Finite Group Databasebeta

← #30760272

Subgroups — #30760272

Order 17152.

494 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 137 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#22565238C211yesmaximal, characteristic
2#22560973C212yesmaximal
3#22560365C211yesmaximal, characteristic
4#22558048C212yesmaximal
5C67×#1296C211yesmaximal, characteristic
6C2×#14524042—22no—
7C2×#14523975—22no—
8C2×#14523974—22no—
9C2×C4×C67⋊C8C2211yescharacteristic
10#16863316—22no—
11#16863292—22no—
12#16863266C2211yescharacteristic
13C4×C4⋊Dic67C2212yes—
14#16863159—22no—
15C2×C4⋊C8×C67C2212yes—
16#384×C67C2211yescharacteristic
17C22×Dic134—22no—
18C22×C67⋊C8—21no—
19#14524042—44no—
20#14524042—44no—
21C2×C4⋊Dic67—22no—
22C2×C4⋊Dic67—21no—
23C2×C4⋊Dic67D412yes—
24C2×C4⋊Dic67D412yes—
25C2×Dic67⋊C4—24no—
26C2×C4×Dic67—22no—
27#14523975—44no—
28#14523974—44no—
29C4×C67⋊C8—21no—
30C4×C67⋊C8Q812yescomplemented
31C67⋊#95—22no—
32C67⋊#95—22no—
33C22×C8×C67—22no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67D412yes—
36C2×C42×C67C2311yescharacteristic
37C4⋊C8×C67—22no—
38C4⋊C8×C67Q814yescomplemented
39C2×Dic134—48no—
40C2×Dic134—82no—
41C22×Dic67—22no—
42C22×Dic67—22no—
43C22×Dic67—41no—
44C2×C67⋊C8—22no—
45C2×C67⋊C8—22no—
46C2×C67⋊C8—22no—
47C4⋊Dic67—22no—
48C4⋊Dic67—22no—
49C4⋊Dic67—42no—
50C4⋊Dic67—42no—
51C4⋊Dic67—42no—
52C4⋊Dic67SD1614yes—
53C4⋊Dic67SD1614yes—
54Dic67⋊C4—48no—
55C4×Dic67—42no—
56C4×Dic67—42no—
57C22×C4×C67—22no—
58C22×C4×C67C2×D412yes—
59C22×C4×C67C2×D411yescharacteristic
60C2×C8×C67—24no—
61C2×C8×C67—42no—
62C2×C8×C67—42no—
63C4⋊C4×C67—42no—
64C4⋊C4×C67—42no—
65C4⋊C4×C67—42no—
66C42×C67C2×Q811yescharacteristic
67C42×C67C2×Q812yes—
68C42×C67#4111yescharacteristic
69C23×C67#12011yescharacteristic
70C2×C4×C67—21no—
71C2×C4×C67—21no—
72C2×C4×C67—44no—
73C2×C4×C67—42no—
74C2×C4×C67C22⋊Q812yes—
75C2×C4×C67C22⋊Q811yescharacteristic
76C2×C4×C67C4⋊Q812yes—
77C2×C4×C67C4⋊Q811yescharacteristic
78C2×C4×C67C2×SD1612yes—
79C2×C4×C67C2×SD1612yes—
80C2×C4×C67AGL(1,ℤ/8)12yes—
81C2×C4×C67#13712yes—
82C2×Dic67—24no—
83C2×Dic67—24no—
84C2×Dic67—42no—
85C2×Dic67—42no—
86C2×Dic67—42no—
87C2×Dic67—42no—
88C2×Dic67—82no—
89C2×Dic67—81no—
90Dic134—88no—
91C8×C67—44no—
92C67⋊C8—24no—
93C22×C67#39511yescharacteristic
94C22×C67#44812yes—
95C22×C67#45012yes—
96C22×C67#45311yescharacteristic
97C22×C67#45611yescharacteristic
98C4×C67—21no—
99C4×C67—21no—
100C4×C67—82no—
101C4×C67#47512yes—
102C4×C67#47612yes—
103C4×C67#49812yes—
104C4×C67#50112yes—
105Dic67—44no—
106Dic67—44no—
107Dic67—161no—
108#10574—671nomaximal, Sylow
109C2×C67#162311yescharacteristic
110C2×C67#162911yescharacteristic
111C2×C67#165112yes—
112C2×C67#205812yes—
113C2×C67#206011yescharacteristic
114#2551—671no—
115#1948—672no—
116#1844—671no—
117#1296D6711yescharacteristic
118#1279—672no—
119C67#1057411yescharacteristic, complemented, Sylow
120C2×C4⋊Q8—1342no—
121C2×#106—1342no—
122C2×#106—1342no—
123C2×C4⋊C8C2×D6712yes—
124C2×C4×C8—671no—
125#395—1342no—
126#389—1342no—
127#384—671no—
128#384C2×D6711yescharacteristic
129C4×C4⋊C4—672no—
130#340—1342no—
131C22×Q8—1342no—
132C22×C8—22no—
133C22×C8—1341no—
134C4⋊Q8—2684no—
135C4⋊Q8—2684no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—672no—
138C2×C4⋊C4—672no—
139C2×C4⋊C4—1344no—
140C2×C4⋊C4—1342no—
141C2×C4⋊C4—1341no—
142C2×C4⋊C4D26812yes—
143C2×C42—1342no—
144C2×C42C22×D6711yescharacteristic
145#106—2684no—
146#106—2684no—
147C4⋊C8—22no—
148C4⋊C8Dic13414yescomplemented
149C4×C8—672no—
150C4×C8—1341no—
151#95—1342no—
152#95—1342no—
153C2×Q8—2688no—
154C2×Q8—5362no—
155C22×C4—22no—
156C22×C4—2681no—
157C22×C4—1342no—
158C22×C4—1342no—
159C22×C4D4×D6712yes—
160C22×C4C2×D26811yescharacteristic
161C2×C8—24no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—1342no—
165C2×C8—1342no—
166C2×C8—1342no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—2682no—
171C4⋊C4—2688no—
172C4⋊C4—2682no—
173C4⋊C4—2682no—
174C4⋊C4—674no—
175C4⋊C4—674no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C42—2682no—
179C42—2682no—
180C42Q8×D6711yescharacteristic
181C42C2×Dic13412yes—
182C42#1176162711yescharacteristic
183C23#1452401311yescharacteristic
184Q8—5368no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—2682no—
190C2×C4—2682no—
191C2×C4—2682no—
192C2×C4—2682no—
193C2×C4—5362no—
194C2×C4—5361no—
195C2×C4—1344no—
196C2×C4—1344no—
197C2×C4#1452402012yes—
198C2×C4#1452402311yescharacteristic
199C2×C4#1452404212yes—
200C2×C4#1452404311yescharacteristic
201C2×C4SD16×D6712yes—
202C2×C4SD16×D6712yes—
203C2×C4#1452406712yes—
204C2×C4#1452407512yes—
205C8—44no—
206C8—1344no—
207C22#1686331711yescharacteristic
208C22#1686353612yes—
209C22#1686354812yes—
210C22#1686356611yescharacteristic
211C22#1686358111yescharacteristic
212C4—21no—
213C4—21no—
214C4—82no—
215C4—2684no—
216C4—2684no—
217C4—10721no—
218C4#1686367912yes—
219C4#1686368412yes—
220C4#1686379312yes—
221C4#1686380412yes—
222C2#2255928811yescharacteristic
223C2#2255932611yescharacteristic
224C2#2255947312yes—
225C2#2256154112yes—
226C2#2256155511yescharacteristic
subgroupG/H
centerC23#14524013
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1296C2
socle, prime radicalC23×C67#120
hypercenter#1296D67
hypocenterC67#10574
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
30. C4×C67⋊C8Q8semidirect
38. C4⋊C8×C67Q8semidirect
119. C67#10574semidirect
148. C4⋊C8Dic134semidirect

Maximal subgroups

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

subgroupindex
#225652382
#225609732
#225603652
#225580482
C67×#12962
#1057467