Finite Group Databasebeta

← #36369707

Subgroups — #36369707

Order 22144.

312 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 70 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#26012494C211yesmaximal, characteristic, complemented
2C2×#19149740C212yesmaximal, complemented
3C2×#19149739C212yesmaximal
4C4×#14583835C211yesmaximal, characteristic, complemented
5C2×#19149665C211yesmaximal, characteristic
6C2×#14583835C2212yes—
7C2×#14583835C2212yes—
8#19149746—21no—
9C4⋊#13063443—21no—
10#19149740—24no—
11#19149739—24no—
12#19149734—21no—
13C4×#13063443—21no—
14C22×Dic346C2211yescharacteristic
15C22×C22⋊D173C2211yescharacteristic
16#19149665—21no—
17#19149665—21no—
18#19149665—21no—
19#19149665—21no—
20C2×C22⋊Dic173C411yescharacteristic
21C2×#14583803C411yescharacteristic
22C2×#14583803C411yescharacteristic
23C2×C4×Dic173C2211yescharacteristic
24C2×C22⋊C4×C173C411yescharacteristic
25#14583835—22no—
26#14583835—22no—
27#14583835—22no—
28#14583835D414yes—
29C2×#13063443—22no—
30C2×#13063443—22no—
31C23×D173C2×C411yescharacteristic
32C2×Dic346—21no—
33C2×Dic346—21no—
34C2×Dic346—22no—
35C2×Dic346—22no—
36C2×C22⋊D173—21no—
37C2×C22⋊D173—21no—
38C2×C22⋊D173—22no—
39C2×C22⋊D173—22no—
40C22×Dic173C2×C411yescharacteristic
41C22×Dic173C2×C411yescharacteristic
42C22×Dic173C2311yescharacteristic
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49C4×Dic173D411yescharacteristic
50C4×Dic173D411yescharacteristic
51C4×Dic173D411yescharacteristic
52C4×Dic173D411yescharacteristic
53C24×C173C2×C411yescharacteristic
54C22×C4×C173C2×C411yescharacteristic
55C22×C4×C173C2×C411yescharacteristic
56C22⋊C4×C173—21no—
57C22⋊C4×C173—21no—
58#13063443—24no—
59#13063443—42no—
60C22×D173—21no—
61C22×D173—22no—
62C22×D173—41no—
63C22×D173—41no—
64Dic346—42no—
65Dic346—41no—
66Dic346—41no—
67C22⋊D173—42no—
68C22⋊D173—41no—
69C22⋊D173—41no—
70C2×Dic173—21no—
71C2×Dic173—22no—
72C2×Dic173—41no—
73C2×Dic173C22⋊C411yescharacteristic
74C2×Dic173C22⋊C411yescharacteristic
75C2×Dic173C2×D411yescharacteristic
76C2×Dic173C2×D412yes—
77C2×Dic173C2×D411yescharacteristic
78C2×Dic173#4112yes—
79C23×C173—21no—
80C23×C173—22no—
81C23×C173—41no—
82C23×C173—41no—
83C23×C173C22×C411yescharacteristic
84C2×C4×C173—21no—
85C2×C4×C173—22no—
86C2×C4×C173—41no—
87C2×C4×C173C22⋊C411yescharacteristic
88C2×C4×C173C22⋊C411yescharacteristic
89C22×C173—21no—
90C22×C173—21no—
91C22×C173—41no—
92C22×C173—41no—
93C22×C173—41no—
94C22×C173—82no—
95C22×C173C2×C22⋊C411yescharacteristic
96C22×C173C4×D412yes—
97C2×D173—41no—
98C2×D173—41no—
99C2×D173—41no—
100C2×D173—82no—
101C4×C173—21no—
102C4×C173—21no—
103C4×C173—41no—
104Dic173—21no—
105Dic173—21no—
106Dic173—41no—
107Dic173#12112yes—
108Dic173#12411yescharacteristic
109Dic173#12711yescharacteristic
110C2×C173—22no—
111C2×C173—81no—
112C2×C173#39011yescharacteristic
113C2×C173#41311yescharacteristic
114C2×C173#41311yescharacteristic
115D173—81no—
116C173#196711yescharacteristic, complemented, Sylow
117#1967—1731nomaximal, Sylow
118C2×#124—1731no—
119#423—1731no—
120C2×#101—1732no—
121C2×#100—1732no—
122C4×M16—1731no—
123C22×Q8—1731no—
124C22×D4—1731no—
125C2×M16—1732no—
126C2×M16—1732no—
127#124—3461no—
128#124—3461no—
129#124—3461no—
130#124—3461no—
131C2×C22⋊C4—1731no—
132C2×C22⋊C4—1731no—
133C2×C22⋊C4—1731no—
134C2×C22⋊C4#1062192411yescharacteristic
135C2×C42—1731no—
136C4⋊C8—3461no—
137C4⋊C8—3461no—
138#101—3464no—
139#100—3464no—
140#97—3461no—
141C4×C8—3461no—
142C24—1731no—
143C24C2×#1062192411yescharacteristic
144C2×Q8—3462no—
145C2×Q8—3461no—
146C2×Q8—3461no—
147C2×Q8—3462no—
148C2×D4—3462no—
149C2×D4—3461no—
150C2×D4—3461no—
151C2×D4—3462no—
152C22×C4—1731no—
153C22×C4—1731no—
154C22×C4—1731no—
155C22×C4C2×#1062192411yescharacteristic
156C22×C4C2×#1062192411yescharacteristic
157M16—3462no—
158M16—3462no—
159M16—3462no—
160M16—1734no—
161C2×C8—3462no—
162C2×C8—3462no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C22⋊C4—3461no—
171C42—1731no—
172C42—1731no—
173C42—1731no—
174C42—1731no—
175C23—21no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—3462no—
180C23—3461no—
181C23—6921no—
182C23—6921no—
183C23C22×#1062192411yescharacteristic
184Q8—6922no—
185Q8—6921no—
186Q8—6921no—
187D4—6922no—
188D4—6921no—
189D4—6921no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—3461no—
194C2×C4—3462no—
195C2×C4—1731no—
196C2×C4—1731no—
197C2×C4—1731no—
198C2×C4—1731no—
199C2×C4—1732no—
200C2×C4—1732no—
201C2×C4—6921no—
202C2×C4C22⋊#1062192411yescharacteristic
203C2×C4C22⋊#1062192411yescharacteristic
204C8—3464no—
205C8—6922no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—82no—
212C22—13842no—
213C22—6921no—
214C22—6921no—
215C22—6921no—
216C22C2×C22⋊#1062192411yescharacteristic
217C22D4×#1062192412yes—
218C4—21no—
219C4—21no—
220C4—41no—
221C4—3461no—
222C4—3461no—
223C4—1732no—
224C4—1731no—
225C4—1731no—
226C4—6921no—
227C2—22no—
228C2—81no—
229C2—13841no—
230C2#2601244311yescharacteristic
231C2#2601247711yescharacteristic
232C2#2601247811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C22
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1967
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #26012494C2semidirect
2. C2×#19149740C2semidirect
4. C4×#14583835C2semidirect
116. C173#1967semidirect

Maximal subgroups

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

subgroupindex
#260124942
C2×#191497402
C2×#191497392
C4×#145838352
C2×#191496652
#1967173