Finite Group Databasebeta

← #30760601

Subgroups — #30760601

Order 17152.

330 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 93 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#22560363C211yesmaximal, characteristic, complemented
3#22558856C212yesmaximal
4#22558105C212yesmaximal, complemented
5C67×#1274C211yesmaximal, characteristic, complemented
6#16863946C2211yescharacteristic
7#16863420C2211yescharacteristic
8C2×C4⋊(C67⋊C8)—21no—
9C2×C4⋊(C67⋊C8)—21no—
10C2×C4⋊(C67⋊C8)C2212yes—
11C2×#14523971—21no—
12C2×#14523971—21no—
13#16863141—22no—
14#16863139—22no—
15#431×C67C2212yes—
16#376×C67C2211yescharacteristic
17C2×#11761628—21no—
18C2×#11761612—21no—
19C22×C67⋊C8—22no—
20#14524043D411yescharacteristic
21#14524040D411yescharacteristic
22#14524031—21no—
23#14524017—22no—
24C4×D268—22no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)—22no—
27C4⋊(C67⋊C8)—22no—
28C4⋊(C67⋊C8)—42no—
29C4⋊(C67⋊C8)—42no—
30#14523971—42no—
31#14523971—42no—
32C2×C42×C67D411yescharacteristic
33C2×C42×C67D411yescharacteristic
34C2×C42×C67C2311yescharacteristic
35C4⋊C8×C67—22no—
36C22⋊C8×C67—22no—
37C4×C8×C67—22no—
38#95×C67—22no—
39#11761628—41no—
40#11761628—41no—
41#11761628—41no—
42#11761628—41no—
43C2×Dic134—21no—
44C2×D268—21no—
45C2×D268—22no—
46C2×C22⋊D67—41no—
47C2×C4×D67—41no—
48#11761612—41no—
49#11761612—41no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—41no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C4⋊Dic67—22no—
55(C2×D67)⋊C4—42no—
56C22×C4×C67—21no—
57C22×C4×C67—21no—
58C22×C4×C67C2×D411yescharacteristic
59C22×C4×C67#4112yes—
60C2×C8×C67—42no—
61C42×C67—21no—
62C42×C67—21no—
63C42×C67D812yes—
64C42×C67SD1612yes—
65C42×C67C2×D411yescharacteristic
66C42×C67#4112yes—
67C42×C67#4111yescharacteristic
68C23×C67#12311yescharacteristic
69C22×D67—41no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—42no—
73C2×C4×C67—42no—
74C2×C4×C67—42no—
75C2×C4×C67#12111yescharacteristic
76C2×C4×C67#12312yes—
77C2×C4×C67#12411yescharacteristic
78C2×C4×C67#12612yes—
79C2×C4×C67C2×D811yescharacteristic
80C2×C4×C67C2×SD1611yescharacteristic
81C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C67#13711yescharacteristic
83C22⋊D67—81no—
84C22⋊D67—81no—
85C2×Dic67—41no—
86D268—41no—
87D268—41no—
88D268—82no—
89C4×D67—81no—
90C4×D67—81no—
91Dic134—41no—
92Dic134—41no—
93C8×C67—82no—
94C67⋊C8—82no—
95C67⋊C8—81no—
96C22×C67—22no—
97C22×C67#39711yescharacteristic
98C22×C67#48011yescharacteristic
99C22×C67#48111yescharacteristic
100C2×D67—81no—
101C2×D67—82no—
102C4×C67—22no—
103C4×C67—22no—
104C4×C67—42no—
105C4×C67—42no—
106C4×C67#45911yescharacteristic
107C4×C67#46111yescharacteristic
108C4×C67#48611yescharacteristic
109C4×C67#48911yescharacteristic
110Dic67—81no—
111#10627—671nomaximal, Sylow
112C2×C67—21no—
113C2×C67—21no—
114C2×C67#207111yescharacteristic
115C2×C67#208012yes—
116D67—161no—
117#1963—671no—
118#1843—671no—
119#1540—672no—
120#1301—672no—
121#1274D6711yescharacteristic, complemented
122C67#1062711yescharacteristic, complemented, Sylow
123#532—671no—
124#431C2×D6712yes—
125#423—671no—
126C2×C4⋊C8—672no—
127C2×C4⋊C8—1341no—
128C2×C4⋊C8—1341no—
129C2×#104—1341no—
130C2×#104—1341no—
131#376C2×D6711yescharacteristic
132#332—1342no—
133#331—1342no—
134C2×#41—1341no—
135C2×M16—1341no—
136C22×C8—1342no—
137C4⋊Q8—671no—
138#127—671no—
139#124—1341no—
140#121—1342no—
141C4×D4—1342no—
142C2×C42D26811yescharacteristic
143C2×C42D26811yescharacteristic
144C2×C42C22×D6711yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—2682no—
147C4⋊C8—2682no—
148C4⋊C8—1342no—
149C4⋊C8—1342no—
150C4⋊C8—1342no—
151#104—2682no—
152#104—2682no—
153C22⋊C8—22no—
154C4×C8—22no—
155#95—22no—
156#41—2681no—
157#41—2681no—
158#41—2681no—
159#41—2681no—
160C2×Q8—1341no—
161C2×D4—2681no—
162C2×D4—1341no—
163C2×D4—1342no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2681no—
167C22×C4C2×D26811yescharacteristic
168C22×C4#1176162712yes—
169M16—2681no—
170M16—2681no—
171C2×C8—42no—
172C2×C8—2682no—
173C2×C8—2681no—
174C2×C8—2682no—
175C2×C8—2682no—
176C4⋊C4—1342no—
177C22⋊C4—2682no—
178C42—21no—
179C42—21no—
180C42D53612yes—
181C42#1176161712yes—
182C42D4×D6711yescharacteristic
183C42#1176162812yes—
184C42#1176162911yescharacteristic
185C23—2681no—
186C23#1452402711yescharacteristic
187Q8—2681no—
188Q8—2681no—
189D4—2681no—
190D4—2681no—
191D4—5362no—
192D4—5361no—
193D4—5361no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—2681no—
200C2×C4—5361no—
201C2×C4—5361no—
202C2×C4#1452401811yescharacteristic
203C2×C4#1452402512yes—
204C2×C4#1452403111yescharacteristic
205C2×C4#1452403812yes—
206C2×C4C2×D53611yescharacteristic
207C2×C4C2×#1176161711yescharacteristic
208C2×C4#1452407111yescharacteristic
209C2×C4#1452407611yescharacteristic
210C8—82no—
211C8—5362no—
212C8—5361no—
213C22—22no—
214C22—5361no—
215C22—5362no—
216C22#1686332311yescharacteristic
217C22#1686371011yescharacteristic
218C22#1686371511yescharacteristic
219C4—22no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—5361no—
224C4#1686359211yescharacteristic
225C4#1686360211yescharacteristic
226C4#1686374311yescharacteristic
227C4#1686375611yescharacteristic
228C2—21no—
229C2—21no—
230C2—10721no—
231C2#2256161411yescharacteristic
232C2#2256166112yes—
subgroupG/H
centerC2×C4#14524031
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1274C2
socleC22×C67#397
prime radical#16863946C22
hypercenter#1274D67
hypocenterC67#10627
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
2. #22560363C2semidirect
4. #22558105C2semidirect
5. C67×#1274C2semidirect
121. #1274D67semidirect
122. C67#10627semidirect

Maximal subgroups

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

subgroupindex
#225610412
#225603632
#225588562
#225581052
C67×#12742
#1062767