Finite Group Databasebeta

← #36352555

Subgroups — #36352555

Order 22144.

228 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012116C211yesmaximal, characteristic
2#26011822C211yesmaximal, characteristic, complemented
3#26011819C211yesmaximal, characteristic, complemented
4#26011418C211yesmaximal, characteristic
5#26011227C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic
7#328×C173C211yesmaximal, characteristic, complemented
8#19149659—21no—
9#19149658C2211yescharacteristic
10#19149657—21no—
11#19149657—21no—
12#19149656—21no—
13#19149655—21no—
14C2×C4⋊Dic173C2211yescharacteristic
15C2×#14583803—21no—
16C2×#14583803C2211yescharacteristic
17C22⋊C4×D173—21no—
18#19149611—21no—
19#19149610—21no—
20#19149604—21no—
21#19149603—21no—
22C22⋊#13063436C2211yescharacteristic
23#19149586—21no—
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×Dic346—21no—
29C2×Dic346D411yescharacteristic
30C22×Dic173—21no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173D411yescharacteristic
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42C22⋊Dic173—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—42no—
48C2×Q8×C173D411yescharacteristic
49C22×C4×C173—21no—
50C22×C4×C173C2311yescharacteristic
51C2×C8×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173D411yescharacteristic
54C22⋊C4×C173—21no—
55C22×D173—21no—
56C22×D173—21no—
57C22×D173—41no—
58C22×D173—41no—
59C22×D173SD1612yes—
60Dic346—41no—
61Dic346—81no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—41no—
66C2×Dic173—41no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C4×D173—81no—
70C4×D173—81no—
71#13063436—41no—
72C23×C173C2×D411yescharacteristic
73Q8×C173—41no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C8×C173—41no—
83C22×C173—21no—
84C22×C173—21no—
85C22×C173#12011yescharacteristic
86C22×C173C2×SD1611yescharacteristic
87C22×C173#13711yescharacteristic
88C2×D173—22no—
89C2×D173—41no—
90C2×D173—41no—
91C2×D173—42no—
92C2×D173—81no—
93C4×C173—21no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173—41no—
97C4×C173—81no—
98Dic173—41no—
99Dic173—41no—
100Dic173—41no—
101Dic173—81no—
102C2×C173—21no—
103C2×C173—21no—
104C2×C173#44811yescharacteristic
105C2×C173#45511yescharacteristic
106C2×C173#45811yescharacteristic
107D173—42no—
108C173#160811yescharacteristic, complemented, Sylow
109#1608—1731nomaximal, Sylow
110#543—1731no—
111#483—1731no—
112#483—1731no—
113#397—1731no—
114#328—1731no—
115#328D17311yescharacteristic, complemented
116#323—1731no—
117C22⋊Q8—3461no—
118C22⋊Q8—3461no—
119C22⋊Q8—3461no—
120C22⋊Q8—3461no—
121C22⋊Q8—3461no—
122C22⋊Q8—1731no—
123C22⋊Q8C2×D17311yescharacteristic
124C2×C4⋊C4—1731no—
125C2×C22⋊C4—3461no—
126C2×C22⋊C4—3461no—
127C2×C22⋊C4—1731no—
128#106—3461no—
129#106—3461no—
130#103—3461no—
131#103—3461no—
132C22⋊C8—1731no—
133C22⋊C8C2×D17311yescharacteristic
134#95—3461no—
135#95C2×D17311yescharacteristic
136C24—1731no—
137C2×Q8—3461no—
138C2×Q8—1731no—
139C2×Q8D69211yescharacteristic
140C22×C4—21no—
141C22×C4—3461no—
142C22×C4—3461no—
143C22×C4—3461no—
144C22×C4—1731no—
145C22×C4C22×D17311yescharacteristic
146C2×C8—21no—
147C2×C8—3461no—
148C4⋊C4—21no—
149C4⋊C4—3461no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—3461no—
154C4⋊C4—3461no—
155C4⋊C4—1731no—
156C4⋊C4D69211yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3461no—
159C22⋊C4—3461no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C22⋊C4—6922no—
164C23—3461no—
165C23—3461no—
166C23—1732no—
167C23—6921no—
168C23—6921no—
169C23D4×D17311yescharacteristic
170Q8—41no—
171Q8—13841no—
172Q8—6921no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—3461no—
180C2×C4—3461no—
181C2×C4—3461no—
182C2×C4—13841no—
183C2×C4—13841no—
184C2×C4—6921no—
185C2×C4—6921no—
186C2×C4—6921no—
187C2×C4—6921no—
188C2×C4D4×D17311yescharacteristic
189C2×C4C2×D69211yescharacteristic
190C8—41no—
191C8—6921no—
192C22—21no—
193C22—21no—
194C22—3462no—
195C22—13841no—
196C22—6922no—
197C22—6921no—
198C22—6921no—
199C22#1914964911yescharacteristic
200C22SD16×D17311yescharacteristic
201C22#1914971111yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—13841no—
208C4—6921no—
209C4—6921no—
210C4—6921no—
211C2—21no—
212C2—21no—
213C2—6922no—
214C2#2601163111yescharacteristic
215C2#2601167011yescharacteristic
216C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radicalC23×D173D4
hypercenter#328D173
hypocenterC173#1608
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26011822C2semidirect
3. #26011819C2semidirect
5. #26011227C2semidirect
7. #328×C173C2semidirect
108. C173#1608semidirect
115. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260121162
#260118222
#260118192
#260114182
#260112272
#260112142
#328×C1732
#1608173