Finite Group Databasebeta

← #46811081

Subgroups — #46811081

Order 34432.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#22575411C211yesmaximal, characteristic
2C2×#22575384C211yesmaximal, characteristic, complemented
3C2×#22575365C211yesmaximal, characteristic
4#31770462C211yesmaximal, characteristic
5#31770368C211yesmaximal, characteristic, complemented
6#31770367C211yesmaximal, characteristic, complemented
7C2×#107×C269C211yesmaximal, characteristic, complemented
8C22×C4×D269C2211yescharacteristic
9C22×#14524130C2211yescharacteristic
10#22575411—22no—
11#22575411—22no—
12C2×C4⋊Dic269C2211yescharacteristic
13C2×C4⋊Dic269C2211yescharacteristic
14C2×#16864379—21no—
15C2×#16864379—21no—
16C2×C4×Dic269—21no—
17#22575384—22no—
18#22575365C414yescomplemented
19#22575360—21no—
20C22×C8×C269C2211yescharacteristic
21C2×C4⋊C4×C269C2211yescharacteristic
22C2×C4⋊C4×C269C2211yescharacteristic
23#107×C269—22no—
24C23×D269D411yescharacteristic
25C22×Dic269—21no—
26C22×Dic269—21no—
27C22×Dic269D411yescharacteristic
28C2×C4×D269—21no—
29C2×C4×D269—21no—
30C2×C4×D269—21no—
31C2×C4×D269—21no—
32C2×C4×D269C2×C412yes—
33C2×C4×D269Q812yes—
34C2×#14524130—21no—
35C2×#14524130—21no—
36C2×#14524130C2×C412yes—
37C4⋊Dic269—21no—
38C4⋊Dic269—21no—
39C4⋊Dic269—21no—
40C4⋊Dic269D412yes—
41#16864379—24no—
42#16864379—42no—
43C4×Dic269—24no—
44C22×C4×C269—21no—
45C22×C4×C269—21no—
46C22×C4×C269C2311yescharacteristic
47C2×C8×C269—21no—
48C2×C8×C269—21no—
49C2×C8×C269C2×C412yes—
50C4⋊C4×C269—21no—
51C4⋊C4×C269—21no—
52C4⋊C4×C269—21no—
53C4⋊C4×C269D412yes—
54C22×D269—21no—
55C22×D269—21no—
56C22×D269—21no—
57C22×D269—21no—
58C22×D269—21no—
59C22×D269—21no—
60C22×D269C4⋊C412yes—
61C2×Dic269—21no—
62C2×Dic269—21no—
63C2×Dic269—22no—
64C2×Dic269—41no—
65C2×Dic269—41no—
66C2×Dic269—41no—
67C2×Dic269—41no—
68C2×Dic269—41no—
69C2×Dic269C4⋊C412yes—
70C4×D269—22no—
71C4×D269—22no—
72C4×D269—22no—
73C4×D269—22no—
74#14524130—22no—
75C23×C269C2×D411yescharacteristic
76C2×C4×C269—22no—
77C2×C4×C269—41no—
78C2×C4×C269—41no—
79C2×C4×C269—41no—
80C2×C4×C269—41no—
81C2×C4×C269—41no—
82C2×C4×C269C22×C411yescharacteristic
83C2×C4×C269C2×D411yescharacteristic
84C2×C4×C269C2×D411yescharacteristic
85C2×C4×C269C2×Q811yescharacteristic
86C2×C4×C269#4111yescharacteristic
87C2×C4×C269#4111yescharacteristic
88C8×C269—22no—
89C22×C269C2×C4⋊C411yescharacteristic
90C22×C269#12011yescharacteristic
91C22×C269#12311yescharacteristic
92C22×C269#13511yescharacteristic
93C22×C269#13511yescharacteristic
94C22×C269AGL(1,ℤ/8)11yescharacteristic
95C22×C269#13711yescharacteristic
96C2×D269—22no—
97C2×D269—22no—
98C2×D269—22no—
99C2×D269—41no—
100C2×D269—41no—
101C2×D269—41no—
102C2×D269—41no—
103C4×C269—42no—
104C4×C269—81no—
105C4×C269C4×D411yescharacteristic
106C4×C269C4×D411yescharacteristic
107C4×C269C22⋊Q811yescharacteristic
108C4×C269C22⋊Q811yescharacteristic
109Dic269—22no—
110Dic269—42no—
111Dic269—81no—
112C2×C269#38511yescharacteristic
113C2×C269#42711yescharacteristic
114C2×C269#42811yescharacteristic
115C2×C269#44911yescharacteristic
116C2×C269#45211yescharacteristic
117C2×C269#48211yescharacteristic
118C2×C269#48511yescharacteristic
119D269—42no—
120C269#185411yescharacteristic, complemented, Sylow
121#1854—2691nomaximal, Sylow
122C2×#117—2691no—
123C2×#107D26911yescharacteristic, complemented
124C2×#106—2691no—
125C2×C22⋊C8—2691no—
126#380—2691no—
127#340—2691no—
128#340—2691no—
129C23×C4—2691no—
130C22×C8—2691no—
131C22×C8C2×D26911yescharacteristic
132#117—5382no—
133#117—5382no—
134C2×C4⋊C4—2691no—
135C2×C4⋊C4—2691no—
136C2×C4⋊C4C2×D26911yescharacteristic
137C2×C4⋊C4C2×D26911yescharacteristic
138C2×C22⋊C4—5381no—
139C2×C22⋊C4—5381no—
140C2×C42—5381no—
141#107—22no—
142#106—5382no—
143C22⋊C8—2694no—
144#95—5381no—
145C24—2691no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—2691no—
149C22×C4—2692no—
150C22×C4—2692no—
151C22×C4—5381no—
152C22×C4—5381no—
153C22×C4—5381no—
154C22×C4—5381no—
155C22×C4—5381no—
156C22×C4—5381no—
157C22×C4C22×D26911yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—2692no—
161C2×C8—5381no—
162C2×C8—5381no—
163C2×C8C4×D26912yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—2692no—
168C4⋊C4—5381no—
169C4⋊C4—5381no—
170C4⋊C4—5381no—
171C4⋊C4C22⋊D26912yes—
172C22⋊C4—5384no—
173C22⋊C4—10762no—
174C42—5384no—
175C23—2692no—
176C23—5381no—
177C23—5381no—
178C23—5381no—
179C23—5381no—
180C23—5381no—
181C23—5381no—
182C23D4×D26911yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2692no—
190C2×C4—5381no—
191C2×C4—5381no—
192C2×C4—5382no—
193C2×C4—5382no—
194C2×C4—5382no—
195C2×C4—5382no—
196C2×C4—5382no—
197C2×C4—10761no—
198C2×C4—10761no—
199C2×C4—10761no—
200C2×C4—10761no—
201C2×C4—10761no—
202C2×C4C2×C4×D26911yescharacteristic
203C2×C4D4×D26911yescharacteristic
204C2×C4C2×C22⋊D26911yescharacteristic
205C2×C4Q8×D26911yescharacteristic
206C2×C4#1686440211yescharacteristic
207C2×C4#1686440311yescharacteristic
208C8—22no—
209C8—5382no—
210C22—5382no—
211C22—5382no—
212C22—5382no—
213C22—10761no—
214C22—10761no—
215C22—10761no—
216C22—10761no—
217C22C4⋊C4×D26911yescharacteristic
218C22#2257542111yescharacteristic
219C22#2257543811yescharacteristic
220C22#2257547211yescharacteristic
221C22#2257547611yescharacteristic
222C22#2257547711yescharacteristic
223C22#2257548511yescharacteristic
224C4—42no—
225C4—81no—
226C4—5382no—
227C4—10762no—
228C4—21521no—
229C4C4×C22⋊D26911yescharacteristic
230C4#2257541511yescharacteristic
231C4#2257542911yescharacteristic
232C4#2257543111yescharacteristic
233C2—10762no—
234C2#3177047611yescharacteristic
235C2#3177064211yescharacteristic
236C2#3177064611yescharacteristic
237C2#3177074711yescharacteristic
238C2#3177076611yescharacteristic
239C2#3177093211yescharacteristic
240C2#3177094811yescharacteristic
subgroupG/H
centerC23D4×D269
commutatorC2×C4×C269C22×C4
FrattiniC22×C4C22×D269
FittingC2×#107×C269C2
socleC23×C269C2×D4
prime radicalC23×D269D4
cosocleC22×C4×C269C23
hypercenterC2×#107D269
hypocenterC269#1854
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#22575384C2semidirect
5. #31770368C2semidirect
6. #31770367C2semidirect
7. C2×#107×C269C2semidirect
18. #22575365C4semidirect
120. C269#1854semidirect
123. C2×#107D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754112
C2×#225753842
C2×#225753652
#317704622
#317703682
#317703672
C2×#107×C2692
#1854269