Finite Group Databasebeta

← #36352486

Subgroups — #36352486

Order 22144.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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#26011641C212yesmaximal, complemented
2#26011400C211yesmaximal, characteristic
3#26011228C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic, complemented
5#543×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7C22×C22⋊D173—21no—
8C2×#14583813—22no—
9C2×#14583803—21no—
10#19149599—22no—
11C22⋊#13063436C2212yes—
12#19149586C2211yescharacteristic
13C22⋊Q8×C173—22no—
14C22⋊Q8×C173—21no—
15C22⋊Q8×C173—22no—
16C22⋊Q8×C173C2212yes—
17C2×C22⋊C4×C173—21no—
18C2×C22⋊C4×C173C2211yescharacteristic
19C23×D173—21no—
20C2×D692—22no—
21C2×D692—42no—
22C2×C22⋊D173—42no—
23C2×C22⋊D173—42no—
24C2×C22⋊D173—42no—
25C22×Dic173—21no—
26#14583813—44no—
27C2×#13063436—22no—
28#14583803—42no—
29C24×C173D411yescharacteristic
30C2×Q8×C173—21no—
31C2×Q8×C173D412yescomplemented
32C22×C4×C173—21no—
33C22×C4×C173—21no—
34C22×C4×C173D411yescharacteristic
35C22×C4×C173C2311yescharacteristic
36C4⋊C4×C173—21no—
37C4⋊C4×C173—22no—
38C4⋊C4×C173—22no—
39C4⋊C4×C173D412yescomplemented
40C22⋊C4×C173—22no—
41C22⋊C4×C173—22no—
42C22⋊C4×C173—22no—
43C22×D173—22no—
44C22×D173—41no—
45C22×D173—41no—
46C22×D173—41no—
47C22×D173—41no—
48C22×D173—41no—
49C22×D173—41no—
50D692—24no—
51D692—42no—
52C22⋊D173—48no—
53C2×Dic173—22no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56#13063436—42no—
57C23×C173—21no—
58C23×C173—21no—
59C23×C173—41no—
60C23×C173—41no—
61C23×C173SD1612yescomplemented
62C23×C173C2×D411yescharacteristic
63Q8×C173—42no—
64Q8×C173—81no—
65C2×C4×C173—22no—
66C2×C4×C173—21no—
67C2×C4×C173—22no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173—81no—
71C2×C4×C173—81no—
72C2×C4×C173C2×D412yes—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173—22no—
76C22×C173—42no—
77C22×C173—41no—
78C22×C173—41no—
79C22×C173—81no—
80C22×C173#12011yescharacteristic
81C22×C173C2×SD1611yescharacteristic
82C22×C173#13711yescharacteristic
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—42no—
86C2×D173—81no—
87C2×D173—81no—
88C2×D173—81no—
89C2×D173—81no—
90C4×C173—22no—
91C4×C173—41no—
92C4×C173—42no—
93C4×C173—42no—
94Dic173—42no—
95C2×C173—21no—
96C2×C173—21no—
97C2×C173—42no—
98C2×C173#44811yescharacteristic
99C2×C173#45311yescharacteristic
100C2×C173B(4,𝔽2)11yescharacteristic
101D173—82no—
102C173#159611yescharacteristic, complemented, Sylow
103#1596—1731nomaximal, Sylow
104#543D17311yescharacteristic, complemented
105#450—1732no—
106#392—1731no—
107#328—1732no—
108#323—1731no—
109C22×D4—3461no—
110C22×D4—1731no—
111C2×SD16—3462no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D17312yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—3461no—
118C2×C22⋊C4C2×D17311yescharacteristic
119#102—3462no—
120C22⋊C8—1732no—
121#95—1731no—
122C24—3461no—
123C24C22⋊D17311yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D17312yescomplemented
126C2×D4—3462no—
127C2×D4—6922no—
128C2×D4—6922no—
129C2×D4—6922no—
130C2×D4—6922no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—3461no—
134C22×C4C22⋊D17311yescharacteristic
135C22×C4C22×D17311yescharacteristic
136SD16—6924no—
137C2×C8—3462no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D17312yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—6922no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—3462no—
151C23—6921no—
152C23—6921no—
153C23—6921no—
154C23—6921no—
155C23—6921no—
156C23—6921no—
157C23#1458381312yescomplemented
158C23C2×C22⋊D17311yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—3464no—
162D4—6922no—
163D4—6928no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—3462no—
172C2×C4—6921no—
173C2×C4—6921no—
174C2×C4C2×C22⋊D17312yes—
175C8—6922no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—13841no—
184C22—13841no—
185C22—13841no—
186C22—13841no—
187C22—6922no—
188C22—6922no—
189C22—6922no—
190C22#1914964811yescharacteristic
191C22C2×#1458381311yescharacteristic
192C22#1914971011yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—6922no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—13842no—
202C2#2601163011yescharacteristic
203C2#2601165911yescharacteristic
204C2#2601167811yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#543×C173C2
socleC22×C173#120
prime radical#26011400C2
hypercenter#543D173
hypocenterC173#1596
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #26011641C2semidirect
3. #26011228C2semidirect
4. #26011215C2semidirect
5. #543×C173C2semidirect
31. C2×Q8×C173D4semidirect
39. C4⋊C4×C173D4semidirect
61. C23×C173SD16semidirect
102. C173#1596semidirect
104. #543D173semidirect
125. C2×Q8C22⋊D173semidirect
141. C4⋊C4C22⋊D173semidirect
157. C23#14583813semidirect

Maximal subgroups

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

subgroupindex
#260116412
#260114002
#260112282
#260112152
#543×C1732
#1596173