Finite Group Databasebeta

← #36352645

Subgroups — #36352645

Order 22144.

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

normal subgroups only

This group has 240 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#26012206C211yesmaximal, characteristic, complemented
2#26011863C211yesmaximal, characteristic, complemented
3#26011683C211yesmaximal, characteristic, complemented
4#26011239C211yesmaximal, characteristic, complemented
5D4⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#459×C173C211yesmaximal, characteristic, complemented
8C2×#14583814—22no—
9C2×#14583811—21no—
10#19149679C2211yescharacteristic
11#19149665—22no—
12#19149653—21no—
13#19149650—21no—
14C4×D692C2211yescharacteristic
15#19149638—21no—
16#19149637—21no—
17C4⋊#13063436C2211yescharacteristic
18#19149600—21no—
19C4×#13063436C2211yescharacteristic
20C2×D8×C173—21no—
21#127×C173C2211yescharacteristic
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24#102×C173—21no—
25C2×Dic346D412yes—
26C2×D692D411yescharacteristic
27C2×C22⋊D173—22no—
28C2×C4×D173—21no—
29#14583814—44no—
30#14583811—42no—
31C2×#13063436—21no—
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173D411yescharacteristic
36C22⋊Dic173—22no—
37#14583803—22no—
38#14583803—21no—
39C4×Dic173—22no—
40C2×D4×C173—21no—
41C2×D4×C173—21no—
42C2×D4×C173—21no—
43C2×D4×C173D411yescharacteristic
44C22×C4×C173—21no—
45D8×C173—42no—
46C2×C8×C173—21no—
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51Dic346—42no—
52D692—21no—
53D692—21no—
54C22⋊D173—82no—
55C2×Dic173—21no—
56C2×Dic173—22no—
57C4×D173—41no—
58C4×D173—41no—
59#13063436—22no—
60#13063436—41no—
61C23×C173—21no—
62C23×C173—41no—
63D4×C173—21no—
64D4×C173—21no—
65D4×C173—42no—
66D4×C173—41no—
67D4×C173—41no—
68D4×C173—41no—
69D4×C173—41no—
70C2×C4×C173—21no—
71C2×C4×C173—41no—
72C2×C4×C173—41no—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C8×C173—41no—
77C22×C173—41no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173—81no—
81C22×C173—81no—
82C22×C173—81no—
83C22×C173#12011yescharacteristic
84C2×D173—41no—
85C2×D173—41no—
86C2×D173—41no—
87C4×C173—21no—
88C4×C173—41no—
89C4×C173#13511yescharacteristic
90C4×C173#13511yescharacteristic
91C4×C173AGL(1,ℤ/8)11yescharacteristic
92C4×C173AGL(1,ℤ/8)11yescharacteristic
93Dic173—41no—
94Dic173—42no—
95C2×C173—81no—
96C2×C173—161no—
97C2×C173#44911yescharacteristic
98C2×C173#44911yescharacteristic
99C2×C173#45311yescharacteristic
100D173—81no—
101C173#162411yescharacteristic, complemented, Sylow
102#1624—1731nomaximal, Sylow
103#555—1731no—
104#492—1731no—
105#459—1731no—
106#459D17311yescharacteristic, complemented
107#333—1731no—
108D4⋊C8—1731no—
109D4⋊C8—1731no—
110C2×SD16—3462no—
111C2×D8—21no—
112C2×D8—3461no—
113C4⋊Q8—1731no—
114#127C2×D17311yescharacteristic
115#124—3462no—
116#121—3461no—
117#121—3461no—
118C4×D4—1731no—
119C4×D4C2×D17311yescharacteristic
120#117—3461no—
121#117—3461no—
122C4⋊C8—1731no—
123C4⋊C8C2×D17311yescharacteristic
124#102—21no—
125#102—3461no—
126C4×C8—1731no—
127C2×Q8—1732no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3462no—
132C2×D4—1731no—
133C2×D4C22⋊D17311yescharacteristic
134C22×C4—21no—
135C22×C4—3461no—
136SD16—6924no—
137D8—42no—
138D8—6922no—
139C2×C8—21no—
140C2×C8—3461no—
141C2×C8—3461no—
142C4⋊C4—3461no—
143C4⋊C4—3461no—
144C4⋊C4—1731no—
145C4⋊C4C22⋊D17311yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—3462no—
148C22⋊C4—3462no—
149C22⋊C4—3461no—
150C42—3462no—
151C42C22×D17311yescharacteristic
152C23—21no—
153C23—41no—
154C23—3461no—
155Q8—6922no—
156D4—21no—
157D4—21no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—3461no—
164D4—3461no—
165D4—13842no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3461no—
170C2×C4—3462no—
171C2×C4—6921no—
172C2×C4—6921no—
173C2×C4D4×D17311yescharacteristic
174C2×C4D4×D17311yescharacteristic
175C2×C4C2×C22⋊D17311yescharacteristic
176C8—41no—
177C8—3462no—
178C8—6921no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—6921no—
186C22—6921no—
187C22—6921no—
188C22#1914964711yescharacteristic
189C4—21no—
190C4—41no—
191C4—6922no—
192C4—6921no—
193C4#1914969811yescharacteristic
194C4#1914969911yescharacteristic
195C4#1914970311yescharacteristic
196C4#1914970511yescharacteristic
197C2—81no—
198C2—161no—
199C2—13841no—
200C2#2601163311yescharacteristic
201C2#2601163411yescharacteristic
202C2#2601165711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#459×C173C2
socleC22×C173#120
hypercenter#459D173
hypocenterC173#1624
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #26012206C2semidirect
2. #26011863C2semidirect
3. #26011683C2semidirect
4. #26011239C2semidirect
5. D4⋊#13063436C2semidirect
6. #26011218C2semidirect
7. #459×C173C2semidirect
101. C173#1624semidirect
106. #459D173semidirect

Maximal subgroups

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

subgroupindex
#260122062
#260118632
#260116832
#260112392
D4⋊#130634362
#260112182
#459×C1732
#1624173