Finite Group Databasebeta

← #16950130

Subgroups — #16950130

Order 4352.

876 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#11762602C211yesmaximal, characteristic
2C2×#11762599C212yesmaximal, complemented
3#14529198C212yesmaximal, complemented
4#14529183C211yesmaximal, characteristic
5C2×C17×#559C211yesmaximal, characteristic
6#11763284C2211yescharacteristic
7#11762814—21no—
8C2×#10587504C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762672—22no—
11#11762602—22no—
12#11762599—24no—
13#11762575—22no—
14#11762569—21no—
15C17×#559—22no—
16C17×#559—24no—
17C17×#559C414yescomplemented
18C2×C17×#127C2211yescharacteristic
19C2×C17×#124—22no—
20C2×C17×#123—21no—
21C2×C17×#123C2212yes—
22C2×C4×Q8×C17—21no—
23C2×C4×D4×C17—22no—
24C22×Q8×C17—21no—
25C22×D4×C17—22no—
26C22×D4×C17D411yescharacteristic
27C22×D4×C17C2311yescharacteristic
28C23×C4×C17—21no—
29C23×C4×C17D412yes—
30C17×#127—22no—
31C17×#127C2×C412yes—
32C17×#127D412yes—
33C17×#124—28no—
34C17×#124—28no—
35C17×#123—24no—
36C17×#123—24no—
37C17×#123—22no—
38C17×#123—22no—
39C17×#123C2×C414yes—
40C17×#123D414yes—
41C4×Q8×C17—24no—
42C4×Q8×C17—24no—
43C4×D4×C17—28no—
44C4×D4×C17—28no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—22no—
47C2×C22⋊C4×C17—22no—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—22no—
50C2×C22⋊C4×C17D412yes—
51C2×C42×C17—22no—
52C2×C42×C17D411yescharacteristic
53C2×C22⋊Dic17—22no—
54C2×C22⋊Dic17—22no—
55D4×Dic17—42no—
56#10587612—21no—
57C2×#63223—21no—
58#10587505—22no—
59#10587505—22no—
60#10587504—22no—
61#10587503—42no—
62#10587502—44no—
63C24×C17—21no—
64C24×C17C2×D412yes—
65C2×Q8×C17—24no—
66C2×Q8×C17—44no—
67C2×D4×C17—24no—
68C2×D4×C17—24no—
69C2×D4×C17—21no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C2×D4×C17—48no—
73C2×D4×C17—41no—
74C2×D4×C17—41no—
75C2×D4×C17C22⋊C412yes—
76C2×D4×C17C22×C411yescharacteristic
77C2×D4×C17C2×D411yescharacteristic
78C2×D4×C17C2×D412yes—
79C22×C4×C17—21no—
80C22×C4×C17—22no—
81C22×C4×C17—22no—
82C22×C4×C17—24no—
83C22×C4×C17—22no—
84C22×C4×C17—42no—
85C22×C4×C17—42no—
86C22×C4×C17—81no—
87C22×C4×C17—81no—
88C22×C4×C17C22⋊C414yes—
89C22×C4×C17C2×D411yescharacteristic
90C22×Dic17—24no—
91C22×Dic17—42no—
92C4⋊C4×C17—24no—
93C4⋊C4×C17—24no—
94C4⋊C4×C17—24no—
95C22⋊C4×C17—24no—
96C22⋊C4×C17—28no—
97C22⋊C4×C17—24no—
98C22⋊C4×C17—22no—
99C22⋊C4×C17—24no—
100C22⋊C4×C17C22⋊C414yes—
101C42×C17—28no—
102C42×C17—21no—
103C42×C17C22⋊C412yes—
104C22⋊Dic17—42no—
105C22⋊Dic17—41no—
106C22⋊Dic17—42no—
107C22⋊Dic17—42no—
108C4⋊Dic17—22no—
109C4×Dic17—41no—
110#63223—41no—
111#63223—41no—
112C2×#3622—41no—
113#12760—171nomaximal, Sylow
114C23×C17—22no—
115C23×C17—22no—
116C23×C17—22no—
117C23×C17—44no—
118C23×C17—44no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17C2×C22⋊C412yes—
122C23×C17#12012yes—
123C23×C17#12011yescharacteristic
124Q8×C17—48no—
125D4×C17—41no—
126D4×C17—42no—
127D4×C17—41no—
128D4×C17—42no—
129D4×C17—48no—
130D4×C17—48no—
131D4×C17—81no—
132C2×C4×C17—24no—
133C2×C4×C17—22no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—24no—
137C2×C4×C17—24no—
138C2×C4×C17—44no—
139C2×C4×C17—42no—
140C2×C4×C17—42no—
141C2×C4×C17—44no—
142C2×C4×C17—82no—
143C2×C4×C17—84no—
144C2×C4×C17C2×C22⋊C411yescharacteristic
145C2×C4×C17#12011yescharacteristic
146C2×Dic17—42no—
147C2×Dic17—84no—
148C2×Dic17—82no—
149C2×Dic17—84no—
150#3622—81no—
151C2×#559D1711yescharacteristic
152C2×#353—171no—
153C2×#352—172no—
154#1890—172no—
155#1887—171no—
156C22×C17—22no—
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—42no—
160C22×C17—42no—
161C22×C17—44no—
162C22×C17—44no—
163C22×C17—42no—
164C22×C17—82no—
165C22×C17—82no—
166C22×C17—82no—
167C22×C17#37911yescharacteristic
168C22×C17B(4,𝔽2)11yescharacteristic
169C22×C17#45811yescharacteristic
170C4×C17—24no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—44no—
174C4×C17—44no—
175Dic17—82no—
176Dic17—162no—
177#559—22no—
178#559—24no—
179#559Dic1714yescomplemented
180C2×#127C2×D1711yescharacteristic
181C2×#124—22no—
182C2×#123—21no—
183C2×#123C2×D1712yes—
184#518—171no—
185C2×C4×Q8—21no—
186C2×C4×D4—22no—
187#418—341no—
188C2×#100—171no—
189C2×#99—172no—
190#381—342no—
191#353—342no—
192#352—344no—
193#342—342no—
194#339—341no—
195C2×C17—21no—
196C2×C17—21no—
197C2×C17—42no—
198C2×C17—42no—
199C2×C17—82no—
200C2×C17#179011yescharacteristic
201C2×C17#219812yes—
202C22×Q8—21no—
203C22×D4—22no—
204C22×D4C22⋊D1711yescharacteristic
205C22×D4C22×D1711yescharacteristic
206C23×C4—21no—
207C23×C4C22⋊D1712yes—
208C2×M16—341no—
209#127—22no—
210#127C2×Dic1712yes—
211#127C22⋊D1712yes—
212#124—28no—
213#124—28no—
214#123—24no—
215#123—24no—
216#123—22no—
217#123—22no—
218#123C2×Dic1714yes—
219#123C22⋊D1714yes—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—682no—
225#117—341no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—342no—
232C2×C22⋊C4—342no—
233C2×C22⋊C4C22⋊D1712yes—
234C2×C42—22no—
235C2×C42C22⋊D1711yescharacteristic
236#102—682no—
237#100—342no—
238#99—342no—
239#99—342no—
240#95—684no—
241C17#1276011yescharacteristic, complemented, Sylow
242C24—21no—
243C24C2×C22⋊D1712yes—
244C2×Q8—24no—
245C2×Q8—44no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—48no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C22⋊Dic1712yes—
255C2×D4C22×Dic1711yescharacteristic
256C2×D4C2×C22⋊D1711yescharacteristic
257C2×D4C2×C22⋊D1712yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—344no—
268C22×C4—682no—
269C22×C4C22⋊Dic1714yes—
270C22×C4C2×C22⋊D1711yescharacteristic
271M16—681no—
272M16—681no—
273C2×C8—681no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—342no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—682no—
284C22⋊C4—681no—
285C22⋊C4—682no—
286C22⋊C4—682no—
287C22⋊C4C22⋊Dic1714yes—
288C42—28no—
289C42—21no—
290C42—681no—
291C42C22⋊Dic1712yes—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—44no—
296C23—44no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic1712yes—
300C23#1058763912yes—
301C23#1058763911yescharacteristic
302Q8—48no—
303D4—41no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—48no—
309D4—81no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—682no—
323C2×C4—1362no—
324C2×C4—1364no—
325C2×C4—1364no—
326C2×C4C2×C22⋊Dic1711yescharacteristic
327C2×C4#1058763911yescharacteristic
328C8—1361no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—44no—
335C22—44no—
336C22—42no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22#1176266511yescharacteristic
341C22#1176299611yescharacteristic
342C22#1176299911yescharacteristic
343C4—24no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—44no—
348C4—2722no—
349C4—1362no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2—82no—
355C2#1452873211yescharacteristic
356C2#1453078012yes—
subgroupG/H
centerC22#11762996
commutatorC2×D4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC2×C17×#559C2
socleC22×C17B(4,𝔽2)
prime radicalC2×C17×#127C22
cosocleC22×D4×C17C23
hypercenterC2×#559D17
hypocenterC17#12760
LCSG ▷ C2×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×#11762599C2semidirect
3. #14529198C2semidirect
17. C17×#559C4semidirect
179. #559Dic17semidirect
241. C17#12760semidirect

Maximal subgroups

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

subgroupindex
C2×#117626022
C2×#117625992
#145291982
#145291832
C2×C17×#5592
#1276017