Finite Group Databasebeta

← #36365018

Subgroups — #36365018

Order 22144.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#26012108C212yesmaximal, complemented
2#26011785C212yesmaximal, complemented
3#26011769C212yesmaximal, complemented
4#26011735C212yesmaximal, complemented
5#26011732C211yesmaximal, characteristic, complemented
6#26011712C212yesmaximal, complemented
7#26011704C212yesmaximal, complemented
8#26011517C211yesmaximal, characteristic, complemented
9#466×C173C211yesmaximal, characteristic
10C2×Dic692C2211yescharacteristic
11C2×#14583814C2212yes—
12C2×#14583810C2211yescharacteristic
13#19149678C2212yes—
14#19149672—22no—
15#19149666—22no—
16#19149666C2212yes—
17#19149660—22no—
18#19149658C2212yes—
19#19149656—22no—
20#19149644C2212yes—
21D4×Dic173C2212yes—
22#19149638C2211yescharacteristic
23#19149613C2211yescharacteristic
24#19149612C2212yes—
25#19149608C2212yes—
26#19149608C2212yes—
27#19149604C2212yes—
28#19149602C2212yes—
29#19149600C2212yes—
30C2×D8×C173C2211yescharacteristic
31C22×C8×C173C2211yescharacteristic
32#121×C173C2212yes—
33#107×C173C2211yescharacteristic
34#102×C173C2212yes—
35C2×Dic346—22no—
36C2×Dic346C2312yes—
37C22×Dic173—22no—
38Dic692—41no—
39#14583814—42no—
40#14583810—41no—
41C2×#13063436C2312yes—
42C4⋊Dic173D411yescharacteristic
43C4⋊Dic173D411yescharacteristic
44C4⋊Dic173C2312yes—
45#14583805—22no—
46#14583805—22no—
47#14583805—22no—
48C22⋊Dic173—22no—
49C22⋊Dic173—22no—
50C22⋊Dic173—22no—
51C22⋊Dic173—22no—
52C22⋊Dic173D412yes—
53C4×Dic173—22no—
54C4×Dic173C2312yes—
55C2×D4×C173—22no—
56C2×D4×C173C2312yes—
57C22×C4×C173C2311yescharacteristic
58D8×C173—41no—
59C2×C8×C173—41no—
60C2×C8×C173C2311yescharacteristic
61C2×C8×C173C2311yescharacteristic
62C4⋊C4×C173C2312yes—
63C22⋊C4×C173—22no—
64Dic346—22no—
65Dic346—22no—
66Dic346—82no—
67C2×Dic173—22no—
68C2×Dic173—22no—
69C2×Dic173—82no—
70C2×Dic173C2×D412yes—
71C2×Dic173C2×D412yes—
72#13063436—22no—
73C23×C173—22no—
74C23×C173C2×D411yescharacteristic
75D4×C173—22no—
76D4×C173—22no—
77D4×C173—82no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—22no—
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2411yescharacteristic
83C8×C173—21no—
84C8×C173—21no—
85C22×C173—21no—
86C22×C173—21no—
87C22×C173—21no—
88C22×C173—42no—
89C22×C173—42no—
90C22×C173—42no—
91C22×C173C22×D411yescharacteristic
92C4×C173—21no—
93C4×C173—42no—
94C4×C173H(5,𝔽2)11yescharacteristic
95C4×C1732−1+411yescharacteristic
96Dic173—22no—
97Dic173—22no—
98Dic173—42no—
99Dic173—42no—
100C2×C173—41no—
101C2×C173—82no—
102C2×C173#53711yescharacteristic
103C2×C173#57611yescharacteristic
104C2×C173#57811yescharacteristic
105C173#326211yescharacteristic, complemented, Sylow
106#3262—1731nomaximal, Sylow
107#542—1732no—
108#477—1732no—
109#474—1732no—
110#468—1732no—
111#467—1731no—
112#466D17311yescharacteristic
113#464—1732no—
114#463—1732no—
115#424—1731no—
116C2×Q16—1731no—
117C2×SD16—1732no—
118C2×D8C2×D17311yescharacteristic
119C2×M16—1731no—
120C22×C8C2×D17311yescharacteristic
121C4⋊Q8—1732no—
122#126—3462no—
123#124—3462no—
124#124—1732no—
125#123—3462no—
126C22⋊Q8—3462no—
127C22⋊Q8—1732no—
128#121C2×D17312yes—
129C4×Q8—1732no—
130C4×D4—1732no—
131#117—1731no—
132#107—1731no—
133#107—1732no—
134#107C2×D17311yescharacteristic
135C4⋊C8—1732no—
136C4⋊C8—1732no—
137#103—1732no—
138#103—1732no—
139#102—1732no—
140#102C2×D17312yes—
141C2×Q8—3462no—
142C2×Q8—1732no—
143C2×D4—22no—
144C2×D4C22×D17312yes—
145C22×C4—3462no—
146C22×C4C22×D17311yescharacteristic
147Q16—6921no—
148SD16—6922no—
149D8—41no—
150M16—6921no—
151C2×C8—41no—
152C2×C8—1732no—
153C2×C8C22×D17311yescharacteristic
154C2×C8C22×D17311yescharacteristic
155C4⋊C4—3462no—
156C4⋊C4—3462no—
157C4⋊C4—3462no—
158C4⋊C4—1732no—
159C4⋊C4—1731no—
160C4⋊C4—1731no—
161C4⋊C4C22×D17312yes—
162C22⋊C4—22no—
163C22⋊C4—3462no—
164C22⋊C4—3462no—
165C22⋊C4—3462no—
166C22⋊C4—3462no—
167C22⋊C4—1732no—
168C42—3462no—
169C42—1732no—
170C23—22no—
171C23D4×D17311yescharacteristic
172Q8—3462no—
173Q8—3462no—
174Q8—13842no—
175D4—22no—
176D4—22no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—3462no—
182C2×C4—3462no—
183C2×C4—13842no—
184C2×C4—1732no—
185C2×C4—1732no—
186C2×C4D4×D17311yescharacteristic
187C2×C4C23×D17311yescharacteristic
188C8—21no—
189C8—21no—
190C8—3462no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22C2×D4×D17311yescharacteristic
198C4—21no—
199C4—42no—
200C4—3462no—
201C4—3462no—
202C4—6922no—
203C4—6922no—
204C4#1914972411yescharacteristic
205C4#1914972611yescharacteristic
206C2—41no—
207C2—82no—
208C2#2601206811yescharacteristic
209C2#2601229211yescharacteristic
210C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#466×C173C2
socleC22×C173C22×D4
hypercenter#466D173
hypocenterC173#3262
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #26012108C2semidirect
2. #26011785C2semidirect
3. #26011769C2semidirect
4. #26011735C2semidirect
5. #26011732C2semidirect
6. #26011712C2semidirect
7. #26011704C2semidirect
8. #26011517C2semidirect
105. C173#3262semidirect

Maximal subgroups

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

subgroupindex
#260121082
#260117852
#260117692
#260117352
#260117322
#260117122
#260117042
#260115172
#466×C1732
#3262173