Finite Group Databasebeta

← #14538436

Subgroups — #14538436

Order 2176.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#63231C211yesmaximal, characteristic, complemented
2C2×#10587634C211yesmaximal, characteristic
3#11763380C211yesmaximal, characteristic
4#11763065C214yesmaximal, complemented
5#11763048C214yesmaximal, complemented
6#11762845C211yesmaximal, characteristic, complemented
7#11762818C211yesmaximal, characteristic, complemented
8#11762817C211yesmaximal, characteristic, complemented
9C17×#536C211yesmaximal, characteristic
10C2×#41×C17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C17×C4⋊Q8—21no—
14C17×#124—21no—
15C17×#123—22no—
16C17×C22⋊Q8C2214yes—
17C17×#117C2211yescharacteristic
18#10587634—24no—
19#10587630—21no—
20C2×#63231—24no—
21C2×#63231C2212yes—
22C2×#63231C2212yes—
23#10587622—21no—
24#10587621—22no—
25#10587612C2211yescharacteristic
26#10587610C2214yes—
27C2×#63223C2212yes—
28C2×#63223C2211yescharacteristic
29C22×#3622C2212yes—
30C22×#3622C2211yescharacteristic
31#10587506C2214yes—
32#10587481C2214yes—
33#10587479C2212yes—
34#10587478C2212yes—
35#41×C17—22no—
36#41×C17—22no—
37C2×Q8×C17—21no—
38C2×Q8×C17—22no—
39C2×Q8×C17—22no—
40C2×Q8×C17D411yescharacteristic
41C2×Q8×C17C2312yes—
42C2×D4×C17D412yes—
43C2×D4×C17D411yescharacteristic
44C22×C4×C17—21no—
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C22×Dic17—21no—
48C2×Dic34—22no—
49C2×Dic34—22no—
50C2×Dic34—21no—
51C2×Dic34C2312yes—
52C4⋊C4×C17—24no—
53C4⋊C4×C17C2312yes—
54C22⋊C4×C17—24no—
55C22⋊C4×C17—21no—
56C42×C17—21no—
57C22⋊Dic17—24no—
58C22⋊Dic17—21no—
59#63231—24no—
60#63231—24no—
61C4⋊Dic17C2312yes—
62#63225—24no—
63C4×Dic17—21no—
64#63223—22no—
65#63223—24no—
66C2×#3622—24no—
67C2×#3622D414yes—
68C2×#3622C2312yes—
69C2×#3622C2311yescharacteristic
70C2×#3622C2311yescharacteristic
71C23×C17C2×D412yes—
72C23×C17C2×D411yescharacteristic
73Q8×C17—22no—
74Q8×C17—22no—
75Q8×C17—22no—
76Q8×C17—42no—
77D4×C17—24no—
78D4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—42no—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2×D412yes—
86C2×C4×C17C2×D411yescharacteristic
87C2×C4×C17#4112yes—
88C2×C4×C17C2411yescharacteristic
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—42no—
92Dic34—22no—
93Dic34—22no—
94Dic34—42no—
95#3622—22no—
96#3622C2×D412yes—
97#3622C2×D412yes—
98#3059—171nomaximal, Sylow
99C22×C17—22no—
100C22×C17—22no—
101C22×C17—24no—
102C22×C17#12112yes—
103C22×C17C22×D411yescharacteristic
104C4×C17—22no—
105C4×C17—42no—
106C4×C17—42no—
107C4×C17#12112yes—
108C4×C17C22×D411yescharacteristic
109C4×C17C2×#4111yescharacteristic
110Dic17—42no—
111Dic17—42no—
112C22×Q16—171no—
113C2×#131—171no—
114#536—171no—
115#536D1711yescharacteristic
116#470—174no—
117#467—174no—
118#427—171no—
119#419—171no—
120#419—171no—
121C2×C17—22no—
122C2×C17—42no—
123C2×C17C2×#12111yescharacteristic
124C2×C17#57812yes—
125C2×#41C2×D1711yescharacteristic
126C22×Q8—171no—
127C22×Q8C2×D1711yescharacteristic
128C2×Q16—172no—
129C2×Q16—172no—
130C2×Q16—344no—
131#131—344no—
132C2×M16—171no—
133C2×M16—172no—
134C22×C8—171no—
135C22×C8—172no—
136C4⋊Q8—21no—
137C4⋊Q8—341no—
138#124—21no—
139#124—341no—
140#123—22no—
141#123—342no—
142C22⋊Q8—174no—
143C22⋊Q8C2×D1714yes—
144#117—171no—
145#117C2×D1711yescharacteristic
146#107—172no—
147#106—172no—
148#103—174no—
149#103—174no—
150C17#305911yescharacteristic, complemented, Sylow
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—172no—
157C2×Q8—341no—
158C2×Q8—342no—
159C2×Q8—342no—
160C2×Q8C22⋊D1711yescharacteristic
161C2×Q8C22×D1712yes—
162C2×D4C22⋊D1712yes—
163C2×D4C22⋊D1711yescharacteristic
164C22×C4—21no—
165C22×C4—341no—
166C22×C4C22×D1712yes—
167C22×C4C22×D1711yescharacteristic
168Q16—344no—
169Q16—344no—
170M16—344no—
171M16—342no—
172C2×C8—174no—
173C2×C8—171no—
174C2×C8—171no—
175C2×C8—172no—
176C2×C8—344no—
177C4⋊C4—24no—
178C4⋊C4—172no—
179C4⋊C4—344no—
180C4⋊C4C22×D1712yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—341no—
184C22⋊C4—344no—
185C42—21no—
186C42—341no—
187C23C2×C22⋊D1712yes—
188C23C2×C22⋊D1711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—342no—
194Q8—342no—
195Q8—682no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—682no—
206C2×C4D4×D1712yes—
207C2×C4#6325412yes—
208C2×C4C2×C22⋊D1712yes—
209C2×C4C2×C22⋊D1711yescharacteristic
210C2×C4C23×D1711yescharacteristic
211C8—172no—
212C8—172no—
213C8—342no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22#1058762512yes—
218C22C22×C22⋊D1711yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—682no—
223C4—682no—
224C4#1058762512yes—
225C4C2×D4×D1711yescharacteristic
226C4C2×#6325411yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1058762511yescharacteristic
230C2#1176362412yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#536C2
socleC22×C17C22×D4
prime radicalC2×#41×C17C22
hypercenter#536D17
hypocenterC17#3059
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#63231C2semidirect
4. #11763065C2semidirect
5. #11763048C2semidirect
6. #11762845C2semidirect
7. #11762818C2semidirect
8. #11762817C2semidirect
150. C17#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#632312
C2×#105876342
#117633802
#117630652
#117630482
#117628452
#117628182
#117628172
C17×#5362
#305917