Finite Group Databasebeta

← #14541693

Subgroups — #14541693

Order 2176.

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

normal subgroups only

This group has 340 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
1Q8×Dic34C211yesmaximal, characteristic, complemented
2#11763520C211yesmaximal, characteristic
3#11763217C212yesmaximal, complemented
4#11763212C211yesmaximal, characteristic, complemented
5#11763114C212yesmaximal, complemented
6#11763100C211yesmaximal, characteristic
7#11763088C212yesmaximal
8#11762936C211yesmaximal, characteristic, complemented
9#11762912C212yesmaximal
10C4×#63230C211yesmaximal, characteristic
11C17×#477C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8C2211yescharacteristic
13C4×Q8×C17C2211yescharacteristic
14#10587632—22no—
15Q8×Dic17C2212yes—
16C2×#63230C2211yescharacteristic
17#10587583—21no—
18#10587579C2212yes—
19C4⋊C4×D17—21no—
20#10587576C2212yes—
21#10587574—22no—
22#10587574C2212yes—
23C4×D68C2211yescharacteristic
24#10587555—21no—
25#10587555C2211yescharacteristic
26C4×Dic34—21no—
27C4×Dic34C2211yescharacteristic
28C17×#107C2212yes—
29C17×C4⋊C8C2211yescharacteristic
30C17×#103C2212yes—
31#10587506C2211yescharacteristic
32#10587492C2212yes—
33#10587488C2212yes—
34#10587486C2212yes—
35#10587485C2212yes—
36#10587480C2211yescharacteristic
37#10587479C2211yescharacteristic
38#10587478C2212yes—
39#10587478C2212yes—
40C4×#3622C2211yescharacteristic
41C2×Q8×C17—21no—
42C2×Q8×C17C2311yescharacteristic
43C2×D68C2311yescharacteristic
44C2×C4×D17—21no—
45C2×C4×D17—22no—
46C2×Dic34—21no—
47C2×Dic34—22no—
48C2×Dic34D411yescharacteristic
49C2×C8×C17C2312yes—
50C4⋊C4×C17—21no—
51C4⋊C4×C17—21no—
52C4⋊C4×C17C2312yes—
53C4⋊C4×C17C2311yescharacteristic
54C42×C17—21no—
55C42×C17C2311yescharacteristic
56#63230Q814yescomplemented
57#63227—21no—
58#63227—22no—
59C4⋊Dic17—21no—
60C4⋊Dic17—22no—
61C4⋊Dic17—22no—
62C4⋊Dic17D411yescharacteristic
63C4⋊Dic17C2312yes—
64C4⋊Dic17C2311yescharacteristic
65#63225—22no—
66#63225—22no—
67#63225D412yes—
68C4×Dic17—22no—
69C4×Dic17C2312yes—
70C2×#3622C2311yescharacteristic
71C2×#3622C2311yescharacteristic
72C22×D17—21no—
73Q8×C17—21no—
74Q8×C17—81no—
75Q8×C17C2×Q812yes—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2411yescharacteristic
82C2×Dic17—22no—
83C2×Dic17—21no—
84C2×Dic17C2×D412yes—
85C2×Dic17C2×D412yes—
86D68—21no—
87D68C2×Q812yes—
88C4×D17—42no—
89C4×D17—82no—
90Dic34—22no—
91Dic34—82no—
92Dic34—81no—
93C8×C17—22no—
94#3622—21no—
95#3622C2×Q812yes—
96#3385—171nomaximal, Sylow
97C22×C17C22×D411yescharacteristic
98C2×D17—22no—
99C2×D17—41no—
100C2×D17—41no—
101C4×C17—22no—
102C4×C17—21no—
103C4×C17—41no—
104C4×C17—42no—
105C4×C17#13712yes—
106C4×C17C22×Q811yescharacteristic
107C4×C172−1+411yescharacteristic
108Dic17—22no—
109Dic17—22no—
110Dic17—41no—
111Dic17—42no—
112Q82—171no—
113#554—171no—
114#501—172no—
115#500—171no—
116#479—172no—
117#477D1711yescharacteristic, complemented
118#476—171no—
119#474—172no—
120#446—171no—
121#440—172no—
122C4×SD16—171no—
123C2×C17D4×Q811yescharacteristic
124C2×C17C2×#13711yescharacteristic
125C2×C17#57611yescharacteristic
126D17—42no—
127C2×SD16—171no—
128C4⋊Q8—172no—
129C4⋊Q8—171no—
130C4⋊Q8—341no—
131C4⋊Q8—342no—
132C4⋊Q8C2×D1711yescharacteristic
133#125—172no—
134C22⋊Q8—341no—
135C22⋊Q8—342no—
136C4×Q8—171no—
137C4×Q8—172no—
138C4×Q8—341no—
139C4×Q8C2×D1711yescharacteristic
140C4×D4—172no—
141C4×D4—171no—
142C2×C4⋊C4—341no—
143#107—172no—
144#107—172no—
145#107C2×D1712yes—
146#106—172no—
147#106—171no—
148C4⋊C8—172no—
149C4⋊C8C2×D1711yescharacteristic
150#103—171no—
151#103C2×D1712yes—
152#102—172no—
153#102—171no—
154#97—172no—
155C4×C8—171no—
156C17#338511yescharacteristic, complemented, Sylow
157C2×Q8—21no—
158C2×Q8—171no—
159C2×Q8—342no—
160C2×Q8—341no—
161C2×Q8C22×D1711yescharacteristic
162C2×D4—171no—
163C22×C4—342no—
164C22×C4—341no—
165SD16—174no—
166C2×C8—171no—
167C2×C8—171no—
168C2×C8C22×D1712yes—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—171no—
172C4⋊C4—172no—
173C4⋊C4—172no—
174C4⋊C4—171no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4—341no—
180C4⋊C4C22×D1711yescharacteristic
181C4⋊C4C22×D1712yes—
182C22⋊C4—342no—
183C22⋊C4—341no—
184C42—21no—
185C42—172no—
186C42—342no—
187C42C22×D1711yescharacteristic
188C23—341no—
189Q8—21no—
190Q8—81no—
191Q8—342no—
192Q8—1361no—
193Q8—1362no—
194Q8Q8×D1712yes—
195D4—172no—
196D4—341no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—172no—
201C2×C4—172no—
202C2×C4—342no—
203C2×C4—341no—
204C2×C4—682no—
205C2×C4—1362no—
206C2×C4D4×D1711yescharacteristic
207C2×C4D4×D1711yescharacteristic
208C2×C4C23×D1711yescharacteristic
209C8—22no—
210C8—172no—
211C8—341no—
212C22—342no—
213C22—681no—
214C22—681no—
215C22C2×D4×D1711yescharacteristic
216C4—22no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—342no—
221C4—342no—
222C4—681no—
223C4—682no—
224C4#1058760412yes—
225C4C2×Q8×D1711yescharacteristic
226C4#1058769711yescharacteristic
227C2—682no—
228C2#1176347211yescharacteristic
229C2C2×#1058760411yescharacteristic
230C2#1176361311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#477C2
socleC22×C17C22×D4
prime radicalC2×D68C23
hypercenter#477D17
hypocenterC17#3385
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. Q8×Dic34C2semidirect
3. #11763217C2semidirect
4. #11763212C2semidirect
5. #11763114C2semidirect
8. #11762936C2semidirect
11. C17×#477C2semidirect
56. #63230Q8semidirect
117. #477D17semidirect
156. C17#3385semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic342
#117635202
#117632172
#117632122
#117631142
#117631002
#117630882
#117629362
#117629122
C4×#632302
C17×#4772
#338517