Finite Group Databasebeta

← #36352609

Subgroups — #36352609

Order 22144.

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

normal subgroups only

This group has 240 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#26012079C211yesmaximal, characteristic
2#26011826C211yesmaximal, characteristic, complemented
3#26011809C211yesmaximal, characteristic, complemented
4#26011227C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011215C211yesmaximal, characteristic, complemented
7#399×C173C211yesmaximal, characteristic, complemented
8#19149672—21no—
9#19149665—21no—
10#19149661—21no—
11#19149658C2211yescharacteristic
12#19149653C2211yescharacteristic
13#19149652—21no—
14#19149648—21no—
15C4×C22⋊D173—21no—
16#19149612—21no—
17#19149610—21no—
18#19149602—21no—
19#19149599—21no—
20C22⋊#13063436C2211yescharacteristic
21C22⋊#13063436C2211yescharacteristic
22C2×C4⋊C4×C173—21no—
23C2×C4⋊C4×C173C2211yescharacteristic
24C2×C22⋊C4×C173—21no—
25C2×C22⋊C4×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C4×D173—21no—
33C2×#13063436—21no—
34C2×#13063436—21no—
35C4⋊Dic173D411yescharacteristic
36C4⋊Dic173D411yescharacteristic
37#14583805—21no—
38#14583805—21no—
39C22⋊Dic173—21no—
40C22⋊Dic173—21no—
41C22⋊Dic173—21no—
42#14583803—21no—
43#14583803—21no—
44C4×Dic173—21no—
45C24×C173D411yescharacteristic
46C22×C4×C173—21no—
47C22×C4×C173—21no—
48C22×C4×C173D411yescharacteristic
49C22×C4×C173C2311yescharacteristic
50C4⋊C4×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173—42no—
53C22⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22⋊C4×C173—42no—
56C22×D173—21no—
57Dic346—41no—
58D692—41no—
59C22⋊D173—41no—
60C22⋊D173—41no—
61C22⋊D173—81no—
62C22⋊D173—81no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C4×D173—81no—
67#13063436—41no—
68#13063436—41no—
69C23×C173—21no—
70C23×C173—21no—
71C23×C173—21no—
72C23×C173—41no—
73C23×C173—41no—
74C23×C173C2×D411yescharacteristic
75C2×C4×C173—22no—
76C2×C4×C173—21no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173—41no—
83C2×C4×C173—41no—
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C22×C173—21no—
87C22×C173—21no—
88C22×C173—42no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173—41no—
92C22×C173—81no—
93C22×C173#12011yescharacteristic
94C22×C173#13511yescharacteristic
95C22×C173#13711yescharacteristic
96C2×D173—41no—
97C2×D173—41no—
98C2×D173—41no—
99C4×C173—21no—
100C4×C173—21no—
101C4×C173—42no—
102C4×C173—41no—
103C4×C173—81no—
104Dic173—41no—
105Dic173—41no—
106Dic173—41no—
107C2×C173—21no—
108C2×C173—21no—
109C2×C173—81no—
110C2×C173#45211yescharacteristic
111C2×C173#45311yescharacteristic
112C2×C173#45811yescharacteristic
113D173—81no—
114C173#161811yescharacteristic, complemented, Sylow
115#1618—1731nomaximal, Sylow
116#538—1731no—
117#484—1731no—
118#481—1731no—
119#399D17311yescharacteristic, complemented
120#328—1731no—
121#327—1731no—
122#323—1731no—
123#126—3461no—
124#124—3461no—
125#123—3461no—
126C22⋊Q8—1731no—
127#121—3461no—
128#121—1731no—
129#120—3461no—
130C4×D4—3461no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2×D17311yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4C2×D17311yescharacteristic
135#107—3461no—
136#106—3461no—
137#103—3461no—
138#102—3461no—
139C22⋊C8—1731no—
140C22⋊C8—1731no—
141#95C2×D17311yescharacteristic
142C24C22⋊D17311yescharacteristic
143C2×Q8—1731no—
144C2×D4—3461no—
145C2×D4—3461no—
146C2×D4—3461no—
147C2×D4—1731no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—3461no—
151C22×C4C22⋊D17311yescharacteristic
152C22×C4C22×D17311yescharacteristic
153C2×C8—3461no—
154C2×C8—3461no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—1731no—
161C4⋊C4—1731no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—42no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C42—3461no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—3461no—
177C23C2×C22⋊D17311yescharacteristic
178Q8—6921no—
179D4—13841no—
180D4—13841no—
181D4—6921no—
182D4—6921no—
183D4—6921no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—3461no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—13841no—
197C2×C4D4×D17311yescharacteristic
198C2×C4D4×D17311yescharacteristic
199C8—6921no—
200C8—6921no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—6921no—
209C22—6921no—
210C22—6921no—
211C22#1914964711yescharacteristic
212C22#1914969911yescharacteristic
213C22#1914971011yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—81no—
219C4—6921no—
220C4—6921no—
221C4—6921no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—13841no—
226C2#2601165111yescharacteristic
227C2#2601165811yescharacteristic
228C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#399×C173C2
socleC22×C173#120
prime radical#26012079C2
hypercenter#399D173
hypocenterC173#1618
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #26011826C2semidirect
3. #26011809C2semidirect
4. #26011227C2semidirect
5. #26011224C2semidirect
6. #26011215C2semidirect
7. #399×C173C2semidirect
114. C173#1618semidirect
119. #399D173semidirect

Maximal subgroups

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

subgroupindex
#260120792
#260118262
#260118092
#260112272
#260112242
#260112152
#399×C1732
#1618173