Finite Group Databasebeta

← #17958353

Subgroups — #17958353

Order 4352.

390 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 92 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#14545146C211yesmaximal, characteristic, complemented
2#14545142C211yesmaximal, characteristic, complemented
3#14543705C212yesmaximal, complemented
4#14543704C212yesmaximal, complemented
5C2×#11762885C211yesmaximal, characteristic
6#11763730—21no—
7#11763730—21no—
8C22⋊#63216C2212yes—
9C22⋊#63216C2212yes—
10#11763698—21no—
11#11763695—21no—
12C22×#63218C411yescharacteristic
13C2×C4×C22⋊D17C411yescharacteristic
14#11762885—21no—
15#11762885—21no—
16#11762885—21no—
17#11762885—21no—
18C2×#10587485C411yescharacteristic
19C2×C22⋊#3622C2211yescharacteristic
20C2×#10587491C2211yescharacteristic
21C2×C8×Dic17C2211yescharacteristic
22C22×M16×C17C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C22×C22⋊D17C811yescharacteristic
25C22×C4×D17C2×C411yescharacteristic
26C2×M16×C17—21no—
27C2×M16×C17—21no—
28C2×M16×C17—42no—
29C22×C8×C17C2×C411yescharacteristic
30C22×C8×C17C2×C411yescharacteristic
31C2×C22⋊Dic17C811yescharacteristic
32C4×C22⋊D17—21no—
33C4×C22⋊D17—21no—
34C4×C22⋊D17—21no—
35C4×C22⋊D17—21no—
36C2×#63227C811yescharacteristic
37C2×#63225C811yescharacteristic
38C2×C4×Dic17C2×C411yescharacteristic
39C22×#3622C2×C411yescharacteristic
40C22×#3622C2311yescharacteristic
41C2×#63218—21no—
42C2×#63218—21no—
43C2×#63218—42no—
44C2×#63216—22no—
45C2×#63216—22no—
46C22⋊#3622—21no—
47C22⋊#3622—21no—
48#10587491—21no—
49#10587491—21no—
50#10587485—21no—
51#10587485—21no—
52C8×Dic17D411yescharacteristic
53C8×Dic17D411yescharacteristic
54C8×Dic17D411yescharacteristic
55C8×Dic17D411yescharacteristic
56C24×C17C2×C811yescharacteristic
57C23×D17C2×C811yescharacteristic
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—41no—
61C22×C4×C17—41no—
62C22×C4×C17C2×C811yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—42no—
67C22×Dic17C2×C811yescharacteristic
68C22×Dic17C2×C811yescharacteristic
69C22×Dic17C2×C811yescharacteristic
70C2×C4×D17—21no—
71C2×C4×D17—21no—
72C2×C4×D17—41no—
73C2×C4×D17—41no—
74M16×C17—42no—
75M16×C17—42no—
76C2×C8×C17—22no—
77C2×C8×C17—21no—
78C2×C8×C17—41no—
79C2×C8×C17C22⋊C411yescharacteristic
80C2×C8×C17C22⋊C411yescharacteristic
81C22⋊Dic17—21no—
82C22⋊Dic17—21no—
83#63227—21no—
84#63227—21no—
85#63225—21no—
86#63225—21no—
87C4×Dic17C22⋊C411yescharacteristic
88C4×Dic17C22⋊C411yescharacteristic
89C4×Dic17M1611yescharacteristic
90C4×Dic17M1611yescharacteristic
91C2×#3622—21no—
92C2×#3622—41no—
93C2×#3622—41no—
94C2×#3622C2×D411yescharacteristic
95C2×#3622C2×D411yescharacteristic
96#63218—42no—
97#63218—42no—
98#63216—42no—
99#63216—42no—
100#11489—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—22no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C22×C811yescharacteristic
106C22×D17—21no—
107C22×D17—22no—
108C22×D17—41no—
109C22×D17—41no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17C2×C22⋊C411yescharacteristic
118C2×C4×C17C2×M1611yescharacteristic
119C22⋊D17—42no—
120C22⋊D17—42no—
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—21no—
124C2×Dic17—41no—
125C2×Dic17C22⋊C811yescharacteristic
126C2×Dic17C22⋊C811yescharacteristic
127C2×Dic17C22⋊C811yescharacteristic
128C2×Dic17C22⋊C811yescharacteristic
129C4×D17—42no—
130C4×D17—41no—
131C4×D17—41no—
132C8×C17—21no—
133C8×C17—41no—
134C8×C17#10012yes—
135#3622—21no—
136#3622—21no—
137#3622—41no—
138C2×#436—171no—
139#2163—171no—
140#2162—171no—
141#1314—172no—
142#1314—172no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—82no—
149C22×C17C2×C22⋊C811yescharacteristic
150C22×C17#50412yes—
151C2×D17—41no—
152C2×D17—41no—
153C2×D17—41no—
154C2×D17—82no—
155C4×C17—21no—
156C4×C17—41no—
157C4×C17#41011yescharacteristic
158C4×C17C2×#10011yescharacteristic
159Dic17—21no—
160Dic17—21no—
161Dic17—21no—
162Dic17—41no—
163Dic17#34912yes—
164C22×M16—171no—
165C22×M16#59511yescharacteristic
166C2×C4×D4—171no—
167#436—341no—
168#436—341no—
169#436—341no—
170#436—341no—
171C2×C4⋊C8—171no—
172C2×C22⋊C8—171no—
173C2×C22⋊C8—171no—
174C2×C4×C8—171no—
175C4⋊C16—341no—
176C4⋊C16—341no—
177C22⋊C16—172no—
178C22⋊C16—172no—
179#346—341no—
180C4×C16—341no—
181C2×C17—22no—
182C2×C17—81no—
183C2×C17#146011yescharacteristic
184C2×C17#211311yescharacteristic
185C2×C17C2×#34911yescharacteristic
186D17—81no—
187C22×D4—171no—
188C23×C4—171no—
189C23×C4C2×#59511yescharacteristic
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—42no—
193C2×M16—341no—
194C2×M16—341no—
195C2×M16—682no—
196C22×C8—171no—
197C22×C8—171no—
198C22×C8C2×#59511yescharacteristic
199C22×C8C2×#59511yescharacteristic
200C4×D4—341no—
201C4×D4—341no—
202C4×D4—341no—
203C4×D4—341no—
204C2×C4⋊C4—171no—
205C2×C22⋊C4—171no—
206C2×C22⋊C4—171no—
207C2×C42—171no—
208C2×C16—342no—
209C2×C16—342no—
210C4⋊C8—341no—
211C4⋊C8—341no—
212C22⋊C8—341no—
213C22⋊C8—341no—
214C22⋊C8—341no—
215C22⋊C8—341no—
216C4×C8—171no—
217C4×C8—171no—
218C4×C8—171no—
219C4×C8—171no—
220C17#1148911yescharacteristic, complemented, Sylow
221C24—171no—
222C24#6324211yescharacteristic
223C2×D4—341no—
224C2×D4—341no—
225C2×D4—682no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—681no—
236C22×C4—681no—
237C22×C4#6324211yescharacteristic
238C22×C4C22×#59511yescharacteristic
239M16—42no—
240M16—42no—
241M16—682no—
242M16—682no—
243C2×C8—22no—
244C2×C8—21no—
245C2×C8—41no—
246C2×C8—171no—
247C2×C8—171no—
248C2×C8—341no—
249C2×C8—681no—
250C2×C8—681no—
251C2×C8C22⋊#59511yescharacteristic
252C2×C8C22⋊#59511yescharacteristic
253C4⋊C4—341no—
254C4⋊C4—341no—
255C22⋊C4—341no—
256C22⋊C4—341no—
257C22⋊C4—341no—
258C22⋊C4—341no—
259C42—171no—
260C42—171no—
261C42—171no—
262C42—171no—
263C16—682no—
264C16—682no—
265C23—21no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—341no—
270C23—342no—
271C23—681no—
272C23—681no—
273C23C2×#6324211yescharacteristic
274D4—682no—
275D4—682no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—171no—
284C2×C4—171no—
285C2×C4—171no—
286C2×C4—171no—
287C2×C4—342no—
288C2×C4—342no—
289C2×C4—341no—
290C2×C4—682no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4D17⋊M1611yescharacteristic
295C2×C4C2×C22⋊#59511yescharacteristic
296C8—21no—
297C8—41no—
298C8—341no—
299C8—341no—
300C8—681no—
301C8#1058755312yes—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—82no—
308C22—681no—
309C22—681no—
310C22—681no—
311C22—1362no—
312C22#1176377711yescharacteristic
313C22#1176386812yes—
314C4—21no—
315C4—41no—
316C4—172no—
317C4—341no—
318C4—341no—
319C4—341no—
320C4—681no—
321C4#1176378811yescharacteristic
322C4C2×#1058755311yescharacteristic
323C2—22no—
324C2—81no—
325C2—1361no—
326C2#1454402311yescharacteristic
327C2#1454501511yescharacteristic
328C2#1454501811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#595
commutatorC23×C17C22×C8
FrattiniC22×C4C22×#595
FittingC22×M16×C17C4
socleC22×C17C2×C22⋊C8
prime radicalC22×C22⋊D17C8
cosocleC22×#3622C23
hypercenterC22×M16#595
hypocenterC17#11489
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #14545146C2semidirect
2. #14545142C2semidirect
3. #14543705C2semidirect
4. #14543704C2semidirect
220. C17#11489semidirect

Maximal subgroups

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

subgroupindex
#145451462
#145451422
#145437052
#145437042
C2×#117628852
#1148917