Finite Group Databasebeta

← #36352500

Subgroups — #36352500

Order 22144.

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

normal subgroups only

This group has 310 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#26012072C211yesmaximal, characteristic, complemented
2#26011627C212yesmaximal, complemented
3#26011408C211yesmaximal, characteristic, complemented
4#26011226C212yesmaximal, complemented
5#324×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7C22×D692C2211yescharacteristic
8C2×D1384—22no—
9#19149659—22no—
10#19149654—22no—
11#19149653—21no—
12#19149653C2212yes—
13C2×#14583803—21no—
14C2×#14583803—21no—
15#19149601—22no—
16#19149586C2211yescharacteristic
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C23×D173—21no—
20#14583826—42no—
21#14583826—42no—
22C2×Dic346—21no—
23C2×D692—21no—
24C2×D692—21no—
25C2×D692—22no—
26C2×D692—42no—
27C2×D692D412yes—
28C2×C22⋊D173—22no—
29C22×Dic173—21no—
30C2×C4×D173—22no—
31D1384—44no—
32C4⋊Dic173—21no—
33C4⋊Dic173D412yes—
34#14583803—22no—
35#14583803—22no—
36#14583803—42no—
37#14583803—42no—
38C22×C4×C173D411yescharacteristic
39C22×C4×C173D411yescharacteristic
40C22×C4×C173C2311yescharacteristic
41C2×C8×C173—22no—
42C4⋊C4×C173—22no—
43C22×D173—22no—
44C22×D173—22no—
45C22×D173—41no—
46C22×D173—41no—
47C22×D173—41no—
48C22×D173—41no—
49C22×D173—41no—
50C22×D173—41no—
51Dic346—42no—
52D692—24no—
53D692—42no—
54D692—42no—
55D692—42no—
56D692—81no—
57C22⋊D173—42no—
58C22⋊D173—42no—
59C2×Dic173—22no—
60C2×Dic173—41no—
61C2×Dic173—41no—
62C2×Dic173—41no—
63C4×D173—44no—
64C23×C173C2×D411yescharacteristic
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173D812yescomplemented
71C2×C4×C173C2×D412yes—
72C8×C173—42no—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173#12011yescharacteristic
76C22×C173C2×D811yescharacteristic
77C22×C173AGL(1,ℤ/8)11yescharacteristic
78C2×D173—42no—
79C2×D173—42no—
80C2×D173—42no—
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—41no—
91Dic173—42no—
92Dic173—81no—
93C2×C173—21no—
94C2×C173—21no—
95C2×C173#44711yescharacteristic
96C2×C173#45411yescharacteristic
97C2×C173B(4,𝔽2)11yescharacteristic
98D173—82no—
99D173—82no—
100C173#159811yescharacteristic, complemented, Sylow
101#1598—1731nomaximal, Sylow
102#537—1731no—
103#447—1732no—
104#394—1731no—
105#327—1732no—
106#324D17311yescharacteristic, complemented
107C2×#41—3461no—
108C22×D4—1731no—
109C2×D8—3462no—
110C22⋊Q8—3462no—
111#121—3462no—
112#121—3461no—
113#121—1732no—
114C2×C4⋊C4C2×D17311yescharacteristic
115C2×C22⋊C4—3461no—
116C2×C22⋊C4—3461no—
117#102—3462no—
118C22⋊C8C2×D17312yes—
119#95—1731no—
120C24—3461no—
121#41—6922no—
122#41—6922no—
123C2×Q8—3461no—
124C2×D4—3461no—
125C2×D4—3461no—
126C2×D4—3462no—
127C2×D4—3462no—
128C2×D4—1732no—
129C2×D4—6922no—
130C22×C4—3462no—
131C22×C4—3461no—
132C22×C4D69211yescharacteristic
133C22×C4D69211yescharacteristic
134C22×C4C22×D17311yescharacteristic
135D8—6924no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—3461no—
139C4⋊C4—1732no—
140C22⋊C4—3462no—
141C22⋊C4—3462no—
142C22⋊C4—6922no—
143C22⋊C4—6922no—
144C23—3462no—
145C23—3462no—
146C23—6921no—
147C23—6921no—
148C23—6921no—
149C23—6921no—
150C23—6921no—
151C23—6921no—
152C23C2×D69211yescharacteristic
153Q8—6922no—
154D4—3464no—
155D4—13841no—
156D4—6922no—
157D4—6922no—
158D4—6922no—
159D4—6922no—
160D4—6922no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—3462no—
167C2×C4—6924no—
168C2×C4—6921no—
169C2×C4—6921no—
170C2×C4—6921no—
171C2×C4D138412yescomplemented
172C2×C4D4×D17312yes—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—13841no—
177C22—13841no—
178C22—13841no—
179C22—13841no—
180C22—6922no—
181C22—6922no—
182C22—6922no—
183C22—6922no—
184C22—6922no—
185C22—6922no—
186C22#1914964911yescharacteristic
187C22C2×D138411yescharacteristic
188C22#1914970711yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—13841no—
193C4—6922no—
194C2—21no—
195C2—21no—
196C2—13842no—
197C2—13842no—
198C2#2601162711yescharacteristic
199C2#2601166611yescharacteristic
200C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173#120
hypercenter#324D173
hypocenterC173#1598
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #26012072C2semidirect
2. #26011627C2semidirect
3. #26011408C2semidirect
4. #26011226C2semidirect
5. #324×C173C2semidirect
70. C2×C4×C173D8semidirect
100. C173#1598semidirect
106. #324D173semidirect
171. C2×C4D1384semidirect

Maximal subgroups

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

subgroupindex
#260120722
#260116272
#260114082
#260112262
#324×C1732
#1598173