Finite Group Databasebeta

← #36352553

Subgroups — #36352553

Order 22144.

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

normal subgroups only

This group has 228 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#26012117C211yesmaximal, characteristic
2#26011821C212yesmaximal, complemented
3#26011227C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic
5#397×C173C211yesmaximal, characteristic
6#19149658—21no—
7#19149658C2212yes—
8#19149656—22no—
9#19149656—22no—
10C2×C22⋊Dic173—21no—
11#19149610—22no—
12#19149602—22no—
13C22⋊#13063436C2212yes—
14C2×C4⋊C4×C173C2211yescharacteristic
15C2×C22⋊C4×C173—21no—
16C2×C22⋊C4×C173C2211yescharacteristic
17#95×C173—21no—
18#95×C173C2211yescharacteristic
19C2×Dic346—21no—
20C2×Dic346D412yes—
21C22×Dic173—21no—
22C22×Dic173—21no—
23C2×#13063436—22no—
24C4⋊Dic173—21no—
25C4⋊Dic173D412yes—
26#14583805—22no—
27#14583805—22no—
28C22⋊Dic173—22no—
29C22⋊Dic173—22no—
30C24×C173D411yescharacteristic
31C22×C4×C173—21no—
32C22×C4×C173—21no—
33C22×C4×C173D411yescharacteristic
34C22×C4×C173C2311yescharacteristic
35C4⋊C4×C173—22no—
36C22⋊C4×C173—22no—
37C22⋊C4×C173—42no—
38Dic346—42no—
39Dic346—81no—
40C2×Dic173—22no—
41C2×Dic173—22no—
42C2×Dic173—81no—
43C2×Dic173—81no—
44#13063436—42no—
45C23×C173—21no—
46C23×C173—21no—
47C23×C173—41no—
48C23×C173—41no—
49C23×C173SD1612yes—
50C23×C173C2×D411yescharacteristic
51C2×C4×C173—21no—
52C2×C4×C173—41no—
53C2×C4×C173—41no—
54C2×C4×C173—41no—
55C2×C4×C173—41no—
56C2×C4×C173—41no—
57C2×C4×C173—41no—
58C2×C4×C173—41no—
59C2×C4×C173—41no—
60C2×C4×C173C2×D412yes—
61C22×C173—21no—
62C22×C173—21no—
63C22×C173—22no—
64C22×C173—42no—
65C22×C173—41no—
66C22×C173—41no—
67C22×C173—81no—
68C22×C173#12011yescharacteristic
69C22×C173C2×SD1611yescharacteristic
70C22×C173#13711yescharacteristic
71C4×C173—22no—
72C4×C173—41no—
73C4×C173—81no—
74C4×C173—81no—
75Dic173—42no—
76Dic173—42no—
77C2×C173—21no—
78C2×C173—21no—
79C2×C173—42no—
80C2×C173#44811yescharacteristic
81C2×C173#45511yescharacteristic
82C2×C173#45811yescharacteristic
83C173#160811yescharacteristic, complemented, Sylow
84#1608—1731nomaximal, Sylow
85#543—1731no—
86#483—1732no—
87#397D17311yescharacteristic
88#328—1732no—
89#323—1731no—
90C22⋊Q8—3462no—
91C22⋊Q8—3462no—
92C22⋊Q8—3461no—
93C22⋊Q8—1732no—
94C2×C4⋊C4C2×D17311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—3461no—
97C2×C22⋊C4C2×D17311yescharacteristic
98#106—3462no—
99#103—3462no—
100C22⋊C8—1732no—
101#95—21no—
102#95C2×D17311yescharacteristic
103C24C22⋊D17311yescharacteristic
104C2×Q8—3461no—
105C2×Q8—1732no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—3461no—
109C22×C4—3461no—
110C22×C4C22⋊D17311yescharacteristic
111C22×C4C22×D17311yescharacteristic
112C2×C8—3462no—
113C4⋊C4—22no—
114C4⋊C4—3462no—
115C4⋊C4—3461no—
116C4⋊C4—3462no—
117C4⋊C4—1732no—
118C22⋊C4—22no—
119C22⋊C4—42no—
120C22⋊C4—3462no—
121C22⋊C4—3462no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1458381412yes—
127C23C2×C22⋊D17311yescharacteristic
128Q8—13841no—
129Q8—6922no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—3462no—
140C2×C4—3462no—
141C2×C4—13841no—
142C2×C4—13841no—
143C2×C4D4×D17312yes—
144C8—6922no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#1914964711yescharacteristic
153C22C2×#1458381411yescharacteristic
154C22#1914970911yescharacteristic
155C4—22no—
156C4—41no—
157C4—81no—
158C4—81no—
159C4—6922no—
160C4—6922no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#2601162911yescharacteristic
165C2#2601166811yescharacteristic
166C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#397×C173C2
socleC22×C173#120
prime radicalC24×C173D4
hypercenter#397D173
hypocenterC173#1608
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #26011821C2semidirect
3. #26011227C2semidirect
83. C173#1608semidirect

Maximal subgroups

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

subgroupindex
#260121172
#260118212
#260112272
#260112152
#397×C1732
#1608173