Finite Group Databasebeta

← #30753445

Subgroups — #30753445

Order 17152.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#22570585C211yesmaximal, characteristic
2#22558947C211yesmaximal, characteristic, complemented
3#22558876C212yesmaximal
4#22558100C212yesmaximal, complemented
5C67×#2986C211yesmaximal, characteristic, complemented
6C2×#14524040C2211yescharacteristic
7#16863762C414yes—
8#16863744—22no—
9#16863420C2212yes—
10#16863400—22no—
11C4×#11761612C2211yescharacteristic
12#16863142—22no—
13#16863139—22no—
14#16863139—22no—
15#533×C67C2211yescharacteristic
16#446×C67—22no—
17C8×Q8×C67—22no—
18#433×C67—22no—
19#431×C67C2212yes—
20C22×D268—21no—
21C22×D268—22no—
22C2×#11761612—21no—
23C2×#11761612—22no—
24#14524040—21no—
25#14524040—22no—
26#14524040C2×C412yes—
27C4⋊(C67⋊C8)—22no—
28C4⋊(C67⋊C8)—22no—
29#14523963—28no—
30#14523954C2×C412yes—
31C4×C67⋊C8—21no—
32C4⋊Q8×C67D412yes—
33#125×C67C2×C412yes—
34C22⋊Q8×C67—22no—
35C4×Q8×C67—22no—
36#117×C67D412yes—
37C2×C42×C67C2311yescharacteristic
38C4⋊C8×C67—24no—
39C4⋊C8×C67D412yes—
40C4⋊C8×C67D412yes—
41C22⋊C8×C67—22no—
42#97×C67—22no—
43C4×C8×C67—22no—
44C4×C8×C67—22no—
45C23×D67—41no—
46C2×D268—22no—
47C2×D268—24no—
48C2×D268—44no—
49C2×D268—41no—
50C2×D268—42no—
51C2×D268—82no—
52C2×D268—82no—
53#11761612—42no—
54#11761612—82no—
55C2×C67⋊C8—22no—
56C2×C67⋊C8—42no—
57C2×Q8×C67—22no—
58C22×C4×C67C2×D411yescharacteristic
59C22×C4×C67#4112yes—
60C2×C8×C67—22no—
61C2×C8×C67—22no—
62C4⋊C4×C67—24no—
63C4⋊C4×C67—22no—
64C4⋊C4×C67C22⋊C414yes—
65C22⋊C4×C67—22no—
66C42×C67—22no—
67C42×C67C22×C411yescharacteristic
68C42×C67C2×D412yes—
69C42×C67C2×D411yescharacteristic
70C23×C67#12311yescharacteristic
71C22×D67—42no—
72C22×D67—82no—
73C22×D67—84no—
74Q8×C67—42no—
75Q8×C67—42no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—24no—
80C2×C4×C67C2×C22⋊C411yescharacteristic
81C2×C4×C67C4×D412yes—
82C2×C4×C67#12011yescharacteristic
83C2×C4×C67#12112yes—
84D268—44no—
85D268—84no—
86D268—82no—
87D268—82no—
88D268—82no—
89C8×C67—42no—
90C8×C67—42no—
91C67⋊C8—42no—
92C67⋊C8—82no—
93C22×C67—21no—
94C22×C67—22no—
95C22×C67#38611yescharacteristic
96C2×D67—82no—
97C2×D67—84no—
98C2×D67—164no—
99C4×C67—21no—
100C4×C67—22no—
101C4×C67—44no—
102C4×C67#45412yes—
103C4×C67#47112yes—
104C4×C67#47212yes—
105#9438—671nomaximal, Sylow
106C2×C67—41no—
107C2×C67#188811yescharacteristic
108C2×C67#190712yes—
109D67—162no—
110#3138—671no—
111#2986D6711yescharacteristic, complemented
112#1561—671no—
113#1545—672no—
114#1299—672no—
115C67#943811yescharacteristic, complemented, Sylow
116#533C2×D6711yescharacteristic
117C2×#127—671no—
118#490—674no—
119#486—1342no—
120#446—22no—
121C8×Q8—22no—
122#433—22no—
123#431C2×D6712yes—
124#423—672no—
125#418—1342no—
126C4×M16—671no—
127#332—1342no—
128#331—1342no—
129#331—1342no—
130C22×D4—1341no—
131C22×D4—1342no—
132C2×M16—1342no—
133C2×M16—1341no—
134C4⋊Q8D26812yes—
135#127—672no—
136#127—1342no—
137#127—1341no—
138#125C4×D6712yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D6712yes—
142C2×C42C22×D6711yescharacteristic
143C4⋊C8—24no—
144C4⋊C8—1342no—
145C4⋊C8—1342no—
146C4⋊C8C22⋊D6712yes—
147C4⋊C8C22⋊D6712yes—
148#102—1348no—
149C22⋊C8—22no—
150#97—22no—
151#97—672no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—1341no—
155C24—2681no—
156C2×Q8—22no—
157C2×D4—2684no—
158C2×D4—2681no—
159C2×D4—2682no—
160C2×D4—5362no—
161C2×D4—5362no—
162C2×D4—1342no—
163C2×D4—1344no—
164C22×C4C2×C22⋊D6711yescharacteristic
165C22×C4#1176162812yes—
166M16—2682no—
167M16—5362no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—2682no—
171C2×C8—1342no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4(C2×D67)⋊C414yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D6711yescharacteristic
178C42C2×C22⋊D6712yes—
179C42C2×D26811yescharacteristic
180C23—2682no—
181C23—5362no—
182C23—5364no—
183C23#1452402611yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—2684no—
187D4—5364no—
188D4—5362no—
189D4—5362no—
190D4—5362no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×(C2×D67)⋊C411yescharacteristic
196C2×C4C4×C22⋊D6712yes—
197C2×C4#1452401211yescharacteristic
198C2×C4#1452401712yes—
199C8—42no—
200C8—42no—
201C8—2682no—
202C8—5362no—
203C22—21no—
204C22—22no—
205C22—5362no—
206C22—5364no—
207C22—10724no—
208C22#1686327511yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1686356912yes—
213C4#1686365912yes—
214C4#1686366612yes—
215C2—41no—
216C2—10722no—
217C2#2256061211yescharacteristic
218C2#2256071712yes—
subgroupG/H
centerC22#16863275
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2986C2
socleC22×C67#386
prime radicalC2×#14524040C22
cosocleC2×C42×C67C23
hypercenter#2986D67
hypocenterC67#9438
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
2. #22558947C2semidirect
4. #22558100C2semidirect
5. C67×#2986C2semidirect
111. #2986D67semidirect
115. C67#9438semidirect

Maximal subgroups

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

subgroupindex
#225705852
#225589472
#225588762
#225581002
C67×#29862
#943867