Finite Group Databasebeta

← #36352667

Subgroups — #36352667

Order 22144.

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

normal subgroups only

This group has 250 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#26012156C211yesmaximal, characteristic, complemented
2#26011839C211yesmaximal, characteristic, complemented
3#26011692C211yesmaximal, characteristic, complemented
4#26011687C211yesmaximal, characteristic
5#26011221C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#332×C173C211yesmaximal, characteristic, complemented
8C2×Q8×D173—21no—
9C2×#14583815—21no—
10C2×D1384—21no—
11#19149678—21no—
12#19149676C2211yescharacteristic
13#19149657—22no—
14#19149655—21no—
15C4×Dic346C2211yescharacteristic
16C4×D692C2211yescharacteristic
17#19149641—21no—
18#19149601—21no—
19#19149601—21no—
20#19149599—22no—
21C4×#13063436C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25C2×Dic346—22no—
26C2×Dic346D411yescharacteristic
27Q8×D173—42no—
28Q8×D173—42no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692—21no—
32C2×D692D411yescharacteristic
33C2×C4×D173—22no—
34C2×C4×D173—21no—
35#14583815—42no—
36D1384—42no—
37C2×#13063436—21no—
38C4⋊Dic173D411yescharacteristic
39C4⋊Dic173D411yescharacteristic
40#14583805—22no—
41#14583805—21no—
42#14583803—22no—
43#14583803—21no—
44C4×Dic173—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—41no—
53Dic346—21no—
54Dic346—44no—
55Dic346D812yescomplemented
56D692—21no—
57D692—42no—
58D692—41no—
59D692—41no—
60D692—41no—
61D692—41no—
62D692SD1612yescomplemented
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C4×D173—44no—
67C4×D173—41no—
68C4×D173—41no—
69#13063436—41no—
70Q8×C173—41no—
71Q8×C173—41no—
72C2×C4×C173—22no—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C8×C173—41no—
77C8×C173—41no—
78C22×C173#12011yescharacteristic
79C2×D173—22no—
80C2×D173—41no—
81C2×D173—41no—
82C2×D173—81no—
83C2×D173—81no—
84C2×D173—81no—
85C4×C173—21no—
86C4×C173—42no—
87C4×C173C2×D811yescharacteristic
88C4×C173C2×SD1611yescharacteristic
89C4×C173AGL(1,ℤ/8)11yescharacteristic
90C4×C173#13711yescharacteristic
91Dic173—22no—
92Dic173—41no—
93Dic173—41no—
94C2×C173#44711yescharacteristic
95C2×C173#44811yescharacteristic
96C2×C173#45411yescharacteristic
97D173—42no—
98D173—161no—
99C173#162711yescharacteristic, complemented, Sylow
100#1627—1731nomaximal, Sylow
101D4×Q8—1731no—
102#486—1731no—
103#460—1731no—
104#459—1731no—
105#332D17311yescharacteristic, complemented
106Q8⋊C8—1731no—
107D4⋊C8—1731no—
108C22×Q8—3461no—
109C2×SD16—3461no—
110C2×D8—3461no—
111C4⋊Q8—3461no—
112C4⋊Q8C2×D17311yescharacteristic
113#127—1731no—
114C22⋊Q8—3461no—
115C22⋊Q8—3462no—
116C4×Q8—1731no—
117C4×D4—3461no—
118C4×D4—1731no—
119C4⋊C8C2×D17311yescharacteristic
120C4⋊C8C2×D17311yescharacteristic
121#102—3461no—
122#102—3461no—
123#102—3462no—
124C4×C8—1731no—
125C2×Q8—21no—
126C2×Q8—3462no—
127C2×Q8—1731no—
128C2×Q8—6922no—
129C2×Q8—6922no—
130C2×D4—3461no—
131C2×D4—3461no—
132C2×D4—3461no—
133C2×D4—1731no—
134C22×C4—3462no—
135C22×C4—3461no—
136SD16—6922no—
137D8—6922no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3461no—
141C4⋊C4—21no—
142C4⋊C4—3462no—
143C4⋊C4—3461no—
144C4⋊C4—1731no—
145C4⋊C4—1731no—
146C4⋊C4D69212yes—
147C22⋊C4—3462no—
148C22⋊C4—3461no—
149C42—3461no—
150C42C22×D17311yescharacteristic
151C23—3461no—
152C23—6921no—
153Q8—41no—
154Q8—41no—
155Q8—3461no—
156Q8—1732no—
157Q8—6924no—
158D4—3461no—
159D4—1732no—
160D4—6922no—
161D4—6921no—
162D4—6921no—
163D4—6921no—
164D4—6921no—
165C2×C4—22no—
166C2×C4—3461no—
167C2×C4—3461no—
168C2×C4—3461no—
169C2×C4—6924no—
170C2×C4—6921no—
171C2×C4—6921no—
172C2×C4D4×D17311yescharacteristic
173C2×C4D4×D17311yescharacteristic
174C2×C4C2×D69211yescharacteristic
175C8—41no—
176C8—41no—
177C8—6921no—
178C22—3462no—
179C22—13841no—
180C22—13841no—
181C22—13841no—
182C22—6921no—
183C22—6921no—
184C22#1914964911yescharacteristic
185C4—21no—
186C4—42no—
187C4—3462no—
188C4—6921no—
189C4—6921no—
190C4D8×D17311yescharacteristic
191C4SD16×D17311yescharacteristic
192C4#1914970611yescharacteristic
193C4#1914971111yescharacteristic
194C2—6922no—
195C2—27681no—
196C2#2601162611yescharacteristic
197C2#2601163111yescharacteristic
198C2#2601166611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#332×C173C2
socleC22×C173#120
prime radical#26011687C2
hypercenter#332D173
hypocenterC173#1627
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. #26012156C2semidirect
2. #26011839C2semidirect
3. #26011692C2semidirect
5. #26011221C2semidirect
6. #26011218C2semidirect
7. #332×C173C2semidirect
55. Dic346D8semidirect
62. D692SD16semidirect
99. C173#1627semidirect
105. #332D173semidirect

Maximal subgroups

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

subgroupindex
#260121562
#260118392
#260116922
#260116872
#260112212
#260112182
#332×C1732
#1627173