Finite Group Databasebeta

← #14529980

Subgroups — #14529980

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×#10587610C211yesmaximal, characteristic
2C2×#10587479C211yesmaximal, characteristic, complemented
3C2×#10587481C211yesmaximal, characteristic, complemented
4C2×#10587481C211yesmaximal, characteristic, complemented
5C2×C22⋊#3622C211yesmaximal, characteristic
6#11762571C211yesmaximal, characteristic, complemented
7C17×#380C211yesmaximal, characteristic
8C23×C4×C17C2211yescharacteristic
9C22×Dic34C2211yescharacteristic
10C2×C4⋊C4×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C22⋊C4×C17—21no—
13C2×C22⋊Dic17—21no—
14C2×C4⋊Dic17C2211yescharacteristic
15#10587610—22no—
16#10587610—22no—
17C2×#63225—21no—
18C22×#3622C2211yescharacteristic
19C22×#3622C2211yescharacteristic
20C17×#95—21no—
21C22⋊#3622—22no—
22#10587481—22no—
23#10587481—22no—
24#10587479—22no—
25C24×C17D411yescharacteristic
26C22×C4×C17—21no—
27C22×C4×C17—21no—
28C22×C4×C17—21no—
29C22×C4×C17—21no—
30C22×C4×C17—21no—
31C22×C4×C17—21no—
32C22×C4×C17—21no—
33C22×C4×C17—21no—
34C22×C4×C17D411yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×Dic17—21no—
37C22×Dic17—21no—
38C2×Dic34—21no—
39C2×Dic34—21no—
40C2×Dic34—21no—
41C2×Dic34—21no—
42C2×Dic34—21no—
43C2×Dic34D412yes—
44C4⋊C4×C17—21no—
45C4⋊C4×C17—21no—
46C4⋊C4×C17—21no—
47C4⋊C4×C17—21no—
48C22⋊C4×C17—42no—
49C22⋊Dic17—24no—
50C4⋊Dic17—21no—
51C4⋊Dic17D412yes—
52#63225—24no—
53C2×#3622—21no—
54C2×#3622—21no—
55C2×#3622—21no—
56C2×#3622—21no—
57C2×#3622—21no—
58C2×#3622D412yes—
59C23×C17—21no—
60C23×C17—21no—
61C23×C17—21no—
62C23×C17—21no—
63C23×C17—21no—
64C23×C17—21no—
65C23×C17SD1612yes—
66C23×C17C2×D411yescharacteristic
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—41no—
71C2×C4×C17—41no—
72C2×C4×C17—41no—
73C2×C4×C17—41no—
74C2×C4×C17—41no—
75C2×C4×C17—41no—
76C2×C4×C17—41no—
77C2×C4×C17—41no—
78C2×C4×C17—41no—
79C2×C4×C17—41no—
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2×D411yescharacteristic
82C2×C4×C17C2×D411yescharacteristic
83C2×C4×C17#4111yescharacteristic
84C2×C4×C17#4111yescharacteristic
85C2×C4×C17#4111yescharacteristic
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—41no—
89C2×Dic17—41no—
90C2×Dic17—41no—
91C2×Dic17—41no—
92Dic34—22no—
93Dic34—22no—
94Dic34—41no—
95Dic34—41no—
96#3622—22no—
97#3622—41no—
98#2035—171nomaximal, Sylow
99C22×C17—22no—
100C22×C17—22no—
101C22×C17—22no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—41no—
106C22×C17#12011yescharacteristic
107C22×C17#12111yescharacteristic
108C22×C17#12311yescharacteristic
109C22×C17C2×SD1611yescharacteristic
110C22×C17#13511yescharacteristic
111C22×C17#13711yescharacteristic
112C22×C17#13711yescharacteristic
113C4×C17—41no—
114C4×C17—81no—
115C4×C17—81no—
116C4×C17#12111yescharacteristic
117C4×C17#12111yescharacteristic
118C4×C17#12311yescharacteristic
119C4×C17#12411yescharacteristic
120Dic17—42no—
121Dic17—42no—
122C2×C22⋊Q8—171no—
123C2×#106—171no—
124C2×#103—171no—
125C2×#103—171no—
126C2×C22⋊C8—171no—
127#380D1711yescharacteristic
128#340—171no—
129C2×C17—42no—
130C2×C17#39411yescharacteristic
131C2×C17#44811yescharacteristic
132C2×C17#45211yescharacteristic
133C2×C17#46511yescharacteristic
134C2×C17#47011yescharacteristic
135C2×C17#48311yescharacteristic
136C2×C17#48511yescharacteristic
137C22×Q8—171no—
138C23×C4C2×D1711yescharacteristic
139C22×C8—171no—
140C22×C8—171no—
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—21no—
148C2×C22⋊C4—341no—
149#106—342no—
150#103—342no—
151#103—342no—
152C22⋊C8—342no—
153#95—21no—
154C17#203511yescharacteristic, complemented, Sylow
155C24C22⋊D1711yescharacteristic
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—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—341no—
171C22×C4—341no—
172C22×C4C22⋊D1711yescharacteristic
173C22×C4C22×D1711yescharacteristic
174C2×C8—172no—
175C2×C8—341no—
176C2×C8—341no—
177C2×C8—341no—
178C2×C8—341no—
179C2×C8—341no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—172no—
185C4⋊C4—344no—
186C4⋊C4—341no—
187C22⋊C4—42no—
188C22⋊C4—344no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#6322912yes—
196C23C2×C22⋊D1711yescharacteristic
197Q8—342no—
198Q8—342no—
199Q8—681no—
200Q8—681no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—681no—
217C2×C4—681no—
218C2×C4—681no—
219C2×C4—681no—
220C2×C4#6325211yescharacteristic
221C2×C4#6325211yescharacteristic
222C2×C4D4×D1711yescharacteristic
223C2×C4D4×D1711yescharacteristic
224C2×C4D4×D1711yescharacteristic
225C2×C4#6325411yescharacteristic
226C8—342no—
227C8—681no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#1058761511yescharacteristic
236C22C2×#6322911yescharacteristic
237C22#1058762311yescharacteristic
238C22#1058762511yescharacteristic
239C22#1058762811yescharacteristic
240C22#1058763611yescharacteristic
241C22#1058763711yescharacteristic
242C4—41no—
243C4—81no—
244C4—81no—
245C4—682no—
246C4—682no—
247C4#1058756911yescharacteristic
248C4#1058757011yescharacteristic
249C4#1058757011yescharacteristic
250C4#1058757111yescharacteristic
251C2—42no—
252C2#1176272411yescharacteristic
253C2#1176294711yescharacteristic
254C2#1176296911yescharacteristic
255C2#1176303611yescharacteristic
256C2#1176306511yescharacteristic
257C2#1176313911yescharacteristic
258C2#1176315011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D17
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#380C2
socleC23×C17C2×D4
prime radicalC24×C17D4
hypercenter#380D17
hypocenterC17#2035
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#10587479C2semidirect
3. C2×#10587481C2semidirect
4. C2×#10587481C2semidirect
6. #11762571C2semidirect
154. C17#2035semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
C2×#105874792
C2×#105874812
C2×C22⋊#36222
#117625712
C17×#3802
#203517