Finite Group Databasebeta

← #36363693

Subgroups — #36363693

Order 22144.

380 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 133 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#26012048C211yesmaximal, characteristic
2#26011821C212yesmaximal
3#26011776C212yesmaximal, complemented
4#26011701C212yesmaximal, complemented
5#26011647C212yesmaximal
6#26011596C214yesmaximal, complemented
7#26011548C211yesmaximal, characteristic
8C2×C4×Q8×C173C211yesmaximal, characteristic
9C2×#14583816C2214yes—
10#19149679C2211yescharacteristic
11#19149679C2211yescharacteristic
12#19149671—21no—
13#19149658—21no—
14#19149658C2212yes—
15C4×Dic346—21no—
16C4×Dic346C2212yes—
17#19149638—21no—
18#19149610C2214yes—
19C4⋊#13063436C2212yes—
20#19149603C2214yes—
21#19149602C2214yes—
22C22⋊#13063436C2212yes—
23#19149590C2212yes—
24C22×Q8×C173C2211yescharacteristic
25C4×Q8×C173—21no—
26C4×Q8×C173—21no—
27C4×Q8×C173C2212yes—
28C4×Q8×C173C2212yes—
29C2×C4⋊C4×C173—21no—
30C2×C4⋊C4×C173C2211yescharacteristic
31C2×C42×C173—21no—
32C2×C42×C173C2211yescharacteristic
33C2×Dic346—21no—
34C2×Dic346C2312yes—
35#14583816—28no—
36C2×#13063436C2314yes—
37C4⋊Dic173—21no—
38C4⋊Dic173—21no—
39C4⋊Dic173—21no—
40C4⋊Dic173C2312yes—
41#14583805—22no—
42#14583805—22no—
43C22⋊Dic173—22no—
44C4×Dic173—22no—
45C2×Q8×C173—24no—
46C2×Q8×C173—21no—
47C2×Q8×C173C2312yes—
48C22×C4×C173—21no—
49C22×C4×C173—21no—
50C22×C4×C173D411yescharacteristic
51C22×C4×C173D411yescharacteristic
52C22×C4×C173C2311yescharacteristic
53C4⋊C4×C173—22no—
54C4⋊C4×C173—22no—
55C4⋊C4×C173—21no—
56C4⋊C4×C173C2312yes—
57C42×C173—22no—
58C42×C173—22no—
59C42×C173D411yescharacteristic
60C42×C173D411yescharacteristic
61C42×C173C2311yescharacteristic
62C42×C173C2311yescharacteristic
63Dic346—24no—
64Dic346—42no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67#13063436—24no—
68C23×C173C2×D411yescharacteristic
69Q8×C173—24no—
70Q8×C173—22no—
71Q8×C173#4114yes—
72C2×C4×C173—24no—
73C2×C4×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173—22no—
76C2×C4×C173—22no—
77C2×C4×C173—22no—
78C2×C4×C173—42no—
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2411yescharacteristic
85C22×C173—21no—
86C22×C173—21no—
87C22×C173#13712yes—
88C22×C173C22×D411yescharacteristic
89C4×C173—24no—
90C4×C173—21no—
91C4×C173—21no—
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—42no—
95C4×C173#13712yes—
96C4×C173C2×#4112yes—
97Dic173—42no—
98Dic173—42no—
99C2×C173—22no—
100C2×C173#52511yescharacteristic
101C2×C173C2×#13711yescharacteristic
102C2×C173C2×#13711yescharacteristic
103C173#311311yescharacteristic, complemented, Sylow
104#3113—1731nomaximal, Sylow
105#533—1731no—
106C2×C4×Q8D17311yescharacteristic
107#483—1732no—
108#475—1732no—
109#462—1732no—
110#451—1732no—
111#441—1734no—
112#432—1731no—
113C22×Q8C2×D17311yescharacteristic
114C2×Q16—1734no—
115C4⋊Q8—1731no—
116C4⋊Q8—1731no—
117#125—3461no—
118C22⋊Q8—3461no—
119C22⋊Q8—1732no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—3461no—
123C4×Q8—1732no—
124C4×Q8C2×D17312yes—
125C4×Q8C2×D17312yes—
126#117—3461no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C2×D17311yescharacteristic
129C2×C42—21no—
130C2×C42C2×D17311yescharacteristic
131#106—1734no—
132C4⋊C8—1732no—
133#103—1734no—
134#103—1734no—
135C22⋊C8—1732no—
136#97—1732no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8—3461no—
140C2×Q8—1732no—
141C2×Q8C22×D17312yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4C22⋊D17311yescharacteristic
145C22×C4C22⋊D17311yescharacteristic
146C22×C4C22×D17311yescharacteristic
147Q16—3468no—
148C2×C8—1734no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—3461no—
153C4⋊C4—3461no—
154C4⋊C4—3461no—
155C4⋊C4—3462no—
156C4⋊C4—3462no—
157C4⋊C4—1732no—
158C4⋊C4C22×D17312yes—
159C22⋊C4—3462no—
160C42—22no—
161C42—22no—
162C42—3462no—
163C42C22⋊D17311yescharacteristic
164C42C22⋊D17311yescharacteristic
165C42C22×D17311yescharacteristic
166C42C22×D17311yescharacteristic
167C23C2×C22⋊D17311yescharacteristic
168Q8—24no—
169Q8—22no—
170Q8—3464no—
171Q8—6922no—
172Q8#1458382614yes—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—3462no—
181C2×C4—3462no—
182C2×C4C2×C22⋊D17311yescharacteristic
183C2×C4C2×C22⋊D17311yescharacteristic
184C2×C4C2×C22⋊D17311yescharacteristic
185C2×C4C2×C22⋊D17311yescharacteristic
186C2×C4C2×C22⋊D17311yescharacteristic
187C2×C4C23×D17311yescharacteristic
188C8—3464no—
189C22—21no—
190C22—21no—
191C22#1914970912yes—
192C22C22×C22⋊D17311yescharacteristic
193C4—24no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—6922no—
200C4—6922no—
201C4#1914971012yes—
202C4C2×#1458382612yes—
203C2—22no—
204C2#2601200211yescharacteristic
205C2C2×#1914970911yescharacteristic
206C2C2×#1914971011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×C4×Q8×C173C2
socleC22×C173C22×D4
prime radicalC23×C173C2×D4
hypercenterC2×C4×Q8D173
hypocenterC173#3113
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #26011776C2semidirect
4. #26011701C2semidirect
6. #26011596C2semidirect
103. C173#3113semidirect

Maximal subgroups

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

subgroupindex
#260120482
#260118212
#260117762
#260117012
#260116472
#260115962
#260115482
C2×C4×Q8×C1732
#3113173