Finite Group Databasebeta

← #30749414

Subgroups — #30749414

Order 17152.

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

normal subgroups only

This group has 378 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#22563174C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558110C212yesmaximal
4#22558069C212yesmaximal
5C67×#1801C211yesmaximal, characteristic
6#16863449C2212yes—
7#16863303—21no—
8#16863282—22no—
9#16863264C412yes—
10#16863262C2211yescharacteristic
11C4⋊C4×Dic67—21no—
12C4×C4⋊Dic67C2211yescharacteristic
13#16863242C412yes—
14#16863122—22no—
15C2×C4×Q8×C67C2211yescharacteristic
16#423×C67C2212yes—
17C2×#101×C67—21no—
18#343×C67—21no—
19C2×C4⋊Dic67—21no—
20C2×C4⋊Dic67—21no—
21C2×C4×Dic67—22no—
22C2×C4×Dic67—21no—
23C4⋊(C67⋊C8)—22no—
24C22⋊(C67⋊C8)—22no—
25C4×C67⋊C8—22no—
26C67⋊#95—22no—
27C67⋊#95—22no—
28C67⋊#95—22no—
29C67⋊#95—21no—
30C67⋊#95C2×C411yescharacteristic
31C67⋊#95C2×C412yes—
32C67⋊#95C2×C411yescharacteristic
33C22×Q8×C67D411yescharacteristic
34C2×M16×C67—22no—
35C4×Q8×C67—22no—
36C4×Q8×C67—22no—
37C2×C4⋊C4×C67D412yes—
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C42×C67C2×C412yes—
40C2×C42×C67C2311yescharacteristic
41C4⋊C8×C67—22no—
42C4⋊C8×C67—22no—
43#101×C67—42no—
44C22×Dic67—21no—
45C22×Dic67—22no—
46C22×Dic67—21no—
47C2×C67⋊C8—42no—
48C4⋊Dic67—42no—
49C4⋊Dic67—42no—
50C4×Dic67—44no—
51C4×Dic67—41no—
52C4×Dic67—41no—
53C2×Q8×C67—22no—
54C2×Q8×C67—22no—
55C2×Q8×C67—21no—
56C2×Q8×C67—21no—
57C22×C4×C67C22⋊C412yes—
58C22×C4×C67C22⋊C412yes—
59C22×C4×C67C22×C411yescharacteristic
60C22×C4×C67C2×D411yescharacteristic
61C22×C4×C67C2×D411yescharacteristic
62M16×C67—42no—
63M16×C67—42no—
64C2×C8×C67—42no—
65C4⋊C4×C67—22no—
66C4⋊C4×C67—24no—
67C42×C67—22no—
68C42×C67—22no—
69C42×C67#4112yes—
70C42×C67#4112yes—
71C23×C67C2×C22⋊C411yescharacteristic
72Q8×C67—42no—
73Q8×C67—42no—
74C2×C4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—41no—
79C2×C4×C67—41no—
80C2×C4×C67—42no—
81C2×C4×C67#11711yescharacteristic
82C2×C4×C67#11711yescharacteristic
83C2×C4×C67#12312yes—
84C2×C4×C67#12312yes—
85C2×C4×C67#13512yes—
86C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C67#13711yescharacteristic
88C2×Dic67—22no—
89C2×Dic67—41no—
90C2×Dic67—41no—
91C2×Dic67—42no—
92C2×Dic67—41no—
93C2×Dic67—41no—
94C2×Dic67—42no—
95C2×Dic67—42no—
96C2×Dic67—41no—
97C8×C67—82no—
98C67⋊C8—82no—
99C22×C67—21no—
100C22×C67—21no—
101C22×C67#38111yescharacteristic
102C22×C67#41611yescharacteristic
103C22×C67#41711yescharacteristic
104C4×C67—22no—
105C4×C67—21no—
106C4×C67—21no—
107C4×C67—42no—
108C4×C67—42no—
109C4×C67#48212yes—
110C4×C67#48512yes—
111Dic67—42no—
112Dic67—81no—
113Dic67—82no—
114#8768—671nomaximal, Sylow
115C2×C67—21no—
116C2×C67—21no—
117C2×C67#182511yescharacteristic
118C2×C67#182711yescharacteristic
119C2×C67#183011yescharacteristic
120#2356—671no—
121#1801D6711yescharacteristic
122#1461—671no—
123#1303—672no—
124#1286—672no—
125C67#876811yescharacteristic, complemented, Sylow
126C2×C4×Q8C2×D6711yescharacteristic
127#431—672no—
128#423C2×D6712yes—
129C2×#101—21no—
130#391—1341no—
131#387—1342no—
132#383—672no—
133#382—671no—
134C4×C4⋊C4—671no—
135C4×C4⋊C4—1341no—
136#376—672no—
137#343—21no—
138#324—1342no—
139C22×Q8C22⋊D6711yescharacteristic
140C2×M16—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—1341no—
144C2×C4⋊C4—1341no—
145C2×C4⋊C4D26812yes—
146C2×C4⋊C4C22⋊D6711yescharacteristic
147C2×C42—1342no—
148C2×C42—1341no—
149C2×C42C4×D6712yes—
150C2×C42C22×D6711yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—1342no—
154#101—42no—
155C22⋊C8—1342no—
156C4×C8—1342no—
157#95—671no—
158#95—672no—
159#95—671no—
160#95—1342no—
161#95—1342no—
162#95—1342no—
163#95—1341no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—1342no—
169C22×C4—1341no—
170C22×C4—1341no—
171C22×C4(C2×D67)⋊C412yes—
172C22×C4(C2×D67)⋊C412yes—
173C22×C4C2×C4×D6711yescharacteristic
174C22×C4C2×C22⋊D6711yescharacteristic
175C22×C4C2×D26811yescharacteristic
176M16—42no—
177M16—42no—
178C2×C8—42no—
179C2×C8—2682no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—2682no—
183C4⋊C4—2682no—
184C42—22no—
185C42—22no—
186C42—2684no—
187C42—2681no—
188C42—2681no—
189C42#1176162712yes—
190C42#1176162712yes—
191C23C2×(C2×D67)⋊C411yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—2681no—
202C2×C4—2681no—
203C2×C4—2682no—
204C2×C4—2681no—
205C2×C4—2681no—
206C2×C4—2682no—
207C2×C4—2682no—
208C2×C4—2681no—
209C2×C4—1342no—
210C2×C4#1452399911yescharacteristic
211C2×C4#1452399911yescharacteristic
212C2×C4#1452402412yes—
213C2×C4#1452402712yes—
214C2×C4#1452406312yes—
215C2×C4#1452406911yescharacteristic
216C2×C4#1452407411yescharacteristic
217C8—82no—
218C8—5362no—
219C22—21no—
220C22—21no—
221C22#1686325711yescharacteristic
222C22#1686339011yescharacteristic
223C22#1686339511yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—2682no—
230C4—5361no—
231C4—5362no—
232C4#1686371712yes—
233C4#1686373412yes—
234C2—21no—
235C2—21no—
236C2#2256030111yescharacteristic
237C2#2256031411yescharacteristic
238C2#2256032111yescharacteristic
subgroupG/H
centerC22#16863257
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1801C2
socleC22×C67#381
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#1801D67
hypocenterC67#8768
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
125. C67#8768semidirect

Maximal subgroups

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

subgroupindex
#225631742
#225585332
#225581102
#225580692
C67×#18012
#876867