Finite Group Databasebeta

← #16947408

Subgroups — #16947408

Order 4352.

330 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 63 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#14530134C211yesmaximal, characteristic
2#14527825C212yesmaximal
3#14526811C212yesmaximal
4#14526728C211yesmaximal, characteristic
5C17×#2844C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11763144—22no—
8#11762849—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C17×#400—22no—
14C17×#400C2212yes—
15C17×#395—22no—
16C17×#395—21no—
17C17×#382—21no—
18C17×#382C2211yescharacteristic
19C22×Q8×C17—21no—
20C2×C4⋊C4×C17—21no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—22no—
23C2×C4⋊C4×C17D411yescharacteristic
24C2×C4⋊C4×C17D412yes—
25C2×C4⋊C4×C17C2311yescharacteristic
26C2×C42×C17D411yescharacteristic
27#10587612—21no—
28#10587610—22no—
29C2×#63223—21no—
30#10587576—41no—
31#10587574—41no—
32C17×#95—22no—
33C17×#95—21no—
34C17×#95—22no—
35C17×#95—22no—
36C17×#95—22no—
37C17×#95D412yes—
38C22⋊#3622—22no—
39#10587481—42no—
40#10587479—41no—
41#10587478—41no—
42#10587478—42no—
43C2×Q8×C17—82no—
44C2×Q8×C17—81no—
45C22×C4×C17—21no—
46C22×C4×C17—22no—
47C22×C4×C17—21no—
48C22×C4×C17—22no—
49C22×C4×C17C2×D411yescharacteristic
50C22×C4×C17C2×D412yes—
51C2×Dic34—22no—
52C4⋊C4×C17—41no—
53C4⋊C4×C17—41no—
54C4⋊C4×C17—42no—
55C4⋊C4×C17—81no—
56C4⋊C4×C17—81no—
57C4⋊C4×C17—82no—
58C42×C17—21no—
59C42×C17Q1612yes—
60C22⋊Dic17—41no—
61C4⋊Dic17—22no—
62C4⋊Dic17—21no—
63C4⋊Dic17—21no—
64#63225—42no—
65C4×Dic17—41no—
66#63223—22no—
67#63223—41no—
68C2×#3622—41no—
69C2×#3622—42no—
70#12261—171nomaximal, Sylow
71C23×C17#12011yescharacteristic
72Q8×C17—84no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—41no—
78C2×C4×C17—41no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17—42no—
82C2×C4×C17—42no—
83C2×C4×C17—42no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17—41no—
87C2×C4×C17—42no—
88C2×C4×C17C2×Q1611yescharacteristic
89C2×C4×C17#13711yescharacteristic
90C2×Dic17—41no—
91C2×Dic17—41no—
92Dic34—82no—
93#3622—42no—
94#3622—82no—
95#2844D1711yescharacteristic
96#2061—171no—
97#1617—172no—
98#1353—172no—
99#1324—171no—
100C22×C17—21no—
101C22×C17—21no—
102C22×C17#45111yescharacteristic
103C22×C17#45611yescharacteristic
104C22×C17B(4,𝔽2)11yescharacteristic
105C4×C17—21no—
106C4×C17—21no—
107C4×C17—42no—
108C4×C17—42no—
109C4×C17—82no—
110C4×C17—81no—
111C4×C17—82no—
112Dic17—81no—
113Dic17—81no—
114#544—171no—
115#484—342no—
116#428—341no—
117#400—22no—
118#400C2×D1712yes—
119#395—22no—
120#395—21no—
121#382—21no—
122#382C2×D1711yescharacteristic
123#343—171no—
124#339—341no—
125#328—342no—
126#324—172no—
127C2×C17—21no—
128C2×C17—21no—
129C2×C17#160511yescharacteristic
130C2×C17#220111yescharacteristic
131C2×C17#220211yescharacteristic
132C22×Q8—21no—
133C2×M16—341no—
134C4⋊Q8—681no—
135#125—681no—
136C22⋊Q8—342no—
137#117—341no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4C22⋊D1711yescharacteristic
142C2×C4⋊C4C22⋊D1712yes—
143C2×C4⋊C4C22×D1711yescharacteristic
144C2×C42C22⋊D1711yescharacteristic
145#107—681no—
146#107—682no—
147#106—681no—
148#103—682no—
149C22⋊C8—342no—
150#95—22no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95C22⋊D1712yes—
156C17#1226111yescharacteristic, complemented, Sylow
157C2×Q8—82no—
158C2×Q8—81no—
159C2×Q8—342no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4C2×C22⋊D1711yescharacteristic
165C22×C4C2×C22⋊D1712yes—
166M16—342no—
167M16—681no—
168C2×C8—681no—
169C2×C8—682no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—82no—
176C4⋊C4—341no—
177C4⋊C4—341no—
178C4⋊C4—342no—
179C4⋊C4—682no—
180C22⋊C4—681no—
181C42—21no—
182C42—681no—
183C42#6323112yes—
184C23#1058763911yescharacteristic
185Q8—84no—
186Q8—1362no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—681no—
203C2×C4—681no—
204C2×C4#1058762811yescharacteristic
205C2×C4C2×#6323111yescharacteristic
206C8—682no—
207C8—1362no—
208C22—21no—
209C22—21no—
210C22#1176296511yescharacteristic
211C22#1176299211yescharacteristic
212C22#1176299611yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4—81no—
220C4—1361no—
221C4—1361no—
222C2—21no—
223C2—21no—
224C2#1452775811yescharacteristic
225C2#1453079711yescharacteristic
226C2#1453080111yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2844C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2844D17
hypocenterC17#12261
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
156. C17#12261semidirect

Maximal subgroups

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

subgroupindex
#145301342
#145278252
#145268112
#145267282
C17×#28442
#1226117