Finite Group Databasebeta

← #36369821

Subgroups — #36369821

Order 22144.

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

normal subgroups only

This group has 280 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
1D173⋊#100C211yesmaximal, characteristic, complemented
2D173⋊#99C211yesmaximal, characteristic, complemented
3#26012393C212yesmaximal, complemented
4#26012392C212yesmaximal, complemented
5#26012052C211yesmaximal, characteristic
6D173⋊M16—21no—
7C2×C22⋊#10621924—21no—
8#19149739—21no—
9#19149739C2212yes—
10#19149738—21no—
11#19149737C2212yes—
12C22×D692C411yescharacteristic, complemented
13C2×D4×D173C2211yescharacteristic
14#19149664—21no—
15#19149662C2211yescharacteristic
16#19149654—21no—
17#19149652—21no—
18#19149649—21no—
19#19149649C2211yescharacteristic
20#19149647C411yescharacteristic, complemented
21C22⋊C4×D173C411yescharacteristic, complemented
22#123×C173C411yescharacteristic, complemented
23C22×#10621924—41no—
24#14583836—41no—
25#14583835—22no—
26D173⋊C8—41no—
27C22⋊#10621924—22no—
28C22⋊#10621924—41no—
29C23×D173C2×C411yescharacteristic
30C23×D173C2×C411yescharacteristic, complemented
31C23×D173D411yescharacteristic
32C2×D692—21no—
33C2×D692—21no—
34C2×D692—21no—
35C2×D692—41no—
36C2×D692—41no—
37C2×D692D411yescharacteristic
38C2×C22⋊D173—21no—
39C2×C22⋊D173D411yescharacteristic
40C2×C22⋊D173C2311yescharacteristic
41D4×D173—41no—
42D4×D173—41no—
43C2×C4×D173—21no—
44C2×C4×D173D411yescharacteristic
45#14583805—21no—
46C22⋊Dic173C2×C411yescharacteristic, complemented
47#14583803—21no—
48#14583803—21no—
49#14583803—21no—
50#14583803—21no—
51C2×D4×C173C2×C411yescharacteristic
52C22×C4×C173C2×C411yescharacteristic, complemented
53C4⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22⋊C4×C173C2×C411yescharacteristic, complemented
56C2×#10621924—42no—
57C2×#10621924—82no—
58#13063443—42no—
59C22×D173—21no—
60C22×D173—21no—
61C22×D173—21no—
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—41no—
65C22×D173—81no—
66C22×D173—81no—
67C22×D173—81no—
68C22×D173C22⋊C411yescharacteristic
69C22×D173C22⋊C411yescharacteristic
70C22×D173C2×D411yescharacteristic
71D692—41no—
72D692—42no—
73D692—41no—
74D692—41no—
75D692—81no—
76C22⋊D173—41no—
77C22⋊D173—41no—
78C22⋊D173—81no—
79C2×Dic173—21no—
80C2×Dic173C2×D411yescharacteristic
81C4×D173—21no—
82C4×D173—21no—
83C4×D173—81no—
84C23×C173C22⋊C411yescharacteristic
85C23×C173C22×C411yescharacteristic
86D4×C173—41no—
87C2×C4×C173—21no—
88C2×C4×C173—21no—
89C2×C4×C173—41no—
90C2×C4×C173C22⋊C411yescharacteristic
91C22×C173—21no—
92C22×C173—21no—
93C22×C173—41no—
94C22×C173—41no—
95C22×C173C2×C22⋊C411yescharacteristic
96C2×D173—21no—
97C2×D173—21no—
98C2×D173—41no—
99C2×D173—42no—
100C2×D173—41no—
101C2×D173—41no—
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—41no—
105C2×D173—81no—
106C2×D173—81no—
107C2×D173—81no—
108C2×D173—82no—
109C2×D173#9911yescharacteristic
110C2×D173#9911yescharacteristic
111#10621924—82no—
112C4×C173—21no—
113C4×C173—41no—
114C4×C173—41no—
115Dic173—21no—
116Dic173—41no—
117C2×C173—21no—
118C2×C173—41no—
119C2×C173—41no—
120C2×C173C2×#9911yescharacteristic
121D173—21no—
122D173—21no—
123D173—41no—
124D173—41no—
125D173—81no—
126C173#212111yescharacteristic, complemented, Sylow
127#2121—1731nomaximal, Sylow
128#534—1731no—
129C2×#100—1731no—
130C2×#99—1731no—
131#352—1732no—
132AΣL(1,𝔽16)—1732no—
133C22×D4—1731no—
134C22×D4—1731no—
135C2×M16—3461no—
136#123—3461no—
137#123—1731no—
138#123#1062192411yescharacteristic, complemented
139#121—3461no—
140#121—3461no—
141#120—3461no—
142#120—1731no—
143#120—1731no—
144C2×C22⋊C4—3461no—
145C2×C22⋊C4—1731no—
146#100—3461no—
147#100—1732no—
148#99—3461no—
149#99—1732no—
150C24—1731no—
151C24—1731no—
152C24—1731no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—3461no—
157C2×D4—1731no—
158C2×D4—1731no—
159C2×D4—1731no—
160C2×D4—6921no—
161C2×D4—6921no—
162C2×D4—6921no—
163C2×D4—6921no—
164C2×D4C2×#1062192411yescharacteristic
165C22×C4—3461no—
166C22×C4—1731no—
167C22×C4—6921no—
168C22×C4C2×#1062192411yescharacteristic, complemented
169M16—3462no—
170M16—6921no—
171C2×C8—6921no—
172C4⋊C4—21no—
173C4⋊C4—3461no—
174C22⋊C4—21no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C22⋊C4—3462no—
180C22⋊C4—1731no—
181C22⋊C4—6921no—
182C22⋊C4C2×#1062192411yescharacteristic, complemented
183C23—3461no—
184C23—3461no—
185C23—3461no—
186C23—3461no—
187C23—3461no—
188C23—13841no—
189C23—13841no—
190C23—13841no—
191C23—1731no—
192C23—1731no—
193C23—1731no—
194C23—6921no—
195C23C22⋊#1062192411yescharacteristic
196C23C22×#1062192411yescharacteristic
197D4—41no—
198D4—13841no—
199D4—13841no—
200D4—6921no—
201D4—6922no—
202D4—6921no—
203D4—6921no—
204D4—6921no—
205D4—6921no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—3461no—
210C2×C4—3461no—
211C2×C4—3461no—
212C2×C4—13841no—
213C2×C4—13842no—
214C2×C4—1731no—
215C2×C4—6922no—
216C2×C4C22⋊#1062192411yescharacteristic
217C8—6922no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—3461no—
223C22—3461no—
224C22—13841no—
225C22—13841no—
226C22—13841no—
227C22—13842no—
228C22—1731no—
229C22—1731no—
230C22—6921no—
231C22—6921no—
232C22—6921no—
233C22—6921no—
234C22—6921no—
235C22—6922no—
236C22—6921no—
237C22C2×C22⋊#1062192411yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—3461no—
242C4—13842no—
243C4—6921no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2—3461no—
248C2—3461no—
249C2—13841no—
250C2—6921no—
251C2—6921no—
252C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#123×C173C4
socleC2×C173C2×#99
prime radical#26012052C2
cosocleC2×C22⋊D173C23
hypercenter#123#10621924
hypocenterC173#2121
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. D173⋊#100C2semidirect
2. D173⋊#99C2semidirect
3. #26012393C2semidirect
4. #26012392C2semidirect
12. C22×D692C4semidirect
20. #19149647C4semidirect
21. C22⋊C4×D173C4semidirect
22. #123×C173C4semidirect
30. C23×D173C2×C4semidirect
46. C22⋊Dic173C2×C4semidirect
52. C22×C4×C173C2×C4semidirect
55. C22⋊C4×C173C2×C4semidirect
126. C173#2121semidirect
138. #123#10621924semidirect
168. C22×C4C2×#10621924semidirect
182. C22⋊C4C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊#1002
D173⋊#992
#260123932
#260123922
#260120522
#2121173