Finite Group Databasebeta

← #30753433

Subgroups — #30753433

Order 17152.

400 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 103 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#22569311C211yesmaximal, characteristic
3#22558895C212yesmaximal
4#22558098C212yesmaximal
5C67×#1561C211yesmaximal, characteristic
6C2×#14524043C2211yescharacteristic
7#16863766C414yes—
8#16863748—22no—
9#16863527—22no—
10Q8×C67⋊C8—22no—
11#16863457—22no—
12#16863449C2212yes—
13#16863404—22no—
14C4×#11761612C2211yescharacteristic
15#16863141—22no—
16#16863141—22no—
17#533×C67C2211yescharacteristic
18#423×C67C2212yes—
19#332×C67—22no—
20C22×Dic134—21no—
21C2×#11761612—21no—
22#14524043—22no—
23#14524043—21no—
24#14524043C2×C412yes—
25C2×C4⋊Dic67—22no—
26C4⋊(C67⋊C8)—24no—
27C4⋊(C67⋊C8)D412yes—
28C4⋊(C67⋊C8)D412yes—
29#14523966—28no—
30C22⋊(C67⋊C8)—22no—
31#14523954—22no—
32#14523954C2×C412yes—
33C4×C67⋊C8—22no—
34C4×C67⋊C8—21no—
35C4×C67⋊C8—22no—
36C2×M16×C67—22no—
37C4⋊Q8×C67D412yes—
38#125×C67C2×C412yes—
39C22⋊Q8×C67—22no—
40C4×Q8×C67—22no—
41#117×C67D412yes—
42C2×C42×C67C2311yescharacteristic
43C4⋊C8×C67—22no—
44C4⋊C8×C67—22no—
45C2×Dic134—22no—
46C2×Dic134—44no—
47C2×Dic134—41no—
48C22×Dic67—41no—
49#11761612—42no—
50C2×C67⋊C8—22no—
51C2×C67⋊C8—22no—
52C2×C67⋊C8—22no—
53C4⋊Dic67—24no—
54C4⋊Dic67—42no—
55C2×Q8×C67—22no—
56C22×C4×C67C2×D411yescharacteristic
57C22×C4×C67#4112yes—
58M16×C67—82no—
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
69Q8×C67—42no—
70Q8×C67—42no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—24no—
75C2×C4×C67C2×C22⋊C411yescharacteristic
76C2×C4×C67C4×D412yes—
77C2×C4×C67#12011yescharacteristic
78C2×C4×C67#12112yes—
79C2×Dic67—42no—
80C2×Dic67—82no—
81Dic134—44no—
82Dic134—82no—
83C8×C67—82no—
84C67⋊C8—42no—
85C67⋊C8—42no—
86C67⋊C8—42no—
87C22×C67—21no—
88C22×C67—22no—
89C22×C67#38611yescharacteristic
90C4×C67—21no—
91C4×C67—22no—
92C4×C67—44no—
93C4×C67#45412yes—
94C4×C67#47112yes—
95C4×C67#47312yes—
96Dic67—82no—
97#9436—671nomaximal, Sylow
98C2×C67—41no—
99C2×C67#188811yescharacteristic
100C2×C67#190812yes—
101#3139—671no—
102#2986—671no—
103#1561D6711yescharacteristic
104#1549—672no—
105#1299—672no—
106C67#943611yescharacteristic, complemented, Sylow
107#533C2×D6711yescharacteristic
108C2×C4⋊Q8—671no—
109#491—674no—
110#487—1342no—
111#446—1342no—
112C8×Q8—1342no—
113#433—1342no—
114#431—672no—
115#423C2×D6712yes—
116#419—1342no—
117C4×M16—671no—
118#332—22no—
119#332—1342no—
120#332—1342no—
121C22×Q8—1341no—
122C2×M16—22no—
123C2×M16—1341no—
124C4⋊Q8—672no—
125C4⋊Q8—1342no—
126C4⋊Q8—1341no—
127C4⋊Q8C22⋊D6712yes—
128#125C4×D6712yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117D26812yes—
132C2×C4⋊C4—1342no—
133C2×C42C22×D6711yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—672no—
137C4⋊C8—672no—
138C4⋊C8—1344no—
139#103—1348no—
140C22⋊C8—1342no—
141#97—672no—
142#97—1342no—
143C4×C8—1342no—
144C4×C8—1342no—
145C4×C8—1341no—
146C2×Q8—22no—
147C2×Q8—2684no—
148C2×Q8—2681no—
149C2×Q8—1342no—
150C22×C4—2681no—
151C22×C4C2×D26811yescharacteristic
152C22×C4#1176162712yes—
153M16—82no—
154M16—2682no—
155C2×C8—42no—
156C2×C8—1342no—
157C2×C8—1342no—
158C2×C8—1342no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—2682no—
162C4⋊C4—1344no—
163C4⋊C4(C2×D67)⋊C414yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D6711yescharacteristic
167C42D4×D6712yes—
168C42C2×C22⋊D6711yescharacteristic
169C23#1452402711yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—2684no—
173Q8—5362no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—2682no—
179C2×C4—5362no—
180C2×C4C2×(C2×D67)⋊C411yescharacteristic
181C2×C4(C22⋊D67)⋊C412yes—
182C2×C4#1452401311yescharacteristic
183C2×C4#1452401512yes—
184C8—82no—
185C8—2682no—
186C8—2682no—
187C8—2682no—
188C22—21no—
189C22—22no—
190C22#1686327311yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—5362no—
195C4#1686357112yes—
196C4#1686365712yes—
197C4#1686366912yes—
198C2—41no—
199C2#2256061011yescharacteristic
200C2#2256072212yes—
subgroupG/H
centerC22#16863273
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1561C2
socleC22×C67#386
prime radicalC23×C67#123
cosocleC2×C42×C67C23
hypercenter#1561D67
hypocenterC67#9436
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
106. C67#9436semidirect

Maximal subgroups

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

subgroupindex
#225705912
#225693112
#225588952
#225580982
C67×#15612
#943667