Finite Group Databasebeta

← #36354703

Subgroups — #36354703

Order 22144.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#19149678C211yesmaximal, characteristic
2C2×#19149604C211yesmaximal, characteristic
3C2×#19149602C211yesmaximal, characteristic
4#26011396C211yesmaximal, characteristic
5#26011253C211yesmaximal, characteristic
6#26011245C211yesmaximal, characteristic
7#340×C173C211yesmaximal, characteristic
8C22×Dic346—21no—
9C22×Dic346C2211yescharacteristic
10C22×#13063436C2211yescharacteristic
11#19149678—22no—
12#19149678—22no—
13C2×C4⋊Dic173C2211yescharacteristic
14C2×#14583805—21no—
15C2×C4×Dic173C2211yescharacteristic
16#19149604—22no—
17#19149602—22no—
18#19149586—21no—
19#19149586—21no—
20C22×C8×C173C2211yescharacteristic
21C2×C4⋊C4×C173C2211yescharacteristic
22C2×C4⋊C4×C173C2211yescharacteristic
23C2×Dic346—24no—
24C2×Dic346—21no—
25C2×Dic346—21no—
26C2×Dic346—21no—
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346—42no—
30C2×Dic346—42no—
31C2×Dic346Q812yescomplemented
32C22×Dic173—21no—
33C22×Dic173—21no—
34C22×Dic173D411yescharacteristic
35C22×Dic173D411yescharacteristic
36C2×#13063436—21no—
37C2×#13063436—21no—
38C2×#13063436—21no—
39C4⋊Dic173—21no—
40C4⋊Dic173D412yes—
41#14583805—24no—
42C4×Dic173—21no—
43C4×Dic173—21no—
44C22×C4×C173—21no—
45C22×C4×C173—21no—
46C22×C4×C173C2311yescharacteristic
47C2×C8×C173—21no—
48C2×C8×C173—21no—
49C2×C8×C173—21no—
50C4⋊C4×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173D412yes—
54Dic346—22no—
55Dic346—22no—
56Dic346—48no—
57Dic346—41no—
58Dic346—41no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—22no—
64C2×Dic173—22no—
65C2×Dic173—41no—
66C2×Dic173—41no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C2×Dic173SD1612yes—
70C2×Dic173Q1612yes—
71#13063436—41no—
72C23×C173C2×D411yescharacteristic
73C2×C4×C173—22no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×Q811yescharacteristic
82C2×C4×C173#4111yescharacteristic
83C2×C4×C173#4111yescharacteristic
84C2×C4×C173#4111yescharacteristic
85C8×C173—41no—
86C22×C173#12011yescharacteristic
87C22×C173C22⋊Q811yescharacteristic
88C22×C173#12411yescharacteristic
89C22×C173C2×SD1611yescharacteristic
90C22×C173C2×Q1611yescharacteristic
91C22×C173AGL(1,ℤ/8)11yescharacteristic
92C22×C173#13711yescharacteristic
93C4×C173—42no—
94C4×C173—81no—
95C4×C173C22⋊Q811yescharacteristic
96C4×C173C22⋊Q811yescharacteristic
97C4×C173#12411yescharacteristic
98C4×C173#12411yescharacteristic
99Dic173—22no—
100Dic173—22no—
101Dic173—42no—
102Dic173—42no—
103C2×C173#39311yescharacteristic
104C2×C173#45011yescharacteristic
105C2×C173#45111yescharacteristic
106C2×C173#47511yescharacteristic
107C2×C173#47711yescharacteristic
108C2×C173#48711yescharacteristic
109C2×C173#48911yescharacteristic
110C173#202411yescharacteristic, complemented, Sylow
111#2024—1731nomaximal, Sylow
112C2×C4⋊Q8—1731no—
113C2×#103—1731no—
114C2×#103—1731no—
115#391—1731no—
116#340—1731no—
117#340D17311yescharacteristic
118#336—1731no—
119C22×Q8—3461no—
120C22×Q8—1731no—
121C22×C8—1731no—
122C22×C8C2×D17311yescharacteristic
123C4⋊Q8—3462no—
124C4⋊Q8—3462no—
125C2×C4⋊C4—3461no—
126C2×C4⋊C4—1731no—
127C2×C4⋊C4C2×D17311yescharacteristic
128C2×C4⋊C4C2×D17311yescharacteristic
129C2×C42—1731no—
130#103—3462no—
131#103—3462no—
132#95—3461no—
133#95—3461no—
134C2×Q8—3464no—
135C2×Q8—3461no—
136C2×Q8—3461no—
137C2×Q8—3461no—
138C2×Q8—3461no—
139C2×Q8—3461no—
140C2×Q8—1732no—
141C2×Q8—6922no—
142C2×Q8—6922no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—3461no—
146C22×C4—3461no—
147C22×C4—1731no—
148C22×C4—1731no—
149C22×C4C22×D17311yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—3461no—
154C2×C8—3461no—
155C2×C8—3461no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—3464no—
160C4⋊C4—3461no—
161C4⋊C4—1732no—
162C4⋊C4D69212yes—
163C42—3461no—
164C42—3461no—
165C23D4×D17311yescharacteristic
166Q8—3462no—
167Q8—3462no—
168Q8—6928no—
169Q8—6921no—
170Q8—6921no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—3461no—
178C2×C4—3461no—
179C2×C4—3461no—
180C2×C4—3461no—
181C2×C4—3462no—
182C2×C4—3462no—
183C2×C4—1732no—
184C2×C4—1732no—
185C2×C4—6921no—
186C2×C4—6921no—
187C2×C4—6921no—
188C2×C4—6921no—
189C2×C4D4×D17311yescharacteristic
190C2×C4C2×D69211yescharacteristic
191C2×C4Q8×D17311yescharacteristic
192C2×C4#1458382511yescharacteristic
193C2×C4#1458382611yescharacteristic
194C2×C4#1458382611yescharacteristic
195C8—41no—
196C8—6921no—
197C22#1914964911yescharacteristic
198C22#1914965711yescharacteristic
199C22#1914966511yescharacteristic
200C22SD16×D17311yescharacteristic
201C22Q16×D17311yescharacteristic
202C22#1914970311yescharacteristic
203C22#1914970811yescharacteristic
204C4—42no—
205C4—81no—
206C4—3462no—
207C4—3462no—
208C4—6922no—
209C4—6922no—
210C4#1914965711yescharacteristic
211C4#1914965911yescharacteristic
212C4#1914966511yescharacteristic
213C4#1914966711yescharacteristic
214C2#2601140411yescharacteristic
215C2#2601164311yescharacteristic
216C2#2601164811yescharacteristic
217C2#2601177511yescharacteristic
218C2#2601178511yescharacteristic
219C2#2601184011yescharacteristic
220C2#2601184911yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#340×C173C2
socle, prime radicalC23×C173C2×D4
hypercenter#340D173
hypocenterC173#2024
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic346Q8semidirect
110. C173#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#191496782
C2×#191496042
C2×#191496022
#260113962
#260112532
#260112452
#340×C1732
#2024173