Finite Group Databasebeta

← #30748346

Subgroups — #30748346

Order 17152.

690 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 131 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#22560292C211yesmaximal, characteristic
2#22560286C211yesmaximal, characteristic
3#22558742C214yesmaximal
4C2×#541×C67C211yesmaximal, characteristic
5#16863360C2212yes—
6#16863259C2212yes—
7#16863165—22no—
8#16863163—22no—
9#16863133—28no—
10#541×C67—28no—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×#126×C67—22no—
13C2×#125×C67—22no—
14C2×#125×C67C411yescharacteristic, complemented
15C2×C22⋊Q8×C67C2212yes—
16C2×C4×Q8×C67C412yescomplemented
17C2×#117×C67C2211yescharacteristic
18C2×#11761612—22no—
19C2×C22⋊Dic67—22no—
20C22⋊(C67⋊C8)—24no—
21C67⋊#95—24no—
22C22×Q8×C67C2×C412yes—
23C23×C4×C67C2311yescharacteristic
24C4⋊Q8×C67—24no—
25#126×C67—216no—
26#125×C67—216no—
27#125×C67—24no—
28C22⋊Q8×C67—24no—
29C22⋊Q8×C67D418yes—
30C4×Q8×C67—24no—
31C4×Q8×C67—24no—
32#117×C67—24no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67C2×C412yes—
38C2×C4⋊C4×C67D412yes—
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67D412yes—
41C2×C42×C67—22no—
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C22×Dic67—42no—
45#11761612—44no—
46C2×C67⋊C8—42no—
47C22⋊Dic67—44no—
48C24×C67C2×D411yescharacteristic
49C2×Q8×C67—22no—
50C2×Q8×C67—44no—
51C2×Q8×C67C22⋊C414yes—
52C22×C4×C67—22no—
53C22×C4×C67—22no—
54C22×C4×C67—42no—
55C22×C4×C67C22⋊C412yes—
56C22×C4×C67C22⋊C412yes—
57C22×C4×C67C22×C411yescharacteristic
58C22×C4×C67C2×D411yescharacteristic
59C22×C4×C67C2×D414yes—
60C4⋊C4×C67—28no—
61C4⋊C4×C67—28no—
62C4⋊C4×C67—28no—
63C4⋊C4×C67—22no—
64C4⋊C4×C67—28no—
65C4⋊C4×C67—24no—
66C4⋊C4×C67C22⋊C414yes—
67C22⋊C4×C67—28no—
68C22⋊C4×C67—24no—
69C42×C67—28no—
70C42×C67—22no—
71C42×C67—22no—
72C23×C67—22no—
73C23×C67—42no—
74C23×C67C2×C22⋊C411yescharacteristic
75C23×C67#12012yes—
76Q8×C67—44no—
77Q8×C67—44no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—24no—
83C2×C4×C67—24no—
84C2×C4×C67—44no—
85C2×C4×C67—44no—
86C2×C4×C67—44no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67C2×C22⋊C412yes—
90C2×C4×C67#12012yes—
91C2×Dic67—82no—
92C2×Dic67—84no—
93C67⋊C8—82no—
94C22×C67—22no—
95C22×C67—24no—
96C22×C67—22no—
97C22×C67—82no—
98C22×C67#37911yescharacteristic
99C22×C67#41012yes—
100C4×C67—22no—
101C4×C67—22no—
102C4×C67—42no—
103C4×C67—42no—
104C4×C67—44no—
105C4×C67—44no—
106Dic67—162no—
107#8521—671nomaximal, Sylow
108C2×C67—22no—
109C2×C67—42no—
110C2×C67#178211yescharacteristic
111C2×C67#179612yes—
112C2×#541D6711yescharacteristic
113#1822—671no—
114#1820—671no—
115#1511—674no—
116C67#852111yescharacteristic, complemented, Sylow
117#541—28no—
118C2×C4⋊Q8Dic6711yescharacteristic, complemented
119C2×#126—22no—
120C2×#125—22no—
121C2×#125Dic6711yescharacteristic, complemented
122C2×C22⋊Q8C2×D6712yes—
123C2×C4×Q8Dic6712yescomplemented
124C2×#117C2×D6711yescharacteristic
125#407—672no—
126#381—672no—
127#343—1342no—
128#342—1342no—
129#328—1348no—
130C22×Q8C2×Dic6712yes—
131C23×C4C22×D6711yescharacteristic
132C2×M16—1342no—
133C4⋊Q8—24no—
134#126—216no—
135#125—216no—
136#125—24no—
137C22⋊Q8—24no—
138C22⋊Q8C22⋊D6718yes—
139C4×Q8—24no—
140C4×Q8—24no—
141#117—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C2×Dic6712yes—
147C2×C4⋊C4C22⋊D6712yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—1342no—
150C2×C22⋊C4C22⋊D6712yes—
151C2×C42—22no—
152C2×C42C2×Dic6711yescharacteristic
153C2×C42C2×Dic6711yescharacteristic
154C22⋊C8—1344no—
155#95—1344no—
156C24C2×C22⋊D6711yescharacteristic
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊Dic6714yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—2682no—
164C22×C4C22⋊Dic6712yes—
165C22×C4C22⋊Dic6712yes—
166C22×C4C22×Dic6711yescharacteristic
167C22×C4C2×C22⋊D6711yescharacteristic
168C22×C4C2×C22⋊D6714yes—
169M16—2684no—
170C2×C8—2682no—
171C4⋊C4—28no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—22no—
175C4⋊C4—28no—
176C4⋊C4—24no—
177C4⋊C4C22⋊Dic6714yes—
178C22⋊C4—28no—
179C22⋊C4—24no—
180C22⋊C4—2684no—
181C42—28no—
182C42—22no—
183C42—22no—
184C23—22no—
185C23—42no—
186C23C2×C22⋊Dic6711yescharacteristic
187C23#1452401212yes—
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—5362no—
202C2×C4—5364no—
203C2×C4C2×C22⋊Dic6712yes—
204C2×C4#1452401212yes—
205C8—5362no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—82no—
210C22#1686325211yescharacteristic
211C22#1686337012yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—44no—
216C4—42no—
217C4—42no—
218C4—10722no—
219C2—22no—
220C2—42no—
221C2#2256012711yescharacteristic
222C2#2256018412yes—
subgroupG/H
centerC22#16863252
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC2×#541×C67C2
socleC22×C67#379
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenterC2×#541D67
hypocenterC67#8521
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C67C4semidirect
14. C2×#125×C67C4semidirect
16. C2×C4×Q8×C67C4semidirect
116. C67#8521semidirect
118. C2×C4⋊Q8Dic67semidirect
121. C2×#125Dic67semidirect
123. C2×C4×Q8Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602922
#225602862
#225587422
C2×#541×C672
#852167