Finite Group Databasebeta

← #30750793

Subgroups — #30750793

Order 17152.

398 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 95 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#22570557C211yesmaximal, characteristic
2#22569315C211yesmaximal, characteristic, complemented
3#22558679C212yesmaximal
4#22558109C212yesmaximal, complemented
5C67×#1544C211yesmaximal, characteristic, complemented
6#16863946C2211yescharacteristic
7#16863766C412yes—
8#16863762C412yes—
9#16863758—22no—
10#16863527—24no—
11#16863457—22no—
12#16863453C2212yes—
13#16863390—22no—
14C2×#14523954C2211yescharacteristic
15#16863125—24no—
16#533×C67C2211yescharacteristic
17#431×C67C2212yes—
18#332×C67—22no—
19C2×#11761628—21no—
20C22×C67⋊C8—21no—
21#14524043C2×C411yescharacteristic
22#14524040C2×C411yescharacteristic
23#14524031—21no—
24#14524017—22no—
25C4×D268—22no—
26C4⋊(C67⋊C8)—24no—
27C4⋊(C67⋊C8)D414yes—
28#14523966—24no—
29#14523963—24no—
30C22⋊(C67⋊C8)—22no—
31#14523954—22no—
32#14523954—21no—
33#14523954—22no—
34#14523954C2×C412yes—
35C4×C67⋊C8—22no—
36C4⋊Q8×C67D412yes—
37#125×C67C2×C412yes—
38C22⋊Q8×C67—22no—
39C4×Q8×C67—22no—
40#117×C67D412yes—
41C2×C42×C67C2311yescharacteristic
42C4⋊C8×C67—22no—
43C22⋊C8×C67—22no—
44C4×C8×C67—22no—
45#11761628—44no—
46C2×Dic134—21no—
47C2×D268—21no—
48C2×D268—22no—
49C2×C22⋊D67—41no—
50C2×C4×D67—41no—
51C2×C67⋊C8—22no—
52C2×C67⋊C8—24no—
53C2×C67⋊C8—42no—
54C4⋊Dic67—22no—
55(C2×D67)⋊C4—42no—
56C2×Q8×C67—22no—
57C22×C4×C67C2×D411yescharacteristic
58C22×C4×C67#4112yes—
59C2×C8×C67—42no—
60C4⋊C4×C67—24no—
61C4⋊C4×C67—22no—
62C4⋊C4×C67C22⋊C414yes—
63C22⋊C4×C67—22no—
64C42×C67—22no—
65C42×C67C22×C411yescharacteristic
66C42×C67C2×D412yes—
67C42×C67C2×D411yescharacteristic
68C23×C67#12311yescharacteristic
69C22×D67—41no—
70Q8×C67—82no—
71C2×C4×C67—22no—
72C2×C4×C67—24no—
73C2×C4×C67—42no—
74C2×C4×C67C2×C22⋊C411yescharacteristic
75C2×C4×C67C4×D412yes—
76C2×C4×C67#12011yescharacteristic
77C2×C4×C67#12112yes—
78C22⋊D67—82no—
79C2×Dic67—41no—
80D268—44no—
81D268—42no—
82C4×D67—82no—
83Dic134—42no—
84C8×C67—82no—
85C67⋊C8—42no—
86C67⋊C8—44no—
87C22×C67—22no—
88C22×C67—21no—
89C22×C67#38611yescharacteristic
90C2×D67—82no—
91C2×D67—81no—
92C4×C67—22no—
93C4×C67—22no—
94C4×C67—21no—
95C4×C67—44no—
96C4×C67#45412yes—
97Dic67—81no—
98#9008—671nomaximal, Sylow
99C2×C67—41no—
100C2×C67#188112yes—
101C2×C67#188811yescharacteristic
102D67—161no—
103#3135—671no—
104#2987—671no—
105#1544D6711yescharacteristic, complemented
106#1496—672no—
107#1302—672no—
108C67#900811yescharacteristic, complemented, Sylow
109#533C2×D6711yescharacteristic
110#532—671no—
111#491—672no—
112#490—672no—
113#489—1342no—
114#446—1344no—
115#433—1342no—
116#432—672no—
117#431C2×D6712yes—
118#416—1342no—
119C2×#97—671no—
120#332—22no—
121D4⋊C8—1344no—
122C2×#41—1341no—
123C22×C8—1341no—
124C4⋊Q8—671no—
125C4⋊Q8C22⋊D6712yes—
126#127—671no—
127#125C4×D6712yes—
128#124—1341no—
129C22⋊Q8—22no—
130#121—1342no—
131C4×Q8—22no—
132C4×D4—1342no—
133#117D26812yes—
134C2×C42C22×D6711yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—674no—
137C4⋊C8—1344no—
138#103—1344no—
139#102—1344no—
140C22⋊C8—22no—
141C22⋊C8—1342no—
142#97—672no—
143#97—1342no—
144#97—1342no—
145#97—1341no—
146C4×C8—22no—
147C4×C8—1342no—
148#41—2684no—
149C2×Q8—22no—
150C2×Q8—1341no—
151C2×D4—2681no—
152C2×D4—1341no—
153C2×D4—1342no—
154C22×C4—2681no—
155C22×C4C2×D26811yescharacteristic
156C22×C4#1176162712yes—
157C2×C8—42no—
158C2×C8—2682no—
159C2×C8—1342no—
160C2×C8—1344no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—1342no—
164C4⋊C4(C2×D67)⋊C414yes—
165C22⋊C4—22no—
166C22⋊C4—2682no—
167C42—22no—
168C42C2×C4×D6711yescharacteristic
169C42D4×D6712yes—
170C42C2×C22⋊D6711yescharacteristic
171C23—2681no—
172C23#1452402711yescharacteristic
173Q8—82no—
174Q8—2682no—
175D4—2682no—
176D4—2684no—
177D4—5362no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—2681no—
182C2×C4—5362no—
183C2×C4C2×(C2×D67)⋊C411yescharacteristic
184C2×C4(C22⋊D67)⋊C412yes—
185C2×C4#1452401311yescharacteristic
186C2×C4#1452401512yes—
187C8—82no—
188C8—2682no—
189C8—2684no—
190C22—22no—
191C22—21no—
192C22—5362no—
193C22—5361no—
194C22#1686327311yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—5361no—
200C4#1686357112yes—
201C2—41no—
202C2—10721no—
203C2#2256056912yes—
204C2#2256061011yescharacteristic
subgroupG/H
centerC2×C4#14524013
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1544C2
socleC22×C67#386
prime radical#16863946C22
cosocleC2×C42×C67C23
hypercenter#1544D67
hypocenterC67#9008
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
2. #22569315C2semidirect
4. #22558109C2semidirect
5. C67×#1544C2semidirect
105. #1544D67semidirect
108. C67#9008semidirect

Maximal subgroups

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

subgroupindex
#225705572
#225693152
#225586792
#225581092
C67×#15442
#900867