Finite Group Databasebeta

← #14529989

Subgroups — #14529989

Order 2176.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#10587583C211yesmaximal, characteristic
2C2×#10587487C211yesmaximal, characteristic, complemented
3C2×#10587481C211yesmaximal, characteristic, complemented
4C2×#10587491C211yesmaximal, characteristic
5#11762666C211yesmaximal, characteristic
6#11762571C211yesmaximal, characteristic, complemented
7C2×C17×#107C211yesmaximal, characteristic, complemented
8C22×C4×D17C2211yescharacteristic
9C22×Dic34C2211yescharacteristic
10C22×C8×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×#63227—21no—
14C2×#63227—21no—
15C2×C4⋊Dic17—21no—
16C2×C4⋊Dic17C2211yescharacteristic
17C22×#3622C2211yescharacteristic
18#10587583—22no—
19#10587583—22no—
20C17×#107—22no—
21#10587502—21no—
22#10587491—22no—
23#10587487—22no—
24#10587481—22no—
25C23×D17D411yescharacteristic
26C22×C4×C17—21no—
27C22×C4×C17—21no—
28C22×C4×C17C2311yescharacteristic
29C22×Dic17—21no—
30C22×Dic17—21no—
31C22×Dic17D411yescharacteristic
32C2×C4×D17—21no—
33C2×C4×D17—21no—
34C2×C4×D17—21no—
35C2×C4×D17—21no—
36C2×C4×D17—21no—
37C2×C4×D17—21no—
38C2×Dic34—21no—
39C2×Dic34—21no—
40C2×Dic34—21no—
41C2×Dic34—21no—
42C2×Dic34—21no—
43C2×Dic34D412yes—
44C2×C8×C17—21no—
45C2×C8×C17—21no—
46C2×C8×C17D412yes—
47C4⋊C4×C17—21no—
48C4⋊C4×C17—21no—
49C4⋊C4×C17—21no—
50C4⋊C4×C17D412yes—
51#63227—24no—
52#63227—42no—
53C4⋊Dic17—24no—
54C4⋊Dic17—21no—
55C4⋊Dic17—21no—
56C2×#3622—21no—
57C2×#3622—21no—
58C2×#3622—21no—
59C23×C17C2×D411yescharacteristic
60C22×D17—21no—
61C22×D17—21no—
62C22×D17—21no—
63C22×D17—21no—
64C22×D17—21no—
65C22×D17—21no—
66C22×D17Q1612yes—
67C2×C4×C17—22no—
68C2×C4×C17—41no—
69C2×C4×C17—41no—
70C2×C4×C17—41no—
71C2×C4×C17—41no—
72C2×C4×C17—41no—
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D411yescharacteristic
76C2×C4×C17#4111yescharacteristic
77C2×C4×C17#4111yescharacteristic
78C2×C4×C17#4111yescharacteristic
79C2×Dic17—22no—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—21no—
83C2×Dic17—41no—
84C2×Dic17—41no—
85C2×Dic17—41no—
86C2×Dic17—41no—
87C2×Dic17—41no—
88C4×D17—41no—
89C4×D17—41no—
90C4×D17—41no—
91C4×D17—41no—
92Dic34—22no—
93Dic34—22no—
94Dic34—41no—
95Dic34—41no—
96C8×C17—22no—
97#3622—41no—
98#2036—171nomaximal, Sylow
99C22×C17#12011yescharacteristic
100C22×C17#12111yescharacteristic
101C22×C17#12311yescharacteristic
102C22×C17C2×Q1611yescharacteristic
103C22×C17#13511yescharacteristic
104C22×C17AGL(1,ℤ/8)11yescharacteristic
105C22×C17#13711yescharacteristic
106C2×D17—22no—
107C2×D17—22no—
108C2×D17—22no—
109C2×D17—41no—
110C2×D17—41no—
111C2×D17—41no—
112C2×D17—41no—
113C4×C17—42no—
114C4×C17—81no—
115C4×C17#12111yescharacteristic
116C4×C17#12111yescharacteristic
117C4×C17#12311yescharacteristic
118C4×C17#12411yescharacteristic
119Dic17—42no—
120Dic17—41no—
121Dic17—81no—
122C2×C22⋊Q8—171no—
123C2×#107D1711yescharacteristic, complemented
124C2×#103—171no—
125C2×#103—171no—
126C2×C22⋊C8—171no—
127#380—171no—
128#340—171no—
129C2×C17#39411yescharacteristic
130C2×C17#44911yescharacteristic
131C2×C17#45111yescharacteristic
132C2×C17#46711yescharacteristic
133C2×C17#46811yescharacteristic
134C2×C17#48411yescharacteristic
135C2×C17#48511yescharacteristic
136D17—42no—
137C22×Q8—171no—
138C23×C4—171no—
139C22×C8—171no—
140C22×C8C2×D1711yescharacteristic
141C22⋊Q8—342no—
142C22⋊Q8—342no—
143C2×C4⋊C4—171no—
144C2×C4⋊C4—341no—
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C22⋊C4—341no—
148C2×C22⋊C4—341no—
149#107—22no—
150#103—342no—
151#103—342no—
152C22⋊C8—342no—
153#95—341no—
154C17#203611yescharacteristic, complemented, Sylow
155C24—171no—
156C2×Q8—172no—
157C2×Q8—341no—
158C2×Q8—341no—
159C2×Q8—341no—
160C2×Q8—341no—
161C2×Q8—341no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—171no—
165C22×C4—341no—
166C22×C4—341no—
167C22×C4—341no—
168C22×C4—341no—
169C22×C4—341no—
170C22×C4—341no—
171C22×C4—341no—
172C22×C4—341no—
173C22×C4C22×D1711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—341no—
177C2×C8—341no—
178C2×C8—341no—
179C2×C8D6812yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—344no—
184C4⋊C4—341no—
185C4⋊C4—341no—
186C4⋊C4C22⋊D1712yes—
187C22⋊C4—344no—
188C22⋊C4—682no—
189C23—172no—
190C23—341no—
191C23—341no—
192C23—341no—
193C23—341no—
194C23—341no—
195C23—341no—
196C23D4×D1711yescharacteristic
197Q8—342no—
198Q8—342no—
199Q8—681no—
200Q8—681no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—341no—
208C2×C4—341no—
209C2×C4—342no—
210C2×C4—341no—
211C2×C4—681no—
212C2×C4—681no—
213C2×C4—681no—
214C2×C4—681no—
215C2×C4—681no—
216C2×C4—681no—
217C2×C4—681no—
218C2×C4—681no—
219C2×C4—681no—
220C2×C4C2×D6811yescharacteristic
221C2×C4#6325211yescharacteristic
222C2×C4D4×D1711yescharacteristic
223C2×C4#6325611yescharacteristic
224C2×C4#6325611yescharacteristic
225C2×C4C2×C22⋊D1711yescharacteristic
226C8—22no—
227C8—681no—
228C22—342no—
229C22—342no—
230C22—342no—
231C22—681no—
232C22—681no—
233C22—681no—
234C22—681no—
235C22#1058758011yescharacteristic
236C22#1058758111yescharacteristic
237C22#1058759711yescharacteristic
238C22#1058759811yescharacteristic
239C22Q16×D1711yescharacteristic
240C22#1058760411yescharacteristic
241C22#1058762311yescharacteristic
242C4—42no—
243C4—81no—
244C4—681no—
245C4—682no—
246C4—1361no—
247C4#1058758011yescharacteristic
248C4#1058758111yescharacteristic
249C4#1058761611yescharacteristic
250C4#1058763311yescharacteristic
251C2—682no—
252C2#1176272611yescharacteristic
253C2#1176295111yescharacteristic
254C2#1176296311yescharacteristic
255C2#1176305111yescharacteristic
256C2#1176305611yescharacteristic
257C2#1176314611yescharacteristic
258C2#1176315211yescharacteristic
subgroupG/H
centerC23D4×D17
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC2×C17×#107C2
socleC23×C17C2×D4
prime radicalC23×D17D4
hypercenterC2×#107D17
hypocenterC17#2036
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#10587487C2semidirect
3. C2×#10587481C2semidirect
6. #11762571C2semidirect
7. C2×C17×#107C2semidirect
123. C2×#107D17semidirect
154. C17#2036semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×#105874872
C2×#105874812
C2×#105874912
#117626662
#117625712
C2×C17×#1072
#203617