Finite Group Databasebeta

← #36355511

Subgroups — #36355511

Order 22144.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#26012291C211yesmaximal, characteristic, complemented
2#26011929C212yesmaximal, complemented
3#26011744C211yesmaximal, characteristic, complemented
4#26011302C212yesmaximal, complemented
5#349×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7#19149712—22no—
8#19149707—22no—
9#19149701—21no—
10#19149701C2212yes—
11C2×D1384C2211yescharacteristic, complemented
12#19149653—21no—
13#19149624—22no—
14D2768—22no—
15#19149611C2211yescharacteristic, complemented
16#19149601—21no—
17C2×M16×C173C2211yescharacteristic
18M32×C173C2212yescomplemented
19#14583826—22no—
20#14583826—22no—
21#14583826—41no—
22#14583826—41no—
23C2×Dic346—21no—
24C2×D692—21no—
25C2×D692—21no—
26C2×C22⋊D173—41no—
27C2×C22⋊D173—41no—
28C2×C4×D173—41no—
29Dic692—21no—
30Dic692D412yescomplemented
31#14583815—22no—
32#14583815—22no—
33D1384—21no—
34D1384—22no—
35D1384D412yescomplemented
36C4⋊Dic173—21no—
37#14583803—41no—
38C22×C4×C173D411yescharacteristic
39M16×C173—22no—
40C2×C8×C173D411yescharacteristic
41C2×C8×C173C2311yescharacteristic
42C16×C173—22no—
43C22×D173—41no—
44C22×D173—41no—
45Dic346—22no—
46Dic346—41no—
47D692—22no—
48D692—41no—
49D692—41no—
50D692—41no—
51C22⋊D173—42no—
52C22⋊D173—42no—
53C22⋊D173—82no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56C4×D173—42no—
57C4×D173—42no—
58C23×C173D811yescharacteristic
59C2×C4×C173—21no—
60C2×C4×C173—21no—
61C2×C4×C173D811yescharacteristic
62C2×C4×C173C2×D411yescharacteristic
63C8×C173—21no—
64C8×C173C2×D412yes—
65C22×C173—21no—
66C22×C173—41no—
67C22×C173C2×D811yescharacteristic
68C2×D173—42no—
69C2×D173—82no—
70C2×D173—81no—
71C2×D173—81no—
72C2×D173—81no—
73C4×C173—21no—
74C4×C173#12011yescharacteristic
75C4×C173AGL(1,ℤ/8)11yescharacteristic
76Dic173—42no—
77Dic173—81no—
78C2×C173—21no—
79C2×C173—41no—
80C2×C173#44711yescharacteristic
81D173—82no—
82D173—161no—
83C173#219011yescharacteristic, complemented, Sylow
84#2190—1731nomaximal, Sylow
85#575—1731no—
86#509—1732no—
87#469—1731no—
88#360—1732no—
89#349D17311yescharacteristic, complemented
90C2×#41—3461no—
91#137—3462no—
92AGL(1,ℤ/8)—3462no—
93#135—3461no—
94#135—1732no—
95C2×D8—1731no—
96C2×M16C2×D17311yescharacteristic
97#121—3461no—
98SD32—3462no—
99D16—3462no—
100M32C2×D17312yescomplemented
101#106—1731no—
102#102—3461no—
103#41—3462no—
104#41—3462no—
105#41—6921no—
106#41—6921no—
107C2×Q8—3461no—
108C2×D4—3461no—
109C2×D4—3461no—
110C2×D4—6921no—
111C2×D4—6921no—
112C22×C4—6921no—
113C22×C4D69211yescharacteristic
114Q16—3461no—
115Q16—1732no—
116SD16—3462no—
117SD16—3462no—
118D8—3461no—
119D8—3462no—
120D8—1732no—
121M16—22no—
122C2×C8D69211yescharacteristic
123C2×C8C22×D17311yescharacteristic
124C4⋊C4—3461no—
125C22⋊C4—6921no—
126C16—22no—
127C23—6921no—
128C23—6921no—
129C23D138411yescharacteristic
130Q8—3462no—
131Q8—6921no—
132D4—3462no—
133D4—13842no—
134D4—6921no—
135D4—6922no—
136D4—6922no—
137D4—6921no—
138D4—6921no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—6922no—
142C2×C4—6921no—
143C2×C4—6921no—
144C2×C4—6922no—
145C2×C4D138411yescharacteristic
146C2×C4C2×D69211yescharacteristic
147C8—21no—
148C8D4×D17312yes—
149C22—21no—
150C22—41no—
151C22—13841no—
152C22—13841no—
153C22—13842no—
154C22—13841no—
155C22—6922no—
156C22C2×D138411yescharacteristic
157C4—21no—
158C4—13841no—
159C4—6922no—
160C4#1914964911yescharacteristic
161C4#1914970711yescharacteristic
162C2—21no—
163C2—41no—
164C2—13842no—
165C2—27681no—
166C2#2601162711yescharacteristic
subgroupG/H
centerC2#26011627
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
Fitting#349×C173C2
socleC2×C173#447
hypercenter#349D173
hypocenterC173#2190
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #26012291C2semidirect
2. #26011929C2semidirect
3. #26011744C2semidirect
4. #26011302C2semidirect
5. #349×C173C2semidirect
11. C2×D1384C22semidirect
15. #19149611C22semidirect
18. M32×C173C22semidirect
30. Dic692D4semidirect
35. D1384D4semidirect
83. C173#2190semidirect
89. #349D173semidirect
100. M32C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122912
#260119292
#260117442
#260113022
#349×C1732
#2190173