Finite Group Databasebeta

← #36369823

Subgroups — #36369823

Order 22144.

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

normal subgroups only

This group has 266 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#26012392C212yesmaximal, complemented
4#26012391C212yesmaximal, complemented
5#26012056C211yesmaximal, characteristic
6C2×#14583835—21no—
7#19149758—21no—
8#19149740—21no—
9#19149739C2212yes—
10#19149737—21no—
11#19149737C2212yes—
12C2×#14583825C2211yescharacteristic
13C22×C22⋊D173C411yescharacteristic, complemented
14#19149675—21no—
15#19149665—21no—
16#19149652—21no—
17#19149651—21no—
18#19149649C2211yescharacteristic
19#19149648C2211yescharacteristic
20#19149647C411yescharacteristic, complemented
21#19149635C411yescharacteristic, complemented
22#120×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⋊D173—21no—
36C2×C22⋊D173—21no—
37C2×C22⋊D173—21no—
38C2×C22⋊D173—41no—
39C2×C22⋊D173—41no—
40C2×C22⋊D173D411yescharacteristic
41C2×C22⋊D173C2311yescharacteristic
42C22×Dic173C2×C411yescharacteristic
43C22×Dic173D411yescharacteristic
44C2×C4×D173D411yescharacteristic
45#14583805—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173C2×C411yescharacteristic
48#14583803—21no—
49#14583803—21no—
50C4×Dic173—21no—
51C24×C173C2×C411yescharacteristic
52C2×D4×C173—21no—
53C2×D4×C173C2×C411yescharacteristic
54C22⋊C4×C173—21no—
55C22⋊C4×C173C2×C411yescharacteristic
56C2×#10621924—42no—
57#13063443—42no—
58C22×D173—21no—
59C22×D173—41no—
60C22×D173—81no—
61C22×D173C2×D411yescharacteristic
62Dic346—41no—
63D692—81no—
64C22⋊D173—41no—
65C22⋊D173—41no—
66C22⋊D173—42no—
67C22⋊D173—41no—
68C22⋊D173—41no—
69C22⋊D173—81no—
70C22⋊D173—81no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173C22⋊C411yescharacteristic
77C2×Dic173C22⋊C411yescharacteristic
78C2×Dic173C2×D411yescharacteristic
79C4×D173—21no—
80C4×D173—21no—
81C23×C173—21no—
82C23×C173—41no—
83C23×C173—81no—
84C23×C173C22⋊C411yescharacteristic
85C23×C173C22×C411yescharacteristic
86D4×C173—41no—
87D4×C173—81no—
88C2×C4×C173—21no—
89C2×C4×C173C22⋊C411yescharacteristic
90C22×C173—21no—
91C22×C173—21no—
92C22×C173—41no—
93C22×C173—41no—
94C22×C173—41no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—82no—
98C22×C173C2×C22⋊C411yescharacteristic
99C2×D173—21no—
100C2×D173—41no—
101C2×D173—41no—
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—82no—
105#10621924—82no—
106C4×C173—21no—
107C4×C173—41no—
108Dic173—21no—
109Dic173—21no—
110Dic173—41no—
111Dic173#9911yescharacteristic
112Dic173#9911yescharacteristic
113C2×C173—21no—
114C2×C173—41no—
115C2×C173—41no—
116C2×C173—81no—
117C2×C173C2×#9911yescharacteristic
118D173—41no—
119D173—81no—
120C173#212211yescharacteristic, complemented, Sylow
121#2122—1731nomaximal, Sylow
122#535—1731no—
123#413—1731no—
124#410—1731no—
125AΣL(1,𝔽16)—1732no—
126AΣL(1,𝔽16)—1732no—
127C2×#41—1731no—
128C22×D4—1731no—
129C2×M16—3461no—
130#127—3461no—
131#124—3461no—
132#121—3461no—
133#121—3461no—
134#120—1731no—
135#120—1731no—
136#120—1731no—
137#120#1062192411yescharacteristic, complemented
138#117—3461no—
139#117—1731no—
140#101—3461no—
141#100—1732no—
142#99—3461no—
143#99—1732no—
144C24—1731no—
145C24C2×#1062192411yescharacteristic
146#41—6921no—
147#41—6921no—
148C2×Q8—1731no—
149C2×D4—21no—
150C2×D4—3461no—
151C2×D4—3461no—
152C2×D4—3461no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—1731no—
156C2×D4—1731no—
157C2×D4—6921no—
158C2×D4—6921no—
159C2×D4C2×#1062192411yescharacteristic
160C22×C4—1731no—
161C22×C4—1731no—
162C22×C4—1731no—
163M16—3462no—
164M16—6921no—
165C2×C8—6921no—
166C4⋊C4—3461no—
167C4⋊C4—6921no—
168C22⋊C4—21no—
169C22⋊C4—3461no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C22⋊C4—3462no—
173C22⋊C4—1731no—
174C22⋊C4C2×#1062192411yescharacteristic
175C42—3461no—
176C42—6921no—
177C23—21no—
178C23—41no—
179C23—81no—
180C23—3461no—
181C23—13841no—
182C23—1731no—
183C23—6921no—
184C23C22⋊#1062192411yescharacteristic
185C23C22×#1062192411yescharacteristic
186Q8—6921no—
187D4—41no—
188D4—81no—
189D4—13841no—
190D4—13841no—
191D4—13841no—
192D4—6921no—
193D4—6921no—
194D4—6922no—
195D4—6921no—
196D4—6921no—
197C2×C4—21no—
198C2×C4—3461no—
199C2×C4—3461no—
200C2×C4—3461no—
201C2×C4—3461no—
202C2×C4—3461no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—1731no—
206C2×C4—1731no—
207C2×C4—1731no—
208C2×C4—6922no—
209C2×C4C22⋊#1062192411yescharacteristic
210C8—6922no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—82no—
219C22—3461no—
220C22—13842no—
221C22—6921no—
222C22—6921no—
223C22—6921no—
224C22—6921no—
225C22C2×C22⋊#1062192411yescharacteristic
226C4—21no—
227C4—41no—
228C4—3461no—
229C4—3461no—
230C4—13842no—
231C4—1731no—
232C4—1731no—
233C4—6921no—
234C2—21no—
235C2—41no—
236C2—41no—
237C2—81no—
238C2—13841no—
239C2—6921no—
240C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#120×C173C4
socleC2×C173C2×#99
prime radical#26012056C2
cosocleC2×C22⋊D173C23
hypercenter#120#10621924
hypocenterC173#2122
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26012478C2semidirect
2. #26012471C2semidirect
3. #26012392C2semidirect
4. #26012391C2semidirect
13. C22×C22⋊D173C4semidirect
20. #19149647C4semidirect
21. #19149635C4semidirect
22. #120×C173C4semidirect
120. C173#2122semidirect
137. #120#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124782
#260124712
#260123922
#260123912
#260120562
#2122173