Finite Group Databasebeta

← #46822347

Subgroups — #46822347

Order 34432.

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#31771228C212yesmaximal, complemented
2#31770898C212yesmaximal, complemented
3#31770884C212yesmaximal, complemented
4#31770853C212yesmaximal, complemented
5#31770841C211yesmaximal, characteristic
6#31770829C212yesmaximal, complemented
7#31770823C212yesmaximal, complemented
8#31770631C211yesmaximal, characteristic, complemented
9#467×C269C211yesmaximal, characteristic, complemented
10C2×#16864389C2212yes—
11C2×D2152C2211yescharacteristic
12C2×#16864386C2211yescharacteristic
13#22575452C2212yes—
14#22575448—22no—
15#22575442C2212yes—
16#22575439—22no—
17#22575435—22no—
18#22575433—22no—
19#22575427C2212yes—
20Q8×Dic269C2212yes—
21#22575415C2212yes—
22#22575412C2211yescharacteristic
23#22575387C2211yescharacteristic
24#22575386C2212yes—
25#22575382C2212yes—
26#22575382C2212yes—
27#22575377C2212yes—
28#22575375C2212yes—
29#22575373C2212yes—
30C2×Q16×C269C2211yescharacteristic
31C22×C8×C269C2211yescharacteristic
32C22⋊Q8×C269C2212yes—
33#107×C269C2211yescharacteristic
34#103×C269C2212yes—
35C2×Dic538—22no—
36C2×D1076C2312yes—
37C2×C22⋊D269—22no—
38C2×C4×D269—22no—
39#16864389—42no—
40D2152—41no—
41#16864386—41no—
42C2×#14524130C2312yes—
43C4⋊Dic269—22no—
44C4⋊Dic269D411yescharacteristic
45C4⋊Dic269D411yescharacteristic
46C4⋊Dic269C2312yes—
47#16864381—22no—
48C22⋊Dic269D412yes—
49#16864379—22no—
50#16864379—22no—
51#16864379—22no—
52#16864379—22no—
53C4×Dic269—22no—
54C4×Dic269C2312yes—
55C2×Q8×C269C2312yes—
56C22×C4×C269C2311yescharacteristic
57Q16×C269—41no—
58C2×C8×C269—41no—
59C2×C8×C269C2311yescharacteristic
60C2×C8×C269C2311yescharacteristic
61C4⋊C4×C269—22no—
62C4⋊C4×C269C2312yes—
63C22⋊C4×C269—22no—
64C22×D269—22no—
65Dic538—82no—
66D1076—22no—
67D1076—22no—
68C22⋊D269—82no—
69C2×Dic269—22no—
70C2×Dic269C2×D412yes—
71C2×Dic269C2×D412yes—
72C4×D269—82no—
73#14524130—22no—
74C23×C269C2×D411yescharacteristic
75Q8×C269—22no—
76Q8×C269—22no—
77C2×C4×C269—21no—
78C2×C4×C269—21no—
79C2×C4×C269—22no—
80C2×C4×C269—22no—
81C2×C4×C269C2×D411yescharacteristic
82C2×C4×C269C2411yescharacteristic
83C8×C269—21no—
84C8×C269—21no—
85C22×C269—21no—
86C22×C269—21no—
87C22×C269—21no—
88C22×C269C22×D411yescharacteristic
89C2×D269—42no—
90C2×D269—42no—
91C2×D269—42no—
92C4×C269—21no—
93C4×C269—42no—
94C4×C269—42no—
95C4×C269H(5,𝔽2)11yescharacteristic
96C4×C2692−1+411yescharacteristic
97Dic269—22no—
98Dic269—22no—
99Dic269—42no—
100C2×C269—41no—
101C2×C269#53711yescharacteristic
102C2×C269#57611yescharacteristic
103C2×C269#57811yescharacteristic
104D269—82no—
105C269#326211yescharacteristic, complemented, Sylow
106#3262—2691nomaximal, Sylow
107#542—2692no—
108#477—2692no—
109#474—2692no—
110#468—2692no—
111#467D26911yescharacteristic, complemented
112#466—2691no—
113#464—2692no—
114#463—2692no—
115#424—2691no—
116C2×Q16C2×D26911yescharacteristic
117C2×SD16—2692no—
118C2×D8—2691no—
119C2×M16—2691no—
120C22×C8C2×D26911yescharacteristic
121C4⋊Q8—2692no—
122#126—5382no—
123#124—2692no—
124#124—5382no—
125#123—5382no—
126C22⋊Q8—5382no—
127C22⋊Q8C2×D26912yes—
128#121—2692no—
129C4×Q8—2692no—
130C4×D4—2692no—
131#117—2691no—
132#107—2691no—
133#107—2692no—
134#107C2×D26911yescharacteristic
135C4⋊C8—2692no—
136C4⋊C8—2692no—
137#103—2692no—
138#103C2×D26912yes—
139#102—2692no—
140#102—2692no—
141C2×Q8—5382no—
142C2×Q8C22×D26912yes—
143C2×D4—2692no—
144C2×D4—5382no—
145C22×C4—5382no—
146C22×C4C22×D26911yescharacteristic
147Q16—41no—
148SD16—10762no—
149D8—10761no—
150M16—10761no—
151C2×C8—41no—
152C2×C8—2692no—
153C2×C8C22×D26911yescharacteristic
154C2×C8C22×D26911yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—2691no—
157C4⋊C4—2691no—
158C4⋊C4—2692no—
159C4⋊C4—5382no—
160C4⋊C4—5382no—
161C4⋊C4C22×D26912yes—
162C22⋊C4—22no—
163C22⋊C4—2692no—
164C22⋊C4—5382no—
165C22⋊C4—5382no—
166C22⋊C4—5382no—
167C22⋊C4—5382no—
168C42—2692no—
169C42—5382no—
170C23—5382no—
171C23D4×D26911yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—21522no—
175D4—5382no—
176D4—5382no—
177D4—21522no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—2692no—
183C2×C4—2692no—
184C2×C4—5382no—
185C2×C4—21522no—
186C2×C4D4×D26911yescharacteristic
187C2×C4C23×D26911yescharacteristic
188C8—21no—
189C8—21no—
190C8—5382no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—10762no—
195C22—10762no—
196C22—10762no—
197C22C2×D4×D26911yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—5382no—
202C4—5382no—
203C4—10762no—
204C4#2257549811yescharacteristic
205C4#2257550011yescharacteristic
206C2—41no—
207C2—21522no—
208C2#3177118211yescharacteristic
209C2#3177140711yescharacteristic
210C2#3177141811yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#467×C269C2
socleC22×C269C22×D4
prime radical#31770841C2
hypercenter#467D269
hypocenterC269#3262
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #31771228C2semidirect
2. #31770898C2semidirect
3. #31770884C2semidirect
4. #31770853C2semidirect
6. #31770829C2semidirect
7. #31770823C2semidirect
8. #31770631C2semidirect
9. #467×C269C2semidirect
105. C269#3262semidirect
111. #467D269semidirect

Maximal subgroups

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

subgroupindex
#317712282
#317708982
#317708842
#317708532
#317708412
#317708292
#317708232
#317706312
#467×C2692
#3262269