Finite Group Databasebeta

← #46822346

Subgroups — #46822346

Order 34432.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#31771226C212yesmaximal, complemented
2#31770901C212yesmaximal, complemented
3#31770886C212yesmaximal, complemented
4#31770851C212yesmaximal, complemented
5#31770846C211yesmaximal, characteristic, complemented
6#31770841C211yesmaximal, characteristic
7#31770828C212yesmaximal, complemented
8#31770821C212yesmaximal, complemented
9#424×C269C211yesmaximal, characteristic, complemented
10C2×Dic1076C2211yescharacteristic
11C2×#16864391C2212yes—
12C2×D2152C2211yescharacteristic
13#22575453C2212yes—
14#22575448—22no—
15#22575441C2212yes—
16#22575439—22no—
17#22575435—22no—
18#22575432C2212yes—
19#22575431—22no—
20#22575427C2212yes—
21C4×Dic538C2212yes—
22C4×D1076C2212yes—
23#22575387C2211yescharacteristic
24#22575387C2212yes—
25#22575387C2211yescharacteristic
26#22575378C2212yes—
27#22575378C2212yes—
28#22575375C2212yes—
29#22575375C2212yes—
30C2×M16×C269C2211yescharacteristic
31C22×C8×C269C2211yescharacteristic
32#117×C269C2211yescharacteristic
33C4⋊C8×C269C2212yes—
34C4⋊C8×C269C2212yes—
35C2×Dic538—22no—
36C2×Dic538C2312yes—
37C2×D1076C2312yes—
38C2×C22⋊D269—22no—
39C2×C4×D269—22no—
40Dic1076—41no—
41#16864391—42no—
42D2152—41no—
43C4⋊Dic269—22no—
44C4⋊Dic269C2312yes—
45C4⋊Dic269C2312yes—
46#16864381—22no—
47#16864381—22no—
48C22⋊Dic269—22no—
49#16864379—22no—
50#16864379—22no—
51#16864379—22no—
52#16864379—22no—
53C4×Dic269—22no—
54C22×C4×C269C2311yescharacteristic
55M16×C269—41no—
56C2×C8×C269—41no—
57C2×C8×C269C2311yescharacteristic
58C2×C8×C269C2312yes—
59C2×C8×C269C2311yescharacteristic
60C4⋊C4×C269D411yescharacteristic
61C4⋊C4×C269D411yescharacteristic
62C22⋊C4×C269D412yes—
63C42×C269C2312yes—
64C22×D269—22no—
65Dic538—22no—
66Dic538—22no—
67Dic538—82no—
68D1076—22no—
69D1076—22no—
70C22⋊D269—82no—
71C2×Dic269—22no—
72C2×Dic269—22no—
73C2×Dic269—22no—
74C4×D269—82no—
75C23×C269C2×D411yescharacteristic
76C2×C4×C269—21no—
77C2×C4×C269—21no—
78C2×C4×C269C2×D412yes—
79C2×C4×C269C2×D412yes—
80C2×C4×C269C2×D411yescharacteristic
81C2×C4×C269C2411yescharacteristic
82C8×C269—21no—
83C8×C269—21no—
84C8×C269—22no—
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—22no—
94C4×C269—22no—
95C4×C269H(5,𝔽2)11yescharacteristic
96C4×C2692−1+411yescharacteristic
97Dic269—42no—
98Dic269—42no—
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#467—2691no—
112#466—2691no—
113#464—2692no—
114#463—2692no—
115#424D26911yescharacteristic, complemented
116C2×Q16—2691no—
117C2×SD16—2692no—
118C2×D8—2691no—
119C2×M16C2×D26911yescharacteristic
120C22×C8C2×D26911yescharacteristic
121C4⋊Q8—2692no—
122#126—5382no—
123#124—2692no—
124#124—5382no—
125#123—5382no—
126C22⋊Q8—2692no—
127C22⋊Q8—5382no—
128#121—2692no—
129C4×Q8—2692no—
130C4×D4—2692no—
131#117C2×D26911yescharacteristic
132#107—2691no—
133#107—2691no—
134#107—2692no—
135C4⋊C8C2×D26912yes—
136C4⋊C8C2×D26912yes—
137#103—2692no—
138#103—2692no—
139#102—2692no—
140#102—2692no—
141C2×Q8—2692no—
142C2×Q8—5382no—
143C2×D4—2692no—
144C2×D4—5382no—
145C22×C4—5382no—
146C22×C4C22×D26911yescharacteristic
147Q16—10761no—
148SD16—10762no—
149D8—10761no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D26911yescharacteristic
153C2×C8C22×D26912yes—
154C2×C8C22×D26911yescharacteristic
155C4⋊C4—2692no—
156C4⋊C4—2692no—
157C4⋊C4—5382no—
158C4⋊C4—5382no—
159C4⋊C4—5382no—
160C4⋊C4D107611yescharacteristic
161C4⋊C4D107611yescharacteristic
162C22⋊C4—5382no—
163C22⋊C4—5382no—
164C22⋊C4—5382no—
165C22⋊C4—5382no—
166C22⋊C4—5382no—
167C22⋊C4D107612yes—
168C42—5382no—
169C42C22×D26912yes—
170C23—5382no—
171C23C2×D107611yescharacteristic
172Q8—5382no—
173Q8—5382no—
174Q8—21522no—
175D4—5382no—
176D4—5382no—
177D4—21522no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—5382no—
181C2×C4—5382no—
182C2×C4—5382no—
183C2×C4—21522no—
184C2×C4C2×D107612yes—
185C2×C4C2×D107612yes—
186C2×C4C2×D107611yescharacteristic
187C2×C4C23×D26911yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—10762no—
195C22—10762no—
196C22—10762no—
197C22C22×D107611yescharacteristic
198C4—22no—
199C4—22no—
200C4—21no—
201C4—10762no—
202C4—10762no—
203C4—10762no—
204C4#2257549811yescharacteristic
205C4#2257550111yescharacteristic
206C2—41no—
207C2—21522no—
208C2#3177118611yescharacteristic
209C2#3177141011yescharacteristic
210C2#3177142211yescharacteristic
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#424×C269C2
socleC22×C269C22×D4
prime radical#31770841C2
hypercenter#424D269
hypocenterC269#3262
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #31771226C2semidirect
2. #31770901C2semidirect
3. #31770886C2semidirect
4. #31770851C2semidirect
5. #31770846C2semidirect
7. #31770828C2semidirect
8. #31770821C2semidirect
9. #424×C269C2semidirect
105. C269#3262semidirect
115. #424D269semidirect

Maximal subgroups

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

subgroupindex
#317712262
#317709012
#317708862
#317708512
#317708462
#317708412
#317708282
#317708212
#424×C2692
#3262269