Finite Group Databasebeta

← #30760332

Subgroups — #30760332

Order 17152.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#22568481C211yesmaximal, characteristic
2#22560378C211yesmaximal, characteristic
3#22560343C212yesmaximal
4#22558091C212yesmaximal
5C67×#1295C211yesmaximal, characteristic
6C2×#14523976—21no—
7C2×#14523974—21no—
8C2×#14523954C2211yescharacteristic
9#16863338—22no—
10#16863335—21no—
11#16863317—22no—
12#16863316—21no—
13#16863266C2211yescharacteristic
14#16863262C2212yes—
15#16863159—22no—
16#16863159—22no—
17C2×#97×C67C2212yes—
18#382×C67C2211yescharacteristic
19C22×Dic134—21no—
20C22×C67⋊C8—21no—
21C2×C4⋊Dic67—21no—
22C2×C4⋊Dic67—22no—
23C2×C4⋊Dic67—21no—
24C2×C4⋊Dic67D412yes—
25C2×C4⋊Dic67D412yes—
26C2×Dic67⋊C4—22no—
27#14523976—42no—
28#14523974—42no—
29#14523954—21no—
30#14523954Q812yescomplemented
31C67⋊#95—21no—
32C67⋊#95—22no—
33C67⋊#95—21no—
34C67⋊#95—22no—
35C67⋊#95—22no—
36C22×C8×C67—22no—
37C2×C4⋊C4×C67D411yescharacteristic
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C42×C67C2311yescharacteristic
40#97×C67—22no—
41#97×C67Q814yescomplemented
42#95×C67—22no—
43C2×Dic134—81no—
44C2×Dic134—82no—
45C22×Dic67—22no—
46C22×Dic67—22no—
47C22×Dic67—41no—
48C2×C67⋊C8—22no—
49C2×C67⋊C8—22no—
50C2×C67⋊C8—22no—
51C4⋊Dic67—42no—
52C4⋊Dic67—42no—
53C4⋊Dic67—41no—
54C4⋊Dic67—41no—
55C4⋊Dic67—44no—
56C4⋊Dic67—81no—
57Dic67⋊C4—82no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67C2×D412yes—
61C22×C4×C67C2×D411yescharacteristic
62C2×C8×C67—24no—
63C2×C8×C67—42no—
64C2×C8×C67—42no—
65C4⋊C4×C67—41no—
66C4⋊C4×C67—41no—
67C42×C67C2×Q811yescharacteristic
68C42×C67C2×Q812yes—
69C42×C67#4111yescharacteristic
70C23×C67#12011yescharacteristic
71C2×C4×C67—21no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—21no—
75C2×C4×C67—41no—
76C2×C4×C67—41no—
77C2×C4×C67—42no—
78C2×C4×C67—42no—
79C2×C4×C67C22⋊Q812yes—
80C2×C4×C67C22⋊Q811yescharacteristic
81C2×C4×C67C4⋊Q812yes—
82C2×C4×C67C4⋊Q811yescharacteristic
83C2×Dic67—42no—
84C2×Dic67—42no—
85C2×Dic67—42no—
86C2×Dic67—42no—
87C2×Dic67—42no—
88C2×Dic67—42no—
89C2×Dic67—81no—
90C2×Dic67—82no—
91Dic134—84no—
92C8×C67—44no—
93C67⋊C8—24no—
94C22×C67#39511yescharacteristic
95C22×C67#45311yescharacteristic
96C22×C67#45412yes—
97C22×C67#45512yes—
98C22×C67#45611yescharacteristic
99C4×C67—21no—
100C4×C67—21no—
101C4×C67—22no—
102C4×C67—22no—
103C4×C67—81no—
104C4×C67—81no—
105Dic67—82no—
106Dic67—82no—
107Dic67—161no—
108#10584—671nomaximal, Sylow
109C2×C67#165711yescharacteristic
110C2×C67#165911yescharacteristic
111C2×C67#166412yes—
112C2×C67#206011yescharacteristic
113C2×C67#206112yes—
114#2844—671no—
115#1847—671no—
116#1836—672no—
117#1295—672no—
118#1295D6711yescharacteristic
119C67#1058411yescharacteristic, complemented, Sylow
120C2×#107—1341no—
121C2×#106—1341no—
122C2×#97—671no—
123C2×#97C2×D6712yes—
124#400—1342no—
125#400—1341no—
126#395—1342no—
127#395—1341no—
128#384—671no—
129#382—672no—
130#382C2×D6711yescharacteristic
131#340—1342no—
132#340—1342no—
133C22×Q8—1341no—
134C22×C8—22no—
135C22×C8—1341no—
136C2×C4⋊C4—672no—
137C2×C4⋊C4—672no—
138C2×C4⋊C4—1342no—
139C2×C4⋊C4—1341no—
140C2×C4⋊C4—1342no—
141C2×C4⋊C4—1341no—
142C2×C4⋊C4D26811yescharacteristic
143C2×C4⋊C4D26811yescharacteristic
144C2×C42C22×D6711yescharacteristic
145#107—2682no—
146#106—2682no—
147#97—22no—
148#97—672no—
149#97—1341no—
150#97Dic13414yescomplemented
151#95—22no—
152#95—1342no—
153#95—1341no—
154#95—1341no—
155#95—1342no—
156#95—1342no—
157C2×Q8—5361no—
158C2×Q8—5362no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—2681no—
162C22×C4—1342no—
163C22×C4—1342no—
164C22×C4D4×D6712yes—
165C22×C4C2×D26811yescharacteristic
166C2×C8—24no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—1342no—
170C2×C8—1342no—
171C2×C8—1342no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—2682no—
175C4⋊C4—2682no—
176C4⋊C4—2681no—
177C4⋊C4—2681no—
178C4⋊C4—2684no—
179C4⋊C4—5362no—
180C4⋊C4—5361no—
181C42Q8×D6711yescharacteristic
182C42C2×Dic13412yes—
183C42#1176162711yescharacteristic
184C23#1452401311yescharacteristic
185Q8—5364no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—2682no—
195C2×C4—2682no—
196C2×C4—2682no—
197C2×C4—2682no—
198C2×C4—2682no—
199C2×C4—2682no—
200C2×C4—5361no—
201C2×C4—5362no—
202C2×C4#1452402012yes—
203C2×C4#1452402311yescharacteristic
204C2×C4#1452404212yes—
205C2×C4#1452404311yescharacteristic
206C8—44no—
207C8—1344no—
208C22#1686331711yescharacteristic
209C22#1686356611yescharacteristic
210C22#1686357012yes—
211C22#1686357512yes—
212C22#1686358111yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—5362no—
220C4—5362no—
221C4—10721no—
222C2#2255950811yescharacteristic
223C2#2255951611yescharacteristic
224C2#2255954512yes—
225C2#2256155511yescharacteristic
226C2#2256155812yes—
subgroupG/H
centerC23#14524013
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1295C2
socle, prime radicalC23×C67#120
hypercenter#1295D67
hypocenterC67#10584
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
30. #14523954Q8semidirect
41. #97×C67Q8semidirect
119. C67#10584semidirect
150. #97Dic134semidirect

Maximal subgroups

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

subgroupindex
#225684812
#225603782
#225603432
#225580912
C67×#12952
#1058467