Finite Group Databasebeta

← #36352524

Subgroups — #36352524

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#26012088C211yesmaximal, characteristic
2#26011647C211yesmaximal, characteristic, complemented
3#26011630C211yesmaximal, characteristic
4#26011227C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic
6#26011217C211yesmaximal, characteristic, complemented
7#395×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—21no—
9C2×#14583813—21no—
10#19149674—21no—
11#19149669—21no—
12#19149664—21no—
13#19149662—21no—
14#19149658C2211yescharacteristic
15#19149657—21no—
16#19149653C2211yescharacteristic
17C4×C22⋊D173—21no—
18#19149603—21no—
19#19149603—21no—
20C22⋊#13063436C2211yescharacteristic
21C22⋊#13063436C2211yescharacteristic
22C22×Q8×C173C2211yescharacteristic
23C2×C4⋊C4×C173—21no—
24C2×C4⋊C4×C173C2211yescharacteristic
25#95×C173—21no—
26#95×C173C2211yescharacteristic
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C4×D173—21no—
31#14583816—42no—
32#14583813—42no—
33C2×#13063436—21no—
34C2×#13063436—21no—
35C4⋊Dic173D411yescharacteristic
36C4⋊Dic173D411yescharacteristic
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41C22⋊Dic173—21no—
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—
50C22×C4×C173—21no—
51C22×C4×C173—21no—
52C22×C4×C173D411yescharacteristic
53C22×C4×C173D411yescharacteristic
54C22×C4×C173C2311yescharacteristic
55C4⋊C4×C173—21no—
56C4⋊C4×C173—21no—
57C4⋊C4×C173—42no—
58C22×D173—21no—
59Dic346—41no—
60D692—41no—
61C22⋊D173—41no—
62C22⋊D173—41no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C4×D173—81no—
67#13063436—41no—
68#13063436—41no—
69C23×C173C2×D411yescharacteristic
70Q8×C173—22no—
71Q8×C173—22no—
72Q8×C173—41no—
73Q8×C173—41no—
74C2×C4×C173—22no—
75C2×C4×C173—22no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173—41no—
83C2×C4×C173—41no—
84C2×C4×C173—41no—
85C2×C4×C173—41no—
86C2×C4×C173C2×D411yescharacteristic
87C2×C4×C173C2×D411yescharacteristic
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—42no—
99C4×C173—41no—
100C4×C173—42no—
101C4×C173—41no—
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#451—1731no—
115#448—1731no—
116#395D17311yescharacteristic, complemented
117#328—1731no—
118#327—1731no—
119#324—1731no—
120C22×Q8C2×D17311yescharacteristic
121C2×Q16—3461no—
122C2×SD16—3461no—
123#126—3461no—
124#125—3461no—
125#123—3461no—
126#123—3461no—
127C22⋊Q8—3461no—
128C22⋊Q8—1731no—
129#121—1731no—
130C4×D4—3461no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2×D17311yescharacteristic
133#103—3461no—
134#103—3461no—
135C22⋊C8—1731no—
136C22⋊C8—1731no—
137#95—21no—
138#95C2×D17311yescharacteristic
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8—1731no—
143C2×D4—3461no—
144C2×D4—1731no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—3461no—
148C22×C4C22⋊D17311yescharacteristic
149C22×C4C22⋊D17311yescharacteristic
150C22×C4C22×D17311yescharacteristic
151Q16—6922no—
152SD16—6922no—
153C2×C8—3461no—
154C2×C8—3461no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—3461no—
162C4⋊C4—1731no—
163C4⋊C4—1731no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C42—3461no—
170C23—3461no—
171C23C2×C22⋊D17311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—6921no—
177D4—6921no—
178D4—6921no—
179D4—6921no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—3461no—
193C2×C4—3461no—
194C2×C4—3461no—
195C2×C4—13841no—
196C2×C4D4×D17311yescharacteristic
197C2×C4D4×D17311yescharacteristic
198C8—6921no—
199C8—6921no—
200C22—21no—
201C22—21no—
202C22—6921no—
203C22—6921no—
204C22—6921no—
205C22#1914964711yescharacteristic
206C22#1914969911yescharacteristic
207C22#1914971011yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—42no—
213C4—41no—
214C4—6921no—
215C4—6921no—
216C4—6921no—
217C2—21no—
218C2—21no—
219C2—13841no—
220C2#2601165111yescharacteristic
221C2#2601167311yescharacteristic
222C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#395×C173C2
socleC22×C173#120
prime radical#19149653C22
hypercenter#395D173
hypocenterC173#1603
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. #26011647C2semidirect
4. #26011227C2semidirect
6. #26011217C2semidirect
7. #395×C173C2semidirect
111. C173#1603semidirect
116. #395D173semidirect

Maximal subgroups

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

subgroupindex
#260120882
#260116472
#260116302
#260112272
#260112242
#260112172
#395×C1732
#1603173