Finite Group Databasebeta

← #36353905

Subgroups — #36353905

Order 22144.

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

normal subgroups only

This group has 338 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×Dic692C211yesmaximal, characteristic
2C2×#19149644C211yesmaximal, characteristic
3C2×#19149602C211yesmaximal, characteristic
4C2×#19149602C211yesmaximal, characteristic
5#26011396C211yesmaximal, characteristic
6#26011245C211yesmaximal, characteristic
7#340×C173C211yesmaximal, characteristic
8C22×Dic346C2211yescharacteristic
9C22×Dic346C2211yescharacteristic
10C2×Dic692—22no—
11C2×Dic692—22no—
12C2×Dic692C414yescomplemented
13C22×#13063436C2211yescharacteristic
14#19149644—22no—
15#19149644—22no—
16C2×#14583805—21no—
17C2×C4×Dic173—21no—
18C2×C4×Dic173C2211yescharacteristic
19#19149602—22no—
20#19149602—22no—
21#19149586—21no—
22#19149586—21no—
23C22×C8×C173C2211yescharacteristic
24C2×C4⋊C4×C173C2211yescharacteristic
25C2×C4⋊C4×C173C2211yescharacteristic
26C2×Dic346—21no—
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346—21no—
31C2×Dic346—21no—
32C2×Dic346—21no—
33C2×Dic346—21no—
34C2×Dic346C2×C412yes—
35C2×Dic346C2×C412yes—
36C2×Dic346D412yes—
37C2×Dic346D412yes—
38C22×Dic173—21no—
39C22×Dic173—21no—
40C22×Dic173D411yescharacteristic
41C22×Dic173D411yescharacteristic
42Dic692—28no—
43C2×#13063436—21no—
44C2×#13063436—21no—
45C2×#13063436—21no—
46#14583805—24no—
47C4×Dic173—24no—
48C4×Dic173—21no—
49C4×Dic173—21no—
50C22×C4×C173—21no—
51C22×C4×C173—21no—
52C22×C4×C173C2311yescharacteristic
53C2×C8×C173—21no—
54C2×C8×C173—21no—
55C2×C8×C173C2×C412yes—
56C4⋊C4×C173—21no—
57C4⋊C4×C173—21no—
58C4⋊C4×C173—21no—
59C4⋊C4×C173D412yes—
60Dic346—22no—
61Dic346—22no—
62Dic346—22no—
63Dic346—22no—
64Dic346—22no—
65Dic346—41no—
66Dic346—41no—
67Dic346C22⋊C414yes—
68C2×Dic173—22no—
69C2×Dic173—22no—
70C2×Dic173—21no—
71C2×Dic173—22no—
72C2×Dic173—21no—
73C2×Dic173—22no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—41no—
78C2×Dic173—41no—
79C2×Dic173Q1612yescomplemented
80#13063436—41no—
81C23×C173C2×D411yescharacteristic
82C2×C4×C173—22no—
83C2×C4×C173—41no—
84C2×C4×C173—41no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173C22×C411yescharacteristic
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D411yescharacteristic
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173#4111yescharacteristic
93C2×C4×C173#4111yescharacteristic
94C8×C173—22no—
95C22×C173C4×D411yescharacteristic
96C22×C173#12011yescharacteristic
97C22×C173#12111yescharacteristic
98C22×C173C2×Q1611yescharacteristic
99C22×C173#13511yescharacteristic
100C22×C173#13711yescharacteristic
101C22×C173#13711yescharacteristic
102C4×C173—42no—
103C4×C173—81no—
104C4×C173C2×C22⋊C411yescharacteristic
105C4×C173C4×D411yescharacteristic
106C4×C173#12111yescharacteristic
107C4×C173#12311yescharacteristic
108Dic173—24no—
109Dic173—22no—
110Dic173—42no—
111Dic173—41no—
112C2×C173#38611yescharacteristic
113C2×C173C4×Q1611yescharacteristic
114C2×C173#44111yescharacteristic
115C2×C173#45111yescharacteristic
116C2×C173#45211yescharacteristic
117C2×C173#46211yescharacteristic
118C2×C173#46411yescharacteristic
119C173#187811yescharacteristic, complemented, Sylow
120#1878—1731nomaximal, Sylow
121C22×Q16—1731no—
122C2×C4×Q8—1731no—
123C2×#103—1731no—
124C2×#103—1731no—
125#391—1731no—
126#340D17311yescharacteristic
127#336—1731no—
128C22×Q8—1731no—
129C22×Q8—1731no—
130C2×Q16—3462no—
131C2×Q16—3462no—
132C2×Q16—1734no—
133C22×C8—1731no—
134C22×C8C2×D17311yescharacteristic
135C4×Q8—3462no—
136C4×Q8—3462no—
137C2×C4⋊C4—3461no—
138C2×C4⋊C4C2×D17311yescharacteristic
139C2×C4⋊C4C2×D17311yescharacteristic
140C2×C42—3461no—
141C2×C42—1731no—
142#103—3462no—
143#103—3462no—
144#95—3461no—
145#95—3461no—
146C2×Q8—3461no—
147C2×Q8—3461no—
148C2×Q8—3461no—
149C2×Q8—3461no—
150C2×Q8—3461no—
151C2×Q8—3461no—
152C2×Q8—3461no—
153C2×Q8—3461no—
154C2×Q8—1732no—
155C2×Q8—1732no—
156C2×Q8—1732no—
157C2×Q8—1732no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—3461no—
161C22×C4—3461no—
162C22×C4—1731no—
163C22×C4—1731no—
164C22×C4C22×D17311yescharacteristic
165Q16—3468no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—3461no—
169C2×C8—3461no—
170C2×C8—3461no—
171C2×C8C4×D17312yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—3464no—
176C4⋊C4D69212yes—
177C42—3464no—
178C42—3461no—
179C42—3461no—
180C23D4×D17311yescharacteristic
181Q8—3462no—
182Q8—3462no—
183Q8—3462no—
184Q8—3462no—
185Q8—3462no—
186Q8—1734no—
187Q8—6921no—
188Q8—6921no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—3462no—
196C2×C4—3461no—
197C2×C4—3462no—
198C2×C4—3462no—
199C2×C4—3461no—
200C2×C4—3462no—
201C2×C4—3461no—
202C2×C4—3461no—
203C2×C4—3461no—
204C2×C4—1732no—
205C2×C4—6921no—
206C2×C4—6921no—
207C2×C4C2×C4×D17311yescharacteristic
208C2×C4D4×D17311yescharacteristic
209C2×C4D4×D17311yescharacteristic
210C2×C4C2×D69211yescharacteristic
211C2×C4#1458382611yescharacteristic
212C2×C4#1458382711yescharacteristic
213C8—22no—
214C8—6921no—
215C22#1914964111yescharacteristic
216C22#1914964911yescharacteristic
217C22#1914965211yescharacteristic
218C22Q16×D17311yescharacteristic
219C22#1914969811yescharacteristic
220C22#1914970811yescharacteristic
221C22#1914970811yescharacteristic
222C4—42no—
223C4—81no—
224C4—3462no—
225C4—3464no—
226C4—6921no—
227C4—6922no—
228C4C22⋊C4×D17311yescharacteristic
229C4C4×D69211yescharacteristic
230C4#1914965411yescharacteristic
231C4#1914966411yescharacteristic
232C2#2601136911yescharacteristic
233C2#2601158511yescharacteristic
234C2#2601159711yescharacteristic
235C2#2601164811yescharacteristic
236C2#2601165411yescharacteristic
237C2#2601170011yescharacteristic
238C2#2601171211yescharacteristic
subgroupG/H
centerC23D4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#340×C173C2
socle, prime radicalC23×C173C2×D4
cosocleC22×C4×C173C23
hypercenter#340D173
hypocenterC173#1878
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
12. C2×Dic692C4semidirect
79. C2×Dic173Q16semidirect
119. C173#1878semidirect

Maximal subgroups

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

subgroupindex
C22×Dic6922
C2×#191496442
C2×#191496022
#260113962
#260112452
#340×C1732
#1878173