Finite Group Databasebeta

← #16923658

Subgroups — #16923658

Order 4352.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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#14536737C211yesmaximal, characteristic
2#14530348C211yesmaximal, characteristic
3#14530347C211yesmaximal, characteristic
4#14530321C211yesmaximal, characteristic
5#14526609C211yesmaximal, characteristic
6#14526608C211yesmaximal, characteristic
7C17×#1271C211yesmaximal, characteristic
8C2×#10587489—21no—
9C2×#10587487—21no—
10#11762748—21no—
11#11762748—21no—
12#11762730—21no—
13#11762715C2211yescharacteristic
14#11762708—21no—
15#11762707C2211yescharacteristic
16#11762696—21no—
17#11762695—21no—
18#11762679C2211yescharacteristic
19#11762572—21no—
20#11762572—21no—
21#11762571—21no—
22#11762571—21no—
23#11762564C2211yescharacteristic
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×Dic34D411yescharacteristic
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⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×C4⋊Dic17D411yescharacteristic
42C2×#63225—21no—
43C2×C4×Dic17—21no—
44C22×#3622—21no—
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#10587489—42no—
55#10587487—42no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17C2×D411yescharacteristic
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×Dic17—21no—
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17—41no—
66C2×Dic34—21no—
67C2×Dic34—21no—
68C2×Dic34—41no—
69C2×Dic34—41no—
70C2×C8×C17—22no—
71C2×C8×C17—41no—
72C2×C8×C17—41no—
73C2×C8×C17—41no—
74C2×C8×C17—41no—
75C2×C8×C17—41no—
76C4⋊C4×C17—21no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17Q1612yes—
81C42×C17—41no—
82C4⋊Dic17—21no—
83C4⋊Dic17—21no—
84C4⋊Dic17—41no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4⋊Dic17—41no—
88C4⋊Dic17—42no—
89C4⋊Dic17—42no—
90C4⋊Dic17—81no—
91C4⋊Dic17—81no—
92#63225—41no—
93#63225—41no—
94C4×Dic17—81no—
95C2×#3622—41no—
96C2×#3622—41no—
97C2×#3622—41no—
98#7835—171nomaximal, Sylow
99C23×C17#12011yescharacteristic
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17C2×Q1611yescharacteristic
113C2×C4×C17#13511yescharacteristic
114C2×C4×C17#13511yescharacteristic
115C2×C4×C17#13711yescharacteristic
116C2×C4×C17#13711yescharacteristic
117C2×C4×C17#13711yescharacteristic
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—81no—
131C2×Dic17—81no—
132C2×Dic17—81no—
133Dic34—42no—
134Dic34—81no—
135C8×C17—42no—
136C8×C17—81no—
137#3622—81no—
138#2805—171no—
139#2100—171no—
140#2100—171no—
141#2095—171no—
142#1272—171no—
143#1271—171no—
144#1271D1711yescharacteristic
145C22×C17#45111yescharacteristic
146C22×C17#45211yescharacteristic
147C22×C17#45211yescharacteristic
148C22×C17#45311yescharacteristic
149C22×C17#45411yescharacteristic
150C22×C17#45411yescharacteristic
151C22×C17#45811yescharacteristic
152C4×C17—21no—
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—42no—
159C4×C17—81no—
160Dic17—81no—
161Dic17—81no—
162Dic17—81no—
163Dic17—81no—
164Dic17—161no—
165C2×#107—341no—
166C2×#103—341no—
167#400—341no—
168#400—341no—
169#395—341no—
170#391—171no—
171#389—171no—
172#389—341no—
173#389C2×D1711yescharacteristic
174#387—341no—
175#387—341no—
176#384—171no—
177#340—341no—
178#340—341no—
179#340—341no—
180#340—341no—
181#336—171no—
182#336C2×D1711yescharacteristic
183#336C2×D1711yescharacteristic
184C2×C17#161511yescharacteristic
185C2×C17#161711yescharacteristic
186C2×C17#161811yescharacteristic
187C2×C17#163211yescharacteristic
188C2×C17#163611yescharacteristic
189C2×C17#163811yescharacteristic
190C2×C17#165711yescharacteristic
191C22×Q8—171no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—341no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—171no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4D6811yescharacteristic
206C2×C4⋊C4D6811yescharacteristic
207C2×C42—341no—
208C2×C42C22×D1711yescharacteristic
209#107—682no—
210#103—682no—
211#95—21no—
212#95—341no—
213#95—341no—
214#95—341no—
215#95—341no—
216#95—341no—
217#95—341no—
218#95—341no—
219#95—341no—
220C17#783511yescharacteristic, complemented, Sylow
221C2×Q8—341no—
222C2×Q8—341no—
223C2×Q8—681no—
224C2×Q8—681no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—681no—
232C22×C4C2×D6811yescharacteristic
233C22×C4D4×D1711yescharacteristic
234C22×C4D4×D1711yescharacteristic
235C2×C8—22no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—681no—
242C2×C8—681no—
243C2×C8—681no—
244C4⋊C4—21no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—341no—
249C4⋊C4—341no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C4⋊C4—682no—
255C4⋊C4—682no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—1361no—
259C4⋊C4—1361no—
260C4⋊C4Dic6812yes—
261C42—41no—
262C42—1361no—
263C23#1058756811yescharacteristic
264Q8—682no—
265Q8—1361no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—681no—
279C2×C4—681no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—1361no—
291C2×C4—1361no—
292C2×C4—1361no—
293C2×C4C2×Dic6811yescharacteristic
294C2×C4#1058759511yescharacteristic
295C2×C4#1058759811yescharacteristic
296C2×C4#1058760111yescharacteristic
297C2×C4#1058760211yescharacteristic
298C2×C4#1058760411yescharacteristic
299C8—42no—
300C8—81no—
301C8—1361no—
302C22#1176296711yescharacteristic
303C22#1176297211yescharacteristic
304C22#1176297411yescharacteristic
305C22#1176297811yescharacteristic
306C22#1176298311yescharacteristic
307C22#1176298411yescharacteristic
308C22#1176300011yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—81no—
317C4—2721no—
318C4—1361no—
319C4—1361no—
320C4—1361no—
321C4—1361no—
322C2#1452781611yescharacteristic
323C2#1452782811yescharacteristic
324C2#1452783511yescharacteristic
325C2#1452792011yescharacteristic
326C2#1452794511yescharacteristic
327C2#1452795711yescharacteristic
328C2#1452808411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socle, prime radicalC23×C17#120
hypercenter#1271D17
hypocenterC17#7835
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
220. C17#7835semidirect

Maximal subgroups

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

subgroupindex
#145367372
#145303482
#145303472
#145303212
#145266092
#145266082
C17×#12712
#783517