Finite Group Databasebeta

← #36352781

Subgroups — #36352781

Order 22144.

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

normal subgroups only

This group has 200 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#26012221C211yesmaximal, characteristic
2#26011773C211yesmaximal, characteristic
3#26011698C211yesmaximal, characteristic
4#26011240C211yesmaximal, characteristic
5#26011223C211yesmaximal, characteristic
6#26011223C211yesmaximal, characteristic
7#487×C173C211yesmaximal, characteristic
8C2×#14583816—21no—
9#19149678—21no—
10#19149678—21no—
11#19149677—22no—
12C4×Dic346C2211yescharacteristic
13C4×Dic346C2211yescharacteristic
14#19149644—21no—
15#19149644—21no—
16#19149610—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149603—21no—
20#19149602—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#103×C173—22no—
24C4×C8×C173C2211yescharacteristic
25C2×Dic346—22no—
26C2×Dic346D411yescharacteristic
27C2×Dic346D411yescharacteristic
28#14583816—42no—
29C2×#13063436—21no—
30C2×#13063436—21no—
31C4⋊Dic173D411yescharacteristic
32C4⋊Dic173D411yescharacteristic
33#14583805—22no—
34#14583805—22no—
35#14583805—21no—
36#14583805—21no—
37C4×Dic173—22no—
38C4×Dic173—21no—
39C4×Dic173—21no—
40C2×Q8×C173—21no—
41C2×Q8×C173D412yes—
42C2×C8×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C42×C173C2311yescharacteristic
48Dic346—21no—
49Dic346—21no—
50Dic346—82no—
51Dic346SD1612yes—
52Dic346Q1612yescomplemented
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57#13063436—41no—
58#13063436—41no—
59Q8×C173—42no—
60Q8×C173—41no—
61Q8×C173—41no—
62C2×C4×C173—22no—
63C2×C4×C173—41no—
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173C2×D411yescharacteristic
66C2×C4×C173C2×D411yescharacteristic
67C8×C173—41no—
68C22×C173#12011yescharacteristic
69C4×C173—21no—
70C4×C173—42no—
71C4×C173—81no—
72C4×C173C2×SD1611yescharacteristic
73C4×C173C2×Q1611yescharacteristic
74C4×C173#13711yescharacteristic
75C4×C173#13711yescharacteristic
76Dic173—22no—
77Dic173—22no—
78Dic173—41no—
79Dic173—41no—
80C2×C173#44811yescharacteristic
81C2×C173#45111yescharacteristic
82C2×C173#45511yescharacteristic
83C173#164511yescharacteristic, complemented, Sylow
84#1645—1731nomaximal, Sylow
85Q82—1731no—
86#487D17311yescharacteristic
87#475—1731no—
88#462—1731no—
89#333—1731no—
90Q8⋊C8—1731no—
91Q8⋊C8—1731no—
92C2×Q16—3461no—
93C4⋊Q8—3461no—
94C4⋊Q8—3461no—
95C4⋊Q8—3462no—
96C4⋊Q8C2×D17311yescharacteristic
97C4⋊Q8C2×D17311yescharacteristic
98C4×Q8—3461no—
99C4×Q8—3461no—
100C4×Q8—1731no—
101C4×Q8—1731no—
102#106—3461no—
103C4⋊C8—1731no—
104C4⋊C8—1731no—
105#103—22no—
106#103—3461no—
107#103—3461no—
108C4×C8C2×D17311yescharacteristic
109C2×Q8—21no—
110C2×Q8—3462no—
111C2×Q8—1731no—
112C2×Q8—1731no—
113C2×Q8C22⋊D17312yes—
114Q16—6922no—
115C2×C8—21no—
116C2×C8—3461no—
117C2×C8—3461no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—3462no—
123C4⋊C4—3462no—
124C4⋊C4—3461no—
125C4⋊C4—3461no—
126C4⋊C4—1731no—
127C4⋊C4—1731no—
128C42—3462no—
129C42—3461no—
130C42—3461no—
131C42C22×D17311yescharacteristic
132Q8—42no—
133Q8—41no—
134Q8—41no—
135Q8—3461no—
136Q8—3461no—
137Q8—13842no—
138Q8—1732no—
139Q8—1732no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—3461no—
143C2×C4—3461no—
144C2×C4—3461no—
145C2×C4—3461no—
146C2×C4D4×D17311yescharacteristic
147C2×C4D4×D17311yescharacteristic
148C2×C4C2×C22⋊D17311yescharacteristic
149C8—41no—
150C8—6921no—
151C8—6921no—
152C22#1914964711yescharacteristic
153C4—21no—
154C4—42no—
155C4—81no—
156C4—3462no—
157C4—3462no—
158C4—6921no—
159C4—6921no—
160C4SD16×D17311yescharacteristic
161C4Q16×D17311yescharacteristic
162C4#1914970811yescharacteristic
163C4#1914971111yescharacteristic
164C2#2601162811yescharacteristic
165C2#2601164511yescharacteristic
166C2#2601166811yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#487×C173C2
socle, prime radicalC22×C173#120
hypercenter#487D173
hypocenterC173#1645
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
52. Dic346Q16semidirect
83. C173#1645semidirect

Maximal subgroups

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

subgroupindex
#260122212
#260117732
#260116982
#260112402
#260112232
#487×C1732
#1645173