Finite Group Databasebeta

← #16947403

Subgroups — #16947403

Order 4352.

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

normal subgroups only

This group has 342 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#14527757C212yesmaximal
3#14526814C212yesmaximal
4#14526728C211yesmaximal, characteristic
5C17×#2844C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11762965—22no—
8#11762849—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C17×#400—21no—
14C17×#400—22no—
15C17×#395—22no—
16C17×#395C2212yes—
17C17×#382—21no—
18C17×#382C2211yescharacteristic
19C22×Q8×C17D412yes—
20C2×C4⋊C4×C17—21no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—22no—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C4⋊C4×C17C2311yescharacteristic
26C2×C42×C17D411yescharacteristic
27C2×#63231—42no—
28#10587612—21no—
29#10587610—22no—
30C2×#63223—21no—
31#10587576—41no—
32#10587574—41no—
33C17×#95—22no—
34C17×#95—22no—
35C17×#95—21no—
36C17×#95—22no—
37C17×#95—22no—
38C17×#95D412yes—
39#10587506—42no—
40C22⋊#3622—22no—
41#10587479—41no—
42#10587478—41no—
43C2×Q8×C17—42no—
44C2×Q8×C17—82no—
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—81no—
55C4⋊C4×C17—81no—
56C4⋊C4×C17—82no—
57C4⋊C4×C17—81no—
58C42×C17—21no—
59C42×C17Q1612yescomplemented
60C22⋊Dic17—41no—
61#63231—84no—
62C4⋊Dic17—21no—
63C4⋊Dic17—21no—
64C4⋊Dic17—22no—
65#63225—42no—
66C4×Dic17—41no—
67#63223—22no—
68#63223—41no—
69C2×#3622—41no—
70C2×#3622—42no—
71#12260—171nomaximal, Sylow
72C23×C17#12011yescharacteristic
73Q8×C17—44no—
74Q8×C17—82no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—42no—
82C2×C4×C17—42no—
83C2×C4×C17—42no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—41no—
89C2×C4×C17—42no—
90C2×C4×C17C2×Q1611yescharacteristic
91C2×C4×C17#13711yescharacteristic
92C2×Dic17—41no—
93C2×Dic17—41no—
94Dic34—82no—
95#3622—42no—
96#3622—82no—
97#2844D1711yescharacteristic
98#2061—171no—
99#1605—172no—
100#1354—172no—
101#1324—171no—
102C22×C17—21no—
103C22×C17—21no—
104C22×C17#45111yescharacteristic
105C22×C17#45611yescharacteristic
106C22×C17B(4,𝔽2)11yescharacteristic
107C4×C17—21no—
108C4×C17—21no—
109C4×C17—42no—
110C4×C17—42no—
111C4×C17—82no—
112C4×C17—82no—
113C4×C17—81no—
114Dic17—81no—
115Dic17—81no—
116#544—171no—
117#451—342no—
118#428—341no—
119#400—21no—
120#400—22no—
121#395—22no—
122#395C2×D1712yes—
123#382—21no—
124#382C2×D1711yescharacteristic
125#343—171no—
126#339—341no—
127#328—342no—
128#324—172no—
129C2×C17—21no—
130C2×C17—21no—
131C2×C17#160511yescharacteristic
132C2×C17#219911yescharacteristic
133C2×C17#220311yescharacteristic
134C22×Q8C22⋊D1712yes—
135C2×Q16—682no—
136C2×M16—341no—
137C4⋊Q8—681no—
138#125—681no—
139C22⋊Q8—342no—
140#117—341no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C22⋊D1711yescharacteristic
146C2×C4⋊C4C22×D1711yescharacteristic
147C2×C42C22⋊D1711yescharacteristic
148#107—681no—
149#106—681no—
150#103—682no—
151C22⋊C8—342no—
152#95—22no—
153#95—22no—
154#95—21no—
155#95—22no—
156#95—22no—
157#95C22⋊D1712yes—
158C17#1226011yescharacteristic, complemented, Sylow
159C2×Q8—42no—
160C2×Q8—82no—
161C2×Q8—342no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×C22⋊D1711yescharacteristic
167C22×C4C2×C22⋊D1712yes—
168Q16—1364no—
169M16—342no—
170M16—681no—
171C2×C8—681no—
172C2×C8—682no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—342no—
180C4⋊C4—341no—
181C4⋊C4—341no—
182C4⋊C4—682no—
183C22⋊C4—681no—
184C42—21no—
185C42—681no—
186C42#6323112yescomplemented
187C23#1058763911yescharacteristic
188Q8—44no—
189Q8—82no—
190Q8—1362no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—681no—
207C2×C4—681no—
208C2×C4#1058762811yescharacteristic
209C2×C4C2×#6323111yescharacteristic
210C8—682no—
211C8—1362no—
212C22—21no—
213C22—21no—
214C22#1176296511yescharacteristic
215C22#1176299211yescharacteristic
216C22#1176299611yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—1361no—
225C4—1361no—
226C2—21no—
227C2—21no—
228C2#1452775811yescharacteristic
229C2#1453078511yescharacteristic
230C2#1453080511yescharacteristic
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#12260
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
59. C42×C17Q16semidirect
158. C17#12260semidirect
186. C42#63231semidirect

Maximal subgroups

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

subgroupindex
#145301342
#145277572
#145268142
#145267282
C17×#28442
#1226017