Finite Group Databasebeta

← #36354728

Subgroups — #36354728

Order 22144.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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, complemented
2#26011346C211yesmaximal, characteristic
3#26011346C212yesmaximal
4#26011257C212yesmaximal, complemented
5#342×C173C211yesmaximal, characteristic, complemented
6C24×D173C2211yescharacteristic
7#19149658—22no—
8#19149656—24no—
9C2×C22⋊Dic173C2211yescharacteristic
10C2×C22⋊Dic173C2212yes—
11C2×#14583803—22no—
12C2×#14583803—24no—
13C2×C22⋊C4×C173C2212yes—
14C2×C22⋊C4×C173C2211yescharacteristic
15C23×D173—22no—
16C23×D173—21no—
17C23×D173—21no—
18C23×D173—22no—
19C23×D173—44no—
20C23×D173Q812yescomplemented
21C2×Dic346—22no—
22C22×Dic173—21no—
23C22×Dic173D414yes—
24C4⋊Dic173—22no—
25#14583805—24no—
26C22⋊Dic173—22no—
27C22⋊Dic173—22no—
28C22⋊Dic173—22no—
29#14583803—44no—
30#14583803—48no—
31C24×C173C2311yescharacteristic
32C22×C4×C173—22no—
33C22×C4×C173D412yes—
34C22⋊C4×C173—22no—
35C22⋊C4×C173—22no—
36C22⋊C4×C173—22no—
37C22×D173—28no—
38C22×D173—24no—
39C22×D173—24no—
40C22×D173—22no—
41C22×D173—44no—
42C22×D173—44no—
43C22×D173—48no—
44C22×D173—44no—
45C22×D173—42no—
46C22×D173—41no—
47C22×D173—42no—
48C22×D173—41no—
49Dic346—82no—
50C2×Dic173—24no—
51C2×Dic173—44no—
52C2×Dic173—42no—
53C2×Dic173—41no—
54C23×C173—42no—
55C23×C173C2×D412yes—
56C23×C173C2×D411yescharacteristic
57C23×C173C2×Q811yescharacteristic
58C23×C173#4111yescharacteristic
59C23×C173#4112yes—
60C2×C4×C173—22no—
61C2×C4×C173—42no—
62C2×C4×C173—44no—
63C2×C4×C173—42no—
64C22×C173—22no—
65C22×C173—24no—
66C22×C173—41no—
67C22×C173—42no—
68C22×C173—41no—
69C22×C173#12011yescharacteristic
70C22×C173C22⋊Q812yes—
71C22×C173C22⋊Q811yescharacteristic
72C22×C173#12412yes—
73C22×C173#12411yescharacteristic
74C2×D173—28no—
75C2×D173—24no—
76C2×D173—44no—
77C2×D173—48no—
78C2×D173—44no—
79C2×D173—42no—
80C2×D173—42no—
81C2×D173—44no—
82C4×C173—42no—
83C4×C173—82no—
84Dic173—44no—
85Dic173—81no—
86C2×C173—21no—
87C2×C173—22no—
88C2×C173—22no—
89C2×C173—21no—
90C2×C173#39311yescharacteristic
91C2×C173B(4,𝔽2)12yes—
92D173—44no—
93C173#202911yescharacteristic, complemented, Sylow
94#2029—1731nomaximal, Sylow
95#543—1731no—
96#379—1732no—
97#379—1731no—
98#342—1732no—
99#342D17311yescharacteristic, complemented
100C25—1731no—
101C22⋊Q8—3464no—
102C22⋊Q8—3462no—
103C2×C22⋊C4—3464no—
104C2×C22⋊C4—3462no—
105C2×C22⋊C4—1731no—
106C2×C22⋊C4—1732no—
107C2×C22⋊C4C2×D17312yes—
108C2×C22⋊C4C2×D17311yescharacteristic
109C24—3462no—
110C24—3461no—
111C24—3461no—
112C24—3462no—
113C24—1732no—
114C24—6924no—
115C24C22×D17311yescharacteristic
116C2×Q8—3462no—
117C22×C4—22no—
118C22×C4—3461no—
119C22×C4—1734no—
120C22×C4D69212yes—
121C4⋊C4—3464no—
122C4⋊C4—3462no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—3462no—
127C22⋊C4—3462no—
128C22⋊C4—3462no—
129C22⋊C4—6924no—
130C22⋊C4—6928no—
131C23—42no—
132C23—3468no—
133C23—3464no—
134C23—3464no—
135C23—3462no—
136C23—6924no—
137C23—6922no—
138C23—6924no—
139C23—6924no—
140C23—6928no—
141C23—6921no—
142C23—6922no—
143C23—6921no—
144C23D4×D17312yes—
145C23C2×D69211yescharacteristic
146C23Q8×D17311yescharacteristic
147C23#1458382511yescharacteristic
148C23#1458382612yes—
149Q8—13842no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—3464no—
155C2×C4—6922no—
156C2×C4—6921no—
157C2×C4—6924no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—3468no—
164C22—3464no—
165C22—6928no—
166C22—6924no—
167C22—6924no—
168C22—6922no—
169C22—6922no—
170C22—6924no—
171C22#1914964911yescharacteristic
172C22#1914965712yes—
173C22#1914965911yescharacteristic
174C22#1914966512yes—
175C22#1914966711yescharacteristic
176C4—42no—
177C4—82no—
178C4—13841no—
179C4—6924no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—6924no—
185C2#2601140411yescharacteristic
186C2#2601167912yes—
subgroupG/H
centerC22#19149649
commutator, cosocleC24×C173C23
FrattiniC24C22×D173
Fitting#342×C173C2
socleC22×C173#120
prime radicalC24×D173C22
hypercenter#342D173
hypocenterC173#2029
LCSG ▷ C24×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #26012117C2semidirect
4. #26011257C2semidirect
5. #342×C173C2semidirect
20. C23×D173Q8semidirect
93. C173#2029semidirect
99. #342D173semidirect

Maximal subgroups

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

subgroupindex
#260121172
#260113462
#260112572
#342×C1732
#2029173