Finite Group Databasebeta

← #14541664

Subgroups — #14541664

Order 2176.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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#11763534C211yesmaximal, characteristic, complemented
2#11763520C211yesmaximal, characteristic
3#11763209C212yesmaximal, complemented
4#11763119C212yesmaximal, complemented
5#11763108C212yesmaximal
6#11763101C211yesmaximal, characteristic, complemented
7#11763086C211yesmaximal, characteristic
8C8×Dic34C211yesmaximal, characteristic, complemented
9C4×#63219C211yesmaximal, characteristic
10#11762896C212yesmaximal
11C17×#498C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8C2211yescharacteristic
13C17×C4⋊Q8C2211yescharacteristic
14#10587632—22no—
15Q8×Dic17—21no—
16#10587630—21no—
17C2×#63219C2211yescharacteristic
18#10587583—21no—
19#10587579C2212yes—
20C4⋊C4×D17—21no—
21#10587576C2212yes—
22#10587576C2212yes—
23#10587575—22no—
24#10587573C2212yes—
25C4×D68C2211yescharacteristic
26C4×Dic34C2211yescharacteristic
27C4×Dic34C2211yescharacteristic
28C17×#106C2212yes—
29C17×#106C2212yes—
30C4×C8×C17C2211yescharacteristic
31#10587492C2211yescharacteristic
32#10587488C2211yescharacteristic
33#10587487C2211yescharacteristic
34#10587485C2212yes—
35C8×Dic17C2212yes—
36#10587481C2212yes—
37#10587480C2212yes—
38#10587478C2212yes—
39#10587478C2212yes—
40C4⋊#3622C2211yescharacteristic
41C2×Q8×C17—21no—
42C2×Q8×C17—21no—
43C2×D68C2311yescharacteristic
44C2×C4×D17—21no—
45C2×C4×D17—22no—
46C2×Dic34D411yescharacteristic
47C2×Dic34C2311yescharacteristic
48C2×C8×C17C2311yescharacteristic
49C2×C8×C17C2311yescharacteristic
50C4⋊C4×C17—21no—
51C4⋊C4×C17—21no—
52C4⋊C4×C17C2312yes—
53C4⋊C4×C17C2312yes—
54C42×C17C2311yescharacteristic
55#63227—21no—
56#63227—22no—
57C4⋊Dic17—22no—
58C4⋊Dic17—22no—
59C4⋊Dic17D411yescharacteristic
60C4⋊Dic17C2311yescharacteristic
61C4⋊Dic17C2311yescharacteristic
62#63225—21no—
63#63225—22no—
64#63225—22no—
65#63225—22no—
66#63225D412yes—
67C4×Dic17—21no—
68C4×Dic17—22no—
69C4×Dic17C2312yes—
70C2×#3622C2312yes—
71#63219Q814yescomplemented
72C22×D17—21no—
73Q8×C17—42no—
74Q8×C17—81no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2411yescharacteristic
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—21no—
83C2×Dic17C2×D412yes—
84C2×Dic17C2×D412yes—
85D68—21no—
86D68C2×Q812yes—
87C4×D17—42no—
88C4×D17—82no—
89Dic34—22no—
90Dic34—21no—
91Dic34C2×Q812yes—
92C8×C17—21no—
93C8×C17C2×Q812yes—
94#3622—22no—
95#3381—171nomaximal, Sylow
96C22×C17C22×D411yescharacteristic
97C2×D17—22no—
98C2×D17—41no—
99C2×D17—41no—
100C4×C17—21no—
101C4×C17—42no—
102C4×C17—42no—
103C4×C17#13512yes—
104C4×C17C22×Q811yescharacteristic
105C4×C172−1+411yescharacteristic
106Dic17—22no—
107Dic17—22no—
108Dic17—22no—
109Dic17—41no—
110Dic17—41no—
111#557—171no—
112#554—171no—
113#499—172no—
114#498D1711yescharacteristic, complemented
115#479—172no—
116#478—172no—
117#477—171no—
118#474—171no—
119C8×Q8—171no—
120C4×SD16—172no—
121C4×SD16—171no—
122C2×C17D4×Q811yescharacteristic
123C2×C17C2×#13511yescharacteristic
124C2×C17#57711yescharacteristic
125D17—42no—
126C2×SD16—171no—
127C4⋊Q8—341no—
128C4⋊Q8C2×D1711yescharacteristic
129C4⋊Q8C2×D1711yescharacteristic
130#125—172no—
131#125—172no—
132#125—342no—
133C22⋊Q8—341no—
134C22⋊Q8—342no—
135C4×Q8—171no—
136C4×Q8—172no—
137C4×Q8—171no—
138C4×Q8—341no—
139C4×D4—172no—
140C4×D4—171no—
141C2×C4⋊C4—341no—
142#107—172no—
143#107—172no—
144#106—171no—
145#106C2×D1712yes—
146#106C2×D1712yes—
147C4⋊C8—172no—
148C4⋊C8—171no—
149#103—171no—
150#103—172no—
151#102—171no—
152#102—172no—
153C4×C8—172no—
154C4×C8C2×D1711yescharacteristic
155C17#338111yescharacteristic, complemented, Sylow
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—171no—
159C2×Q8—171no—
160C2×D4—171no—
161C22×C4—342no—
162C22×C4—341no—
163SD16—174no—
164C2×C8—172no—
165C2×C8C22×D1711yescharacteristic
166C2×C8C22×D1711yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—171no—
170C4⋊C4—172no—
171C4⋊C4—171no—
172C4⋊C4—171no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—341no—
179C4⋊C4C22×D1712yes—
180C4⋊C4C22×D1712yes—
181C22⋊C4—342no—
182C22⋊C4—341no—
183C42—172no—
184C42—342no—
185C42—341no—
186C42C22×D1711yescharacteristic
187C23—341no—
188Q8—42no—
189Q8—81no—
190Q8—172no—
191Q8—341no—
192Q8—342no—
193D4—172no—
194D4—341no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—172no—
198C2×C4—172no—
199C2×C4—341no—
200C2×C4—341no—
201C2×C4—341no—
202C2×C4—682no—
203C2×C4—1362no—
204C2×C4D4×D1711yescharacteristic
205C2×C4D4×D1711yescharacteristic
206C2×C4C23×D1711yescharacteristic
207C8—21no—
208C8—342no—
209C8Q8×D1712yes—
210C22—342no—
211C22—681no—
212C22—681no—
213C22C2×D4×D1711yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—342no—
218C4—342no—
219C4—342no—
220C4—681no—
221C4—681no—
222C4#1058760212yes—
223C4C2×Q8×D1711yescharacteristic
224C4#1058769411yescharacteristic
225C2—682no—
226C2#1176347411yescharacteristic
227C2C2×#1058760211yescharacteristic
228C2#1176361511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#498C2
socleC22×C17C22×D4
prime radicalC2×D68C23
hypercenter#498D17
hypocenterC17#3381
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #11763534C2semidirect
3. #11763209C2semidirect
4. #11763119C2semidirect
6. #11763101C2semidirect
8. C8×Dic34C2semidirect
11. C17×#498C2semidirect
71. #63219Q8semidirect
114. #498D17semidirect
155. C17#3381semidirect

Maximal subgroups

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

subgroupindex
#117635342
#117635202
#117632092
#117631192
#117631082
#117631012
#117630862
C8×Dic342
C4×#632192
#117628962
C17×#4982
#338117