Finite Group Databasebeta

← #16947604

Subgroups — #16947604

Order 4352.

354 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 71 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#14530967C212yesmaximal, complemented
2#14530277C211yesmaximal, characteristic, complemented
3#14526793C212yesmaximal, complemented
4#14526724C211yesmaximal, characteristic, complemented
5C17×#3245C211yesmaximal, characteristic
6C2×#10587489C2211yescharacteristic
7#11762707—21no—
8#11762636—22no—
9#11762610—22no—
10C22⋊#63214C2212yes—
11#11762572—21no—
12#11762572C2211yescharacteristic
13C17×#532—21no—
14C17×#474—22no—
15C17×#466—21no—
16C17×#466C2212yes—
17C17×#459—22no—
18C2×C17×C4⋊C8C2211yescharacteristic
19C2×#41×C17—41no—
20C2×D8×C17—21no—
21C2×D8×C17D412yes—
22C22×C8×C17D411yescharacteristic
23C22×C8×C17C2311yescharacteristic
24C17×C4⋊Q8—21no—
25C17×#127—21no—
26C17×#124—41no—
27C17×#121—22no—
28C17×#121—41no—
29C4×D4×C17—22no—
30C4×D4×C17—41no—
31C2×C42×C17D411yescharacteristic
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×#63225—41no—
35C17×#107—21no—
36C17×#107D412yes—
37C17×C4⋊C8—22no—
38C17×#102—22no—
39C17×#102—22no—
40#10587502—41no—
41#10587489—22no—
42C2×#63214—22no—
43#41×C17—82no—
44#41×C17—82no—
45C2×Q8×C17—41no—
46C2×D4×C17—22no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17—42no—
50C22×C4×C17—42no—
51C22×C4×C17D811yescharacteristic
52C22×C4×C17D811yescharacteristic
53C22×C4×C17C2×D411yescharacteristic
54C22×Dic17—41no—
55C22×Dic17—41no—
56D8×C17—42no—
57D8×C17—81no—
58C2×C8×C17—21no—
59C2×C8×C17—41no—
60C2×C8×C17—41no—
61C2×C8×C17C2×D412yes—
62C4⋊C4×C17—22no—
63C4⋊C4×C17—41no—
64C22⋊C4×C17—42no—
65C22⋊C4×C17—42no—
66C42×C17—21no—
67C42×C17—21no—
68C4⋊Dic17—22no—
69C4⋊Dic17—41no—
70C4⋊Dic17—41no—
71C4⋊Dic17—41no—
72#63225—82no—
73#63214—42no—
74#12291—171nomaximal, Sylow
75C23×C17—42no—
76C23×C17C2×D811yescharacteristic
77Q8×C17—82no—
78D4×C17—42no—
79D4×C17—42no—
80D4×C17—82no—
81D4×C17—82no—
82D4×C17—82no—
83D4×C17—82no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—41no—
90C2×C4×C17—42no—
91C2×C4×C17—84no—
92C2×C4×C17SD3212yes—
93C2×C4×C17#12011yescharacteristic
94C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
95C2×Dic17—42no—
96C2×Dic17—81no—
97C2×Dic17—81no—
98C2×Dic17—81no—
99C2×Dic17—81no—
100C2×Dic17—81no—
101C8×C17—22no—
102C8×C17—41no—
103#3245D1711yescharacteristic
104#2233—172no—
105#2087—171no—
106#1347—172no—
107#1322—171no—
108C22×C17—21no—
109C22×C17—21no—
110C22×C17—82no—
111C22×C17—82no—
112C22×C17—82no—
113C22×C17#44711yescharacteristic
114C22×C17C2×SD3211yescharacteristic
115C22×C17#50911yescharacteristic
116C4×C17—21no—
117C4×C17—22no—
118C4×C17—41no—
119C4×C17—82no—
120C4×C17#45611yescharacteristic
121C4×C17#45811yescharacteristic
122Dic17—82no—
123Dic17—161no—
124#532—21no—
125#474—22no—
126#466—21no—
127#466C2×D1712yes—
128#459—22no—
129C2×#107—171no—
130C2×C4⋊C8C2×D1711yescharacteristic
131#389—341no—
132#367—342no—
133#357—342no—
134C22⋊C16—172no—
135#340—171no—
136#340—341no—
137C2×C17—21no—
138C2×C17—21no—
139C2×C17—162no—
140C2×C17#161011yescharacteristic
141C2×C17#218711yescharacteristic
142C2×C17#219011yescharacteristic
143C2×#41—41no—
144C2×D8—21no—
145C2×D8C22⋊D1712yes—
146C22×C8C22⋊D1711yescharacteristic
147C22×C8C22×D1711yescharacteristic
148C4⋊Q8—21no—
149#127—21no—
150#124—41no—
151#121—22no—
152#121—41no—
153C4×D4—22no—
154C4×D4—41no—
155C2×C4⋊C4—341no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4—681no—
158C2×C42C22⋊D1711yescharacteristic
159C2×C16—342no—
160#107—21no—
161#107—342no—
162#107C22⋊D1712yes—
163C4⋊C8—22no—
164#102—22no—
165#102—22no—
166#95—681no—
167C17#1229111yescharacteristic, complemented, Sylow
168#41—82no—
169#41—82no—
170C2×Q8—41no—
171C2×D4—22no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—42no—
176C22×C4—681no—
177C22×C4—681no—
178C22×C4#6322811yescharacteristic
179C22×C4#6322811yescharacteristic
180C22×C4C2×C22⋊D1711yescharacteristic
181D8—42no—
182D8—81no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8C2×C22⋊D1712yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—342no—
190C4⋊C4—681no—
191C4⋊C4—681no—
192C4⋊C4—681no—
193C4⋊C4—1362no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C16—682no—
199C23—42no—
200C23C2×#6322811yescharacteristic
201Q8—82no—
202D4—42no—
203D4—42no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4—682no—
217C2×C4—1361no—
218C2×C4—1361no—
219C2×C4—1361no—
220C2×C4—1361no—
221C2×C4—1361no—
222C2×C4#1058749812yes—
223C2×C4#1058761811yescharacteristic
224C2×C4#1058763911yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22#1176294311yescharacteristic
233C22C2×#1058749811yescharacteristic
234C22#1176324511yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—82no—
239C4—2721no—
240C4—1362no—
241C4#1176299211yescharacteristic
242C4#1176299911yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2#1452778711yescharacteristic
247C2#1453071311yescharacteristic
248C2#1453073011yescharacteristic
subgroupG/H
centerC22#11762943
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
Fitting, prime radicalC17×#3245C2
socleC22×C17#447
hypercenter#3245D17
hypocenterC17#12291
LCSG ▷ C22×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #14530967C2semidirect
2. #14530277C2semidirect
3. #14526793C2semidirect
4. #14526724C2semidirect
167. C17#12291semidirect

Maximal subgroups

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

subgroupindex
#145309672
#145302772
#145267932
#145267242
C17×#32452
#1229117