Finite Group Databasebeta

← #36352487

Subgroups — #36352487

Order 22144.

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

normal subgroups only

This group has 316 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, complemented
2#26011644C211yesmaximal, characteristic, complemented
3#26011641C211yesmaximal, characteristic, complemented
4#26011401C211yesmaximal, characteristic
5#26011227C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8C22×D692—21no—
9C22×D692C2211yescharacteristic
10C2×#14583815—21no—
11C2×#14583813—21no—
12#19149659—21no—
13#19149658C2211yescharacteristic
14#19149657—21no—
15#19149657—21no—
16#19149656—21no—
17#19149655—21no—
18C2×#14583803—21no—
19C2×#14583803C2211yescharacteristic
20C22⋊C4×D173—21no—
21#19149601—21no—
22#19149599—21no—
23C22⋊#13063436C2211yescharacteristic
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173—21no—
28C23×D173D411yescharacteristic
29C2×Dic346—21no—
30C2×Dic346D411yescharacteristic, complemented
31C2×D692—21no—
32C2×D692—21no—
33C2×D692—42no—
34C2×D692—42no—
35C2×D692—42no—
36C2×D692—42no—
37C22×Dic173D411yescharacteristic
38C2×C4×D173—21no—
39C2×C4×D173—21no—
40#14583815—42no—
41#14583813—42no—
42C2×#13063436—21no—
43C4⋊Dic173—21no—
44C4⋊Dic173D411yescharacteristic, complemented
45#14583805—21no—
46#14583805—21no—
47#14583805—21no—
48C22⋊Dic173—21no—
49#14583803—21no—
50#14583803—21no—
51#14583803—21no—
52#14583803—21no—
53#14583803—42no—
54C2×Q8×C173D411yescharacteristic, complemented
55C22×C4×C173—21no—
56C22×C4×C173C2311yescharacteristic
57C2×C8×C173—21no—
58C4⋊C4×C173—21no—
59C4⋊C4×C173D411yescharacteristic, complemented
60C22⋊C4×C173—21no—
61C22×D173—21no—
62C22×D173—21no—
63C22×D173—22no—
64C22×D173—41no—
65C22×D173—41no—
66C22×D173—41no—
67C22×D173—41no—
68C22×D173—41no—
69C22×D173—41no—
70C22×D173—41no—
71C22×D173—41no—
72C22×D173SD1612yescomplemented
73Dic346—41no—
74Dic346—81no—
75D692—22no—
76D692—22no—
77D692—41no—
78D692—41no—
79D692—48no—
80C2×Dic173—21no—
81C2×Dic173—21no—
82C2×Dic173—21no—
83C2×Dic173—41no—
84C4×D173—81no—
85C4×D173—81no—
86#13063436—41no—
87C23×C173C2×D411yescharacteristic
88Q8×C173—41no—
89C2×C4×C173—22no—
90C2×C4×C173—21no—
91C2×C4×C173—21no—
92C2×C4×C173—41no—
93C2×C4×C173—41no—
94C2×C4×C173—41no—
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C8×C173—41no—
98C22×C173—21no—
99C22×C173—21no—
100C22×C173#12011yescharacteristic
101C22×C173C2×SD1611yescharacteristic
102C22×C173#13711yescharacteristic
103C2×D173—22no—
104C2×D173—41no—
105C2×D173—41no—
106C2×D173—42no—
107C2×D173—42no—
108C2×D173—42no—
109C2×D173—42no—
110C2×D173—81no—
111C2×D173—81no—
112C2×D173—81no—
113C2×D173—81no—
114C2×D173—81no—
115C4×C173—21no—
116C4×C173—21no—
117C4×C173—42no—
118C4×C173—41no—
119C4×C173—41no—
120Dic173—41no—
121Dic173—41no—
122Dic173—41no—
123C2×C173—21no—
124C2×C173—21no—
125C2×C173#44811yescharacteristic
126C2×C173#45311yescharacteristic
127C2×C173B(4,𝔽2)11yescharacteristic
128D173—42no—
129D173—82no—
130C173#159611yescharacteristic, complemented, Sylow
131#1596—1731nomaximal, Sylow
132#543—1731no—
133#450—1731no—
134#450—1731no—
135#392—1731no—
136#328—1731no—
137#328D17311yescharacteristic, complemented
138#323—1731no—
139C22×D4—3461no—
140C22×D4—1731no—
141C2×SD16—3461no—
142C2×SD16—3461no—
143C22⋊Q8—3461no—
144C22⋊Q8—3461no—
145C22⋊Q8—3461no—
146C22⋊Q8—3461no—
147C22⋊Q8—3461no—
148C22⋊Q8—1731no—
149C22⋊Q8C2×D17311yescharacteristic
150C2×C22⋊C4—3461no—
151C2×C22⋊C4—3461no—
152C2×C22⋊C4—1731no—
153#102—3461no—
154#102—3461no—
155C22⋊C8—1731no—
156C22⋊C8C2×D17311yescharacteristic
157#95C2×D17311yescharacteristic
158C24—3461no—
159C24—1731no—
160C2×Q8—3461no—
161C2×Q8—1731no—
162C2×Q8D69211yescharacteristic, complemented
163C2×D4—3461no—
164C2×D4—3461no—
165C2×D4—6922no—
166C2×D4—6922no—
167C2×D4—6922no—
168C2×D4—6922no—
169C22×C4—21no—
170C22×C4—3461no—
171C22×C4—3461no—
172C22×C4—1731no—
173C22×C4C22×D17311yescharacteristic
174SD16—6922no—
175SD16—6922no—
176C2×C8—21no—
177C2×C8—3461no—
178C4⋊C4—21no—
179C4⋊C4—3461no—
180C4⋊C4—3461no—
181C4⋊C4—3461no—
182C4⋊C4—3461no—
183C4⋊C4—1731no—
184C4⋊C4D69211yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—3461no—
187C22⋊C4—3461no—
188C22⋊C4—3461no—
189C22⋊C4—3461no—
190C22⋊C4—3461no—
191C22⋊C4—6922no—
192C23—3462no—
193C23—3461no—
194C23—3461no—
195C23—1732no—
196C23—6921no—
197C23—6921no—
198C23—6921no—
199C23—6921no—
200C23—6921no—
201C23—6921no—
202C23—6921no—
203C23—6921no—
204C23D4×D17311yescharacteristic
205Q8—41no—
206Q8—13841no—
207Q8—6921no—
208D4—3462no—
209D4—3462no—
210D4—6921no—
211D4—6921no—
212D4—6928no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—3461no—
220C2×C4—3461no—
221C2×C4—3461no—
222C2×C4—13841no—
223C2×C4—13841no—
224C2×C4—6921no—
225C2×C4D4×D17311yescharacteristic
226C2×C4C2×D69211yescharacteristic
227C8—41no—
228C8—6921no—
229C22—21no—
230C22—21no—
231C22—3462no—
232C22—13841no—
233C22—13841no—
234C22—13841no—
235C22—13841no—
236C22—13841no—
237C22—6922no—
238C22—6922no—
239C22—6922no—
240C22—6922no—
241C22—6921no—
242C22—6921no—
243C22#1914964911yescharacteristic
244C22SD16×D17311yescharacteristic
245C22#1914971111yescharacteristic
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—6921no—
252C4—6921no—
253C4—6921no—
254C2—21no—
255C2—21no—
256C2—13842no—
257C2—6922no—
258C2#2601163111yescharacteristic
259C2#2601166011yescharacteristic
260C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radical#26011401C2
hypercenter#328D173
hypocenterC173#1596
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
1. #26012116C2semidirect
2. #26011644C2semidirect
3. #26011641C2semidirect
5. #26011227C2semidirect
6. #26011214C2semidirect
7. #328×C173C2semidirect
30. C2×Dic346D4semidirect
44. C4⋊Dic173D4semidirect
54. C2×Q8×C173D4semidirect
59. C4⋊C4×C173D4semidirect
72. C22×D173SD16semidirect
130. C173#1596semidirect
137. #328D173semidirect
162. C2×Q8D692semidirect
184. C4⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260121162
#260116442
#260116412
#260114012
#260112272
#260112142
#328×C1732
#1596173