Finite Group Databasebeta

← #36352653

Subgroups — #36352653

Order 22144.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012144C211yesmaximal, characteristic, complemented
2#26011839C211yesmaximal, characteristic, complemented
3#26011687C211yesmaximal, characteristic, complemented
4#26011687C211yesmaximal, characteristic, complemented
5#26011218C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#332×C173C211yesmaximal, characteristic, complemented
8C2×#14583827—21no—
9C2×D1384—21no—
10C2×D1384—21no—
11#19149676C2211yescharacteristic
12#19149664—22no—
13#19149655—21no—
14#19149654—21no—
15C4×D692C2211yescharacteristic
16C4×D692C2211yescharacteristic
17C4⋊C4×D173—21no—
18#19149601—21no—
19#19149601—21no—
20#19149599—22no—
21C4×#13063436C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25#14583827—42no—
26#14583827—42no—
27C2×D692—22no—
28C2×D692—21no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692D411yescharacteristic
32C2×D692D411yescharacteristic
33C2×C4×D173—22no—
34C2×C4×D173—21no—
35C2×C4×D173—21no—
36D1384—42no—
37D1384—42no—
38C2×#13063436—21no—
39C4⋊Dic173D411yescharacteristic
40C4⋊Dic173D411yescharacteristic
41#14583805—22no—
42#14583803—22no—
43#14583803—21no—
44#14583803—21no—
45C2×Q8×C173—21no—
46C2×C8×C173—21no—
47C2×C8×C173—21no—
48C4⋊C4×C173—21no—
49C4⋊C4×C173D412yes—
50C42×C173C2311yescharacteristic
51C22×D173—21no—
52C22×D173—21no—
53C22×D173—41no—
54D692—21no—
55D692—21no—
56D692—21no—
57D692—42no—
58D692—44no—
59D692—41no—
60D692—41no—
61D692—41no—
62D692—41no—
63D692D812yescomplemented
64C2×Dic173—21no—
65C2×Dic173—21no—
66C4×D173—44no—
67C4×D173—41no—
68C4×D173—41no—
69C4×D173—41no—
70C4×D173—41no—
71#13063436—41no—
72Q8×C173—41no—
73Q8×C173—41no—
74C2×C4×C173—22no—
75C2×C4×C173C2×D411yescharacteristic
76C2×C4×C173C2×D411yescharacteristic
77C2×C4×C173C2×D411yescharacteristic
78C8×C173—41no—
79C8×C173—41no—
80C22×C173#12011yescharacteristic
81C2×D173—22no—
82C2×D173—41no—
83C2×D173—41no—
84C2×D173—41no—
85C2×D173—41no—
86C2×D173—41no—
87C2×D173—81no—
88C2×D173—81no—
89C2×D173—81no—
90C4×C173—21no—
91C4×C173—42no—
92C4×C173C2×D811yescharacteristic
93C4×C173#13511yescharacteristic
94C4×C173AGL(1,ℤ/8)11yescharacteristic
95C4×C173AGL(1,ℤ/8)11yescharacteristic
96Dic173—41no—
97Dic173—41no—
98C2×C173#44711yescharacteristic
99C2×C173#44911yescharacteristic
100C2×C173#45411yescharacteristic
101D173—42no—
102D173—81no—
103D173—161no—
104C173#162511yescharacteristic, complemented, Sylow
105#1625—1731nomaximal, Sylow
106#547—1731no—
107#486—1731no—
108#459—1731no—
109#459—1731no—
110#332D17311yescharacteristic, complemented
111D4⋊C8—1731no—
112D4⋊C8—1731no—
113C2×#41—3461no—
114C2×D8—3461no—
115C2×D8—3461no—
116C4⋊Q8C2×D17311yescharacteristic
117#127—1731no—
118#123—3462no—
119C22⋊Q8—3461no—
120#121—3461no—
121C4×D4—1731no—
122C4×D4—1731no—
123C2×C4⋊C4—3461no—
124C4⋊C8C2×D17311yescharacteristic
125C4⋊C8C2×D17311yescharacteristic
126#102—3461no—
127#102—3461no—
128#102—3462no—
129C4×C8—1731no—
130#41—6922no—
131#41—6922no—
132C2×Q8—21no—
133C2×D4—3462no—
134C2×D4—3461no—
135C2×D4—3461no—
136C2×D4—3461no—
137C2×D4—1731no—
138C2×D4—1731no—
139C22×C4—3462no—
140C22×C4—3461no—
141C22×C4—3461no—
142D8—6922no—
143D8—6922no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—3461no—
147C4⋊C4—21no—
148C4⋊C4—3462no—
149C4⋊C4—1731no—
150C4⋊C4—1731no—
151C4⋊C4D69212yes—
152C22⋊C4—3462no—
153C22⋊C4—3461no—
154C22⋊C4—3461no—
155C42C22×D17311yescharacteristic
156C23—3461no—
157C23—3461no—
158C23—6921no—
159Q8—41no—
160Q8—41no—
161D4—3461no—
162D4—3461no—
163D4—3461no—
164D4—1732no—
165D4—6922no—
166D4—6924no—
167D4—6921no—
168D4—6921no—
169D4—6921no—
170D4—6921no—
171C2×C4—22no—
172C2×C4—3461no—
173C2×C4—3461no—
174C2×C4—6924no—
175C2×C4—6921no—
176C2×C4—6921no—
177C2×C4—6921no—
178C2×C4—6921no—
179C2×C4D4×D17311yescharacteristic
180C2×C4D4×D17311yescharacteristic
181C2×C4C2×D69211yescharacteristic
182C8—41no—
183C8—41no—
184C8—6921no—
185C22—3462no—
186C22—13841no—
187C22—13841no—
188C22—13841no—
189C22—6921no—
190C22—6921no—
191C22—6921no—
192C22—6921no—
193C22—6921no—
194C22#1914964911yescharacteristic
195C4—21no—
196C4—42no—
197C4—6921no—
198C4—6921no—
199C4D8×D17311yescharacteristic
200C4#1914970211yescharacteristic
201C4#1914970611yescharacteristic
202C4#1914970611yescharacteristic
203C2—13841no—
204C2—6922no—
205C2—27681no—
206C2#2601162611yescharacteristic
207C2#2601163911yescharacteristic
208C2#2601166611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#332×C173C2
socleC22×C173#120
hypercenter#332D173
hypocenterC173#1625
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #26012144C2semidirect
2. #26011839C2semidirect
3. #26011687C2semidirect
4. #26011687C2semidirect
5. #26011218C2semidirect
6. #26011218C2semidirect
7. #332×C173C2semidirect
63. D692D8semidirect
104. C173#1625semidirect
110. #332D173semidirect

Maximal subgroups

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

subgroupindex
#260121442
#260118392
#260116872
#260112182
#332×C1732
#1625173