Finite Group Databasebeta

← #36352718

Subgroups — #36352718

Order 22144.

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

normal subgroups only

This group has 188 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#26012216C211yesmaximal, characteristic
2#26011783C211yesmaximal, characteristic
3#26011783C211yesmaximal, characteristic
4#26011237C211yesmaximal, characteristic
5#26011223C211yesmaximal, characteristic
6#26011223C211yesmaximal, characteristic
7#487×C173C211yesmaximal, characteristic
8#19149677—21no—
9#19149670—21no—
10#19149669—22no—
11Q8×Dic173—21no—
12C4×Dic346C2211yescharacteristic
13C4×Dic346C2211yescharacteristic
14#19149644—21no—
15#19149612—21no—
16#19149612—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149602—21no—
20#19149602—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#103×C173—22no—
24C4×C8×C173C2211yescharacteristic
25C2×Dic346D411yescharacteristic
26C2×Dic346D411yescharacteristic
27C2×#13063436—21no—
28C2×#13063436—21no—
29C4⋊Dic173—22no—
30C4⋊Dic173D411yescharacteristic
31C4⋊Dic173D411yescharacteristic
32#14583805—22no—
33#14583805—22no—
34#14583805—21no—
35#14583805—21no—
36C4×Dic173—22no—
37C4×Dic173—21no—
38C4×Dic173—21no—
39C2×Q8×C173—21no—
40C2×Q8×C173—21no—
41C2×C8×C173—21no—
42C4⋊C4×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173D412yes—
46C42×C173C2311yescharacteristic
47Dic346—21no—
48Dic346—21no—
49Dic346—21no—
50Dic346Q1612yes—
51C2×Dic173—21no—
52C2×Dic173—21no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55#13063436—41no—
56#13063436—41no—
57Q8×C173—42no—
58Q8×C173—41no—
59Q8×C173—41no—
60C2×C4×C173—22no—
61C2×C4×C173—41no—
62C2×C4×C173C2×D411yescharacteristic
63C2×C4×C173C2×D411yescharacteristic
64C2×C4×C173C2×D411yescharacteristic
65C8×C173—41no—
66C22×C173#12011yescharacteristic
67C4×C173—21no—
68C4×C173—42no—
69C4×C173—81no—
70C4×C173C2×Q1611yescharacteristic
71C4×C173#13511yescharacteristic
72C4×C173#13711yescharacteristic
73C4×C173#13711yescharacteristic
74Dic173—22no—
75Dic173—41no—
76Dic173—41no—
77Dic173—41no—
78C2×C173#45111yescharacteristic
79C2×C173#45211yescharacteristic
80C2×C173#45411yescharacteristic
81C173#163611yescharacteristic, complemented, Sylow
82#1636—1731nomaximal, Sylow
83#557—1731no—
84#487D17311yescharacteristic
85#477—1731no—
86#477—1731no—
87#332—1731no—
88Q8⋊C8—1731no—
89Q8⋊C8—1731no—
90C4⋊Q8—3461no—
91C4⋊Q8C2×D17311yescharacteristic
92C4⋊Q8C2×D17311yescharacteristic
93#125—3461no—
94#125—3462no—
95C4×Q8—3461no—
96C4×Q8—3461no—
97C4×Q8—1731no—
98C4×Q8—1731no—
99#107—3461no—
100#107—3461no—
101C4⋊C8—1731no—
102C4⋊C8—1731no—
103#103—22no—
104#103—3461no—
105#103—3461no—
106C4×C8C2×D17311yescharacteristic
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—1731no—
110C2×Q8—1731no—
111C2×C8—21no—
112C2×C8—3461no—
113C2×C8—3461no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—3462no—
118C4⋊C4—3462no—
119C4⋊C4—3462no—
120C4⋊C4—3461no—
121C4⋊C4—3461no—
122C4⋊C4—1731no—
123C4⋊C4—1731no—
124C4⋊C4C22⋊D17312yes—
125C42—3462no—
126C42—3461no—
127C42—3461no—
128C42C22×D17311yescharacteristic
129Q8—42no—
130Q8—41no—
131Q8—41no—
132Q8—3461no—
133Q8—3461no—
134Q8—3461no—
135Q8—1732no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—3461no—
139C2×C4—3461no—
140C2×C4—3461no—
141C2×C4—3461no—
142C2×C4D4×D17311yescharacteristic
143C2×C4D4×D17311yescharacteristic
144C2×C4C2×C22⋊D17311yescharacteristic
145C8—41no—
146C8—6921no—
147C8—6921no—
148C22#1914964711yescharacteristic
149C4—21no—
150C4—42no—
151C4—81no—
152C4—3462no—
153C4—6921no—
154C4—6921no—
155C4—6921no—
156C4Q16×D17311yescharacteristic
157C4#1914970011yescharacteristic
158C4#1914970811yescharacteristic
159C4#1914971111yescharacteristic
160C2#2601164511yescharacteristic
161C2#2601165011yescharacteristic
162C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#487×C173C2
socle, prime radicalC22×C173#120
hypercenter#487D173
hypocenterC173#1636
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
81. C173#1636semidirect

Maximal subgroups

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

subgroupindex
#260122162
#260117832
#260112372
#260112232
#487×C1732
#1636173