Finite Group Databasebeta

← #36368925

Subgroups — #36368925

Order 22144.

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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#26012512C211yesmaximal, characteristic
2#26012507C211yesmaximal, characteristic
3#26012341C212yesmaximal, complemented
4Q8⋊#13063443C212yesmaximal, complemented
5#26012043C211yesmaximal, characteristic
6C4⋊#13063443—21no—
7C4⋊#13063443C2212yes—
8C22⋊#13063443—21no—
9C22⋊#13063443—21no—
10#19149734—21no—
11C4×#13063443C2212yes—
12#19149678C411yescharacteristic
13#19149677C411yescharacteristic
14#19149669—21no—
15#19149658C411yescharacteristic
16#19149656—21no—
17Q8×Dic173C2211yescharacteristic
18#19149644—21no—
19#19149644C2211yescharacteristic
20#19149635—21no—
21C2×C4×Dic173C2211yescharacteristic
22C22⋊Q8×C173C411yescharacteristic
23C2×#13063443—22no—
24C2×#13063443—22no—
25C2×Dic346—21no—
26C2×Dic346C2×C411yescharacteristic
27C22×Dic173D411yescharacteristic
28C22×Dic173D411yescharacteristic
29C4⋊Dic173—21no—
30C4⋊Dic173C2×C411yescharacteristic
31#14583805—21no—
32#14583805—21no—
33#14583805—21no—
34#14583805—21no—
35#14583805—21no—
36C22⋊Dic173—21no—
37C4×Dic173—21no—
38C4×Dic173—21no—
39C4×Dic173C2×C411yescharacteristic
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173C2311yescharacteristic
43C2×Q8×C173C2×C411yescharacteristic
44C22×C4×C173C2×C411yescharacteristic
45C4⋊C4×C173—21no—
46C4⋊C4×C173C2×C411yescharacteristic
47C22⋊C4×C173—21no—
48#13063443—42no—
49#13063443—42no—
50Dic346—22no—
51Dic346—42no—
52C2×Dic173—22no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—41no—
56C2×Dic173C22⋊C411yescharacteristic
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C2×D411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C23×C173C22⋊C411yescharacteristic
61Q8×C173—22no—
62C2×C4×C173—22no—
63C2×C4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173C22⋊C411yescharacteristic
66C2×C4×C173C22×C411yescharacteristic
67C22×C173—21no—
68C22×C173—21no—
69C22×C173—21no—
70C22×C173C2×C22⋊C411yescharacteristic
71C4×C173—21no—
72C4×C173—41no—
73C4×C173—41no—
74C4×C173#13112yes—
75Dic173—21no—
76Dic173—21no—
77Dic173—21no—
78Dic173—41no—
79Dic173—41no—
80Dic173#13712yes—
81C2×C173—41no—
82C2×C173#40811yescharacteristic
83C2×C173#41911yescharacteristic
84C2×C173#42111yescharacteristic
85C173#149711yescharacteristic, complemented, Sylow
86#1497—1731nomaximal, Sylow
87#533—1731no—
88#432—1731no—
89#431—1731no—
90Q8⋊C8—1732no—
91Q8⋊C8—1732no—
92C4⋊Q8—1731no—
93C4⋊Q8—1731no—
94#125—3461no—
95C22⋊Q8—3461no—
96C22⋊Q8—1731no—
97C22⋊Q8#1062192411yescharacteristic
98C4×Q8—3461no—
99C4×Q8—1731no—
100C4×Q8—1731no—
101#117—3461no—
102C2×C42—1731no—
103C4⋊C8—3461no—
104C4⋊C8—1732no—
105C22⋊C8—3461no—
106C22⋊C8—3461no—
107#97—3461no—
108C4×C8—1732no—
109C2×Q8—3461no—
110C2×Q8—1731no—
111C2×Q8C2×#1062192411yescharacteristic
112C22×C4—1731no—
113C22×C4—1731no—
114C22×C4C2×#1062192411yescharacteristic
115C2×C8—3462no—
116C2×C8—3462no—
117C4⋊C4—21no—
118C4⋊C4—3461no—
119C4⋊C4—3461no—
120C4⋊C4—3461no—
121C4⋊C4—3461no—
122C4⋊C4—3461no—
123C4⋊C4—3461no—
124C4⋊C4—1731no—
125C4⋊C4C2×#1062192411yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—3461no—
128C42—3461no—
129C42—3461no—
130C42—1731no—
131C42—1731no—
132C42—1731no—
133C42—1731no—
134C23C22⋊#1062192411yescharacteristic
135Q8—22no—
136Q8—3462no—
137Q8—6922no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—3462no—
142C2×C4—3461no—
143C2×C4—3461no—
144C2×C4—1731no—
145C2×C4—1731no—
146C2×C4—1731no—
147C2×C4—1731no—
148C2×C4—6921no—
149C2×C4C22⋊#1062192411yescharacteristic
150C2×C4C22×#1062192411yescharacteristic
151C8—6922no—
152C8—6922no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22C2×C22⋊#1062192411yescharacteristic
157C4—21no—
158C4—41no—
159C4—41no—
160C4—3461no—
161C4—3461no—
162C4—3461no—
163C4—1732no—
164C4—6921no—
165C4—6921no—
166C4#1914976612yes—
167C2—41no—
168C2#2601246411yescharacteristic
169C2#2601248411yescharacteristic
170C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×C173C22⋊C4
cosocleC4×Dic173C23
hypercenterC22⋊Q8#10621924
hypocenterC173#1497
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #26012341C2semidirect
4. Q8⋊#13063443C2semidirect
85. C173#1497semidirect

Maximal subgroups

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

subgroupindex
#260125122
#260125072
#260123412
Q8⋊#130634432
#260120432
#1497173