Finite Group Databasebeta

← #30770268

Subgroups — #30770268

Order 17152.

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

normal subgroups only

This group has 224 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#22562436C211yesmaximal, characteristic, complemented
2#22562028C211yesmaximal, characteristic
3#22559599C211yesmaximal, characteristic
4#22559571C211yesmaximal, characteristic
5#22558282C211yesmaximal, characteristic, complemented
6#22558263C211yesmaximal, characteristic, complemented
7C67×#1362C211yesmaximal, characteristic, complemented
8C2×C16×D67—21no—
9#16863801C2211yescharacteristic
10#16863673—21no—
11#16863602—21no—
12C8×D268C2211yescharacteristic
13#16863200—21no—
14#16863200—21no—
15(C2×D67)⋊C16—21no—
16C4×C67⋊C16C2211yescharacteristic
17#16863149C2211yescharacteristic
18#16863143—21no—
19#16863142—21no—
20#494×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×#11761617—41no—
24C2×C8×D67—21no—
25#14524043—21no—
26C4×D268D411yescharacteristic
27C2×C67⋊C16D411yescharacteristic
28C2×C67⋊C16D411yescharacteristic
29C16×D67—42no—
30#14523977—21no—
31#14523977—21no—
32#14523976—41no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)D411yescharacteristic
35#14523968—41no—
36#14523963—41no—
37(C2×D67)⋊C8—21no—
38C2×Q16×C67—21no—
39C2×Q16×C67—21no—
40C4⋊Q8×C67—21no—
41C2×C16×C67—21no—
42C4⋊C8×C67—21no—
43C4×C8×C67C2311yescharacteristic
44C2×Dic134—41no—
45C2×D268D811yescharacteristic
46C2×C4×D67—21no—
47#11761617—82no—
48C2×C67⋊C8—21no—
49C2×C67⋊C8—41no—
50C8×D67—41no—
51C8×D67—41no—
52C4⋊Dic67—41no—
53C4⋊Dic67—41no—
54C4⋊Dic67D811yescharacteristic
55(C2×D67)⋊C4—21no—
56C67⋊C16—21no—
57C67⋊C16—21no—
58C2×Q8×C67—41no—
59Q16×C67—41no—
60Q16×C67—41no—
61Q16×C67—81no—
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—
70Q8×C67—81no—
71Q8×C67—81no—
72C2×C4×C67—81no—
73C2×C4×C67#12111yescharacteristic
74C2×C4×C67C2×D811yescharacteristic
75C2×C4×C67#13511yescharacteristic
76C2×Dic67—21no—
77C2×Dic67—81no—
78D268—21no—
79D268SD3212yes—
80C4×D67—41no—
81C4×D67—41no—
82Dic134—81no—
83Dic134—81no—
84C8×C67—21no—
85C8×C67—81no—
86C8×C67#13511yescharacteristic
87C8×C67AGL(1,ℤ/8)11yescharacteristic
88C67⋊C8—41no—
89C67⋊C8—81no—
90C22×C67#46611yescharacteristic
91C2×D67—22no—
92C2×D67—41no—
93C2×D67—41no—
94C4×C67—21no—
95C4×C67—161no—
96C4×C67#46311yescharacteristic
97C4×C67#47111yescharacteristic
98C4×C67C2×SD3211yescharacteristic
99C4×C67#50811yescharacteristic
100Dic67—41no—
101Dic67—161no—
102#12470—671nomaximal, Sylow
103C2×C67#168111yescharacteristic
104C2×C67#221711yescharacteristic
105C2×C67#222211yescharacteristic
106D67—42no—
107#2241—671no—
108D4×C16—671no—
109#1673—671no—
110#1669—671no—
111#1372—671no—
112#1364—671no—
113#1362D6711yescharacteristic, complemented
114C67#1247011yescharacteristic, complemented, Sylow
115C22×C16—1341no—
116#500—671no—
117#494C2×D6711yescharacteristic
118#474—1341no—
119#461—1341no—
120C8×D4—671no—
121C4⋊C16C2×D6711yescharacteristic
122#358—1341no—
123#358—1341no—
124C22⋊C16—1341no—
125C4×C16—671no—
126#335—671no—
127#335C2×D6711yescharacteristic
128#332—1341no—
129#332—1341no—
130C2×Q16—21no—
131C2×Q16—21no—
132C2×SD16—2681no—
133C22×C8—1341no—
134C4⋊Q8—21no—
135C4⋊Q8—1341no—
136C4×D4—671no—
137C2×C16—21no—
138C2×C16—2682no—
139C2×C16—671no—
140C2×C16—671no—
141#107—2681no—
142#107—1341no—
143#107—1341no—
144C4⋊C8—21no—
145C4⋊C8—671no—
146C4⋊C8—1341no—
147#103—2681no—
148#102—2681no—
149C22⋊C8—1341no—
150C4×C8C22×D6711yescharacteristic
151C2×Q8—41no—
152C2×Q8—2681no—
153C2×D4—671no—
154C22×C4—1341no—
155Q16—41no—
156Q16—41no—
157Q16—81no—
158SD16—5362no—
159C2×C8—41no—
160C2×C8—2681no—
161C2×C8—2681no—
162C2×C8—2681no—
163C2×C8—1341no—
164C2×C8D4×D6711yescharacteristic
165C2×C8#1176162811yescharacteristic
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—2681no—
169C4⋊C4—2681no—
170C4⋊C4—671no—
171C22⋊C4—1341no—
172C42D4×D6711yescharacteristic
173C16—41no—
174C16—1341no—
175C16—1341no—
176C23—1341no—
177Q8—81no—
178Q8—81no—
179Q8—5361no—
180Q8—5361no—
181D4—672no—
182D4—1341no—
183C2×C4—81no—
184C2×C4—2681no—
185C2×C4—2681no—
186C2×C4—5361no—
187C2×C4—1341no—
188C2×C4#1452401611yescharacteristic
189C2×C4D8×D6711yescharacteristic
190C2×C4#1452406411yescharacteristic
191C8—21no—
192C8—81no—
193C8—2681no—
194C8—5361no—
195C8#1452406411yescharacteristic
196C8#1452406711yescharacteristic
197C22—2681no—
198C22—2681no—
199C22—1342no—
200C22#1686363111yescharacteristic
201C4—21no—
202C4—2681no—
203C4—161no—
204C4—10721no—
205C4#1686361011yescharacteristic
206C4#1686365811yescharacteristic
207C4SD32×D6711yescharacteristic
208C4#1686382911yescharacteristic
209C2—2682no—
210C2#2255964511yescharacteristic
211C2#2256230811yescharacteristic
212C2#2256233311yescharacteristic
subgroupG/H
centerC22#16863631
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1362C2
socleC22×C67#466
prime radicalC2×D268D8
hypercenter#1362D67
hypocenterC67#12470
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #22562436C2semidirect
5. #22558282C2semidirect
6. #22558263C2semidirect
7. C67×#1362C2semidirect
113. #1362D67semidirect
114. C67#12470semidirect

Maximal subgroups

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

subgroupindex
#225624362
#225620282
#225595992
#225595712
#225582822
#225582632
C67×#13622
#1247067