Finite Group Databasebeta

← #30760610

Subgroups — #30760610

Order 17152.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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#22561041C211yesmaximal, characteristic
2#22558859C212yesmaximal
3#22558106C212yesmaximal, complemented
4#22558038C211yesmaximal, characteristic, complemented
5C67×#1832C211yesmaximal, characteristic, complemented
6#16863946C2211yescharacteristic
7#16863449C2212yes—
8#16863420C2211yescharacteristic
9C2×#14523971—22no—
10#16863143—22no—
11#16863139—22no—
12C2×C4⋊C8×C67C2212yes—
13#376×C67C2211yescharacteristic
14#336×C67—22no—
15C2×#11761628—21no—
16C2×#11761612—21no—
17#14524043D411yescharacteristic
18#14524040D411yescharacteristic
19#14524031—21no—
20#14524017—22no—
21C4×D268—22no—
22C4⋊(C67⋊C8)—22no—
23C4⋊(C67⋊C8)—22no—
24#14523971—44no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—22no—
27C22×C8×C67—22no—
28C2×C42×C67D412yes—
29C2×C42×C67C2311yescharacteristic
30C4⋊C8×C67—22no—
31C4⋊C8×C67—22no—
32#95×C67—22no—
33#11761628—42no—
34#11761628—42no—
35C2×Dic134—21no—
36C2×D268—21no—
37C2×D268—22no—
38C2×C22⋊D67—41no—
39C2×C4×D67—41no—
40#11761612—42no—
41C2×C67⋊C8—42no—
42C2×C67⋊C8—41no—
43C4⋊Dic67—22no—
44(C2×D67)⋊C4—42no—
45C22×C4×C67—22no—
46C22×C4×C67C2×D411yescharacteristic
47C22×C4×C67#4112yes—
48C2×C8×C67—42no—
49C2×C8×C67—42no—
50C2×C8×C67—42no—
51C42×C67—22no—
52C42×C67—22no—
53C42×C67C2×D411yescharacteristic
54C42×C67#4111yescharacteristic
55C42×C67#4112yes—
56C23×C67#12311yescharacteristic
57C22×D67—41no—
58C2×C4×C67—42no—
59C2×C4×C67—44no—
60C2×C4×C67—42no—
61C2×C4×C67#12111yescharacteristic
62C2×C4×C67#12312yes—
63C2×C4×C67#12411yescharacteristic
64C2×C4×C67#12612yes—
65C2×C4×C67#13512yes—
66C2×C4×C67AGL(1,ℤ/8)12yes—
67C22⋊D67—82no—
68C2×Dic67—41no—
69D268—42no—
70D268—82no—
71C4×D67—81no—
72C4×D67—81no—
73Dic134—42no—
74C8×C67—82no—
75C67⋊C8—82no—
76C67⋊C8—81no—
77C22×C67—22no—
78C22×C67#39711yescharacteristic
79C22×C67#48212yes—
80C2×D67—81no—
81C2×D67—82no—
82C4×C67—22no—
83C4×C67—22no—
84C4×C67—82no—
85C4×C67#46312yes—
86C4×C67#48811yescharacteristic
87C4×C67#49011yescharacteristic
88Dic67—81no—
89#10629—671nomaximal, Sylow
90C2×C67—22no—
91C2×C67#207311yescharacteristic
92C2×C67#208012yes—
93D67—161no—
94#1963—671no—
95#1832D6711yescharacteristic, complemented
96#1540—672no—
97#1301—672no—
98#1274—671no—
99C67#1062911yescharacteristic, complemented, Sylow
100#532—671no—
101#431—672no—
102#423—671no—
103C2×C4⋊C8C2×D6712yes—
104C2×#104—1342no—
105#376C2×D6711yescharacteristic
106#336—22no—
107#332—1342no—
108#331—1342no—
109C2×#41—1341no—
110C2×M16—1341no—
111C22×C8—22no—
112C4⋊Q8—671no—
113#127—671no—
114#124—1341no—
115#121—1342no—
116C4×D4—1342no—
117C2×C42C22⋊D6712yes—
118C2×C42C22×D6711yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—1342no—
122C4⋊C8—1342no—
123#104—2684no—
124C22⋊C8—1342no—
125C4×C8—1342no—
126#95—22no—
127#41—2682no—
128#41—2682no—
129C2×Q8—1341no—
130C2×D4—2681no—
131C2×D4—1341no—
132C2×D4—1342no—
133C22×C4—22no—
134C22×C4—2681no—
135C22×C4C2×C22⋊D6711yescharacteristic
136C22×C4#1176162812yes—
137M16—2682no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—2682no—
142C2×C8—2681no—
143C4⋊C4—1342no—
144C22⋊C4—2682no—
145C42—22no—
146C42—22no—
147C42D4×D6711yescharacteristic
148C42#1176162711yescharacteristic
149C42#1176162912yes—
150C23—2681no—
151C23#1452402611yescharacteristic
152Q8—2682no—
153D4—2682no—
154D4—5362no—
155D4—5362no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—2681no—
160C2×C4—5361no—
161C2×C4—5361no—
162C2×C4#1452401411yescharacteristic
163C2×C4#1452402812yes—
164C2×C4#1452403211yescharacteristic
165C2×C4#1452403812yes—
166C2×C4#1452406312yes—
167C2×C4#1452406812yes—
168C8—82no—
169C8—5362no—
170C8—5361no—
171C22—22no—
172C22—5361no—
173C22—5362no—
174C22#1686332211yescharacteristic
175C22#1686372012yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—5361no—
180C4#1686360812yes—
181C4#1686375311yescharacteristic
182C4#1686376211yescharacteristic
183C2—22no—
184C2—10721no—
185C2#2256162311yescharacteristic
186C2#2256166012yes—
subgroupG/H
centerC2×C4#14524032
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1832C2
socleC22×C67#397
prime radical#16863946C22
hypercenter#1832D67
hypocenterC67#10629
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
3. #22558106C2semidirect
4. #22558038C2semidirect
5. C67×#1832C2semidirect
95. #1832D67semidirect
99. C67#10629semidirect

Maximal subgroups

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

subgroupindex
#225610412
#225588592
#225581062
#225580382
C67×#18322
#1062967