—3462no—
207C22—3461no—
208C22—6921no—
209C22—6921no—
210C22—6922no—
211C22—6922no—
212C22—6921no—
213C22C2×C22⋊#1062192411yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—3461no—
218C4—3461no—
219C4—13842no—
220C4—1731no—
221C4—1731no—
222C4—6921no—
223C2—21no—
224C2—41no—
225C2—41no—
226C2—6921no—
227C2—6922no—
228C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#123×C173C4
socleC2×C173C2×#99
prime radical#26011999C2
cosocleC2×C22⋊D173C23
hypercenter#123#10621924
hypocenterC173#2120
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #26012478C2semidirect
2. #26012471C2semidirect
3. #26012393C2semidirect
4. #26012392C2semidirect
13. C22×C4×D173C4semidirect
18. #19149647C4semidirect
21. #19149635C4semidirect
22. #123×C173C4semidirect
114. C173#2120semidirect
125. #123#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124782
#260124712
#260123932
#260123922
#260119992
#2120173