Finite Group Databasebeta

← #14541659

Subgroups — #14541659

Order 2176.

338 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 133 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#11763537C211yesmaximal, characteristic, complemented
2#11763208C212yesmaximal, complemented
3#11763205C211yesmaximal, characteristic, complemented
4#11763113C212yesmaximal, complemented
5#11763109C212yesmaximal
6#11763101C211yesmaximal, characteristic, complemented
7#11763089C211yesmaximal, characteristic
8Q8×#3622C211yesmaximal, characteristic, complemented
9C4×#63229C211yesmaximal, characteristic
10C4×#63229C212yesmaximal
11C17×#554C211yesmaximal, characteristic
12C17×C4⋊Q8C2211yescharacteristic
13C17×#125C2212yes—
14C17×C22⋊Q8—21no—
15C17×C22⋊Q8—22no—
16C4×Q8×C17C2211yescharacteristic
17C4×D4×C17C2211yescharacteristic
18C4×D4×C17C2212yes—
19C2×C4⋊C4×C17—21no—
20#10587630—21no—
21C2×#63229C2211yescharacteristic
22#10587575—22no—
23#10587573—21no—
24#10587556C2212yes—
25#10587555C2211yescharacteristic
26C4×Dic34C2211yescharacteristic
27C4×Dic34C2212yes—
28#10587503C2212yes—
29#10587503C2211yescharacteristic
30#10587481C2212yes—
31#10587481C2211yescharacteristic
32#10587479C2212yes—
33#10587479C2212yes—
34#10587479C2211yescharacteristic
35#10587478C2212yes—
36#10587478C2212yes—
37C4⋊#3622C2211yescharacteristic
38C4⋊#3622C2212yes—
39C4×#3622C2211yescharacteristic
40C4×#3622C2212yes—
41C2×Q8×C17—21no—
42C2×Q8×C17D411yescharacteristic
43C2×D4×C17C2311yescharacteristic
44C22×C4×C17—21no—
45C22×C4×C17—22no—
46C2×Dic34—21no—
47C2×Dic34C2311yescharacteristic
48C4⋊C4×C17—22no—
49C4⋊C4×C17—22no—
50C4⋊C4×C17—21no—
51C4⋊C4×C17D412yes—
52C4⋊C4×C17D411yescharacteristic
53C4⋊C4×C17C2311yescharacteristic
54C4⋊C4×C17C2312yes—
55C22⋊C4×C17—21no—
56C22⋊C4×C17—22no—
57C42×C17C2312yes—
58C42×C17C2311yescharacteristic
59#63229Q814yescomplemented
60C4⋊Dic17—21no—
61C4⋊Dic17C2311yescharacteristic
62C4⋊Dic17C2312yes—
63#63225—21no—
64#63225—22no—
65#63225—22no—
66#63225—22no—
67C4×Dic17—22no—
68C4×Dic17—21no—
69C2×#3622C2312yes—
70C2×#3622C2311yescharacteristic
71C2×#3622C2311yescharacteristic
72C23×C17—21no—
73Q8×C17—22no—
74Q8×C17—81no—
75D4×C17—21no—
76D4×C17C2×Q812yes—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—42no—
80C2×C4×C17—82no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17C2×D411yescharacteristic
83C2×C4×C17C2×D412yes—
84C2×C4×C17C2×D411yescharacteristic
85C2×C4×C17C2411yescharacteristic
86C2×Dic17—21no—
87C2×Dic17—22no—
88C2×Dic17—21no—
89Dic34—21no—
90Dic34—42no—
91Dic34C2×Q812yes—
92#3622—22no—
93#3622—21no—
94#3622C2×Q812yes—
95#3381—171nomaximal, Sylow
96C22×C17—22no—
97C22×C17—41no—
98C22×C17—41no—
99C22×C17C22×D411yescharacteristic
100C4×C17—22no—
101C4×C17—22no—
102C4×C17—21no—
103C4×C17—41no—
104C4×C17—42no—
105C4×C17#13512yes—
106C4×C17C22×Q811yescharacteristic
107C4×C172−1+411yescharacteristic
108Dic17—22no—
109Dic17—42no—
110Dic17—41no—
111#557—171no—
112#554D1711yescharacteristic
113#499—172no—
114#498—171no—
115#479—172no—
116#478—172no—
117#477—171no—
118#474—171no—
119C8×Q8—171no—
120C4×SD16—171no—
121C4×SD16—172no—
122C2×C17—42no—
123C2×C17D4×Q811yescharacteristic
124C2×C17C2×#13511yescharacteristic
125C2×C17#57711yescharacteristic
126C2×SD16—171no—
127C4⋊Q8—171no—
128C4⋊Q8—341no—
129C4⋊Q8C2×D1711yescharacteristic
130#125—172no—
131#125—342no—
132#125C2×D1712yes—
133C22⋊Q8—21no—
134C22⋊Q8—22no—
135C4×Q8—171no—
136C4×Q8—172no—
137C4×Q8—341no—
138C4×Q8C2×D1711yescharacteristic
139C4×D4C2×D1711yescharacteristic
140C4×D4C2×D1712yes—
141C2×C4⋊C4—21no—
142#107—172no—
143#107—172no—
144#106—172no—
145#106—171no—
146#106—172no—
147C4⋊C8—171no—
148C4⋊C8—172no—
149#103—172no—
150#103—171no—
151#102—172no—
152#102—171no—
153C4×C8—171no—
154C4×C8—172no—
155C17#338111yescharacteristic, complemented, Sylow
156C2×Q8—21no—
157C2×Q8—171no—
158C2×Q8—341no—
159C2×Q8C22⋊D1711yescharacteristic
160C2×D4C22×D1711yescharacteristic
161C22×C4—21no—
162C22×C4—22no—
163SD16—174no—
164C2×C8—171no—
165C2×C8—171no—
166C2×C8—172no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—172no—
171C4⋊C4—171no—
172C4⋊C4—341no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—341no—
177C4⋊C4C22⋊D1711yescharacteristic
178C4⋊C4C22⋊D1712yes—
179C4⋊C4C22×D1712yes—
180C4⋊C4C22×D1711yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—22no—
183C42—342no—
184C42—341no—
185C42C22×D1712yes—
186C42C22×D1711yescharacteristic
187C23—21no—
188Q8—22no—
189Q8—81no—
190Q8—172no—
191Q8—341no—
192Q8—682no—
193D4—21no—
194D4Q8×D1712yes—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—341no—
200C2×C4—341no—
201C2×C4—342no—
202C2×C4C2×C22⋊D1712yes—
203C2×C4C2×C22⋊D1712yes—
204C2×C4C2×C22⋊D1711yescharacteristic
205C2×C4C2×C22⋊D1711yescharacteristic
206C2×C4C23×D1711yescharacteristic
207C8—172no—
208C8—341no—
209C8—342no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C22×C22⋊D1711yescharacteristic
214C4—22no—
215C4—22no—
216C4—21no—
217C4—41no—
218C4—42no—
219C4—342no—
220C4—682no—
221C4—681no—
222C4#1058763612yes—
223C4C2×Q8×D1711yescharacteristic
224C4#1058769411yescharacteristic
225C2—42no—
226C2Q8×C22⋊D1711yescharacteristic
227C2C2×#1058763611yescharacteristic
228C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#554C2
socleC22×C17C22×D4
prime radicalC2×D4×C17C23
hypercenter#554D17
hypocenterC17#3381
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #11763537C2semidirect
2. #11763208C2semidirect
3. #11763205C2semidirect
4. #11763113C2semidirect
6. #11763101C2semidirect
8. Q8×#3622C2semidirect
59. #63229Q8semidirect
155. C17#3381semidirect

Maximal subgroups

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

subgroupindex
#117635372
#117632082
#117632052
#117631132
#117631092
#117631012
#117630892
Q8×#36222
C4×#632292
C17×#5542
#338117