Finite Group Databasebeta

← #16923625

Subgroups — #16923625

Order 4352.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 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
1#14535816C211yesmaximal, characteristic
2#14530348C212yesmaximal
3#14530322C211yesmaximal, characteristic
4#14526606C212yesmaximal
5C17×#1272C211yesmaximal, characteristic
6C2×#10587630—21no—
7C2×#10587478—21no—
8C2×#10587506—21no—
9#11762739—21no—
10#11762730—22no—
11#11762707C2212yes—
12#11762694—22no—
13#11762679C2211yescharacteristic
14#11762572—22no—
15#11762572—22no—
16#11762564C2211yescharacteristic
17C17×#391C2211yescharacteristic
18C17×#336C2212yes—
19C22×Q8×C17D411yescharacteristic
20C22×Dic34—21no—
21C22×C8×C17—22no—
22C2×C4⋊C4×C17D411yescharacteristic
23C2×C42×C17C2311yescharacteristic
24#10587630—42no—
25#10587630—42no—
26C2×C4⋊Dic17—21no—
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17—22no—
29C2×C4⋊Dic17D412yes—
30C2×C4⋊Dic17D412yes—
31C2×#63225—22no—
32C2×#63225—22no—
33C2×C4×Dic17—21no—
34C22×#3622—21no—
35C17×#95—22no—
36#10587506—42no—
37#10587502—21no—
38#10587502—22no—
39#10587502—22no—
40#10587478—42no—
41C2×Q8×C17—21no—
42C2×Q8×C17—41no—
43C2×Q8×C17—41no—
44C2×Q8×C17SD1612yes—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C2×D411yescharacteristic
48C22×C4×C17C2×D412yes—
49C22×Dic17—22no—
50C22×Dic17—22no—
51C22×Dic17—41no—
52C2×Dic34—42no—
53C2×Dic34—81no—
54C2×Dic34—81no—
55C2×C8×C17—42no—
56C2×C8×C17—42no—
57C2×C8×C17—42no—
58C4⋊C4×C17—41no—
59C42×C17—41no—
60C4⋊Dic17—22no—
61C4⋊Dic17—42no—
62C4⋊Dic17—41no—
63C4⋊Dic17—41no—
64C4⋊Dic17—44no—
65C4⋊Dic17—81no—
66C4⋊Dic17Q1614yes—
67#63225—44no—
68#63225—42no—
69#63225—42no—
70C4×Dic17—42no—
71C2×#3622—22no—
72C2×#3622—41no—
73C2×#3622—41no—
74#7829—171nomaximal, Sylow
75C23×C17#12011yescharacteristic
76Q8×C17—42no—
77Q8×C17—81no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17—41no—
86C2×C4×C17—42no—
87C2×C4×C17C2×SD1611yescharacteristic
88C2×C4×C17C2×Q1612yes—
89C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C17#13712yes—
91C2×Dic17—24no—
92C2×Dic17—42no—
93C2×Dic17—42no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17—42no—
97C2×Dic17—81no—
98C2×Dic17—82no—
99Dic34—84no—
100C8×C17—82no—
101#3622—42no—
102#2734—171no—
103#2100—172no—
104#2095—171no—
105#1272D1711yescharacteristic
106#1271—172no—
107C22×C17#45011yescharacteristic
108C22×C17#45112yes—
109C22×C17#45311yescharacteristic
110C22×C17#45412yes—
111C22×C17#45811yescharacteristic
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—22no—
115C4×C17—22no—
116C4×C17—42no—
117C4×C17—81no—
118Dic17—44no—
119Dic17—82no—
120Dic17—161no—
121C2×C4⋊Q8—341no—
122C2×#107—341no—
123C2×#103—341no—
124#398—341no—
125#395—342no—
126#391C2×D1711yescharacteristic
127#389—172no—
128#387—342no—
129#384—171no—
130#340—342no—
131#340—342no—
132#336—171no—
133#336C2×D1712yes—
134C2×C17#161311yescharacteristic
135C2×C17#161712yes—
136C2×C17#163311yescharacteristic
137C2×C17#163712yes—
138C2×C17#165711yescharacteristic
139C22×Q8—341no—
140C22×Q8D6811yescharacteristic
141C22×C8—22no—
142C22×C8—341no—
143C4⋊Q8—682no—
144C4⋊Q8—682no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—342no—
148C2×C4⋊C4—342no—
149C2×C4⋊C4—341no—
150C2×C4⋊C4—342no—
151C2×C4⋊C4—341no—
152C2×C4⋊C4D6811yescharacteristic
153C2×C42—341no—
154C2×C42C22×D1711yescharacteristic
155#107—682no—
156#103—682no—
157#95—22no—
158#95—341no—
159#95—342no—
160#95—342no—
161C17#782911yescharacteristic, complemented, Sylow
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—682no—
166C2×Q8—1361no—
167C2×Q8—1361no—
168C2×Q8#6321912yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—342no—
172C22×C4—342no—
173C22×C4—681no—
174C22×C4C2×D6811yescharacteristic
175C22×C4D4×D1712yes—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—342no—
180C2×C8—681no—
181C2×C8—681no—
182C4⋊C4—41no—
183C4⋊C4—174no—
184C4⋊C4—342no—
185C4⋊C4—682no—
186C4⋊C4—684no—
187C4⋊C4—681no—
188C4⋊C4—681no—
189C4⋊C4—684no—
190C4⋊C4—682no—
191C4⋊C4—682no—
192C4⋊C4—1361no—
193C42—41no—
194C42—682no—
195C23#1058756811yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—1364no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—344no—
209C2×C4—682no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4—682no—
214C2×C4—1361no—
215C2×C4—1362no—
216C2×C4C2×#6321911yescharacteristic
217C2×C4#1058759411yescharacteristic
218C2×C4#1058760112yes—
219C2×C4Q16×D1712yes—
220C8—82no—
221C8—682no—
222C22#1176296211yescharacteristic
223C22#1176296612yes—
224C22#1176297911yescharacteristic
225C22#1176298312yes—
226C22#1176300011yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—22no—
231C4—42no—
232C4—81no—
233C4—2721no—
234C4—684no—
235C4—1362no—
236C2#1452780611yescharacteristic
237C2#1452782712yes—
238C2#1452792511yescharacteristic
239C2#1452795112yes—
240C2#1452808311yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1272C2
socle, prime radicalC23×C17#120
hypercenter#1272D17
hypocenterC17#7829
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
161. C17#7829semidirect

Maximal subgroups

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

subgroupindex
#145358162
#145303482
#145303222
#145266062
C17×#12722
#782917