Finite Group Databasebeta

← #16923219

Subgroups — #16923219

Order 4352.

416 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 85 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#14535735C211yesmaximal, characteristic
2#14530114C211yesmaximal, characteristic
3#14530114C211yesmaximal, characteristic
4#14526607C211yesmaximal, characteristic
5#14526606C211yesmaximal, characteristic
6#14526606C211yesmaximal, characteristic
7C17×#2043C211yesmaximal, characteristic
8C2×#10587576—21no—
9C2×#10587479—21no—
10C2×#10587479—21no—
11#11762742—21no—
12#11762742—21no—
13#11762741—21no—
14#11762707C2211yescharacteristic
15#11762707C2211yescharacteristic
16#11762706—21no—
17#11762695—21no—
18#11762571—21no—
19#11762571—21no—
20#11762564C2211yescharacteristic
21#11762564C2211yescharacteristic
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C17×#103—21no—
24C2×C17×#103—21no—
25C17×#389C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×Q8×C17—21no—
28C22×C8×C17—21no—
29C17×C4⋊Q8—42no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C42×C17C2311yescharacteristic
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17D411yescharacteristic
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×C4⋊Dic17D411yescharacteristic
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×#63225—21no—
46C2×C4×Dic17—21no—
47C22×#3622—21no—
48C22×#3622—21no—
49#10587576—42no—
50#10587576—42no—
51C17×#103—42no—
52C17×#103—42no—
53C17×#95—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59#10587479—42no—
60#10587479—42no—
61C2×Q8×C17—22no—
62C2×Q8×C17—41no—
63C2×Q8×C17—41no—
64C2×Q8×C17—41no—
65C2×Q8×C17—41no—
66C2×Q8×C17—41no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—41no—
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C2×C8×C17—41no—
78C2×C8×C17—41no—
79C2×C8×C17—41no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—22no—
82C4⋊C4×C17—22no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17SD1612yes—
89C42×C17—41no—
90C4⋊Dic17—21no—
91C4⋊Dic17—21no—
92C4⋊Dic17—21no—
93C4⋊Dic17—42no—
94C4⋊Dic17—42no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97C4⋊Dic17SD1612yes—
98#63225—42no—
99#63225—41no—
100#63225—41no—
101#63225—41no—
102#63225—41no—
103#63225—81no—
104C4×Dic17—42no—
105C2×#3622—22no—
106C2×#3622—22no—
107C2×#3622—41no—
108C2×#3622—41no—
109C2×#3622—41no—
110C2×#3622—41no—
111#7768—171nomaximal, Sylow
112C23×C17#12011yescharacteristic
113Q8×C17—42no—
114Q8×C17—42no—
115Q8×C17—81no—
116Q8×C17—81no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—22no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17C2×SD1611yescharacteristic
133C2×C4×C17C2×SD1611yescharacteristic
134C2×C4×C17#13511yescharacteristic
135C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C17#13711yescharacteristic
137C2×C4×C17#13711yescharacteristic
138C2×Dic17—22no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C8×C17—81no—
151#3622—42no—
152#3622—42no—
153#2726—171no—
154#2058—171no—
155#2058—171no—
156#2043D1711yescharacteristic
157#1271—171no—
158#1271—171no—
159#1271—171no—
160C22×C17#44811yescharacteristic
161C22×C17#45011yescharacteristic
162C22×C17#45211yescharacteristic
163C22×C17#45411yescharacteristic
164C22×C17#45611yescharacteristic
165C22×C17#45611yescharacteristic
166C22×C17B(4,𝔽2)11yescharacteristic
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—42no—
174C4×C17—82no—
175C4×C17—81no—
176Dic17—42no—
177Dic17—81no—
178Dic17—81no—
179Dic17—81no—
180C2×C4⋊Q8C2×D1711yescharacteristic
181C2×#125—341no—
182C2×#106—341no—
183C2×#106—341no—
184C2×#103—21no—
185C2×#103—21no—
186#398—341no—
187#398—341no—
188#398—341no—
189#389—171no—
190#389—171no—
191#389—341no—
192#389C2×D1711yescharacteristic
193#387—341no—
194#340—341no—
195#340—341no—
196#336—171no—
197#336—171no—
198#336C2×D1711yescharacteristic
199C2×C17#159911yescharacteristic
200C2×C17#160311yescharacteristic
201C2×C17#160411yescharacteristic
202C2×C17#163511yescharacteristic
203C2×C17#165111yescharacteristic
204C2×C17#165311yescharacteristic
205C2×C17#166311yescharacteristic
206C22×Q8—21no—
207C22×C8—21no—
208C22×C8—341no—
209C22×C8—341no—
210C4⋊Q8—42no—
211#125—682no—
212#125—682no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4C22⋊D1711yescharacteristic
227C2×C4⋊C4C22⋊D1711yescharacteristic
228C2×C42—341no—
229C2×C42C22×D1711yescharacteristic
230#106—682no—
231#106—682no—
232#103—42no—
233#103—42no—
234#95—21no—
235#95—341no—
236#95—341no—
237#95—341no—
238#95—341no—
239#95—341no—
240C17#776811yescharacteristic, complemented, Sylow
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—341no—
251C22×C4—341no—
252C22×C4—341no—
253C22×C4—341no—
254C22×C4D4×D1711yescharacteristic
255C22×C4D4×D1711yescharacteristic
256C22×C4C2×C22⋊D1711yescharacteristic
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—342no—
261C2×C8—342no—
262C2×C8—681no—
263C2×C8—681no—
264C2×C8—681no—
265C2×C8—681no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—172no—
275C4⋊C4—341no—
276C4⋊C4—341no—
277C4⋊C4—341no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C4⋊C4—682no—
281C4⋊C4—682no—
282C4⋊C4—682no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C4⋊C4—681no—
286C4⋊C4—681no—
287C4⋊C4—1361no—
288C4⋊C4#6322912yes—
289C42—41no—
290C42—682no—
291C23#1058762311yescharacteristic
292Q8—42no—
293Q8—42no—
294Q8—81no—
295Q8—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—342no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4SD16×D1711yescharacteristic
324C2×C4#1058760011yescharacteristic
325C2×C4#1058760411yescharacteristic
326C2×C4#1058760511yescharacteristic
327C2×C4C2×#6322911yescharacteristic
328C2×C4#1058763711yescharacteristic
329C8—81no—
330C8—682no—
331C8—682no—
332C22#1176294711yescharacteristic
333C22#1176295811yescharacteristic
334C22#1176297011yescharacteristic
335C22#1176298111yescharacteristic
336C22#1176299011yescharacteristic
337C22#1176299011yescharacteristic
338C22#1176299511yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—81no—
347C4—82no—
348C4—682no—
349C4—1361no—
350C4—1361no—
351C4—1361no—
352C2#1452772311yescharacteristic
353C2#1452774611yescharacteristic
354C2#1452775111yescharacteristic
355C2#1452793211yescharacteristic
356C2#1452804511yescharacteristic
357C2#1452805911yescharacteristic
358C2#1452811011yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2043C2
socle, prime radicalC23×C17#120
hypercenter#2043D17
hypocenterC17#7768
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
240. C17#7768semidirect

Maximal subgroups

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

subgroupindex
#145357352
#145301142
#145266072
#145266062
C17×#20432
#776817