Finite Group Databasebeta

← #46822331

Subgroups — #46822331

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#31771217C211yesmaximal, characteristic, complemented
2#31770909C212yesmaximal, complemented
3#31770905C212yesmaximal, complemented
4#31770849C212yesmaximal, complemented
5#31770832C212yesmaximal, complemented
6#31770810C212yesmaximal, complemented
7#31770802C212yesmaximal, complemented
8#31770632C211yesmaximal, characteristic, complemented
9#540×C269C211yesmaximal, characteristic
10C2×#16864390C2212yes—
11C2×#16864390C2212yes—
12C2×#16864386C2211yescharacteristic
13C22×#14524130C2211yescharacteristic
14#22575453C2211yescharacteristic
15#22575446—22no—
16#22575445C2211yescharacteristic
17#22575443—22no—
18#22575432C2212yes—
19C4×Dic538C2212yes—
20#22575386C2212yes—
21#22575384C2212yes—
22C4⋊#14524130C2212yes—
23C4⋊#14524130C2212yes—
24#22575376C2212yes—
25#22575376C2212yes—
26#22575374C2212yes—
27#22575374C2212yes—
28#127×C269C2211yescharacteristic
29#125×C269C2211yescharacteristic
30#123×C269—22no—
31#121×C269—22no—
32#121×C269C2212yes—
33C4×D4×C269C2212yes—
34#117×C269C2211yescharacteristic
35C2×Dic538C2312yes—
36#16864390—42no—
37#16864390—42no—
38#16864386—41no—
39C2×#14524130—41no—
40C2×#14524130C2312yes—
41C2×#14524130C2312yes—
42C4⋊Dic269—22no—
43C4⋊Dic269C2312yes—
44#16864381—22no—
45#16864381—22no—
46#16864381—22no—
47C22⋊Dic269—22no—
48C4×Dic269—22no—
49C2×D4×C269—22no—
50C2×D4×C269—22no—
51C2×D4×C269C2312yes—
52C22×C4×C269—22no—
53C22×C4×C269C2311yescharacteristic
54C4⋊C4×C269—22no—
55C4⋊C4×C269D412yes—
56C4⋊C4×C269C2312yes—
57C22⋊C4×C269—22no—
58C22⋊C4×C269—22no—
59C22⋊C4×C269D412yes—
60C42×C269C2311yescharacteristic
61C42×C269C2311yescharacteristic
62Dic538—22no—
63Dic538—22no—
64C2×Dic269—22no—
65C2×Dic269—22no—
66#14524130—22no—
67#14524130—22no—
68C23×C269—22no—
69C23×C269C2×D411yescharacteristic
70D4×C269—22no—
71D4×C269—22no—
72D4×C269—82no—
73D4×C269—82no—
74C2×C4×C269—21no—
75C2×C4×C269—21no—
76C2×C4×C269—22no—
77C2×C4×C269—82no—
78C2×C4×C269C2×D412yes—
79C2×C4×C269C2×D412yes—
80C2×C4×C269C2×D411yescharacteristic
81C2×C4×C269C2411yescharacteristic
82C22×C269—21no—
83C22×C269—22no—
84C22×C269—42no—
85C22×C269—42no—
86C22×C269—42no—
87C22×C269C22×D411yescharacteristic
88C4×C269—21no—
89C4×C269—22no—
90C4×C269—22no—
91C4×C269—42no—
92C4×C269H(5,𝔽2)11yescharacteristic
93C4×C2692−1+411yescharacteristic
94Dic269—42no—
95Dic269—42no—
96C2×C269—41no—
97C2×C269—82no—
98C2×C269#53711yescharacteristic
99C2×C269#57712yes—
100C269#326011yescharacteristic, complemented, Sylow
101#3260—2691nomaximal, Sylow
102#541—2691no—
103#540D26911yescharacteristic
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⋊Q8—2691no—
116#127C2×D26911yescharacteristic
117#126—5382no—
118#125—2691no—
119#125—5382no—
120#125C2×D26911yescharacteristic
121#123—22no—
122C22⋊Q8—2692no—
123#121—22no—
124#121C2×D26912yes—
125C4×Q8—2692no—
126C4×D4C2×D26912yes—
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×Q8—2692no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4C22×D26912yes—
140C22×C4—22no—
141C22×C4C22×D26911yescharacteristic
142SD16—10762no—
143SD16—10762no—
144M16—10761no—
145C2×C8—2692no—
146C2×C8—2692no—
147C2×C8—10761no—
148C4⋊C4—22no—
149C4⋊C4—2692no—
150C4⋊C4—5382no—
151C4⋊C4—5382no—
152C4⋊C4—5382no—
153C4⋊C4—5382no—
154C4⋊C4C22⋊D26912yes—
155C4⋊C4C22×D26912yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—5382no—
159C22⋊C4C22⋊D26912yes—
160C42—5382no—
161C42C22×D26911yescharacteristic
162C42C22×D26911yescharacteristic
163C23—22no—
164C23C2×C22⋊D26911yescharacteristic
165Q8—5382no—
166Q8—5382no—
167D4—22no—
168D4—22no—
169D4—82no—
170D4—82no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—82no—
175C2×C4—5382no—
176C2×C4—5382no—
177C2×C4C2×C22⋊D26912yes—
178C2×C4C2×C22⋊D26912yes—
179C2×C4C2×C22⋊D26911yescharacteristic
180C2×C4C23×D26911yescharacteristic
181C8—5382no—
182C8—5382no—
183C22—22no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22C22×C22⋊D26911yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—10762no—
194C4—10762no—
195C4#2257549911yescharacteristic
196C4#2257550011yescharacteristic
197C2—41no—
198C2—82no—
199C2#3177118311yescharacteristic
200C2#3177141612yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting, prime radical#540×C269C2
socleC22×C269C22×D4
hypercenter#540D269
hypocenterC269#3260
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #31771217C2semidirect
2. #31770909C2semidirect
3. #31770905C2semidirect
4. #31770849C2semidirect
5. #31770832C2semidirect
6. #31770810C2semidirect
7. #31770802C2semidirect
8. #31770632C2semidirect
100. C269#3260semidirect

Maximal subgroups

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

subgroupindex
#317712172
#317709092
#317709052
#317708492
#317708322
#317708102
#317708022
#317706322
#540×C2692
#3260269