Finite Group Databasebeta

← #16923274

Subgroups — #16923274

Order 4352.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#14536775C211yesmaximal, characteristic
2#14530118C211yesmaximal, characteristic
3#14530112C211yesmaximal, characteristic
4#14530031C211yesmaximal, characteristic
5#14526608C211yesmaximal, characteristic
6#14526606C211yesmaximal, characteristic
7C17×#1271C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9C2×#10587478—21no—
10C2×#10587488—21no—
11C2×#10587487—21no—
12C2×#10587487—21no—
13#11762749—21no—
14#11762748—21no—
15#11762747—21no—
16#11762707C2211yescharacteristic
17#11762707C2211yescharacteristic
18#11762696—21no—
19#11762695—21no—
20#11762690—21no—
21#11762572—21no—
22#11762571—21no—
23#11762564C2211yescharacteristic
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×Dic34—21no—
28C22×C8×C17—21no—
29C22×C8×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C42×C17C2311yescharacteristic
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17D411yescharacteristic
37C2×C4⋊Dic17D411yescharacteristic
38C2×C4⋊Dic17D411yescharacteristic
39C2×C4⋊Dic17D411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C22×#3622—21no—
44#10587555—42no—
45C17×#95—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587488—42no—
56#10587487—42no—
57#10587487—42no—
58#10587478—42no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—41no—
69C2×Dic34—22no—
70C2×Dic34—41no—
71C2×Dic34—41no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C2×Dic34—41no—
75C2×C8×C17—22no—
76C2×C8×C17—41no—
77C2×C8×C17—41no—
78C2×C8×C17—41no—
79C2×C8×C17—41no—
80C2×C8×C17—41no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C42×C17—41no—
87C4⋊Dic17—22no—
88C4⋊Dic17—22no—
89C4⋊Dic17—21no—
90C4⋊Dic17—21no—
91C4⋊Dic17—21no—
92C4⋊Dic17—21no—
93C4⋊Dic17—41no—
94C4⋊Dic17—41no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97#63225—41no—
98#63225—41no—
99#63225—41no—
100#63225—41no—
101C4×Dic17—81no—
102C2×#3622—22no—
103C2×#3622—41no—
104C2×#3622—41no—
105#7775—171nomaximal, Sylow
106C23×C17#12011yescharacteristic
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17#13511yescharacteristic
120C2×C4×C17#13511yescharacteristic
121C2×C4×C17#13511yescharacteristic
122C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C17#13711yescharacteristic
124C2×C4×C17#13711yescharacteristic
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—42no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—81no—
139C2×Dic17—81no—
140Dic34—42no—
141Dic34—42no—
142Dic34—81no—
143Dic34—81no—
144C8×C17—42no—
145C8×C17—81no—
146#3622—42no—
147#2810—171no—
148#2058—171no—
149#2057—171no—
150#2043—171no—
151#1271—171no—
152#1271—171no—
153#1271D1711yescharacteristic
154C22×C17#44911yescharacteristic
155C22×C17#45211yescharacteristic
156C22×C17#45211yescharacteristic
157C22×C17#45411yescharacteristic
158C22×C17#45611yescharacteristic
159C22×C17#45611yescharacteristic
160C22×C17B(4,𝔽2)11yescharacteristic
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—81no—
168C4×C17—81no—
169Dic17—82no—
170Dic17—81no—
171Dic17—81no—
172Dic17—81no—
173Dic17—81no—
174C2×C4⋊Q8—171no—
175C2×#107—341no—
176C2×#106—341no—
177C2×#103—341no—
178C2×#103—341no—
179#400—341no—
180#400—341no—
181#400—341no—
182#389—171no—
183#389—171no—
184#389C2×D1711yescharacteristic
185#387—341no—
186#387—341no—
187#387—341no—
188#340—341no—
189#340—341no—
190#336—171no—
191#336C2×D1711yescharacteristic
192#336C2×D1711yescharacteristic
193C2×C17#159711yescharacteristic
194C2×C17#160311yescharacteristic
195C2×C17#160311yescharacteristic
196C2×C17#163811yescharacteristic
197C2×C17#165011yescharacteristic
198C2×C17#165211yescharacteristic
199C2×C17#166311yescharacteristic
200C22×Q8—341no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—341no—
204C4⋊Q8—682no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—171no—
208C2×C4⋊C4—171no—
209C2×C4⋊C4—171no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4D6811yescharacteristic
215C2×C4⋊C4D6811yescharacteristic
216C2×C42—341no—
217C2×C42C22×D1711yescharacteristic
218#107—682no—
219#106—682no—
220#103—682no—
221#103—682no—
222#95—21no—
223#95—341no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230#95—341no—
231#95—341no—
232C17#777511yescharacteristic, complemented, Sylow
233C2×Q8—342no—
234C2×Q8—681no—
235C2×Q8—681no—
236C2×Q8—681no—
237C2×Q8—681no—
238C2×Q8—681no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—681no—
246C22×C4C2×D6811yescharacteristic
247C22×C4D4×D1711yescharacteristic
248C22×C4D4×D1711yescharacteristic
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—342no—
256C2×C8—681no—
257C2×C8—681no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—342no—
264C4⋊C4—342no—
265C4⋊C4—341no—
266C4⋊C4—341no—
267C4⋊C4—341no—
268C4⋊C4—341no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—681no—
277C42—41no—
278C42—1361no—
279C23#1058756811yescharacteristic
280Q8—682no—
281Q8—682no—
282Q8—1361no—
283Q8—1361no—
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—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
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—682no—
309C2×C4—1361no—
310C2×C4—1361no—
311C2×C4#1058759011yescharacteristic
312C2×C4#1058759511yescharacteristic
313C2×C4#1058759711yescharacteristic
314C2×C4#1058760211yescharacteristic
315C2×C4#1058760211yescharacteristic
316C2×C4#1058760411yescharacteristic
317C8—42no—
318C8—81no—
319C8—682no—
320C22#1176295511yescharacteristic
321C22#1176297311yescharacteristic
322C22#1176297411yescharacteristic
323C22#1176298311yescharacteristic
324C22#1176299311yescharacteristic
325C22#1176299411yescharacteristic
326C22#1176299711yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—81no—
334C4—81no—
335C4—1361no—
336C4—1361no—
337C4—1361no—
338C4—1361no—
339C4—1362no—
340C2#1452771411yescharacteristic
341C2#1452774911yescharacteristic
342C2#1452775011yescharacteristic
343C2#1452795911yescharacteristic
344C2#1452804111yescharacteristic
345C2#1452805711yescharacteristic
346C2#1452811411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socle, prime radicalC23×C17#120
hypercenter#1271D17
hypocenterC17#7775
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
232. C17#7775semidirect

Maximal subgroups

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

subgroupindex
#145367752
#145301182
#145301122
#145300312
#145266082
#145266062
C17×#12712
#777517