Finite Group Databasebeta

← #36352873

Subgroups — #36352873

Order 22144.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012245C211yesmaximal, characteristic
2#26011897C211yesmaximal, characteristic, complemented
3#26011883C211yesmaximal, characteristic
4#26011233C211yesmaximal, characteristic, complemented
5#26011232C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic
7#501×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—21no—
9C2×#14583813—21no—
10#19149678—21no—
11#19149674—21no—
12#19149674—21no—
13#19149671C2211yescharacteristic
14#19149667C2211yescharacteristic
15#19149664—21no—
16#19149659—21no—
17#19149657—21no—
18#19149612—21no—
19#19149610—21no—
20#19149590C2211yescharacteristic
21#19149590C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#125×C173C2211yescharacteristic
24#107×C173—21no—
25#106×C173—21no—
26#97×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C4×D173—21no—
31#14583816—42no—
32#14583813—42no—
33C2×#13063436—21no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173D411yescharacteristic
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C2×Q8×C173—21no—
48C2×C8×C173—21no—
49C4⋊C4×C173—21no—
50C4⋊C4×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173D411yescharacteristic
54C4⋊C4×C173D411yescharacteristic
55C42×C173C2311yescharacteristic
56C22×D173—21no—
57Dic346—41no—
58Dic346—81no—
59D692—41no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C4×D173—81no—
64#13063436—22no—
65#13063436—22no—
66Q8×C173—41no—
67Q8×C173—41no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—41no—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C8×C173—22no—
75C22×C173#12011yescharacteristic
76C2×D173—41no—
77C2×D173—41no—
78C2×D173—41no—
79C4×C173—21no—
80C4×C173—21no—
81C4×C173—21no—
82C4×C173—41no—
83C4×C173—41no—
84C4×C173—81no—
85Dic173—41no—
86Dic173—41no—
87Dic173—41no—
88C2×C173#45311yescharacteristic
89C2×C173#45611yescharacteristic
90C2×C173#45611yescharacteristic
91D173—81no—
92C173#166011yescharacteristic, complemented, Sylow
93#1660—1731nomaximal, Sylow
94#563—1731no—
95#501—1731no—
96#501D17311yescharacteristic, complemented
97#497—1731no—
98#330—1731no—
99#330—1731no—
100#329—1731no—
101C2×Q16—3461no—
102C2×SD16—3461no—
103C4⋊Q8—3461no—
104C4⋊Q8C2×D17311yescharacteristic
105#126—3461no—
106#126—3461no—
107#125—1731no—
108#125C2×D17311yescharacteristic
109#124—1731no—
110#123—3461no—
111C22⋊Q8—3461no—
112C22⋊Q8—3461no—
113#107—21no—
114#107—3461no—
115#106—21no—
116#106—3461no—
117#97—1731no—
118#97—1731no—
119#97C2×D17311yescharacteristic
120C2×Q8—21no—
121C2×Q8—3461no—
122C2×Q8—1731no—
123C2×D4—1731no—
124C22×C4—3461no—
125Q16—6922no—
126SD16—6922no—
127C2×C8—21no—
128C2×C8—3461no—
129C2×C8—3461no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3461no—
135C4⋊C4—3461no—
136C4⋊C4—3461no—
137C4⋊C4—3461no—
138C4⋊C4—1731no—
139C4⋊C4—1731no—
140C4⋊C4C22⋊D17311yescharacteristic
141C4⋊C4C22⋊D17311yescharacteristic
142C22⋊C4—3461no—
143C22⋊C4—3461no—
144C22⋊C4—3461no—
145C22⋊C4—3461no—
146C22⋊C4—3461no—
147C42—3461no—
148C42C22×D17311yescharacteristic
149C23—3461no—
150Q8—41no—
151Q8—41no—
152Q8—13841no—
153Q8—6921no—
154D4—6921no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—3461no—
159C2×C4—3461no—
160C2×C4—3461no—
161C2×C4—13841no—
162C2×C4D4×D17311yescharacteristic
163C2×C4D4×D17311yescharacteristic
164C2×C4C2×C22⋊D17311yescharacteristic
165C8—22no—
166C8—3462no—
167C8—3462no—
168C22—6921no—
169C22—6921no—
170C22—6921no—
171C22#1914964711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—6921no—
179C4—6921no—
180C4—6921no—
181C2—13841no—
182C2#2601165711yescharacteristic
183C2#2601167211yescharacteristic
184C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#501×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#501D173
hypocenterC173#1660
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #26011897C2semidirect
4. #26011233C2semidirect
5. #26011232C2semidirect
7. #501×C173C2semidirect
92. C173#1660semidirect
96. #501D173semidirect

Maximal subgroups

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

subgroupindex
#260122452
#260118972
#260118832
#260112332
#260112322
#260112302
#501×C1732
#1660173