Finite Group Databasebeta

← #36363096

Subgroups — #36363096

Order 22144.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#19149709C211yesmaximal, characteristic
2C22×Dic692C211yesmaximal, characteristic, complemented
3#26012062C211yesmaximal, characteristic
4#26011968C211yesmaximal, characteristic, complemented
5#26011712C214yesmaximal, complemented
6#26011700C214yesmaximal, complemented
7#26011517C211yesmaximal, characteristic, complemented
8#26011499C211yesmaximal, characteristic, complemented
9#416×C173C211yesmaximal, characteristic
10C22×Dic346C2211yescharacteristic
11C22×Dic346C2211yescharacteristic
12#19149709—24no—
13C2×Dic692—24no—
14C2×Dic692C2214yes—
15C2×#14583816C2212yes—
16C2×#14583814C2212yes—
17C2×#14583810C2211yescharacteristic
18#19149677C2212yes—
19#19149666C2212yes—
20#19149660—22no—
21#19149658—22no—
22C4×Dic346—22no—
23#19149644C2214yes—
24#19149638C2211yescharacteristic
25#19149635—22no—
26#19149608C2214yes—
27#19149602C2214yes—
28C2×#41×C173C2211yescharacteristic
29C22×C8×C173C2211yescharacteristic
30#117×C173C2211yescharacteristic
31#103×C173C2212yes—
32#102×C173C2212yes—
33C2×Dic346—21no—
34C2×Dic346—21no—
35C2×Dic346—22no—
36C2×Dic346—22no—
37C2×Dic346—22no—
38C2×Dic346D414yes—
39C2×Dic346C2312yes—
40C2×Dic346C2312yes—
41C22×Dic173—21no—
42C22×Dic173—21no—
43Dic692—28no—
44#14583816—24no—
45#14583814—24no—
46#14583810—22no—
47C2×#13063436C2312yes—
48C4⋊Dic173D412yes—
49#14583805—24no—
50#14583805—24no—
51C22⋊Dic173—24no—
52C22⋊Dic173D412yes—
53C4×Dic173—24no—
54C4×Dic173C2312yes—
55#41×C173—22no—
56#41×C173—22no—
57C2×Q8×C173C2311yescharacteristic
58C2×D4×C173—21no—
59C2×D4×C173C2311yescharacteristic
60C22×C4×C173—21no—
61C22×C4×C173C2311yescharacteristic
62C2×C8×C173—22no—
63C2×C8×C173C2312yes—
64C4⋊C4×C173C2312yes—
65C22⋊C4×C173—21no—
66C42×C173—21no—
67Dic346—24no—
68Dic346—22no—
69Dic346—22no—
70Dic346—22no—
71Dic346—42no—
72Dic346C2×D414yes—
73C2×Dic173—24no—
74C2×Dic173—22no—
75C2×Dic173—22no—
76C2×Dic173—42no—
77C2×Dic173C2×D414yes—
78#13063436—22no—
79C23×C173—21no—
80C23×C173C2×D411yescharacteristic
81Q8×C173—21no—
82Q8×C173—21no—
83D4×C173—21no—
84D4×C173—21no—
85D4×C173—42no—
86C2×C4×C173—22no—
87C2×C4×C173—21no—
88C2×C4×C173—42no—
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D412yes—
91C2×C4×C173#4112yes—
92C2×C4×C173C2411yescharacteristic
93C8×C173—22no—
94C22×C173—22no—
95C22×C173—41no—
96C22×C173—42no—
97C22×C173#12112yes—
98C22×C173C22×D411yescharacteristic
99C4×C173—41no—
100C4×C173—42no—
101C4×C173#12112yes—
102C4×C173C22×D411yescharacteristic
103C4×C173C2×#4111yescharacteristic
104Dic173—24no—
105Dic173—24no—
106Dic173—42no—
107C2×C173—22no—
108C2×C173—81no—
109C2×C173C2×#12111yescharacteristic
110C2×C173#57812yes—
111C173#304411yescharacteristic, complemented, Sylow
112#3044—1731nomaximal, Sylow
113C2×#137—1731no—
114C22×Q16—1731no—
115#536—1731no—
116#519—1731no—
117#464—1734no—
118#462—1734no—
119#424—1731no—
120#419—1731no—
121#416D17311yescharacteristic
122C2×#41C2×D17311yescharacteristic
123C22×Q8—1731no—
124C22×Q8—1731no—
125#137—3464no—
126C2×Q16—3464no—
127C2×Q16—1732no—
128C2×Q16—1734no—
129C2×SD16—1732no—
130C2×M16—1731no—
131C22×C8C2×D17311yescharacteristic
132C4⋊Q8—1732no—
133#124—1732no—
134#123—3462no—
135C22⋊Q8—3462no—
136C4×Q8—3462no—
137C4×Q8—1734no—
138#117—3462no—
139#117—1731no—
140#117C2×D17311yescharacteristic
141C4⋊C8—1734no—
142#103—1734no—
143#103C2×D17312yes—
144#102C2×D17312yes—
145#41—22no—
146#41—22no—
147C2×Q8—3461no—
148C2×Q8—3461no—
149C2×Q8—3462no—
150C2×Q8—3462no—
151C2×Q8—3462no—
152C2×Q8—1734no—
153C2×Q8—1732no—
154C2×Q8—1732no—
155C2×Q8C22×D17311yescharacteristic
156C2×D4—21no—
157C2×D4C22×D17311yescharacteristic
158C22×C4—21no—
159C22×C4—3461no—
160C22×C4—3461no—
161C22×C4C22×D17311yescharacteristic
162Q16—3468no—
163Q16—3464no—
164SD16—3464no—
165M16—3462no—
166C2×C8—22no—
167C2×C8—1732no—
168C2×C8C22×D17312yes—
169C4⋊C4—3464no—
170C4⋊C4—3464no—
171C4⋊C4—1732no—
172C4⋊C4C22×D17312yes—
173C22⋊C4—21no—
174C22⋊C4—3464no—
175C22⋊C4—1732no—
176C42—21no—
177C42—3464no—
178C42—1732no—
179C23—21no—
180C23D4×D17311yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—3464no—
184Q8—3462no—
185Q8—3462no—
186Q8—3462no—
187Q8—1734no—
188Q8—6922no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—3464no—
196C2×C4—3462no—
197C2×C4—3462no—
198C2×C4—1734no—
199C2×C4—6922no—
200C2×C4D4×D17311yescharacteristic
201C2×C4D4×D17312yes—
202C2×C4#1458382612yes—
203C2×C4C23×D17311yescharacteristic
204C8—22no—
205C8—3462no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1914965212yes—
210C22C2×D4×D17311yescharacteristic
211C4—41no—
212C4—42no—
213C4—3464no—
214C4—3464no—
215C4—6922no—
216C4#1914965212yes—
217C4C2×D4×D17311yescharacteristic
218C4C2×#1458382611yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1914965211yescharacteristic
222C2#2601230512yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
prime radicalC2×#41×C173C22
hypercenter#416D173
hypocenterC173#3044
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic692C2semidirect
4. #26011968C2semidirect
5. #26011712C2semidirect
6. #26011700C2semidirect
7. #26011517C2semidirect
8. #26011499C2semidirect
111. C173#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#191497092
C22×Dic6922
#260120622
#260119682
#260117122
#260117002
#260115172
#260114992
#416×C1732
#3044173