Finite Group Databasebeta

← #36352498

Subgroups — #36352498

Order 22144.

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

normal subgroups only

This group has 310 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#26011624C212yesmaximal, complemented
2#26011408C211yesmaximal, characteristic, complemented
3#26011225C212yesmaximal, complemented
4#26011217C211yesmaximal, characteristic, complemented
5#537×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7C2×#14583811—22no—
8C2×#14583803—21no—
9C2×#14583803—21no—
10#19149599—22no—
11C22⋊#13063436C2212yes—
12#19149586C2211yescharacteristic
13C2×#41×C173—21no—
14C22⋊Q8×C173—22no—
15#121×C173—22no—
16#121×C173—21no—
17#121×C173C2212yes—
18C2×C4⋊C4×C173C2211yescharacteristic
19C23×D173—21no—
20C2×D692—22no—
21C2×D692—42no—
22C22×Dic173—21no—
23#14583811—44no—
24C2×#13063436—22no—
25#14583803—42no—
26#14583803—42no—
27#41×C173—42no—
28#41×C173—42no—
29C2×Q8×C173—21no—
30C2×D4×C173—21no—
31C2×D4×C173—22no—
32C2×D4×C173—21no—
33C2×D4×C173D412yes—
34C22×C4×C173—22no—
35C22×C4×C173D411yescharacteristic
36C22×C4×C173D411yescharacteristic
37C22×C4×C173C2311yescharacteristic
38C4⋊C4×C173—21no—
39C4⋊C4×C173—22no—
40C4⋊C4×C173D412yes—
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—
52C2×Dic173—41no—
53C2×Dic173—41no—
54C2×Dic173—41no—
55#13063436—42no—
56C23×C173—22no—
57C23×C173C2×D411yescharacteristic
58Q8×C173—42no—
59D4×C173—42no—
60D4×C173—42no—
61D4×C173—42no—
62D4×C173—42no—
63D4×C173—81no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—22no—
68C2×C4×C173—41no—
69C2×C4×C173—44no—
70C2×C4×C173—41no—
71C2×C4×C173D812yescomplemented
72C2×C4×C173C2×D412yes—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173—42no—
78C22×C173#12011yescharacteristic
79C22×C173C2×D811yescharacteristic
80C22×C173AGL(1,ℤ/8)11yescharacteristic
81C2×D173—42no—
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—81no—
85C2×D173—81no—
86C2×D173—81no—
87C2×D173—81no—
88C4×C173—22no—
89C4×C173—22no—
90C4×C173—42no—
91C4×C173—41no—
92Dic173—81no—
93C2×C173—21no—
94C2×C173—21no—
95C2×C173—82no—
96C2×C173#44711yescharacteristic
97C2×C173#45411yescharacteristic
98C2×C173B(4,𝔽2)11yescharacteristic
99D173—82no—
100C173#159811yescharacteristic, complemented, Sylow
101#1598—1731nomaximal, Sylow
102#537D17311yescharacteristic, complemented
103#447—1732no—
104#394—1731no—
105#327—1732no—
106#324—1731no—
107C2×#41—21no—
108C22×D4—1731no—
109C2×D8—3462no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D17312yes—
114C2×C4⋊C4C2×D17311yescharacteristic
115C2×C22⋊C4—3461no—
116C2×C22⋊C4—3461no—
117#102—3462no—
118C22⋊C8—1732no—
119#95—1731no—
120C24—3461no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—3462no—
128C2×D4—6922no—
129C2×D4C22⋊D17312yes—
130C22×C4—22no—
131C22×C4—3461no—
132C22×C4C22⋊D17311yescharacteristic
133C22×C4C22⋊D17311yescharacteristic
134C22×C4C22×D17311yescharacteristic
135D8—6924no—
136C2×C8—3462no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D17312yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—6922no—
143C22⋊C4—6922no—
144C23—22no—
145C23—3462no—
146C23—6921no—
147C23—6921no—
148C23—6921no—
149C23—6921no—
150C23—6921no—
151C23—6921no—
152C23C2×C22⋊D17311yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—3464no—
160D4—6922no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—6921no—
169C2×C4—6921no—
170C2×C4—6921no—
171C2×C4#1458381112yescomplemented
172C2×C4C2×C22⋊D17312yes—
173C8—6922no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—13841no—
180C22—13841no—
181C22—13841no—
182C22—13841no—
183C22—6922no—
184C22—6922no—
185C22—6922no—
186C22#1914964811yescharacteristic
187C22C2×#1458381111yescharacteristic
188C22#1914970511yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—42no—
193C4—13841no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—13842no—
198C2#2601162511yescharacteristic
199C2#2601166411yescharacteristic
200C2#2601167811yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#537×C173C2
socleC22×C173#120
hypercenter#537D173
hypocenterC173#1598
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #26011624C2semidirect
2. #26011408C2semidirect
3. #26011225C2semidirect
4. #26011217C2semidirect
5. #537×C173C2semidirect
71. C2×C4×C173D8semidirect
100. C173#1598semidirect
102. #537D173semidirect
171. C2×C4#14583811semidirect

Maximal subgroups

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

subgroupindex
#260116242
#260114082
#260112252
#260112172
#537×C1732
#1598173