Finite Group Databasebeta

← #36353919

Subgroups — #36353919

Order 22144.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14583815C211yesmaximal, characteristic
2C2×#19149644C211yesmaximal, characteristic, complemented
3C2×#19149602C211yesmaximal, characteristic, complemented
4C2×#19149599C211yesmaximal, characteristic
5#26011392C211yesmaximal, characteristic
6#26011245C211yesmaximal, characteristic, complemented
7#340×C173C211yesmaximal, characteristic, complemented
8C22×Dic346C2211yescharacteristic
9C22×D692C2211yescharacteristic
10C2×#14583815—22no—
11C2×#14583815—22no—
12C2×#14583815C414yescomplemented
13C22×#13063436C2211yescharacteristic
14#19149644—22no—
15#19149644—22no—
16C2×#14583805—21no—
17C2×#14583803—21no—
18C2×#14583803—21no—
19C2×C4×Dic173—21no—
20C2×C4×Dic173C2211yescharacteristic
21#19149602—22no—
22#19149599—22no—
23C22×C8×C173C2211yescharacteristic
24C2×C4⋊C4×C173C2211yescharacteristic
25C2×C4⋊C4×C173C2211yescharacteristic
26C23×D173—21no—
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346C2×C412yes—
31C2×Dic346D412yes—
32C2×Dic346D412yes—
33C2×D692—21no—
34C2×D692—21no—
35C2×D692—21no—
36C2×D692—21no—
37C2×D692—21no—
38C2×D692C2×C412yes—
39C22×Dic173—21no—
40C22×Dic173D411yescharacteristic
41C22×Dic173D411yescharacteristic
42#14583815—28no—
43C2×#13063436—21no—
44C2×#13063436—21no—
45C2×#13063436—21no—
46#14583805—24no—
47#14583803—42no—
48#14583803—42no—
49C4×Dic173—24no—
50C4×Dic173—21no—
51C4×Dic173—21no—
52C22×C4×C173—21no—
53C22×C4×C173—21no—
54C22×C4×C173C2311yescharacteristic
55C2×C8×C173—21no—
56C2×C8×C173—21no—
57C2×C8×C173C2×C412yes—
58C4⋊C4×C173—21no—
59C4⋊C4×C173—21no—
60C4⋊C4×C173—21no—
61C4⋊C4×C173D412yes—
62C22×D173—22no—
63C22×D173—41no—
64C22×D173—41no—
65C22×D173—41no—
66C22×D173—41no—
67C22×D173—41no—
68C22×D173—41no—
69Dic346—22no—
70Dic346—22no—
71Dic346—22no—
72Dic346C22⋊C414yescomplemented
73D692—22no—
74D692—22no—
75D692—41no—
76D692—41no—
77C2×Dic173—22no—
78C2×Dic173—22no—
79C2×Dic173—21no—
80C2×Dic173—22no—
81C2×Dic173—21no—
82C2×Dic173—21no—
83C2×Dic173—21no—
84C2×Dic173—21no—
85C2×Dic173SD1612yescomplemented
86#13063436—41no—
87C23×C173C2×D411yescharacteristic
88C2×C4×C173—22no—
89C2×C4×C173—41no—
90C2×C4×C173—41no—
91C2×C4×C173—41no—
92C2×C4×C173—41no—
93C2×C4×C173—41no—
94C2×C4×C173C22×C411yescharacteristic
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173#4111yescharacteristic
99C2×C4×C173#4111yescharacteristic
100C8×C173—22no—
101C22×C173C4×D411yescharacteristic
102C22×C173#12011yescharacteristic
103C22×C173#12111yescharacteristic
104C22×C173C2×SD1611yescharacteristic
105C22×C173#13511yescharacteristic
106C22×C173AGL(1,ℤ/8)11yescharacteristic
107C22×C173#13711yescharacteristic
108C2×D173—42no—
109C2×D173—42no—
110C2×D173—42no—
111C2×D173—81no—
112C2×D173—81no—
113C2×D173—81no—
114C2×D173—81no—
115C4×C173—42no—
116C4×C173—81no—
117C4×C173C2×C22⋊C411yescharacteristic
118C4×C173C4×D411yescharacteristic
119C4×C173#12111yescharacteristic
120C4×C173#12311yescharacteristic
121Dic173—24no—
122Dic173—22no—
123Dic173—41no—
124C2×C173#38611yescharacteristic
125C2×C173C4×SD1611yescharacteristic
126C2×C173#44011yescharacteristic
127C2×C173#44811yescharacteristic
128C2×C173#44911yescharacteristic
129C2×C173#46011yescharacteristic
130C2×C173#46411yescharacteristic
131D173—82no—
132C173#188011yescharacteristic, complemented, Sylow
133#1880—1731nomaximal, Sylow
134C22×SD16—1731no—
135C2×C4×Q8—1731no—
136C2×#103—1731no—
137C2×#102—1731no—
138#390—1731no—
139#340D17311yescharacteristic, complemented
140#336—1731no—
141C22×Q8—1731no—
142C22×D4—1731no—
143C2×SD16—3462no—
144C2×SD16—3462no—
145C2×SD16—1734no—
146C22×C8—1731no—
147C22×C8C2×D17311yescharacteristic
148C4×Q8—3462no—
149C4×Q8—3462no—
150C2×C4⋊C4—3461no—
151C2×C4⋊C4C2×D17311yescharacteristic
152C2×C4⋊C4C2×D17311yescharacteristic
153C2×C22⋊C4—3461no—
154C2×C22⋊C4—3461no—
155C2×C42—3461no—
156C2×C42—1731no—
157#103—3462no—
158#102—3462no—
159C24—3461no—
160C2×Q8—3461no—
161C2×Q8—3461no—
162C2×Q8—3461no—
163C2×Q8—1732no—
164C2×Q8—1732no—
165C2×Q8—1732no—
166C2×D4—3461no—
167C2×D4—3461no—
168C2×D4—3461no—
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—1732no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—3461no—
175C22×C4—1731no—
176C22×C4—1731no—
177C22×C4C22×D17311yescharacteristic
178SD16—3468no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—3461no—
182C2×C8—3461no—
183C2×C8—3461no—
184C2×C8C4×D17312yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3464no—
189C4⋊C4D69212yes—
190C22⋊C4—6922no—
191C22⋊C4—6922no—
192C42—3464no—
193C42—3461no—
194C42—3461no—
195C23—3462no—
196C23—6921no—
197C23—6921no—
198C23—6921no—
199C23—6921no—
200C23—6921no—
201C23—6921no—
202C23D4×D17311yescharacteristic
203Q8—3462no—
204Q8—3462no—
205Q8—3462no—
206Q8—1734no—
207D4—3462no—
208D4—3462no—
209D4—6921no—
210D4—6921no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—3462no—
218C2×C4—3461no—
219C2×C4—3462no—
220C2×C4—3462no—
221C2×C4—3461no—
222C2×C4—3461no—
223C2×C4—3461no—
224C2×C4—3461no—
225C2×C4—1732no—
226C2×C4C2×C4×D17311yescharacteristic
227C2×C4D4×D17311yescharacteristic
228C2×C4D4×D17311yescharacteristic
229C2×C4C2×D69211yescharacteristic
230C2×C4#1458382611yescharacteristic
231C2×C4#1458382711yescharacteristic
232C8—22no—
233C8—6921no—
234C22—13841no—
235C22—13841no—
236C22—13841no—
237C22—13841no—
238C22—6922no—
239C22—6922no—
240C22—6922no—
241C22#1914964111yescharacteristic
242C22#1914964911yescharacteristic
243C22#1914965211yescharacteristic
244C22SD16×D17311yescharacteristic
245C22#1914970011yescharacteristic
246C22#1914970311yescharacteristic
247C22#1914971111yescharacteristic
248C4—42no—
249C4—81no—
250C4—3464no—
251C4—3462no—
252C4—6921no—
253C4C22⋊C4×D17311yescharacteristic
254C4C4×D69211yescharacteristic
255C4#1914965411yescharacteristic
256C4#1914966411yescharacteristic
257C2—13842no—
258C2#2601136911yescharacteristic
259C2#2601157811yescharacteristic
260C2#2601159011yescharacteristic
261C2#2601163111yescharacteristic
262C2#2601163711yescharacteristic
263C2#2601169011yescharacteristic
264C2#2601171611yescharacteristic
subgroupG/H
centerC23D4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#340×C173C2
socleC23×C173C2×D4
prime radicalC22×D692C22
cosocleC22×C4×C173C23
hypercenter#340D173
hypocenterC173#1880
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#19149644C2semidirect
3. C2×#19149602C2semidirect
6. #26011245C2semidirect
7. #340×C173C2semidirect
12. C2×#14583815C4semidirect
72. Dic346C22⋊C4semidirect
85. C2×Dic173SD16semidirect
132. C173#1880semidirect
139. #340D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838152
C2×#191496442
C2×#191496022
C2×#191495992
#260113922
#260112452
#340×C1732
#1880173