Finite Group Databasebeta

← #36352660

Subgroups — #36352660

Order 22144.

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

normal subgroups only

This group has 210 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#26012216C211yesmaximal, characteristic, complemented
2#26011839C211yesmaximal, characteristic
3#26011692C211yesmaximal, characteristic
4#26011692C211yesmaximal, characteristic
5#26011221C211yesmaximal, characteristic, complemented
6#26011221C211yesmaximal, characteristic, complemented
7#332×C173C211yesmaximal, characteristic, complemented
8C2×#14583815—21no—
9C2×#14583815—21no—
10#19149677—21no—
11#19149676C2211yescharacteristic
12#19149670—21no—
13#19149669—22no—
14Q8×Dic173—21no—
15C4×Dic346C2211yescharacteristic
16C4×Dic346C2211yescharacteristic
17#19149644—21no—
18#19149601—21no—
19#19149601—21no—
20#19149599—22no—
21C4×#13063436C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25C2×Dic346D411yescharacteristic
26C2×Dic346D411yescharacteristic
27C2×D692—21no—
28C2×D692—21no—
29C2×D692—21no—
30#14583815—42no—
31#14583815—42no—
32C2×#13063436—21no—
33C4⋊Dic173—22no—
34C4⋊Dic173D411yescharacteristic
35C4⋊Dic173D411yescharacteristic
36#14583805—22no—
37#14583805—22no—
38#14583805—21no—
39#14583805—21no—
40C4×Dic173—22no—
41C4×Dic173—21no—
42C4×Dic173—21no—
43C2×Q8×C173—21no—
44C2×C8×C173—21no—
45C2×C8×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D412yes—
48C42×C173C2311yescharacteristic
49C22×D173—41no—
50Dic346—21no—
51Dic346—21no—
52Dic346—21no—
53Dic346SD1612yescomplemented
54D692—42no—
55D692—41no—
56D692—41no—
57D692—41no—
58D692—41no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63#13063436—41no—
64Q8×C173—41no—
65Q8×C173—41no—
66C2×C4×C173—22no—
67C2×C4×C173C2×D411yescharacteristic
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C8×C173—41no—
71C8×C173—41no—
72C22×C173#12011yescharacteristic
73C2×D173—81no—
74C2×D173—81no—
75C2×D173—81no—
76C4×C173—21no—
77C4×C173—42no—
78C4×C173C2×SD1611yescharacteristic
79C4×C173#13511yescharacteristic
80C4×C173AGL(1,ℤ/8)11yescharacteristic
81C4×C173#13711yescharacteristic
82Dic173—22no—
83Dic173—41no—
84Dic173—41no—
85Dic173—41no—
86C2×C173#44811yescharacteristic
87C2×C173#44911yescharacteristic
88C2×C173#45411yescharacteristic
89D173—161no—
90C173#162611yescharacteristic, complemented, Sylow
91#1626—1731nomaximal, Sylow
92#557—1731no—
93#486—1731no—
94#460—1731no—
95#460—1731no—
96#332D17311yescharacteristic, complemented
97Q8⋊C8—1731no—
98Q8⋊C8—1731no—
99C2×SD16—3461no—
100C2×SD16—3461no—
101C4⋊Q8—3461no—
102C4⋊Q8C2×D17311yescharacteristic
103#127—1731no—
104#125—3461no—
105#125—3462no—
106C4×Q8—3461no—
107C4×Q8—3461no—
108C4×Q8—1731no—
109C4×Q8—1731no—
110C4⋊C8C2×D17311yescharacteristic
111C4⋊C8C2×D17311yescharacteristic
112#102—3461no—
113#102—3461no—
114#102—3462no—
115C4×C8—1731no—
116C2×Q8—21no—
117C2×Q8—1731no—
118C2×Q8—1731no—
119C2×D4—3461no—
120C2×D4—3461no—
121C2×D4—3461no—
122SD16—6922no—
123SD16—6922no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—3461no—
127C4⋊C4—21no—
128C4⋊C4—3462no—
129C4⋊C4—3462no—
130C4⋊C4—3462no—
131C4⋊C4—3461no—
132C4⋊C4—3461no—
133C4⋊C4—1731no—
134C4⋊C4—1731no—
135C4⋊C4D69212yes—
136C42—3462no—
137C42—3461no—
138C42—3461no—
139C42C22×D17311yescharacteristic
140C23—6921no—
141Q8—41no—
142Q8—41no—
143Q8—3461no—
144Q8—3461no—
145Q8—3461no—
146Q8—1732no—
147D4—6922no—
148D4—6921no—
149D4—6921no—
150D4—6921no—
151D4—6921no—
152C2×C4—22no—
153C2×C4—3461no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4D4×D17311yescharacteristic
158C2×C4D4×D17311yescharacteristic
159C2×C4C2×D69211yescharacteristic
160C8—41no—
161C8—41no—
162C8—6921no—
163C22—13841no—
164C22—13841no—
165C22—13841no—
166C22#1914964911yescharacteristic
167C4—21no—
168C4—42no—
169C4—3462no—
170C4—6921no—
171C4—6921no—
172C4—6921no—
173C4SD16×D17311yescharacteristic
174C4#1914970011yescharacteristic
175C4#1914970311yescharacteristic
176C4#1914971111yescharacteristic
177C2—27681no—
178C2#2601163111yescharacteristic
179C2#2601163711yescharacteristic
180C2#2601166611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#332×C173C2
socleC22×C173#120
prime radical#19149676C22
hypercenter#332D173
hypocenterC173#1626
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. #26012216C2semidirect
5. #26011221C2semidirect
6. #26011221C2semidirect
7. #332×C173C2semidirect
53. Dic346SD16semidirect
90. C173#1626semidirect
96. #332D173semidirect

Maximal subgroups

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

subgroupindex
#260122162
#260118392
#260116922
#260112212
#332×C1732
#1626173