Finite Group Databasebeta

← #36352696

Subgroups — #36352696

Order 22144.

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

normal subgroups only

This group has 214 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#26012201C211yesmaximal, characteristic
2#26011870C211yesmaximal, characteristic, complemented
3#26011778C211yesmaximal, characteristic
4#26011239C211yesmaximal, characteristic, complemented
5Q8⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic
7#477×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—22no—
9#19149679C2211yescharacteristic
10#19149671—21no—
11#19149657—22no—
12#19149655—21no—
13C4×D692—21no—
14C4×D692C2211yescharacteristic
15C4⋊C4×D173—21no—
16#19149610—21no—
17C4⋊#13063436C2211yescharacteristic
18#19149599—21no—
19C4×#13063436C2211yescharacteristic
20C4⋊Q8×C173C2211yescharacteristic
21C4×Q8×C173C2211yescharacteristic
22#107×C173—21no—
23C4⋊C8×C173C2211yescharacteristic
24#103×C173—21no—
25C2×Dic346D412yes—
26C2×D692D411yescharacteristic
27C2×C4×D173—22no—
28C2×C4×D173—21no—
29#14583816—44no—
30C2×#13063436—21no—
31C2×#13063436—21no—
32C4⋊Dic173—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173D411yescharacteristic
35#14583805—22no—
36#14583805—22no—
37#14583803—22no—
38#14583803—21no—
39C2×Q8×C173—21no—
40C2×Q8×C173D411yescharacteristic
41C2×C8×C173—21no—
42C4⋊C4×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173D411yescharacteristic
46C42×C173—21no—
47C42×C173C2311yescharacteristic
48C22×D173—21no—
49Dic346—42no—
50D692—21no—
51D692Q1612yescomplemented
52C2×Dic173—22no—
53C2×Dic173—21no—
54C4×D173—41no—
55C4×D173—41no—
56C4×D173—82no—
57#13063436—22no—
58#13063436—41no—
59Q8×C173—21no—
60Q8×C173—21no—
61Q8×C173—42no—
62C2×C4×C173—21no—
63C2×C4×C173—21no—
64C2×C4×C173—41no—
65C2×C4×C173C2×D411yescharacteristic
66C2×C4×C173C2×D411yescharacteristic
67C2×C4×C173C2×D411yescharacteristic
68C8×C173—41no—
69C22×C173#12011yescharacteristic
70C2×D173—22no—
71C2×D173—41no—
72C2×D173—41no—
73C4×C173—21no—
74C4×C173—41no—
75C4×C173—41no—
76C4×C173—81no—
77C4×C173C2×Q1611yescharacteristic
78C4×C173#13511yescharacteristic
79C4×C173#13711yescharacteristic
80C4×C173#13711yescharacteristic
81Dic173—41no—
82Dic173—42no—
83C2×C173#45111yescharacteristic
84C2×C173#45211yescharacteristic
85C2×C173#45311yescharacteristic
86D173—42no—
87C173#163211yescharacteristic, complemented, Sylow
88#1632—1731nomaximal, Sylow
89#554—1731no—
90#494—1731no—
91#477D17311yescharacteristic, complemented
92#476—1731no—
93#333—1731no—
94Q8⋊C8—1731no—
95D4⋊C8—1731no—
96C2×Q16—3462no—
97C4⋊Q8—1731no—
98C4⋊Q8C2×D17311yescharacteristic
99#125—3461no—
100C22⋊Q8—3462no—
101C22⋊Q8—3461no—
102C4×Q8C2×D17311yescharacteristic
103C4×D4—3461no—
104C4×D4—1731no—
105C2×C4⋊C4—3461no—
106#107—21no—
107#106—3461no—
108C4⋊C8—1731no—
109C4⋊C8C2×D17311yescharacteristic
110#103—21no—
111#102—3461no—
112C4×C8—1731no—
113C2×Q8—21no—
114C2×Q8—1732no—
115C2×Q8C22⋊D17311yescharacteristic
116C2×D4—1731no—
117C22×C4—3462no—
118C22×C4—3461no—
119Q16—6924no—
120C2×C8—21no—
121C2×C8—3461no—
122C2×C8—3461no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3462no—
127C4⋊C4—3462no—
128C4⋊C4—3461no—
129C4⋊C4—3461no—
130C4⋊C4—1731no—
131C4⋊C4C22⋊D17311yescharacteristic
132C22⋊C4—3462no—
133C22⋊C4—3461no—
134C42—21no—
135C42C22×D17311yescharacteristic
136C23—3461no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—6922no—
141D4—3461no—
142D4—1732no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—3461no—
147C2×C4—3462no—
148C2×C4—13842no—
149C2×C4—6921no—
150C2×C4—6921no—
151C2×C4D4×D17311yescharacteristic
152C2×C4D4×D17311yescharacteristic
153C2×C4C2×C22⋊D17311yescharacteristic
154C8—41no—
155C8—3462no—
156C8—6921no—
157C22—3462no—
158C22—6921no—
159C22—6921no—
160C22#1914964711yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—6921no—
166C4—6922no—
167C4Q16×D17311yescharacteristic
168C4#1914969911yescharacteristic
169C4#1914971011yescharacteristic
170C4#1914971111yescharacteristic
171C2—6922no—
172C2#2601164511yescharacteristic
173C2#2601165111yescharacteristic
174C2#2601165711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#477×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#477D173
hypocenterC173#1632
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #26011870C2semidirect
4. #26011239C2semidirect
5. Q8⋊#13063436C2semidirect
7. #477×C173C2semidirect
51. D692Q16semidirect
87. C173#1632semidirect
91. #477D173semidirect

Maximal subgroups

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

subgroupindex
#260122012
#260118702
#260117782
#260112392
Q8⋊#130634362
#260112182
#477×C1732
#1632173