Finite Group Databasebeta

← #46822326

Subgroups — #46822326

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#31771210C211yesmaximal, characteristic
2#31770912C212yesmaximal, complemented
3#31770902C212yesmaximal, complemented
4#31770853C212yesmaximal, complemented
5#31770836C212yesmaximal, complemented
6#31770807C212yesmaximal, complemented
7#31770805C212yesmaximal, complemented
8#31770632C211yesmaximal, characteristic, complemented
9#541×C269C211yesmaximal, characteristic, complemented
10C2×#16864389C2212yes—
11C2×#16864389C2212yes—
12C2×#16864386C2211yescharacteristic
13C22×#14524130C2211yescharacteristic
14#22575450C2211yescharacteristic
15#22575445C2211yescharacteristic
16#22575438—22no—
17#22575427C2212yes—
18#22575426—22no—
19C4×D1076C2212yes—
20#22575386C2212yes—
21#22575384C2212yes—
22C4⋊#14524130C2212yes—
23C4⋊#14524130C2212yes—
24#22575377C2212yes—
25#22575377C2212yes—
26#22575373C2212yes—
27#22575373C2212yes—
28C4⋊Q8×C269C2211yescharacteristic
29#126×C269—22no—
30#125×C269—22no—
31#125×C269C2211yescharacteristic
32C22⋊Q8×C269C2212yes—
33C4×Q8×C269C2212yes—
34#117×C269C2211yescharacteristic
35C2×D1076—22no—
36C2×D1076C2312yes—
37C2×C22⋊D269—22no—
38C2×C4×D269—22no—
39#16864389—42no—
40#16864389—42no—
41#16864386—41no—
42C2×#14524130—41no—
43C2×#14524130C2312yes—
44C2×#14524130C2312yes—
45C4⋊Dic269C2312yes—
46#16864381—22no—
47#16864379—22no—
48#16864379—22no—
49C2×Q8×C269C2312yes—
50C22×C4×C269C2311yescharacteristic
51C4⋊C4×C269—22no—
52C4⋊C4×C269—22no—
53C4⋊C4×C269—22no—
54C4⋊C4×C269—22no—
55C4⋊C4×C269D412yes—
56C4⋊C4×C269C2312yes—
57C22⋊C4×C269—22no—
58C22⋊C4×C269D412yes—
59C42×C269—22no—
60C42×C269C2311yescharacteristic
61C42×C269C2311yescharacteristic
62C22×D269—22no—
63D1076—22no—
64D1076—22no—
65D1076—82no—
66C22⋊D269—82no—
67C2×Dic269—22no—
68C4×D269—82no—
69#14524130—22no—
70#14524130—22no—
71C23×C269C2×D411yescharacteristic
72Q8×C269—22no—
73Q8×C269—22no—
74C2×C4×C269—21no—
75C2×C4×C269—21no—
76C2×C4×C269—22no—
77C2×C4×C269—22no—
78C2×C4×C269C2×D412yes—
79C2×C4×C269C2×D412yes—
80C2×C4×C269C2×D411yescharacteristic
81C2×C4×C269C2411yescharacteristic
82C22×C269—21no—
83C22×C269—22no—
84C22×C269C22×D411yescharacteristic
85C2×D269—42no—
86C2×D269—42no—
87C2×D269—42no—
88C4×C269—21no—
89C4×C269—22no—
90C4×C269—22no—
91C4×C269—42no—
92C4×C269—42no—
93C4×C269H(5,𝔽2)11yescharacteristic
94C4×C2692−1+411yescharacteristic
95Dic269—42no—
96C2×C269—41no—
97C2×C269#53711yescharacteristic
98C2×C269#57712yes—
99D269—82no—
100C269#326011yescharacteristic, complemented, Sylow
101#3260—2691nomaximal, Sylow
102#541D26911yescharacteristic, complemented
103#540—2691no—
104#479—2692no—
105#478—2692no—
106#468—2692no—
107#465—2692no—
108#461—2692no—
109#460—2692no—
110#424—2691no—
111C2×SD16—2692no—
112C2×SD16—2692no—
113C2×M16—2691no—
114C22×C8—2691no—
115C4⋊Q8C2×D26911yescharacteristic
116#127—2691no—
117#126—22no—
118#125—22no—
119#125—2691no—
120#125C2×D26911yescharacteristic
121#123—5382no—
122C22⋊Q8C2×D26912yes—
123#121—2692no—
124#121—5382no—
125C4×Q8C2×D26912yes—
126C4×D4—2692no—
127#117C2×D26911yescharacteristic
128#107—2692no—
129#106—2692no—
130C4⋊C8—2692no—
131C4⋊C8—2692no—
132#103—2692no—
133#103—2692no—
134#102—2692no—
135#102—2692no—
136C2×Q8C22×D26912yes—
137C2×D4—2692no—
138C2×D4—5382no—
139C2×D4—5382no—
140C22×C4—5382no—
141C22×C4C22×D26911yescharacteristic
142SD16—10762no—
143SD16—10762no—
144M16—10761no—
145C2×C8—2692no—
146C2×C8—2692no—
147C2×C8—10761no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—2692no—
153C4⋊C4—5382no—
154C4⋊C4C22⋊D26912yes—
155C4⋊C4C22×D26912yes—
156C22⋊C4—22no—
157C22⋊C4—5382no—
158C22⋊C4—5382no—
159C22⋊C4C22⋊D26912yes—
160C42—22no—
161C42C22×D26911yescharacteristic
162C42C22×D26911yescharacteristic
163C23—5382no—
164C23C2×C22⋊D26911yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—5382no—
168D4—5382no—
169D4—21522no—
170D4—21522no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—5382no—
176C2×C4—21522no—
177C2×C4C2×C22⋊D26912yes—
178C2×C4C2×C22⋊D26912yes—
179C2×C4C2×C22⋊D26911yescharacteristic
180C2×C4C23×D26911yescharacteristic
181C8—5382no—
182C8—5382no—
183C22—22no—
184C22—21no—
185C22—10762no—
186C22—10762no—
187C22—10762no—
188C22C22×C22⋊D26911yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—10762no—
195C4#2257549911yescharacteristic
196C4#2257550011yescharacteristic
197C2—41no—
198C2—21522no—
199C2#3177118311yescharacteristic
200C2#3177141312yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#541×C269C2
socleC22×C269C22×D4
prime radical#31771210C2
hypercenter#541D269
hypocenterC269#3260
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #31770912C2semidirect
3. #31770902C2semidirect
4. #31770853C2semidirect
5. #31770836C2semidirect
6. #31770807C2semidirect
7. #31770805C2semidirect
8. #31770632C2semidirect
9. #541×C269C2semidirect
100. C269#3260semidirect
102. #541D269semidirect

Maximal subgroups

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

subgroupindex
#317712102
#317709122
#317709022
#317708532
#317708362
#317708072
#317708052
#317706322
#541×C2692
#3260269