Finite Group Databasebeta

← #36352590

Subgroups — #36352590

Order 22144.

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

normal subgroups only

This group has 236 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#26011827C212yesmaximal, complemented
3#26011427C211yesmaximal, characteristic
4#26011229C212yesmaximal, complemented
5#323×C173C211yesmaximal, characteristic
6#19149658—21no—
7#19149658C2212yes—
8#19149656—22no—
9#19149656—22no—
10C2×C4⋊Dic173C2211yescharacteristic
11C2×#14583805—21no—
12C2×C22⋊Dic173—21no—
13C2×C22⋊Dic173—21no—
14#19149613—22no—
15#19149604—22no—
16#19149586C2211yescharacteristic
17C2×C22⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C2×Dic346—21no—
20C2×Dic346D412yes—
21C22×Dic173—21no—
22C22×Dic173—21no—
23C22×Dic173—21no—
24C22×Dic173—21no—
25C4⋊Dic173—21no—
26C4⋊Dic173—22no—
27C4⋊Dic173D412yes—
28#14583805—22no—
29#14583805—22no—
30#14583805—42no—
31C22⋊Dic173—22no—
32C22⋊Dic173—22no—
33C22⋊Dic173—42no—
34C24×C173D411yescharacteristic
35C22×C4×C173D411yescharacteristic
36C22×C4×C173C2311yescharacteristic
37C2×C8×C173—22no—
38C22⋊C4×C173—22no—
39Dic346—42no—
40Dic346—81no—
41C2×Dic173—22no—
42C2×Dic173—22no—
43C2×Dic173—22no—
44C2×Dic173—41no—
45C2×Dic173—41no—
46C2×Dic173—41no—
47C2×Dic173—41no—
48C2×Dic173—41no—
49C2×Dic173—81no—
50C2×Dic173—81no—
51C23×C173—21no—
52C23×C173—21no—
53C23×C173—41no—
54C23×C173—41no—
55C23×C173Q1612yes—
56C23×C173C2×D411yescharacteristic
57C2×C4×C173—21no—
58C2×C4×C173—41no—
59C2×C4×C173—41no—
60C2×C4×C173C2×D412yes—
61C8×C173—42no—
62C22×C173—21no—
63C22×C173—21no—
64C22×C173—22no—
65C22×C173—42no—
66C22×C173—41no—
67C22×C173—41no—
68C22×C173—81no—
69C22×C173#12011yescharacteristic
70C22×C173C2×Q1611yescharacteristic
71C22×C173#13711yescharacteristic
72C4×C173—22no—
73C4×C173—41no—
74Dic173—42no—
75Dic173—42no—
76Dic173—42no—
77Dic173—81no—
78C2×C173—21no—
79C2×C173—21no—
80C2×C173—42no—
81C2×C173#45111yescharacteristic
82C2×C173#45311yescharacteristic
83C2×C173#45811yescharacteristic
84C173#161411yescharacteristic, complemented, Sylow
85#1614—1731nomaximal, Sylow
86#543—1731no—
87#484—1732no—
88#399—1731no—
89#328—1732no—
90#323D17311yescharacteristic
91C22⋊Q8—3462no—
92C22⋊Q8—3462no—
93C22⋊Q8—3461no—
94C22⋊Q8—1732no—
95C2×C4⋊C4—3461no—
96C2×C4⋊C4—1731no—
97C2×C22⋊C4—3461no—
98C2×C22⋊C4—3461no—
99C2×C22⋊C4C2×D17311yescharacteristic
100#107—3462no—
101#103—3462no—
102C22⋊C8C2×D17312yes—
103#95—1731no—
104C24D69211yescharacteristic
105C2×Q8—3461no—
106C2×Q8—1732no—
107C22×C4—3461no—
108C22×C4—3461no—
109C22×C4—3461no—
110C22×C4—3461no—
111C22×C4D69211yescharacteristic
112C22×C4C22×D17311yescharacteristic
113C2×C8—22no—
114C4⋊C4—3462no—
115C4⋊C4—3461no—
116C4⋊C4—3462no—
117C4⋊C4—3462no—
118C4⋊C4—1732no—
119C4⋊C4—6922no—
120C22⋊C4—22no—
121C22⋊C4—3462no—
122C22⋊C4—3462no—
123C22⋊C4—6922no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic69212yes—
129C23C2×D69211yescharacteristic
130Q8—13841no—
131Q8—6922no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—3462no—
136C2×C4—3462no—
137C2×C4—3462no—
138C2×C4—13841no—
139C2×C4—13841no—
140C2×C4—6921no—
141C2×C4—6921no—
142C2×C4—6921no—
143C2×C4—6921no—
144C2×C4—6921no—
145C2×C4D4×D17312yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1914964911yescharacteristic
155C22C2×Dic69211yescharacteristic
156C22#1914971211yescharacteristic
157C4—22no—
158C4—41no—
159C4—13841no—
160C4—6922no—
161C4—6922no—
162C4—6922no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2601164911yescharacteristic
167C2#2601166111yescharacteristic
168C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173#120
prime radicalC24×C173D4
hypercenter#323D173
hypocenterC173#1614
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26011827C2semidirect
4. #26011229C2semidirect
84. C173#1614semidirect

Maximal subgroups

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

subgroupindex
#260121172
#260118272
#260114272
#260112292
#323×C1732
#1614173