Finite Group Databasebeta

← #30748822

Subgroups — #30748822

Order 17152.

388 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 93 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#22568920C211yesmaximal, characteristic
2#22558970C212yesmaximal, complemented
3#22558843C212yesmaximal
4#22558120C211yesmaximal, characteristic, complemented
5C67×#1307C211yesmaximal, characteristic, complemented
6C2×#14524051C412yes—
7#16863946C2211yescharacteristic
8#16863473—22no—
9#16863461—22no—
10#16863453C2212yes—
11#16863364C412yes—
12C2×C4×C67⋊C8C2211yescharacteristic
13#16863148—22no—
14#16863146—22no—
15#16863135—24no—
16#432×C67C2212yes—
17#423×C67C2211yescharacteristic
18C2×#11761628C2×C412yes—
19#14524051—44no—
20C2×#11761611—22no—
21C22×C67⋊C8C2×C411yescharacteristic
22C22×C67⋊C8C2×C411yescharacteristic
23C2×C8×D67—22no—
24#14524043Q811yescharacteristic
25#14524040Q811yescharacteristic
26#14524031D412yes—
27#14524017—22no—
28C4×D268—22no—
29C4⋊(C67⋊C8)—22no—
30C22⋊(C67⋊C8)—22no—
31(C2×D67)⋊C8—24no—
32#14523954—22no—
33C4×C67⋊C8D412yes—
34C4×C67⋊C8D412yes—
35C2×M16×C67C2×C412yes—
36C2×C42×C67C2311yescharacteristic
37C4⋊C8×C67—22no—
38C4⋊C8×C67—22no—
39C22⋊C8×C67—22no—
40#97×C67—22no—
41#11761628—42no—
42#11761628—42no—
43C2×Dic134C4⋊C412yes—
44C2×D268—22no—
45C2×D268C4⋊C412yes—
46C2×C22⋊D67—22no—
47C2×C4×D67—22no—
48#11761611—44no—
49C2×C67⋊C8—22no—
50C2×C67⋊C8—41no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8C22⋊C412yes—
53C2×C67⋊C8C22⋊C411yescharacteristic
54C2×C67⋊C8C22⋊C411yescharacteristic
55C8×D67—44no—
56C4⋊Dic67—22no—
57(C2×D67)⋊C4—24no—
58C22×C4×C67C22×C411yescharacteristic
59C22×C4×C67#4112yes—
60M16×C67—42no—
61C2×C8×C67—22no—
62C2×C8×C67—42no—
63C42×C67C2×D412yes—
64C42×C67C2×D411yescharacteristic
65C42×C67C2×Q811yescharacteristic
66C23×C67#11711yescharacteristic
67C22×D67—22no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—42no—
71C2×C4×C67C2×C22⋊C411yescharacteristic
72C2×C4×C67C2×C4⋊C411yescharacteristic
73C2×C4×C67#12112yes—
74C2×C4×C67C22⋊Q812yes—
75C22⋊D67—44no—
76C2×Dic67—22no—
77D268—42no—
78D268—44no—
79C4×D67—42no—
80C4×D67—42no—
81Dic134—42no—
82C8×C67—42no—
83C8×C67—82no—
84C67⋊C8—21no—
85C67⋊C8—22no—
86C67⋊C8—82no—
87C67⋊C8#10012yes—
88C22×C67—21no—
89C22×C67—22no—
90C22×C67#38011yescharacteristic
91C2×D67—42no—
92C2×D67—44no—
93C4×C67—21no—
94C4×C67—22no—
95C4×C67—22no—
96C4×C67C2×#10011yescharacteristic
97C4×C67#41311yescharacteristic
98Dic67—42no—
99#8630—671nomaximal, Sylow
100C2×C67—41no—
101C2×C67#180011yescharacteristic
102C2×C67#181412yes—
103D67—82no—
104#2931—671no—
105#1568—672no—
106#1536—672no—
107#1307—671no—
108#1307D6711yescharacteristic, complemented
109C67#863011yescharacteristic, complemented, Sylow
110C2×#131—672no—
111#532—671no—
112#436—1342no—
113C8×D4—1342no—
114#432—672no—
115#432C2×D6712yes—
116#423C2×D6711yescharacteristic
117#408—672no—
118C2×C4×C8—671no—
119#335—1342no—
120#334—1342no—
121#329—1344no—
122C2×#41—672no—
123#131—2684no—
124C2×M16—1342no—
125C2×M16C4×D6712yes—
126C22×C8—671no—
127C22×C8—671no—
128C22×C8—1342no—
129C4⋊Q8—671no—
130#127—671no—
131#124—672no—
132#121—1342no—
133C4×D4—1342no—
134C2×C42C22×D6711yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—1342no—
138C22⋊C8—22no—
139C22⋊C8—1344no—
140C22⋊C8—1342no—
141#97—22no—
142#97—1342no—
143C4×C8—672no—
144C4×C8—672no—
145#41—2682no—
146#41—2682no—
147C2×Q8—672no—
148C2×D4—672no—
149C2×D4—1342no—
150C2×D4—1342no—
151C22×C4—1342no—
152C22×C4C2×C4×D6711yescharacteristic
153C22×C4#1176162812yes—
154M16—42no—
155M16—2684no—
156C2×C8—22no—
157C2×C8—42no—
158C2×C8—2684no—
159C2×C8—2682no—
160C2×C8—2681no—
161C2×C8—671no—
162C2×C8—672no—
163C2×C8—671no—
164C2×C8—1342no—
165C4⋊C4—1342no—
166C22⋊C4—1344no—
167C42D4×D6712yes—
168C42D4×D6711yescharacteristic
169C42Q8×D6711yescharacteristic
170C23—1342no—
171C23#1452400011yescharacteristic
172Q8—2682no—
173D4—2682no—
174D4—2684no—
175D4—2684no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—2682no—
180C2×C4—2682no—
181C2×C4—1342no—
182C2×C4C22⋊C4×D6711yescharacteristic
183C2×C4C4⋊C4×D6711yescharacteristic
184C2×C4#1452401612yes—
185C2×C4#1452402112yes—
186C8—42no—
187C8—82no—
188C8—5362no—
189C8—672no—
190C8—1341no—
191C8—1342no—
192C22—22no—
193C22—21no—
194C22—2684no—
195C22—2682no—
196C22#1686325511yescharacteristic
197C4—22no—
198C4—22no—
199C4—21no—
200C4—2682no—
201C4#100×D6711yescharacteristic
202C4#1686337911yescharacteristic
203C2—41no—
204C2—5362no—
205C2#2256020011yescharacteristic
206C2#2256026212yes—
subgroupG/H
centerC22#16863255
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1307C2
socleC22×C67#380
prime radical#16863946C22
cosocleC2×C42×C67C23
hypercenter#1307D67
hypocenterC67#8630
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #22558970C2semidirect
4. #22558120C2semidirect
5. C67×#1307C2semidirect
108. #1307D67semidirect
109. C67#8630semidirect

Maximal subgroups

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

subgroupindex
#225689202
#225589702
#225588432
#225581202
C67×#13072
#863067