Finite Group Databasebeta

← #36355514

Subgroups — #36355514

Order 22144.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#26011931C212yesmaximal, complemented
2#26011749C211yesmaximal, characteristic, complemented
3#26011301C212yesmaximal, complemented
4#26011273C211yesmaximal, characteristic, complemented
5#575×C173C211yesmaximal, characteristic
6C2×Dic692C2211yescharacteristic, complemented
7#19149658—21no—
8#19149625—22no—
9#19149623—22no—
10#19149619C2212yescomplemented
11#19149611C2211yescharacteristic, complemented
12#19149604—21no—
13C2×#41×C173—21no—
14#137×C173—22no—
15AGL(1,ℤ/8)×C173—22no—
16#135×C173—21no—
17#135×C173C2212yes—
18C2×M16×C173C2211yescharacteristic
19C2×Dic346—21no—
20Dic692—22no—
21C4⋊Dic173—21no—
22#14583805—41no—
23C22⋊Dic173—41no—
24#14583801—22no—
25#41×C173—22no—
26#41×C173—22no—
27#41×C173—41no—
28#41×C173—41no—
29C2×Q8×C173—21no—
30C2×D4×C173—21no—
31C2×D4×C173—41no—
32C22×C4×C173—41no—
33C22×C4×C173D411yescharacteristic
34Q16×C173—21no—
35Q16×C173D412yes—
36SD16×C173—22no—
37SD16×C173—22no—
38D8×C173—21no—
39D8×C173D412yes—
40M16×C173—22no—
41C2×C8×C173D411yescharacteristic
42C2×C8×C173C2311yescharacteristic
43Dic346—41no—
44Dic346—41no—
45C2×Dic173—41no—
46C2×Dic173—41no—
47C23×C173—41no—
48C23×C173D811yescharacteristic
49Q8×C173—22no—
50Q8×C173—41no—
51D4×C173—22no—
52D4×C173—41no—
53D4×C173—42no—
54D4×C173—42no—
55C2×C4×C173—21no—
56C2×C4×C173—21no—
57C2×C4×C173—42no—
58C2×C4×C173—41no—
59C2×C4×C173—42no—
60C2×C4×C173D811yescharacteristic
61C2×C4×C173C2×D411yescharacteristic
62C8×C173—21no—
63C8×C173C2×D412yes—
64C22×C173—21no—
65C22×C173—41no—
66C22×C173—42no—
67C22×C173—82no—
68C22×C173C2×D811yescharacteristic
69C4×C173—21no—
70C4×C173—42no—
71C4×C173#12011yescharacteristic
72C4×C173AGL(1,ℤ/8)11yescharacteristic
73Dic173—81no—
74Dic173—81no—
75C2×C173—21no—
76C2×C173—41no—
77C2×C173—82no—
78C2×C173#44711yescharacteristic
79C173#219111yescharacteristic, complemented, Sylow
80#2191—1731nomaximal, Sylow
81#575D17311yescharacteristic
82#510—1732no—
83#470—1731no—
84#360—1732no—
85#349—1731no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D17312yes—
91C2×Q16—1731no—
92C2×M16C2×D17311yescharacteristic
93C22⋊Q8—3461no—
94Q32—3462no—
95SD32—3462no—
96M32—1732no—
97#106—1731no—
98#103—3461no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—3461no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D17311yescharacteristic
109Q16—21no—
110Q16—3462no—
111Q16C22⋊D17312yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D17312yes—
116M16—22no—
117C2×C8C22⋊D17311yescharacteristic
118C2×C8C22×D17311yescharacteristic
119C4⋊C4—3461no—
120C4⋊C4—6921no—
121C22⋊C4—6921no—
122C16—3462no—
123C23—41no—
124C23#1458381111yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—6921no—
128Q8—6921no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—6921no—
139C2×C4—6921no—
140C2×C4#1458381111yescharacteristic
141C2×C4C2×C22⋊D17311yescharacteristic
142C8—21no—
143C8C2×C22⋊D17312yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1458381111yescharacteristic
149C4—21no—
150C4—42no—
151C4—13841no—
152C4—13841no—
153C4#1914964811yescharacteristic
154C4#1914970511yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2601162511yescharacteristic
subgroupG/H
centerC2#26011625
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
Fitting, prime radical#575×C173C2
socleC2×C173#447
hypercenter#575D173
hypocenterC173#2191
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #26011931C2semidirect
2. #26011749C2semidirect
3. #26011301C2semidirect
4. #26011273C2semidirect
6. C2×Dic692C22semidirect
10. #19149619C22semidirect
11. #19149611C22semidirect
79. C173#2191semidirect

Maximal subgroups

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

subgroupindex
#260119312
#260117492
#260113012
#260112732
#575×C1732
#2191173