Finite Group Databasebeta

← #16922958

Subgroups — #16922958

Order 4352.

388 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 101 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#14537143C211yesmaximal, characteristic
2#14528926C211yesmaximal, characteristic
3#14528899C211yesmaximal, characteristic
4#14526608C211yesmaximal, characteristic
5#14526607C211yesmaximal, characteristic
6#14526606C211yesmaximal, characteristic
7C17×#1271C211yesmaximal, characteristic
8C2×#10587575—21no—
9C2×#10587485—21no—
10#11762708C411yescharacteristic
11#11762707C2211yescharacteristic
12#11762706—21no—
13#11762705C411yescharacteristic
14#11762704—21no—
15#11762678C411yescharacteristic
16#11762678C411yescharacteristic
17#11762673—21no—
18C42×Dic17C2211yescharacteristic
19#11762564C2211yescharacteristic
20#11762564C2211yescharacteristic
21#11762563—21no—
22#11762563—21no—
23#11762563—21no—
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×C8×C17—21no—
28C22×C8×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C42×C17C2311yescharacteristic
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17C2×C411yescharacteristic
43C2×C4×Dic17C2×C411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C22×#3622—21no—
49C22×#3622—21no—
50#10587575—42no—
51#10587575—42no—
52C17×#95—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587485—42no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C22⋊C411yescharacteristic
68C2×C8×C17—41no—
69C2×C8×C17—41no—
70C2×C8×C17—41no—
71C2×C8×C17—41no—
72C2×C8×C17—41no—
73C2×C8×C17—41no—
74C4⋊C4×C17—21no—
75C4⋊C4×C17—21no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C42×C17—41no—
80C4⋊Dic17—21no—
81C4⋊Dic17—21no—
82C4⋊Dic17—42no—
83C4⋊Dic17—41no—
84#63225—42no—
85#63225—42no—
86#63225—42no—
87#63225—42no—
88#63225—42no—
89#63225—41no—
90#63225—41no—
91C4×Dic17—21no—
92C4×Dic17—21no—
93C4×Dic17—21no—
94C4×Dic17—21no—
95C4×Dic17—22no—
96C4×Dic17—41no—
97C4×Dic17—41no—
98C4×Dic17SD1612yes—
99C4×Dic17Q1612yes—
100C2×#3622—41no—
101C2×#3622—41no—
102C2×#3622—41no—
103C2×#3622—41no—
104C2×#3622—41no—
105C2×#3622—41no—
106#7720—171nomaximal, Sylow
107C23×C17C2×C22⋊C411yescharacteristic
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17#13111yescharacteristic
121C2×C4×C17#13111yescharacteristic
122C2×C4×C17C2×SD1611yescharacteristic
123C2×C4×C17C2×Q1611yescharacteristic
124C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17#10112yes—
142C2×Dic17#10312yes—
143C2×Dic17#10312yes—
144C8×C17—81no—
145C8×C17—81no—
146#3622—81no—
147#3622—81no—
148#2859—171no—
149#1839—171no—
150#1831—171no—
151#1271—171no—
152#1271—171no—
153#1271—171no—
154#1271D1711yescharacteristic
155C22×C17#40811yescharacteristic
156C22×C17#41011yescharacteristic
157C22×C17C2×#10111yescharacteristic
158C22×C17C2×#10311yescharacteristic
159C22×C17#41811yescharacteristic
160C22×C17#42111yescharacteristic
161C22×C17#42111yescharacteristic
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—81no—
169C4×C17—81no—
170Dic17—22no—
171Dic17—22no—
172Dic17—22no—
173Dic17—41no—
174Dic17—81no—
175Dic17—81no—
176C2×#125—341no—
177C2×C4⋊C8—341no—
178#389—171no—
179#389—171no—
180#389—171no—
181#389—341no—
182#389—341no—
183#389C2×D1711yescharacteristic
184#384—171no—
185#384—171no—
186#382—341no—
187C43—171no—
188#336—171no—
189#336—171no—
190#336—341no—
191#336—341no—
192#336—341no—
193#336C2×D1711yescharacteristic
194#336C2×D1711yescharacteristic
195C2×C17#147011yescharacteristic
196C2×C17#148111yescharacteristic
197C2×C17#149611yescharacteristic
198C2×C17#151411yescharacteristic
199C2×C17#151811yescharacteristic
200C2×C17#153411yescharacteristic
201C2×C17#154911yescharacteristic
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—341no—
205C22×C8—341no—
206#125—682no—
207#125—682no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—171no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4C4×D1711yescharacteristic
219C2×C4⋊C4C4×D1711yescharacteristic
220C2×C42—171no—
221C2×C42—171no—
222C2×C42—171no—
223C2×C42—171no—
224C2×C42—171no—
225C2×C42—171no—
226C2×C42C22×D1711yescharacteristic
227C4⋊C8—682no—
228#95—21no—
229#95—341no—
230#95—341no—
231#95—341no—
232C17#772011yescharacteristic, complemented, Sylow
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4C2×C4×D1711yescharacteristic
242C22×C4D4×D1711yescharacteristic
243C22×C4D4×D1711yescharacteristic
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—681no—
251C2×C8—681no—
252C2×C8—681no—
253C2×C8—681no—
254C2×C8—681no—
255C2×C8—681no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—341no—
262C4⋊C4—341no—
263C4⋊C4—681no—
264C4⋊C4—682no—
265C4⋊C4—682no—
266C4⋊C4—682no—
267C4⋊C4—682no—
268C4⋊C4—682no—
269C4⋊C4—682no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C42—41no—
273C42—172no—
274C42—172no—
275C42—342no—
276C42—341no—
277C42—341no—
278C42—341no—
279C42—341no—
280C42—681no—
281C42—681no—
282C23C22⋊C4×D1711yescharacteristic
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—172no—
296C2×C4—172no—
297C2×C4—172no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4#1058758711yescharacteristic
314C2×C4#1058759311yescharacteristic
315C2×C4#1058759711yescharacteristic
316C2×C4SD16×D1711yescharacteristic
317C2×C4#1058760011yescharacteristic
318C2×C4Q16×D1711yescharacteristic
319C8—81no—
320C8—81no—
321C8—1361no—
322C8—1361no—
323C22#1176277411yescharacteristic
324C22#1176278111yescharacteristic
325C22#101×D1711yescharacteristic
326C22#103×D1711yescharacteristic
327C22#1176281211yescharacteristic
328C22#1176282411yescharacteristic
329C22#1176282411yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—81no—
338C4—342no—
339C4—342no—
340C4—342no—
341C4—681no—
342C4—1361no—
343C4—1361no—
344C2#1452713811yescharacteristic
345C2#1452718511yescharacteristic
346C2#1452725111yescharacteristic
347C2#1452732911yescharacteristic
348C2#1452734611yescharacteristic
349C2#1452740911yescharacteristic
350C2#1452746711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1271D17
hypocenterC17#7720
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
232. C17#7720semidirect

Maximal subgroups

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

subgroupindex
#145371432
#145289262
#145288992
#145266082
#145266072
#145266062
C17×#12712
#772017