Finite Group Databasebeta

← #16923482

Subgroups — #16923482

Order 4352.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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#14530342C211yesmaximal, characteristic
2#14530255C211yesmaximal, characteristic
3#14530192C211yesmaximal, characteristic
4#14526610C211yesmaximal, characteristic
5#14526607C211yesmaximal, characteristic
6#14526607C211yesmaximal, characteristic
7C17×#2805C211yesmaximal, characteristic
8C2×#10587478—21no—
9C2×#10587506—21no—
10#11762679C2211yescharacteristic
11#11762571—21no—
12#11762571—21no—
13#11762571—21no—
14#11762571—21no—
15#11762564C2211yescharacteristic
16#11762564C2211yescharacteristic
17#11762564C2211yescharacteristic
18C17×#400—21no—
19C17×#400—21no—
20C17×#395—21no—
21C17×#391C2211yescharacteristic
22C17×#389—21no—
23C17×#389C2211yescharacteristic
24C17×#389C2211yescharacteristic
25C17×#387—21no—
26C17×#387—21no—
27C22×Q8×C17D411yescharacteristic
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C42×C17—21no—
38C2×C42×C17C2311yescharacteristic
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C22×#3622—21no—
43C22×#3622—21no—
44C22×#3622—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54#10587506—42no—
55#10587478—42no—
56C2×Q8×C17—21no—
57C2×Q8×C17—21no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—41no—
70C4⋊C4×C17—21no—
71C4⋊C4×C17—21no—
72C4⋊C4×C17—21no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—81no—
81C4⋊C4×C17—81no—
82C4⋊C4×C17Q1612yes—
83C42×C17—41no—
84C42×C17—81no—
85C4⋊Dic17—42no—
86C4⋊Dic17—41no—
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90C2×#3622—22no—
91C2×#3622—41no—
92C2×#3622—41no—
93C2×#3622—41no—
94C2×#3622—41no—
95C2×#3622—41no—
96C2×#3622—41no—
97C2×#3622—41no—
98C2×#3622—41no—
99#7806—171nomaximal, Sylow
100C23×C17#12011yescharacteristic
101Q8×C17—42no—
102Q8×C17—81no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
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—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17C2×Q1611yescharacteristic
128C2×C4×C17#13511yescharacteristic
129C2×C4×C17#13511yescharacteristic
130C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C17#13711yescharacteristic
133C2×Dic17—81no—
134C2×Dic17—81no—
135C2×Dic17—81no—
136#3622—42no—
137#3622—81no—
138#3622—81no—
139#2805D1711yescharacteristic
140#2099—171no—
141#2083—171no—
142#2072—171no—
143#1272—171no—
144#1271—171no—
145#1271—171no—
146C22×C17#44911yescharacteristic
147C22×C17#44911yescharacteristic
148C22×C17#45111yescharacteristic
149C22×C17#45411yescharacteristic
150C22×C17#45511yescharacteristic
151C22×C17#45611yescharacteristic
152C22×C17#45811yescharacteristic
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—42no—
160C4×C17—81no—
161C4×C17—81no—
162C4×C17—81no—
163C4×C17—81no—
164C4×C17—81no—
165Dic17—161no—
166C2×#107—341no—
167C2×#103—341no—
168#400—21no—
169#400—21no—
170#395—21no—
171#391C2×D1711yescharacteristic
172#389—21no—
173#389C2×D1711yescharacteristic
174#389C2×D1711yescharacteristic
175#387—21no—
176#387—21no—
177#384—171no—
178#340—341no—
179#340—341no—
180#340—341no—
181#340—341no—
182#336—171no—
183#336—171no—
184#336—171no—
185C2×C17#160911yescharacteristic
186C2×C17#161211yescharacteristic
187C2×C17#161711yescharacteristic
188C2×C17#162811yescharacteristic
189C2×C17#164411yescharacteristic
190C2×C17#164811yescharacteristic
191C2×C17#166411yescharacteristic
192C22×Q8C22⋊D1711yescharacteristic
193C22×C8—341no—
194C22×C8—341no—
195C22×C8—341no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4C22⋊D1711yescharacteristic
204C2×C4⋊C4C22⋊D1711yescharacteristic
205C2×C4⋊C4C22⋊D1711yescharacteristic
206C2×C4⋊C4C22⋊D1711yescharacteristic
207C2×C4⋊C4C22⋊D1711yescharacteristic
208C2×C42—21no—
209C2×C42C22×D1711yescharacteristic
210#107—682no—
211#103—682no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221C17#780611yescharacteristic, complemented, Sylow
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—41no—
225C2×Q8—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—681no—
233C22×C4C2×C22⋊D1711yescharacteristic
234C22×C4C2×C22⋊D1711yescharacteristic
235C22×C4C2×C22⋊D1711yescharacteristic
236C2×C8—342no—
237C2×C8—681no—
238C2×C8—681no—
239C2×C8—681no—
240C2×C8—681no—
241C2×C8—681no—
242C2×C8—681no—
243C2×C8—681no—
244C2×C8—681no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—682no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C4⋊C4—681no—
262C4⋊C4#6323112yes—
263C42—41no—
264C42—81no—
265C23#1058763911yescharacteristic
266Q8—42no—
267Q8—81no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—1361no—
293C2×C4—1361no—
294C2×C4—1361no—
295C2×C4#1058762811yescharacteristic
296C2×C4C2×#6323111yescharacteristic
297C2×C4#1058763511yescharacteristic
298C2×C4#1058763511yescharacteristic
299C2×C4#1058763611yescharacteristic
300C2×C4#1058763611yescharacteristic
301C8—682no—
302C8—1361no—
303C8—1361no—
304C22#1176295411yescharacteristic
305C22#1176295411yescharacteristic
306C22#1176296511yescharacteristic
307C22#1176298211yescharacteristic
308C22#1176298711yescharacteristic
309C22#1176299211yescharacteristic
310C22#1176299911yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—2721no—
324C2#1452778111yescharacteristic
325C2#1452779811yescharacteristic
326C2#1452782611yescharacteristic
327C2#1452789411yescharacteristic
328C2#1452799811yescharacteristic
329C2#1452802611yescharacteristic
330C2#1452811811yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2805C2
socle, prime radicalC23×C17#120
hypercenter#2805D17
hypocenterC17#7806
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

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

NH ≅ G/Ntype
221. C17#7806semidirect

Maximal subgroups

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

subgroupindex
#145303422
#145302552
#145301922
#145266102
#145266072
C17×#28052
#780617