Finite Group Databasebeta

← #36352505

Subgroups — #36352505

Order 22144.

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

normal subgroups only

This group has 266 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#26012122C211yesmaximal, characteristic, complemented
2#26011644C212yesmaximal
3#26011408C211yesmaximal, characteristic
4#26011229C212yesmaximal, complemented
5#324×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7C2×#14583815—22no—
8#19149678—22no—
9#19149670—22no—
10#19149658—21no—
11#19149658C2212yes—
12#19149638—21no—
13C2×#14583803—21no—
14C2×#14583803—21no—
15#19149601—22no—
16#19149586C2211yescharacteristic
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C23×D173—21no—
20C2×Dic346—21no—
21C2×Dic346D412yes—
22C2×D692—22no—
23C2×D692—42no—
24C22×Dic173—21no—
25#14583815—44no—
26C4⋊Dic173—21no—
27C4⋊Dic173—21no—
28C4⋊Dic173—21no—
29C4⋊Dic173D412yes—
30#14583805—22no—
31#14583805—22no—
32C22⋊Dic173—22no—
33#14583803—42no—
34#14583803—42no—
35C4×Dic173—22no—
36C22×C4×C173D411yescharacteristic
37C22×C4×C173D411yescharacteristic
38C22×C4×C173C2311yescharacteristic
39C2×C8×C173—22no—
40C4⋊C4×C173—22no—
41C22×D173—22no—
42C22×D173—41no—
43C22×D173—41no—
44C22×D173—41no—
45C22×D173—41no—
46C22×D173—41no—
47C22×D173—41no—
48Dic346—42no—
49Dic346—81no—
50D692—24no—
51D692—42no—
52C2×Dic173—22no—
53C2×Dic173—22no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56C2×Dic173—41no—
57C23×C173C2×D411yescharacteristic
58C2×C4×C173—21no—
59C2×C4×C173—21no—
60C2×C4×C173—21no—
61C2×C4×C173—41no—
62C2×C4×C173—41no—
63C2×C4×C173SD1612yescomplemented
64C2×C4×C173C2×D412yes—
65C8×C173—42no—
66C22×C173—21no—
67C22×C173—21no—
68C22×C173#12011yescharacteristic
69C22×C173C2×SD1611yescharacteristic
70C22×C173#13711yescharacteristic
71C2×D173—42no—
72C2×D173—42no—
73C2×D173—42no—
74C2×D173—81no—
75C2×D173—81no—
76C2×D173—81no—
77C2×D173—81no—
78C4×C173—22no—
79C4×C173—22no—
80C4×C173—41no—
81Dic173—42no—
82Dic173—42no—
83Dic173—81no—
84C2×C173—21no—
85C2×C173—21no—
86C2×C173#44811yescharacteristic
87C2×C173#45411yescharacteristic
88C2×C173B(4,𝔽2)11yescharacteristic
89D173—82no—
90C173#159911yescharacteristic, complemented, Sylow
91#1599—1731nomaximal, Sylow
92#544—1731no—
93#450—1732no—
94#394—1731no—
95#328—1732no—
96#324D17311yescharacteristic, complemented
97C22×D4—1731no—
98C2×SD16—3462no—
99C4⋊Q8—3462no—
100#125—3462no—
101C22⋊Q8—3461no—
102C22⋊Q8—1732no—
103#117—3461no—
104C2×C4⋊C4C2×D17311yescharacteristic
105C2×C22⋊C4—3461no—
106C2×C22⋊C4—3461no—
107#102—3462no—
108C22⋊C8C2×D17312yes—
109#95—1731no—
110C24—3461no—
111C2×Q8—3461no—
112C2×Q8—1732no—
113C2×D4—3462no—
114C2×D4—6922no—
115C22×C4—3461no—
116C22×C4D69211yescharacteristic
117C22×C4D69211yescharacteristic
118C22×C4C22×D17311yescharacteristic
119SD16—6924no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—3461no—
123C4⋊C4—3461no—
124C4⋊C4—3461no—
125C4⋊C4—3462no—
126C4⋊C4—3462no—
127C4⋊C4—1732no—
128C22⋊C4—3462no—
129C22⋊C4—6922no—
130C22⋊C4—6922no—
131C42—3462no—
132C23—3462no—
133C23—6921no—
134C23—6921no—
135C23—6921no—
136C23—6921no—
137C23—6921no—
138C23—6921no—
139C23C2×D69211yescharacteristic
140Q8—13841no—
141Q8—6922no—
142D4—3464no—
143D4—6922no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—3462no—
150C2×C4—3462no—
151C2×C4—6921no—
152C2×C4—6921no—
153C2×C4—6921no—
154C2×C4#1458381512yescomplemented
155C2×C4D4×D17312yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—13841no—
160C22—13841no—
161C22—13841no—
162C22—13841no—
163C22—6922no—
164C22—6922no—
165C22—6922no—
166C22#1914964911yescharacteristic
167C22C2×#1458381511yescharacteristic
168C22#1914971211yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—13841no—
173C4—6922no—
174C4—6922no—
175C2—21no—
176C2—21no—
177C2—13842no—
178C2#2601163211yescharacteristic
179C2#2601166611yescharacteristic
180C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173#120
prime radicalC22×D692C22
hypercenter#324D173
hypocenterC173#1599
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #26012122C2semidirect
4. #26011229C2semidirect
5. #324×C173C2semidirect
63. C2×C4×C173SD16semidirect
90. C173#1599semidirect
96. #324D173semidirect
154. C2×C4#14583815semidirect

Maximal subgroups

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

subgroupindex
#260121222
#260116442
#260114082
#260112292
#324×C1732
#1599173