Finite Group Databasebeta

← #30770258

Subgroups — #30770258

Order 17152.

234 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 63 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#22562434C211yesmaximal, characteristic, complemented
2#22562028C211yesmaximal, characteristic, complemented
3#22559605C211yesmaximal, characteristic, complemented
4#22559565C211yesmaximal, characteristic
5#22558282C211yesmaximal, characteristic, complemented
6#22558256C211yesmaximal, characteristic, complemented
7C67×#1365C211yesmaximal, characteristic, complemented
8C2×C16×D67—21no—
9#16863773C2211yescharacteristic
10#16863683—21no—
11#16863592—21no—
12C8×D268C2211yescharacteristic
13#16863196—21no—
14#16863196—21no—
15(C2×D67)⋊C16—21no—
16C4×C67⋊C16C2211yescharacteristic
17#16863149C2211yescharacteristic
18#16863145—21no—
19#16863139—21no—
20#500×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×D536—21no—
24C2×D536—21no—
25C2×D536—41no—
26C2×C8×D67—21no—
27#14524040—21no—
28C4×D268D411yescharacteristic
29C2×C67⋊C16D411yescharacteristic
30C2×C67⋊C16D411yescharacteristic
31C16×D67—42no—
32#14523974—41no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)D411yescharacteristic
35#14523965—41no—
36#14523963—41no—
37(C2×D67)⋊C8—21no—
38C4⋊Q8×C67—21no—
39C2×C16×C67—21no—
40#107×C67—21no—
41#107×C67—21no—
42C4⋊C8×C67—21no—
43C4×C8×C67C2311yescharacteristic
44C2×D268—41no—
45C2×D268—41no—
46C2×D268—41no—
47C2×D268D811yescharacteristic
48C2×C4×D67—21no—
49D536—41no—
50D536—41no—
51D536—81no—
52D536—82no—
53C2×C67⋊C8—21no—
54C2×C67⋊C8—41no—
55C8×D67—41no—
56C8×D67—41no—
57C4⋊Dic67D811yescharacteristic
58(C2×D67)⋊C4—21no—
59C67⋊C16—21no—
60C67⋊C16—21no—
61C2×Q8×C67—41no—
62C2×C8×C67—41no—
63C2×C8×C67C2×D411yescharacteristic
64C2×C8×C67#4111yescharacteristic
65C4⋊C4×C67—41no—
66C4⋊C4×C67—41no—
67C42×C67C2×D411yescharacteristic
68C16×C67—41no—
69C22×D67—21no—
70C22×D67—81no—
71Q8×C67—161no—
72C2×C4×C67—81no—
73C2×C4×C67#12111yescharacteristic
74C2×C4×C67C2×D811yescharacteristic
75C2×C4×C67#13511yescharacteristic
76C2×Dic67—21no—
77D268—21no—
78D268—81no—
79D268—81no—
80D268—81no—
81D268—81no—
82D268—161no—
83D268SD3212yes—
84C4×D67—41no—
85C4×D67—41no—
86C8×C67—21no—
87C8×C67—81no—
88C8×C67#13511yescharacteristic
89C8×C67AGL(1,ℤ/8)11yescharacteristic
90C67⋊C8—41no—
91C67⋊C8—81no—
92C22×C67#46611yescharacteristic
93C2×D67—22no—
94C2×D67—41no—
95C2×D67—41no—
96C2×D67—161no—
97C2×D67—161no—
98C2×D67—161no—
99C4×C67—21no—
100C4×C67—161no—
101C4×C67#46311yescharacteristic
102C4×C67#47111yescharacteristic
103C4×C67C2×SD3211yescharacteristic
104C4×C67#50811yescharacteristic
105Dic67—41no—
106#12468—671nomaximal, Sylow
107C2×C67#168111yescharacteristic
108C2×C67#221711yescharacteristic
109C2×C67#222111yescharacteristic
110D67—42no—
111D67—321no—
112#2240—671no—
113D4×C16—671no—
114#1675—671no—
115#1667—671no—
116#1372—671no—
117#1365D6711yescharacteristic, complemented
118#1361—671no—
119C67#1246811yescharacteristic, complemented, Sylow
120C22×C16—1341no—
121#500C2×D6711yescharacteristic
122#493—671no—
123#476—1341no—
124#459—1341no—
125C8×D4—671no—
126C4⋊C16C2×D6711yescharacteristic
127#357—1341no—
128#357—1341no—
129C22⋊C16—1341no—
130C4×C16—671no—
131#335—671no—
132#335C2×D6711yescharacteristic
133#333—1341no—
134#331—1341no—
135C2×D8—2681no—
136C2×D8—1341no—
137C2×D8—1341no—
138C22×C8—1341no—
139C4⋊Q8—21no—
140#127—1341no—
141C4×D4—671no—
142C2×C16—21no—
143C2×C16—2682no—
144C2×C16—671no—
145C2×C16—671no—
146#107—21no—
147#107—21no—
148#106—2681no—
149C4⋊C8—21no—
150C4⋊C8—671no—
151C4⋊C8—1341no—
152#102—2681no—
153#102—2681no—
154C22⋊C8—1341no—
155C4×C8C22×D6711yescharacteristic
156C2×Q8—41no—
157C2×D4—2681no—
158C2×D4—2681no—
159C2×D4—2681no—
160C2×D4—671no—
161C22×C4—1341no—
162D8—2681no—
163D8—2681no—
164D8—5361no—
165D8—5362no—
166C2×C8—41no—
167C2×C8—2681no—
168C2×C8—2681no—
169C2×C8—2681no—
170C2×C8—1341no—
171C2×C8D4×D6711yescharacteristic
172C2×C8#1176162811yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—671no—
176C22⋊C4—1341no—
177C42D4×D6711yescharacteristic
178C16—41no—
179C16—1341no—
180C16—1341no—
181C23—5361no—
182C23—1341no—
183Q8—161no—
184D4—5361no—
185D4—5361no—
186D4—5361no—
187D4—5361no—
188D4—10721no—
189D4—672no—
190D4—1341no—
191C2×C4—81no—
192C2×C4—2681no—
193C2×C4—2681no—
194C2×C4—1341no—
195C2×C4#1452401611yescharacteristic
196C2×C4D8×D6711yescharacteristic
197C2×C4#1452406411yescharacteristic
198C8—21no—
199C8—81no—
200C8—2681no—
201C8—5361no—
202C8#1452406611yescharacteristic
203C8#1452407011yescharacteristic
204C22—2681no—
205C22—2681no—
206C22—10721no—
207C22—10721no—
208C22—10721no—
209C22—1342no—
210C22#1686363111yescharacteristic
211C4—21no—
212C4—2681no—
213C4—161no—
214C4#1686361411yescharacteristic
215C4#1686366211yescharacteristic
216C4SD32×D6711yescharacteristic
217C4#1686382911yescharacteristic
218C2—2682no—
219C2—21441no—
220C2#2255964911yescharacteristic
221C2#2256230811yescharacteristic
222C2#2256232811yescharacteristic
subgroupG/H
centerC22#16863631
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1365C2
socleC22×C67#466
prime radical#22559565C2
hypercenter#1365D67
hypocenterC67#12468
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
1. #22562434C2semidirect
2. #22562028C2semidirect
3. #22559605C2semidirect
5. #22558282C2semidirect
6. #22558256C2semidirect
7. C67×#1365C2semidirect
117. #1365D67semidirect
119. C67#12468semidirect

Maximal subgroups

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

subgroupindex
#225624342
#225620282
#225596052
#225595652
#225582822
#225582562
C67×#13652
#1246867