Finite Group Databasebeta

← #36363095

Subgroups — #36363095

Order 22144.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14583816C211yesmaximal, characteristic, complemented
2#26012062C211yesmaximal, characteristic
3#26011969C211yesmaximal, characteristic, complemented
4#26011711C214yesmaximal, complemented
5#26011699C214yesmaximal, complemented
6#26011519C211yesmaximal, characteristic, complemented
7#26011501C211yesmaximal, characteristic, complemented
8#26011487C211yesmaximal, characteristic
9C2×#137×C173C211yesmaximal, characteristic
10C22×Dic346C2211yescharacteristic
11C2×#14583816—24no—
12C2×#14583816C2214yes—
13C22×#13063436C2211yescharacteristic
14#19149677C2212yes—
15#19149666C2212yes—
16#19149660—22no—
17#19149658—22no—
18Q8×Dic173—22no—
19Q8×Dic173C2214yes—
20#19149638—22no—
21#19149638C2211yescharacteristic
22#19149638C2211yescharacteristic
23#19149608C2214yes—
24#19149604C2214yes—
25#19149603C2212yes—
26#19149600C2212yes—
27C2×#41×C173C2211yescharacteristic
28C22×Q8×C173C2211yescharacteristic
29#137×C173—24no—
30C2×Q16×C173C2212yes—
31C2×SD16×C173C2212yes—
32C2×M16×C173C2211yescharacteristic
33C2×Dic346—21no—
34C2×Dic346—22no—
35C2×Dic346—22no—
36C2×Dic346C2312yes—
37C22×Dic173—21no—
38#14583816—28no—
39C2×#13063436—22no—
40C2×#13063436C2312yes—
41C4⋊Dic173—24no—
42C4⋊Dic173D412yes—
43C4⋊Dic173C2312yes—
44#14583805—24no—
45C22⋊Dic173—24no—
46C22⋊Dic173—21no—
47C22⋊Dic173D412yes—
48C4×Dic173—24no—
49C4×Dic173—21no—
50C4×Dic173C2312yes—
51#41×C173—22no—
52#41×C173—22no—
53C2×Q8×C173—21no—
54C2×Q8×C173—22no—
55C2×Q8×C173D414yes—
56C2×Q8×C173C2311yescharacteristic
57C2×Q8×C173C2312yes—
58C2×D4×C173—21no—
59C2×D4×C173C2311yescharacteristic
60C22×C4×C173—21no—
61C22×C4×C173—21no—
62C22×C4×C173C2311yescharacteristic
63Q16×C173—24no—
64SD16×C173—24no—
65M16×C173—22no—
66C2×C8×C173C2312yes—
67Dic346—22no—
68Dic346—22no—
69Dic346—42no—
70C2×Dic173—22no—
71C2×Dic173—22no—
72C2×Dic173—42no—
73C2×Dic173C2×D414yes—
74#13063436—22no—
75C23×C173—21no—
76C23×C173C2×D411yescharacteristic
77Q8×C173—24no—
78Q8×C173—22no—
79Q8×C173—21no—
80Q8×C173—21no—
81Q8×C173C2×D414yes—
82D4×C173—21no—
83D4×C173—21no—
84D4×C173—42no—
85C2×C4×C173—24no—
86C2×C4×C173—22no—
87C2×C4×C173—21no—
88C2×C4×C173—42no—
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D412yes—
91C2×C4×C173#4112yes—
92C2×C4×C173C2411yescharacteristic
93C8×C173—22no—
94C22×C173—22no—
95C22×C173—41no—
96C22×C173—42no—
97C22×C173#12112yes—
98C22×C173C22×D411yescharacteristic
99C4×C173—24no—
100C4×C173—41no—
101C4×C173#12112yes—
102C4×C173C22×D411yescharacteristic
103C4×C173C2×#4111yescharacteristic
104Dic173—24no—
105Dic173—42no—
106Dic173—42no—
107C2×C173—22no—
108C2×C173—81no—
109C2×C173C2×#12111yescharacteristic
110C2×C173#57812yes—
111C173#304411yescharacteristic, complemented, Sylow
112#3044—1731nomaximal, Sylow
113C2×#137D17311yescharacteristic
114C22×Q16—1731no—
115#536—1731no—
116#519—1731no—
117#464—1734no—
118#462—1734no—
119#424—1731no—
120#419—1731no—
121#416—1731no—
122C2×#41C2×D17311yescharacteristic
123C22×Q8—1731no—
124C22×Q8C2×D17311yescharacteristic
125#137—24no—
126C2×Q16—3464no—
127C2×Q16—1734no—
128C2×Q16C2×D17312yes—
129C2×SD16C2×D17312yes—
130C2×M16C2×D17311yescharacteristic
131C22×C8—1731no—
132C4⋊Q8—1732no—
133#124—1732no—
134#123—3462no—
135C22⋊Q8—3462no—
136C4×Q8—3462no—
137C4×Q8—1734no—
138#117—3462no—
139#117—1731no—
140#117—1731no—
141C4⋊C8—1734no—
142#103—1732no—
143#103—1734no—
144#102—1732no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3461no—
150C2×Q8—3462no—
151C2×Q8—3462no—
152C2×Q8—1732no—
153C2×Q8C22⋊D17314yes—
154C2×Q8C22×D17311yescharacteristic
155C2×Q8C22×D17312yes—
156C2×D4—21no—
157C2×D4C22×D17311yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—3461no—
161C22×C4C22×D17311yescharacteristic
162Q16—24no—
163Q16—3468no—
164SD16—24no—
165M16—22no—
166C2×C8—3462no—
167C2×C8—1732no—
168C2×C8C22×D17312yes—
169C4⋊C4—3464no—
170C4⋊C4—3464no—
171C4⋊C4—1732no—
172C4⋊C4—1732no—
173C22⋊C4—3461no—
174C22⋊C4—3464no—
175C22⋊C4—1732no—
176C42—3464no—
177C42—3461no—
178C42—1732no—
179C23—21no—
180C23D4×D17311yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—3462no—
186Q8—3462no—
187Q8—6922no—
188Q8C2×C22⋊D17314yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—3462no—
197C2×C4—3462no—
198C2×C4—1734no—
199C2×C4—6922no—
200C2×C4D4×D17311yescharacteristic
201C2×C4C2×C22⋊D17312yes—
202C2×C4#1458382512yes—
203C2×C4C23×D17311yescharacteristic
204C8—22no—
205C8—3462no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1914965112yes—
210C22C2×D4×D17311yescharacteristic
211C4—24no—
212C4—41no—
213C4—3464no—
214C4—6922no—
215C4—6922no—
216C4#1914965112yes—
217C4C22×C22⋊D17311yescharacteristic
218C4C2×#1458382511yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1914965111yescharacteristic
222C2#2601230712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
prime radicalC2×#41×C173C22
hypercenterC2×#137D173
hypocenterC173#3044
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14583816C2semidirect
3. #26011969C2semidirect
4. #26011711C2semidirect
5. #26011699C2semidirect
6. #26011519C2semidirect
7. #26011501C2semidirect
111. C173#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#145838162
#260120622
#260119692
#260117112
#260116992
#260115192
#260115012
#260114872
C2×#137×C1732
#3044173