Finite Group Databasebeta

← #30753351

Subgroups — #30753351

Order 17152.

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

normal subgroups only

This group has 384 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#22570591C211yesmaximal, characteristic
2#22569315C211yesmaximal, characteristic
3#22558925C212yesmaximal
4#22558082C212yesmaximal
5C67×#1537C211yesmaximal, characteristic
6C2×#14524043C2211yescharacteristic
7#16863766—22no—
8#16863748C414yes—
9#16863527—24no—
10#16863457—22no—
11#16863453C2212yes—
12#16863404—22no—
13C4×#11761612C2211yescharacteristic
14#16863141—24no—
15#533×C67C2211yescharacteristic
16#432×C67C2212yes—
17#330×C67—22no—
18C22×Dic134—21no—
19C2×#11761612—21no—
20#14524043—22no—
21#14524043—21no—
22#14524043C2×C412yes—
23C2×C4⋊Dic67—22no—
24C4⋊(C67⋊C8)—24no—
25C4⋊(C67⋊C8)D414yes—
26#14523966—28no—
27C22⋊(C67⋊C8)—22no—
28#14523954—22no—
29#14523954—21no—
30#14523954—22no—
31C4×C67⋊C8—22no—
32C4×C67⋊C8C2×C412yes—
33C4⋊Q8×C67C2×C412yes—
34#125×C67D412yes—
35C22⋊Q8×C67—22no—
36C4×Q8×C67—22no—
37#117×C67D412yes—
38C2×C42×C67C2311yescharacteristic
39C4⋊C8×C67—22no—
40C22⋊C8×C67—22no—
41#97×C67—22no—
42C2×Dic134—22no—
43C2×Dic134—44no—
44C2×Dic134—41no—
45C22×Dic67—41no—
46#11761612—42no—
47C2×C67⋊C8—22no—
48C2×C67⋊C8—24no—
49C4⋊Dic67—24no—
50C4⋊Dic67—42no—
51C2×Q8×C67—22no—
52C22×C4×C67C2×D411yescharacteristic
53C22×C4×C67#4112yes—
54C2×C8×C67—42no—
55C4⋊C4×C67—22no—
56C4⋊C4×C67—24no—
57C4⋊C4×C67C22⋊C414yes—
58C22⋊C4×C67—22no—
59C42×C67—22no—
60C42×C67C22×C411yescharacteristic
61C42×C67C2×D412yes—
62C42×C67C2×D411yescharacteristic
63C23×C67#12311yescharacteristic
64Q8×C67—82no—
65C2×C4×C67—22no—
66C2×C4×C67—24no—
67C2×C4×C67—42no—
68C2×C4×C67C2×C22⋊C411yescharacteristic
69C2×C4×C67C4×D412yes—
70C2×C4×C67#12011yescharacteristic
71C2×C4×C67#12112yes—
72C2×Dic67—42no—
73C2×Dic67—82no—
74Dic134—44no—
75Dic134—82no—
76C8×C67—82no—
77C67⋊C8—42no—
78C67⋊C8—44no—
79C22×C67—21no—
80C22×C67—22no—
81C22×C67#38611yescharacteristic
82C4×C67—21no—
83C4×C67—22no—
84C4×C67—22no—
85C4×C67—44no—
86C4×C67#45412yes—
87Dic67—82no—
88#9422—671nomaximal, Sylow
89C2×C67—41no—
90C2×C67#188811yescharacteristic
91C2×C67#190212yes—
92#3139—671no—
93#2987—671no—
94#1556—672no—
95#1537D6711yescharacteristic
96#1291—672no—
97C67#942211yescharacteristic, complemented, Sylow
98#533C2×D6711yescharacteristic
99C2×C4⋊Q8—671no—
100#491—1342no—
101#487—674no—
102#446—1344no—
103#433—1342no—
104#432—672no—
105#432C2×D6712yes—
106#419—1342no—
107C4×M16—671no—
108#332—1344no—
109#330—22no—
110C22×Q8—1341no—
111C2×M16—1341no—
112C4⋊Q8—672no—
113C4⋊Q8—1342no—
114C4⋊Q8—1341no—
115C4⋊Q8C4×D6712yes—
116#125C22⋊D6712yes—
117C22⋊Q8—22no—
118C4×Q8—22no—
119#117D26812yes—
120C2×C4⋊C4—1342no—
121C2×C42C22×D6711yescharacteristic
122C4⋊C8—22no—
123C4⋊C8—674no—
124C4⋊C8—1344no—
125#103—1348no—
126C22⋊C8—22no—
127C22⋊C8—1342no—
128#97—22no—
129#97—1342no—
130#97—1342no—
131#97—1341no—
132C4×C8—672no—
133C4×C8—1342no—
134C2×Q8—22no—
135C2×Q8—2684no—
136C2×Q8—2681no—
137C2×Q8—1342no—
138C22×C4—2681no—
139C22×C4C2×D26811yescharacteristic
140C22×C4#1176162712yes—
141M16—2682no—
142C2×C8—42no—
143C2×C8—1342no—
144C2×C8—1344no—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—2682no—
148C4⋊C4—1344no—
149C4⋊C4(C2×D67)⋊C414yes—
150C22⋊C4—22no—
151C42—22no—
152C42C2×C4×D6711yescharacteristic
153C42D4×D6712yes—
154C42C2×C22⋊D6711yescharacteristic
155C23#1452402711yescharacteristic
156Q8—82no—
157Q8—2684no—
158Q8—5362no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—42no—
162C2×C4—2682no—
163C2×C4—5362no—
164C2×C4C2×(C2×D67)⋊C411yescharacteristic
165C2×C4(C22⋊D67)⋊C412yes—
166C2×C4#1452401311yescharacteristic
167C2×C4#1452401512yes—
168C8—82no—
169C8—2682no—
170C8—2684no—
171C22—21no—
172C22—22no—
173C22#1686327311yescharacteristic
174C4—21no—
175C4—22no—
176C4—22no—
177C4—44no—
178C4—5362no—
179C4#1686357112yes—
180C2—41no—
181C2#2256061011yescharacteristic
182C2#2256068012yes—
subgroupG/H
centerC22#16863273
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1537C2
socleC22×C67#386
prime radicalC23×C67#123
cosocleC2×C42×C67C23
hypercenter#1537D67
hypocenterC67#9422
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
97. C67#9422semidirect

Maximal subgroups

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

subgroupindex
#225705912
#225693152
#225589252
#225580822
C67×#15372
#942267