Finite Group Databasebeta

← #16947589

Subgroups — #16947589

Order 4352.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#14540094C211yesmaximal, characteristic
2#14530965C212yesmaximal, complemented
3#14530272C211yesmaximal, characteristic, complemented
4#14526794C212yesmaximal, complemented
5C17×#1322C211yesmaximal, characteristic, complemented
6#11763359—21no—
7#11763090—22no—
8#11763045—21no—
9#11763045C2212yes—
10#11763005—22no—
11C2×#10587489C2211yescharacteristic
12#11762707—21no—
13#11762635—22no—
14#11762611—22no—
15#11762572—21no—
16#11762572C2211yescharacteristic
17C2×C17×C4⋊C8C2211yescharacteristic
18C17×C22⋊C16C2212yes—
19C2×#63252—41no—
20C22×C8×C17D411yescharacteristic
21C22×C8×C17C2311yescharacteristic
22C2×C42×C17D411yescharacteristic
23#10587616—22no—
24#10587616—41no—
25C2×C4⋊Dic17—21no—
26C2×C4⋊Dic17—21no—
27C2×#63225—41no—
28C2×D136—21no—
29C2×D136D412yes—
30#10587561—41no—
31#10587560—21no—
32C4×D68—22no—
33C4×D68—41no—
34#10587555—21no—
35C2×C16×C17—22no—
36C17×C4⋊C8—22no—
37#10587502—41no—
38#10587492—22no—
39#10587492—22no—
40#10587489—21no—
41#10587489—22no—
42#10587489D412yes—
43C22×C4×C17D811yescharacteristic
44C22×C4×C17D811yescharacteristic
45C22×C4×C17C2×D411yescharacteristic
46C2×C22⋊D17—42no—
47C22×Dic17—41no—
48C22×Dic17—41no—
49#63252—82no—
50#63252—82no—
51C2×D68—22no—
52C2×D68—41no—
53C2×D68—41no—
54C2×C4×D17—42no—
55C2×Dic34—41no—
56C2×C8×C17—21no—
57C2×C8×C17—41no—
58C2×C8×C17—41no—
59C2×C8×C17C2×D412yes—
60C42×C17—21no—
61C42×C17—21no—
62#63227—42no—
63#63227—42no—
64C4⋊Dic17—22no—
65C4⋊Dic17—22no—
66C4⋊Dic17—41no—
67C4⋊Dic17—41no—
68C4⋊Dic17—41no—
69C4⋊Dic17—41no—
70#63225—82no—
71D136—42no—
72D136—81no—
73C16×C17—42no—
74#12288—171nomaximal, Sylow
75C23×C17C2×D811yescharacteristic
76C22×D17—42no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—41no—
83C2×C4×C17D1612yes—
84C2×C4×C17#12011yescharacteristic
85C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
86C22⋊D17—82no—
87C22⋊D17—82no—
88C2×Dic17—42no—
89C2×Dic17—42no—
90C2×Dic17—81no—
91C2×Dic17—81no—
92C2×Dic17—81no—
93C2×Dic17—81no—
94C2×Dic17—81no—
95D68—42no—
96D68—42no—
97D68—82no—
98D68—82no—
99C4×D17—84no—
100Dic34—82no—
101C8×C17—22no—
102C8×C17—41no—
103#3245—171no—
104#2232—172no—
105#2086—171no—
106#1347—172no—
107#1322D1711yescharacteristic, complemented
108C22×C17—21no—
109C22×C17—21no—
110C22×C17#44711yescharacteristic
111C22×C17C2×D1611yescharacteristic
112C22×C17#50911yescharacteristic
113C2×D17—82no—
114C2×D17—82no—
115C2×D17—82no—
116C4×C17—21no—
117C4×C17—22no—
118C4×C17—41no—
119C4×C17#45611yescharacteristic
120C4×C17#45811yescharacteristic
121Dic17—82no—
122Dic17—82no—
123Dic17—161no—
124#532—341no—
125#474—342no—
126#466—172no—
127#466—341no—
128#459—342no—
129C2×#107—171no—
130C2×C4⋊C8C2×D1711yescharacteristic
131#389—341no—
132#366—342no—
133#357—342no—
134C22⋊C16C2×D1712yes—
135#340—171no—
136#340—341no—
137C2×C17—21no—
138C2×C17—21no—
139C2×C17#161011yescharacteristic
140C2×C17#218411yescharacteristic
141C2×C17#219111yescharacteristic
142D17—162no—
143C2×#41—681no—
144C2×D8—172no—
145C2×D8—341no—
146C22×C8D6811yescharacteristic
147C22×C8C22×D1711yescharacteristic
148C4⋊Q8—341no—
149#127—341no—
150#124—681no—
151#121—342no—
152#121—681no—
153C4×D4—342no—
154C4×D4—681no—
155C2×C4⋊C4—341no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4—681no—
158C2×C42D6811yescharacteristic
159C2×C16—22no—
160#107—172no—
161#107—341no—
162#107—342no—
163C4⋊C8—22no—
164#102—342no—
165#102—342no—
166#95—681no—
167C17#1228811yescharacteristic, complemented, Sylow
168#41—1362no—
169#41—1362no—
170C2×Q8—681no—
171C2×D4—342no—
172C2×D4—681no—
173C2×D4—681no—
174C2×D4—682no—
175C22×C4—682no—
176C22×C4—681no—
177C22×C4—681no—
178C22×C4D13611yescharacteristic
179C22×C4D13611yescharacteristic
180C22×C4C2×D6811yescharacteristic
181D8—682no—
182D8—1361no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D1712yes—
187C4⋊C4—342no—
188C4⋊C4—342no—
189C4⋊C4—681no—
190C4⋊C4—681no—
191C4⋊C4—681no—
192C4⋊C4—681no—
193C4⋊C4—1362no—
194C22⋊C4—682no—
195C22⋊C4—682no—
196C42—21no—
197C42—21no—
198C16—42no—
199C23—682no—
200C23C2×D13611yescharacteristic
201Q8—1362no—
202D4—682no—
203D4—682no—
204D4—1362no—
205D4—1362no—
206D4—1362no—
207D4—1362no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—682no—
215C2×C4—682no—
216C2×C4—1364no—
217C2×C4—1361no—
218C2×C4—1361no—
219C2×C4—1361no—
220C2×C4—1361no—
221C2×C4—1361no—
222C2×C4D27212yes—
223C2×C4#1058756811yescharacteristic
224C2×C4#1058759411yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—1362no—
230C22—1362no—
231C22—1362no—
232C22#1176294511yescharacteristic
233C22C2×D27211yescharacteristic
234C22#1176324711yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—2721no—
239C4—1362no—
240C4—1362no—
241C4#1176299411yescharacteristic
242C4#1176300011yescharacteristic
243C2—21no—
244C2—21no—
245C2—2722no—
246C2#1452778911yescharacteristic
247C2#1453069811yescharacteristic
248C2#1453073711yescharacteristic
subgroupG/H
centerC22#11762945
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC17×#1322C2
socleC22×C17#447
prime radical#14540094C2
hypercenter#1322D17
hypocenterC17#12288
LCSG ▷ C22×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #14530965C2semidirect
3. #14530272C2semidirect
4. #14526794C2semidirect
5. C17×#1322C2semidirect
107. #1322D17semidirect
167. C17#12288semidirect

Maximal subgroups

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

subgroupindex
#145400942
#145309652
#145302722
#145267942
C17×#13222
#1228817