Finite Group Databasebeta

← #16946919

Subgroups — #16946919

Order 4352.

372 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 63 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#14530013C211yesmaximal, characteristic
2#14527752C212yesmaximal
3#14526814C212yesmaximal, complemented
4#14526728C211yesmaximal, characteristic, complemented
5C17×#2844C211yesmaximal, characteristic, complemented
6#11763390C2211yescharacteristic
7#11762948—22no—
8C2×#10587476—21no—
9#11762808—21no—
10#11762578C2211yescharacteristic
11#11762542—22no—
12#11762535C2212yes—
13C17×#400—21no—
14C17×#400—22no—
15C17×#395—22no—
16C17×#395C2212yes—
17C17×#382—21no—
18C17×#382C2211yescharacteristic
19C22×Q8×C17D412yes—
20C2×#63252—21no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17D411yescharacteristic
28C2×#63230—42no—
29#10587616—22no—
30C2×#63223—21no—
31#10587583—41no—
32#10587581—41no—
33C17×#95—22no—
34C17×#95—22no—
35C17×#95—21no—
36C17×#95—22no—
37C17×#95—22no—
38C17×#95D412yes—
39#10587506—42no—
40C22⋊#3622—22no—
41#10587481—41no—
42#10587480—41no—
43#10587476—42no—
44C2×Q8×C17—42no—
45C2×Q8×C17—82no—
46C22×C4×C17—21no—
47C22×C4×C17—22no—
48C22×C4×C17—21no—
49C22×C4×C17—22no—
50C22×C4×C17C2×D411yescharacteristic
51C22×C4×C17C2×D412yes—
52C2×C22⋊D17—41no—
53#63252—41no—
54#63252—41no—
55#63252—41no—
56#63252—41no—
57C2×D68—21no—
58C2×D68—22no—
59C2×C4×D17—41no—
60C2×Dic34—21no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—81no—
64C4⋊C4×C17—81no—
65C4⋊C4×C17—82no—
66C4⋊C4×C17—81no—
67C42×C17—21no—
68C42×C17SD1612yescomplemented
69#63230—84no—
70#63227—42no—
71C4⋊Dic17—22no—
72#63223—41no—
73#63223—41no—
74C2×#3622—41no—
75C2×#3622—42no—
76#12184—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78C22×D17—41no—
79Q8×C17—44no—
80Q8×C17—82no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—41no—
86C2×C4×C17—41no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17C2×SD1611yescharacteristic
97C2×C4×C17#13711yescharacteristic
98C22⋊D17—81no—
99C22⋊D17—81no—
100C2×Dic17—41no—
101D68—41no—
102D68—41no—
103D68—82no—
104C4×D17—81no—
105C4×D17—81no—
106Dic34—41no—
107Dic34—41no—
108#3622—81no—
109#3622—82no—
110#2844D1711yescharacteristic, complemented
111#2040—171no—
112#1604—172no—
113#1354—172no—
114#1324—171no—
115C22×C17—21no—
116C22×C17—21no—
117C22×C17#44811yescharacteristic
118C22×C17#45411yescharacteristic
119C22×C17B(4,𝔽2)11yescharacteristic
120C2×D17—81no—
121C2×D17—81no—
122C2×D17—81no—
123C4×C17—21no—
124C4×C17—21no—
125C4×C17—42no—
126C4×C17—42no—
127C4×C17—82no—
128C4×C17—82no—
129C4×C17—81no—
130Dic17—81no—
131#537—171no—
132#448—342no—
133C2×#104—341no—
134#417—341no—
135#400—21no—
136#400—22no—
137#395—22no—
138#395C2×D1712yes—
139#382—21no—
140#382C2×D1711yescharacteristic
141#343—171no—
142#327—342no—
143#324—172no—
144C2×C17—21no—
145C2×C17—21no—
146C2×C17#159911yescharacteristic
147C2×C17#219911yescharacteristic
148C2×C17#220311yescharacteristic
149D17—161no—
150C2×#41—341no—
151C22×Q8C22⋊D1712yes—
152C2×SD16—682no—
153C2×M16—341no—
154C22⋊Q8—681no—
155#121—342no—
156#121—681no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22⋊D1711yescharacteristic
162C2×C4⋊C4C22×D1711yescharacteristic
163C2×C42C22⋊D1711yescharacteristic
164#104—682no—
165#103—681no—
166#103—682no—
167#102—681no—
168C22⋊C8—342no—
169#95—22no—
170#95—22no—
171#95—21no—
172#95—22no—
173#95—22no—
174#95C22⋊D1712yes—
175C17#1218411yescharacteristic, complemented, Sylow
176#41—681no—
177#41—681no—
178#41—681no—
179#41—681no—
180C2×Q8—42no—
181C2×Q8—82no—
182C2×Q8—341no—
183C2×D4—341no—
184C2×D4—342no—
185C2×D4—681no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—681no—
191C22×C4C2×C22⋊D1711yescharacteristic
192C22×C4C2×C22⋊D1712yes—
193SD16—1364no—
194M16—681no—
195M16—681no—
196C2×C8—682no—
197C2×C8—681no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—82no—
203C4⋊C4—81no—
204C4⋊C4—342no—
205C22⋊C4—682no—
206C42—21no—
207C42#6323012yescomplemented
208C23—681no—
209C23#1058763911yescharacteristic
210Q8—44no—
211Q8—82no—
212Q8—681no—
213Q8—681no—
214D4—681no—
215D4—681no—
216D4—1362no—
217D4—1361no—
218D4—1361no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—681no—
235C2×C4—1361no—
236C2×C4—1361no—
237C2×C4C2×#6323011yescharacteristic
238C2×C4#1058762811yescharacteristic
239C8—1361no—
240C8—1362no—
241C22—21no—
242C22—21no—
243C22—1361no—
244C22—1361no—
245C22—1361no—
246C22#1176294811yescharacteristic
247C22#1176298211yescharacteristic
248C22#1176299611yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—1361no—
257C2—21no—
258C2—21no—
259C2—2721no—
260C2#1452772411yescharacteristic
261C2#1453078511yescharacteristic
262C2#1453080511yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2844C2
socleC22×C17B(4,𝔽2)
prime radical#11763390C22
hypercenter#2844D17
hypocenterC17#12184
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
4. #14526728C2semidirect
5. C17×#2844C2semidirect
68. C42×C17SD16semidirect
110. #2844D17semidirect
175. C17#12184semidirect
207. C42#63230semidirect

Maximal subgroups

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

subgroupindex
#145300132
#145277522
#145268142
#145267282
C17×#28442
#1218417