Finite Group Databasebeta

← #30758961

Subgroups — #30758961

Order 17152.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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#22561190C211yesmaximal, characteristic
2#22558707C212yesmaximal
3#22558112C212yesmaximal
4#22558038C211yesmaximal, characteristic
5C67×#2399C211yesmaximal, characteristic
6#16863953C2211yescharacteristic
7#16863449C2212yes—
8#16863420C2211yescharacteristic
9#16863156—21no—
10#16863156—21no—
11#16863132—22no—
12#16863132—22no—
13C2×C4×Q8×C67—21no—
14#398×C67—22no—
15#395×C67—22no—
16C4×C4⋊C4×C67—21no—
17C4×C4⋊C4×C67C2212yes—
18#376×C67C2211yescharacteristic
19C2×#11761612—21no—
20#14524043Q812yescomplemented
21#14524035—21no—
22#14524022—22no—
23C4×Dic134—22no—
24#14524002—21no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)—22no—
27C22⋊(C67⋊C8)—22no—
28C4×C67⋊C8—22no—
29C22×Q8×C67—21no—
30C4×Q8×C67—42no—
31C4×Q8×C67—42no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67—22no—
36C2×C42×C67—22no—
37C2×C42×C67D411yescharacteristic
38C2×C42×C67D411yescharacteristic
39C2×C42×C67C2311yescharacteristic
40#95×C67—21no—
41#95×C67—21no—
42#95×C67—22no—
43#95×C67D412yes—
44#95×C67D412yes—
45C2×Dic134—22no—
46#11761612—41no—
47#11761612—41no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—41no—
50C4⋊Dic67—22no—
51C4⋊Dic67—22no—
52Dic67⋊C4—42no—
53C22⋊Dic67—41no—
54C4×Dic67—41no—
55C2×Q8×C67—42no—
56C2×Q8×C67—44no—
57C22×C4×C67—22no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67C2×D412yes—
62C22×C4×C67C2×D411yescharacteristic
63C4⋊C4×C67—44no—
64C4⋊C4×C67—44no—
65C4⋊C4×C67—44no—
66C4⋊C4×C67—42no—
67C42×C67—21no—
68C42×C67—21no—
69C42×C67—42no—
70C42×C67—42no—
71C42×C67SD1612yes—
72C42×C67Q1612yes—
73C42×C67C2×Q811yescharacteristic
74C42×C67#4112yes—
75C42×C67#4111yescharacteristic
76C23×C67#12011yescharacteristic
77Q8×C67—48no—
78C2×C4×C67—24no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—42no—
82C2×C4×C67—42no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67C22⋊Q812yes—
90C2×C4×C67C22⋊Q811yescharacteristic
91C2×C4×C67#12412yes—
92C2×C4×C67#12411yescharacteristic
93C2×C4×C67C2×SD1611yescharacteristic
94C2×C4×C67C2×Q1611yescharacteristic
95C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C67#13711yescharacteristic
97C2×Dic67—42no—
98Dic134—82no—
99C67⋊C8—82no—
100C67⋊C8—81no—
101C22×C67—22no—
102C22×C67#39311yescharacteristic
103C22×C67#45011yescharacteristic
104C22×C67#45111yescharacteristic
105C4×C67—22no—
106C4×C67—22no—
107C4×C67—42no—
108C4×C67—44no—
109C4×C67—42no—
110C4×C67—82no—
111C4×C67#47511yescharacteristic
112C4×C67#47711yescharacteristic
113C4×C67#48711yescharacteristic
114C4×C67#48911yescharacteristic
115Dic67—82no—
116#10371—671nomaximal, Sylow
117C2×C67—21no—
118C2×C67—21no—
119C2×C67#202411yescharacteristic
120C2×C67#203112yes—
121#2399D6711yescharacteristic
122#1995—671no—
123#1503—672no—
124#1303—672no—
125#1274—671no—
126C67#1037111yescharacteristic, complemented, Sylow
127#533—671no—
128C2×C4×Q8—21no—
129#431—672no—
130#423—671no—
131#398—22no—
132#395—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×D6712yes—
135#376C2×D6711yescharacteristic
136#339—1341no—
137#339—1341no—
138#328—1342no—
139#328—1342no—
140C22×Q8—21no—
141C2×M16—1341no—
142C4⋊Q8—672no—
143#125—1341no—
144C22⋊Q8—1342no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—1342no—
148#117—1341no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C22⋊D6711yescharacteristic
155C2×C42C22⋊D6711yescharacteristic
156C2×C42C22×D6711yescharacteristic
157C4⋊C8—1342no—
158C4⋊C8—1342no—
159C22⋊C8—1342no—
160C4×C8—1342no—
161#95—21no—
162#95—21no—
163#95—22no—
164#95C22⋊D6712yes—
165#95C22⋊D6712yes—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—1342no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4C2×C22⋊D6712yes—
174C22×C4C2×C22⋊D6711yescharacteristic
175M16—2681no—
176M16—2681no—
177C2×C8—2682no—
178C2×C8—2681no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—2682no—
184C4⋊C4—1342no—
185C4⋊C4—1342no—
186C22⋊C4—2681no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42—2681no—
192C42#1176161612yes—
193C42#1176161812yes—
194C42Q8×D6711yescharacteristic
195C42#1176162911yescharacteristic
196C42#1176162912yes—
197C23#1452401211yescharacteristic
198Q8—48no—
199Q8—5362no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—2682no—
212C2×C4#1452401912yes—
213C2×C4#1452401911yescharacteristic
214C2×C4#1452403212yes—
215C2×C4#1452403211yescharacteristic
216C2×C4C2×#1176161611yescharacteristic
217C2×C4C2×#1176161811yescharacteristic
218C2×C4#1452406911yescharacteristic
219C2×C4#1452407411yescharacteristic
220C8—5362no—
221C8—5361no—
222C22—22no—
223C22#1686330711yescharacteristic
224C22#1686354711yescharacteristic
225C22#1686355211yescharacteristic
226C4—22no—
227C4—22no—
228C4—44no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4—5362no—
233C4#1686367811yescharacteristic
234C4#1686368811yescharacteristic
235C4#1686374811yescharacteristic
236C4#1686375811yescharacteristic
237C2—21no—
238C2—21no—
239C2#2256134711yescharacteristic
240C2#2256137812yes—
subgroupG/H
centerC2×C4#14524032
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2399C2
socleC22×C67#393
prime radicalC23×C67#120
hypercenter#2399D67
hypocenterC67#10371
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
20. #14524043Q8semidirect
126. C67#10371semidirect

Maximal subgroups

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

subgroupindex
#225611902
#225587072
#225581122
#225580382
C67×#23992
#1037167