Finite Group Databasebeta

← #36352525

Subgroups — #36352525

Order 22144.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#26012090C211yesmaximal, characteristic
2#26011630C211yesmaximal, characteristic
3#26011410C211yesmaximal, characteristic, complemented
4#26011228C211yesmaximal, characteristic, complemented
5#26011226C211yesmaximal, characteristic
6#26011216C211yesmaximal, characteristic, complemented
7#451×C173C211yesmaximal, characteristic, complemented
8C2×#14583813—21no—
9#19149674—21no—
10#19149670—21no—
11#19149664—21no—
12#19149663—21no—
13#19149655—21no—
14#19149653C2211yescharacteristic
15#19149639—21no—
16C2×#14583805—21no—
17C2×#14583805C2211yescharacteristic
18#19149603—21no—
19C22⋊#13063436C2211yescharacteristic
20#19149586—21no—
21#19149586C2211yescharacteristic
22C22×Q8×C173C2211yescharacteristic
23C2×Q16×C173—21no—
24C22⋊Q8×C173C2211yescharacteristic
25#103×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27C2×D692D411yescharacteristic
28C2×C22⋊D173—21no—
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33#14583813—42no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583805—42no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C2×Q8×C173—21no—
48C2×Q8×C173—21no—
49C2×Q8×C173—42no—
50C2×Q8×C173D411yescharacteristic
51C22×C4×C173—21no—
52C22×C4×C173C2311yescharacteristic
53Q16×C173—42no—
54C2×C8×C173—21no—
55C4⋊C4×C173—21no—
56C4⋊C4×C173D411yescharacteristic
57C22⋊C4×C173—21no—
58C22×D173—21no—
59D692—41no—
60C22⋊D173—41no—
61C22⋊D173—41no—
62C2×Dic173—22no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C4×D173—81no—
72#13063436—41no—
73C23×C173C2×D411yescharacteristic
74Q8×C173—22no—
75Q8×C173—22no—
76Q8×C173—41no—
77Q8×C173—41no—
78Q8×C173—41no—
79C2×C4×C173—22no—
80C2×C4×C173—21no—
81C2×C4×C173—21no—
82C2×C4×C173—41no—
83C2×C4×C173—41no—
84C2×C4×C173—41no—
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173C2×D411yescharacteristic
87C8×C173—41no—
88C22×C173—21no—
89C22×C173—21no—
90C22×C173#12011yescharacteristic
91C22×C173#13511yescharacteristic
92C22×C173#13711yescharacteristic
93C2×D173—41no—
94C2×D173—41no—
95C2×D173—41no—
96C4×C173—21no—
97C4×C173—21no—
98C4×C173—41no—
99C4×C173—42no—
100C4×C173—41no—
101Dic173—42no—
102Dic173—41no—
103Dic173—41no—
104Dic173—41no—
105C2×C173—21no—
106C2×C173—21no—
107C2×C173#45211yescharacteristic
108C2×C173#45611yescharacteristic
109C2×C173B(4,𝔽2)11yescharacteristic
110D173—81no—
111C173#160311yescharacteristic, complemented, Sylow
112#1603—1731nomaximal, Sylow
113#539—1731no—
114#451D17311yescharacteristic, complemented
115#448—1731no—
116#395—1731no—
117#328—1731no—
118#327—1731no—
119#324—1731no—
120C22×Q8C2×D17311yescharacteristic
121C2×Q16—21no—
122C2×SD16—3461no—
123#126—3461no—
124#125—3461no—
125#123—3461no—
126#123—3461no—
127C22⋊Q8—3461no—
128C22⋊Q8C2×D17311yescharacteristic
129#121—1731no—
130C4×D4—3461no—
131C2×C4⋊C4—3461no—
132C2×C4⋊C4—1731no—
133#103—21no—
134#103—3461no—
135C22⋊C8—1731no—
136C22⋊C8C2×D17311yescharacteristic
137#95—3461no—
138#95—1731no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D17311yescharacteristic
143C2×D4—3461no—
144C2×D4—1731no—
145C22×C4—21no—
146C22×C4—3461no—
147C22×C4—3461no—
148C22×C4—1731no—
149C22×C4—1731no—
150C22×C4C22×D17311yescharacteristic
151Q16—42no—
152SD16—6922no—
153C2×C8—21no—
154C2×C8—3461no—
155C4⋊C4—21no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—1731no—
162C4⋊C4—6922no—
163C4⋊C4C22⋊D17311yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C42—3461no—
170C23—3461no—
171C23D4×D17311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—6921no—
178D4—6921no—
179D4—6921no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3462no—
187C2×C4—3461no—
188C2×C4—3461no—
189C2×C4—3461no—
190C2×C4—3461no—
191C2×C4—3461no—
192C2×C4—13841no—
193C2×C4—6921no—
194C2×C4—6921no—
195C2×C4—6921no—
196C2×C4D4×D17311yescharacteristic
197C2×C4C2×C22⋊D17311yescharacteristic
198C8—41no—
199C8—6921no—
200C22—21no—
201C22—21no—
202C22—6921no—
203C22—6921no—
204C22—6921no—
205C22#1914964711yescharacteristic
206C22#1914970211yescharacteristic
207C22#1914971111yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—6922no—
214C4—6921no—
215C4—6921no—
216C4—6921no—
217C2—21no—
218C2—21no—
219C2—13841no—
220C2#2601165211yescharacteristic
221C2#2601167211yescharacteristic
222C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#451×C173C2
socleC22×C173#120
prime radical#19149653C22
hypercenter#451D173
hypocenterC173#1603
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #26011410C2semidirect
4. #26011228C2semidirect
6. #26011216C2semidirect
7. #451×C173C2semidirect
111. C173#1603semidirect
114. #451D173semidirect

Maximal subgroups

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

subgroupindex
#260120902
#260116302
#260114102
#260112282
#260112262
#260112162
#451×C1732
#1603173