Finite Group Databasebeta

← #17958626

Subgroups — #17958626

Order 4352.

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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#14544047C211yesmaximal, characteristic, complemented
2#14544027C211yesmaximal, characteristic, complemented
3#14543715C212yesmaximal, complemented
4#14543712C212yesmaximal
5#14533763C211yesmaximal, characteristic
6#11763829—21no—
7#11763825—21no—
8#11763654—21no—
9#11763654—21no—
10#11763654C2212yes—
11#11763654C2212yes—
12C2×#10587630C411yescharacteristic
13C2×#10587622C411yescharacteristic
14C2×Q8×Dic17C2211yescharacteristic
15#11762732—21no—
16#11762732—21no—
17#11762715C2211yescharacteristic
18#11762712C2211yescharacteristic
19#11762703—21no—
20#11762703—21no—
21C4×#63225C411yescharacteristic
22C2×C17×#124C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17C2×C411yescharacteristic
25C22×Dic34C2×C411yescharacteristic
26C17×#124—42no—
27C2×C22⋊C4×C17—21no—
28C2×C22⋊C4×C17—21no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34Q8×Dic17—41no—
35Q8×Dic17—41no—
36#10587630—42no—
37#10587622—41no—
38#10587622—41no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×C4⋊Dic17C2311yescharacteristic
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17C2×C411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C22⋊#3624—22no—
50C22⋊#3624—22no—
51#10587544—41no—
52#10587542—41no—
53C4⋊#3624—41no—
54C4×#3624—41no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58C24×C17—21no—
59C2×Q8×C17—22no—
60C2×Q8×C17—22no—
61C2×Q8×C17—41no—
62C2×D4×C17—22no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C22×C4×C17—21no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×Dic34—22no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×#3624—42no—
79C2×#3624—42no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—42no—
82C42×C17—41no—
83C22⋊Dic17—42no—
84C22⋊Dic17—42no—
85C22⋊Dic17—42no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91#63225—42no—
92#63225—42no—
93#63225—42no—
94C4×Dic17—21no—
95C4×Dic17—21no—
96C4×Dic17—21no—
97C4×Dic17—21no—
98C4×Dic17—21no—
99C4×Dic17—21no—
100C4×Dic17—41no—
101#11565—171nomaximal, Sylow
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17C2×C22⋊C411yescharacteristic
110Q8×C17—44no—
111D4×C17—82no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17#10012yes—
123C2×C4×C17#13112yes—
124C2×Dic17—22no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17#9911yescharacteristic
138C2×Dic17#9911yescharacteristic
139Dic34—82no—
140#3624—82no—
141#3624—82no—
142#2545—171no—
143#1467—171no—
144#1461—171no—
145#1319—172no—
146#1318—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—81no—
150C22×C17—81no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17—81no—
154C22×C17—81no—
155C22×C17—81no—
156C22×C17#40811yescharacteristic
157C22×C17C2×#9911yescharacteristic
158C22×C17C2×#10011yescharacteristic
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—41no—
162C4×C17—81no—
163Dic17—21no—
164Dic17—21no—
165Dic17—42no—
166Dic17—41no—
167Dic17—41no—
168Dic17—41no—
169Dic17—81no—
170C2×C4⋊Q8—171no—
171C2×#124—171no—
172C2×#124#59511yescharacteristic
173C2×C4×Q8—171no—
174#432—341no—
175#431—341no—
176#396—341no—
177#396—341no—
178#391—171no—
179#390—171no—
180#388—341no—
181#388—341no—
182C4×C4⋊C4—171no—
183#324—172no—
184#324—172no—
185#324—341no—
186#324—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—161no—
190C2×C17#147111yescharacteristic
191C2×C17#212611yescharacteristic
192C2×C17#213311yescharacteristic
193C22×Q8—171no—
194C22×Q8C2×#59511yescharacteristic
195C22×D4C2×#59511yescharacteristic
196C4⋊Q8—682no—
197#124—42no—
198#124—681no—
199#124—681no—
200C4×Q8—681no—
201C4×Q8—681no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42—171no—
217C2×C42—171no—
218C2×C42C2×#59511yescharacteristic
219C4⋊C8—681no—
220C4⋊C8—681no—
221C22⋊C8—342no—
222C22⋊C8—342no—
223#97—681no—
224C4×C8—681no—
225#95—341no—
226#95—341no—
227#95—341no—
228C17#1156511yescharacteristic, complemented, Sylow
229C24—21no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—342no—
234C2×Q8—681no—
235C2×Q8—681no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4C22⋊#59511yescharacteristic
247C22×C4C22⋊#59511yescharacteristic
248C22×C4C22×#59511yescharacteristic
249C2×C8—682no—
250C2×C8—682no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C4⋊C4—682no—
255C4⋊C4—682no—
256C4⋊C4—682no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—681no—
260C22⋊C4—681no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C22⋊C4—682no—
264C42—41no—
265C42—341no—
266C42—341no—
267C42—341no—
268C42—341no—
269C42—341no—
270C42—341no—
271C42—681no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C22⋊#59511yescharacteristic
280Q8—44no—
281Q8—1362no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—171no—
294C2×C4—171no—
295C2×C4—342no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4#1058755312yes—
309C2×C4#1058767812yes—
310C8—1362no—
311C8—1362no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#1176378211yescharacteristic
322C22D17⋊#9911yescharacteristic
323C22C2×#1058755311yescharacteristic
324C4—22no—
325C4—22no—
326C4—41no—
327C4—81no—
328C4—341no—
329C4—341no—
330C4—682no—
331C4—681no—
332C4—681no—
333C4—681no—
334C4—1361no—
335C2—21no—
336C2—21no—
337C2—161no—
338C2#1454406011yescharacteristic
339C2#1454505611yescharacteristic
340C2#1454508011yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×#124#595
hypocenterC17#11565
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #14544047C2semidirect
2. #14544027C2semidirect
3. #14543715C2semidirect
228. C17#11565semidirect

Maximal subgroups

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

subgroupindex
#145440472
#145440272
#145437152
#145437122
#145337632
#1156517