Finite Group Databasebeta

← #16951118

Subgroups — #16951118

Order 4352.

672 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 99 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
2#14529183C211yesmaximal, characteristic
3#14529064C212yesmaximal, complemented
4#14526902C212yesmaximal, complemented
5C17×#2366C211yesmaximal, characteristic
6#11763284C2211yescharacteristic
7#11762814—21no—
8C2×#10587504C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762682—22no—
11#11762602C414yescomplemented
12#11762575—22no—
13#11762569—21no—
14C2×C17×#127C2211yescharacteristic
15C2×C4×D4×C17—22no—
16C17×#390—22no—
17C17×#386—21no—
18C17×#386C2212yes—
19C4×C4⋊C4×C17—21no—
20C22×D4×C17—22no—
21C22×D4×C17D411yescharacteristic
22C22×D4×C17C2311yescharacteristic
23C23×C4×C17—21no—
24C23×C4×C17D412yes—
25C17×#127—22no—
26C17×#127C2×C412yes—
27C17×#127Q812yes—
28C4×D4×C17—48no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17D412yes—
35C2×C42×C17—22no—
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—22no—
38C2×C22⋊Dic17—22no—
39D4×Dic17—41no—
40D4×Dic17—41no—
41#10587612—21no—
42C2×#63225—42no—
43C2×#63223—21no—
44C17×#95—22no—
45#10587505—22no—
46#10587505C2×C414yescomplemented
47#10587504—22no—
48#10587503—41no—
49#10587503—41no—
50#10587502—42no—
51C24×C17—21no—
52C24×C17C2×D412yes—
53C2×D4×C17—24no—
54C2×D4×C17—24no—
55C2×D4×C17—21no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17—44no—
59C2×D4×C17—44no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17C4⋊C412yescomplemented
63C2×D4×C17C22×C411yescharacteristic
64C2×D4×C17C2×Q811yescharacteristic
65C2×D4×C17#4112yes—
66C22×C4×C17—21no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—44no—
70C22×C4×C17—44no—
71C22×C4×C17—42no—
72C22×C4×C17—81no—
73C22×C4×C17C2×D411yescharacteristic
74C22×Dic17—22no—
75C22×Dic17—22no—
76C22×Dic17—42no—
77C4⋊C4×C17—24no—
78C4⋊C4×C17—42no—
79C22⋊C4×C17—44no—
80C22⋊C4×C17—44no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—81no—
83C42×C17—21no—
84C42×C17—44no—
85C42×C17C4⋊C412yescomplemented
86C22⋊Dic17—24no—
87C22⋊Dic17—42no—
88C22⋊Dic17—42no—
89C22⋊Dic17—41no—
90C4⋊Dic17—21no—
91C4⋊Dic17—21no—
92#63225—84no—
93C4×Dic17—41no—
94#63223—41no—
95#63223—41no—
96C2×#3622—41no—
97#12964—171nomaximal, Sylow
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—22no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—42no—
104C23×C17—42no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17C4×D412yes—
108C23×C17#12011yescharacteristic
109C23×C17C22⋊Q812yes—
110D4×C17—41no—
111D4×C17—41no—
112D4×C17—42no—
113D4×C17—42no—
114D4×C17—48no—
115D4×C17—48no—
116D4×C17—81no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—24no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—44no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—82no—
130C2×C4×C17—84no—
131C2×C4×C17C2×C4⋊C411yescharacteristic
132C2×C4×C17#12311yescharacteristic
133C2×Dic17—44no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—82no—
137C2×Dic17—82no—
138C2×Dic17—82no—
139C2×Dic17—82no—
140#3622—81no—
141#2366D1711yescharacteristic
142C2×#353—171no—
143#1887—171no—
144#1867—172no—
145#1390—172no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—22no—
149C22×C17—22no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—44no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—82no—
157C22×C17—82no—
158C22×C17—81no—
159C22×C17—81no—
160C22×C17#38511yescharacteristic
161C22×C17B(4,𝔽2)11yescharacteristic
162C22×C17#45811yescharacteristic
163C4×C17—24no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—44no—
167C4×C17—82no—
168Dic17—84no—
169Dic17—81no—
170Dic17—81no—
171C2×#127C2×D1711yescharacteristic
172#518—171no—
173C2×C4×D4—22no—
174#418—341no—
175C2×#100—171no—
176C2×#99—172no—
177#390—22no—
178#386—21no—
179#386C2×D1712yes—
180#385—342no—
181C4×C4⋊C4—21no—
182#353—174no—
183#342—342no—
184#339—341no—
185C2×C17—21no—
186C2×C17—21no—
187C2×C17—42no—
188C2×C17—42no—
189C2×C17—82no—
190C2×C17#186711yescharacteristic
191C2×C17#219611yescharacteristic
192C2×C17#219811yescharacteristic
193C22×D4—22no—
194C22×D4C22⋊D1711yescharacteristic
195C22×D4C22×D1711yescharacteristic
196C23×C4—21no—
197C23×C4C22⋊D1712yes—
198C2×M16—341no—
199#127—22no—
200#127Dic3412yes—
201#127C4×D1712yes—
202C4×D4—48no—
203C4×D4—681no—
204C4×D4—681no—
205#117—341no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—682no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4C22⋊D1712yes—
215C2×C42—22no—
216C2×C42C22⋊D1711yescharacteristic
217#102—681no—
218#102—681no—
219#100—342no—
220#99—174no—
221#99—342no—
222#95—22no—
223#95—682no—
224C17#1296411yescharacteristic, complemented, Sylow
225C24—21no—
226C24C2×C22⋊D1712yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4#6322512yescomplemented
237C2×D4C2×Dic3411yescharacteristic
238C2×D4C2×C4×D1711yescharacteristic
239C2×D4#6325212yes—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4—682no—
250C22×C4C2×C22⋊D1711yescharacteristic
251M16—681no—
252M16—681no—
253C2×C8—681no—
254C4⋊C4—24no—
255C4⋊C4—42no—
256C4⋊C4—341no—
257C4⋊C4—341no—
258C4⋊C4—1364no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—81no—
263C22⋊C4—344no—
264C22⋊C4—682no—
265C22⋊C4—682no—
266C22⋊C4—681no—
267C42—21no—
268C42—44no—
269C42—681no—
270C42#6322512yescomplemented
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23#1058761012yes—
281C23C4×C22⋊D1712yes—
282C23#1058763911yescharacteristic
283D4—41no—
284D4—41no—
285D4—42no—
286D4—42no—
287D4—48no—
288D4—48no—
289D4—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—684no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—1362no—
308C2×C4—1362no—
309C2×C4—1362no—
310C2×C4—1362no—
311C2×C4C2×#6322511yescharacteristic
312C2×C4#1058761511yescharacteristic
313C8—1361no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1176268211yescharacteristic
329C22#1176299611yescharacteristic
330C22#1176299911yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—82no—
336C4—1364no—
337C4—1361no—
338C4—1361no—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2#1452906511yescharacteristic
345C2#1453076811yescharacteristic
346C2#1453078011yescharacteristic
subgroupG/H
centerC22#11762996
commutatorC2×D4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2366C2
socleC22×C17B(4,𝔽2)
prime radicalC2×C17×#127C22
cosocleC22×D4×C17C23
hypercenter#2366D17
hypocenterC17#12964
LCSG ▷ C2×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2366

Complements

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

NH ≅ G/Ntype
3. #14529064C2semidirect
4. #14526902C2semidirect
11. #11762602C4semidirect
46. #10587505C2×C4semidirect
62. C2×D4×C17C4⋊C4semidirect
85. C42×C17C4⋊C4semidirect
224. C17#12964semidirect
236. C2×D4#63225semidirect
270. C42#63225semidirect

Maximal subgroups

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

subgroupindex
C2×#117626022
#145291832
#145290642
#145269022
C17×#23662
#1296417