Finite Group Databasebeta

← #36352480

Subgroups — #36352480

Order 22144.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#26012053C211yesmaximal, characteristic, complemented
2#26011625C212yesmaximal, complemented
3#26011224C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic, complemented
5#392×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173—21no—
7C2×#14583811—22no—
8#19149663—22no—
9#19149653—21no—
10#19149653C2212yes—
11#19149649—22no—
12#19149600—22no—
13C22⋊#13063436C2212yescomplemented
14C22×D4×C173—21no—
15C22×D4×C173C2211yescharacteristic, complemented
16C2×C22⋊C4×C173—21no—
17C2×C22⋊C4×C173C2211yescharacteristic
18#95×C173C2211yescharacteristic, complemented
19C23×D173—21no—
20C2×D692—21no—
21C2×D692D412yescomplemented
22C2×C22⋊D173—22no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—42no—
25C2×C22⋊D173—42no—
26C22×Dic173—21no—
27#14583811—44no—
28C2×#13063436—22no—
29C4⋊Dic173—21no—
30C4⋊Dic173D412yescomplemented
31#14583803—22no—
32#14583803—22no—
33C24×C173—21no—
34C24×C173D411yescharacteristic
35C2×D4×C173—22no—
36C2×D4×C173—42no—
37C2×D4×C173—42no—
38C2×D4×C173—42no—
39C2×D4×C173—42no—
40C22×C4×C173—21no—
41C22×C4×C173D411yescharacteristic
42C22×C4×C173C2311yescharacteristic
43C22⋊C4×C173—22no—
44C22⋊C4×C173—42no—
45C22×D173—22no—
46C22×D173—42no—
47C22×D173—81no—
48C22×D173—81no—
49D692—42no—
50D692—81no—
51C22⋊D173—44no—
52C22⋊D173—42no—
53C22⋊D173—42no—
54C2×Dic173—22no—
55C2×Dic173—42no—
56#13063436—42no—
57C23×C173—21no—
58C23×C173—21no—
59C23×C173—22no—
60C23×C173—41no—
61C23×C173—41no—
62C23×C173—41no—
63C23×C173—41no—
64C23×C173—41no—
65C23×C173—41no—
66C23×C173—41no—
67C23×C173—41no—
68C23×C173D812yescomplemented
69C23×C173C2×D411yescharacteristic
70D4×C173—24no—
71D4×C173—42no—
72D4×C173—48no—
73C2×C4×C173—22no—
74C2×C4×C173—21no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173C2×D412yes—
80C22×C173—21no—
81C22×C173—21no—
82C22×C173—22no—
83C22×C173—42no—
84C22×C173—42no—
85C22×C173—42no—
86C22×C173—41no—
87C22×C173—41no—
88C22×C173—42no—
89C22×C173—81no—
90C22×C173—81no—
91C22×C173—81no—
92C22×C173—81no—
93C22×C173—81no—
94C22×C173#12011yescharacteristic
95C22×C173C2×D811yescharacteristic
96C22×C173AGL(1,ℤ/8)11yescharacteristic
97C2×D173—42no—
98C2×D173—42no—
99C2×D173—42no—
100C2×D173—84no—
101C4×C173—22no—
102C4×C173—42no—
103C4×C173—41no—
104Dic173—42no—
105C2×C173—21no—
106C2×C173—21no—
107C2×C173—42no—
108C2×C173—82no—
109C2×C173#44711yescharacteristic
110C2×C173#45311yescharacteristic
111C2×C173B(4,𝔽2)11yescharacteristic
112D173—82no—
113C173#159511yescharacteristic, complemented, Sylow
114#1595—1731nomaximal, Sylow
115#534—1731no—
116#447—1732no—
117#392D17311yescharacteristic, complemented
118#327—1732no—
119#323—1731no—
120C22×D4—21no—
121C22×D4—3461no—
122C22×D4C2×D17311yescharacteristic, complemented
123C2×D8—3462no—
124#123—3462no—
125#121—3461no—
126#121—1732no—
127#120—3462no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4C2×D17311yescharacteristic
130#102—3462no—
131C22⋊C8—1732no—
132#95C2×D17311yescharacteristic, complemented
133C24—21no—
134C24—3461no—
135C24C22⋊D17311yescharacteristic
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—3461no—
142C2×D4—3462no—
143C2×D4—3462no—
144C2×D4—1732no—
145C2×D4—6922no—
146C2×D4—6922no—
147C22×C4—21no—
148C22×C4—3461no—
149C22×C4C22⋊D17311yescharacteristic
150C22×C4C22×D17311yescharacteristic
151D8—6924no—
152C2×C8—3462no—
153C4⋊C4—3461no—
154C4⋊C4—1732no—
155C22⋊C4—22no—
156C22⋊C4—42no—
157C22⋊C4—3462no—
158C22⋊C4—3462no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—3462no—
171C23—13841no—
172C23—13841no—
173C23—6922no—
174C23#1458381112yescomplemented
175C23C2×C22⋊D17311yescharacteristic
176D4—24no—
177D4—42no—
178D4—48no—
179D4—13841no—
180D4—6922no—
181D4—6924no—
182D4—6922no—
183D4—6922no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3462no—
191C2×C4—6922no—
192C2×C4D4×D17312yes—
193C8—6922no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—13844no—
209C22—6922no—
210C22—6922no—
211C22—6922no—
212C22#1914964711yescharacteristic
213C22C2×#1458381111yescharacteristic
214C22#1914970411yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—6922no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—13842no—
224C2#2601162411yescharacteristic
225C2#2601165811yescharacteristic
226C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#392×C173C2
socleC22×C173#120
hypercenter#392D173
hypocenterC173#1595
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #26012053C2semidirect
2. #26011625C2semidirect
3. #26011224C2semidirect
4. #26011215C2semidirect
5. #392×C173C2semidirect
13. C22⋊#13063436C22semidirect
15. C22×D4×C173C22semidirect
18. #95×C173C22semidirect
21. C2×D692D4semidirect
30. C4⋊Dic173D4semidirect
68. C23×C173D8semidirect
113. C173#1595semidirect
117. #392D173semidirect
122. C22×D4C2×D173semidirect
132. #95C2×D173semidirect
174. C23#14583811semidirect

Maximal subgroups

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

subgroupindex
#260120532
#260116252
#260112242
#260112152
#392×C1732
#1595173