Finite Group Databasebeta

← #16923391

Subgroups — #16923391

Order 4352.

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

normal subgroups only

This group has 374 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#14530342C211yesmaximal, characteristic
3#14530257C211yesmaximal, characteristic
4#14530257C211yesmaximal, characteristic
5#14526608C211yesmaximal, characteristic
6#14526608C211yesmaximal, characteristic
7C17×#1271C211yesmaximal, characteristic
8C2×#10587576—21no—
9C2×#10587489—21no—
10C2×#10587489—21no—
11#11762742—21no—
12#11762742—21no—
13#11762741—21no—
14#11762707C2211yescharacteristic
15#11762707C2211yescharacteristic
16#11762706—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×C8×C17—21no—
28C22×C8×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C42×C17C2311yescharacteristic
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
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—
48#10587576—42no—
49#10587576—42no—
50C17×#95—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587489—42no—
57#10587489—42no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—41no—
68C2×C8×C17—22no—
69C2×C8×C17—22no—
70C2×C8×C17—41no—
71C2×C8×C17—41no—
72C2×C8×C17—41no—
73C2×C8×C17—41no—
74C4⋊C4×C17—21no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17Q1612yes—
79C42×C17—41no—
80C4⋊Dic17—21no—
81C4⋊Dic17—21no—
82C4⋊Dic17—21no—
83C4⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4⋊Dic17—42no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17D812yes—
93#63225—42no—
94#63225—41no—
95#63225—41no—
96#63225—41no—
97#63225—41no—
98#63225—81no—
99C4×Dic17—42no—
100C2×#3622—41no—
101C2×#3622—41no—
102C2×#3622—41no—
103#7792—171nomaximal, Sylow
104C23×C17#12011yescharacteristic
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17C2×D811yescharacteristic
118C2×C4×C17C2×Q1611yescharacteristic
119C2×C4×C17#13511yescharacteristic
120C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C17#13711yescharacteristic
123C2×Dic17—22no—
124C2×Dic17—41no—
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—81no—
136C2×Dic17—81no—
137C2×Dic17—81no—
138C8×C17—42no—
139C8×C17—42no—
140#3622—81no—
141#2726—171no—
142#2099—171no—
143#2083—171no—
144#2083—171no—
145#1271—171no—
146#1271—171no—
147#1271D1711yescharacteristic
148C22×C17#44711yescharacteristic
149C22×C17#44911yescharacteristic
150C22×C17#45111yescharacteristic
151C22×C17#45411yescharacteristic
152C22×C17#45611yescharacteristic
153C22×C17#45611yescharacteristic
154C22×C17#45811yescharacteristic
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—42no—
162C4×C17—81no—
163Dic17—42no—
164Dic17—81no—
165Dic17—81no—
166Dic17—81no—
167Dic17—161no—
168C2×#125—341no—
169C2×#107—341no—
170C2×#107—341no—
171#398—341no—
172#398—341no—
173#398—341no—
174#389—171no—
175#389—171no—
176#389—341no—
177#389C2×D1711yescharacteristic
178#387—341no—
179#384—171no—
180#340—341no—
181#340—341no—
182#340—341no—
183#340—341no—
184#336—171no—
185#336C2×D1711yescharacteristic
186#336C2×D1711yescharacteristic
187C2×C17#160911yescharacteristic
188C2×C17#161011yescharacteristic
189C2×C17#161711yescharacteristic
190C2×C17#162511yescharacteristic
191C2×C17#164811yescharacteristic
192C2×C17#164911yescharacteristic
193C2×C17#166311yescharacteristic
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—341no—
197#125—682no—
198#125—682no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4D6811yescharacteristic
214C2×C4⋊C4D6811yescharacteristic
215C2×C42—341no—
216C2×C42C22×D1711yescharacteristic
217#107—682no—
218#107—682no—
219#95—21no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95—341no—
225C17#779211yescharacteristic, complemented, Sylow
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—341no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—341no—
232C22×C4—681no—
233C22×C4C2×D6811yescharacteristic
234C22×C4D4×D1711yescharacteristic
235C22×C4D4×D1711yescharacteristic
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—681no—
243C2×C8—681no—
244C2×C8—681no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—172no—
250C4⋊C4—341no—
251C4⋊C4—341no—
252C4⋊C4—341no—
253C4⋊C4—681no—
254C4⋊C4—681no—
255C4⋊C4—682no—
256C4⋊C4—682no—
257C4⋊C4—682no—
258C4⋊C4—682no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C4⋊C4—681no—
262C4⋊C4—681no—
263C4⋊C4—681no—
264C4⋊C4—681no—
265C4⋊C4—681no—
266C4⋊C4—681no—
267C4⋊C4—1361no—
268C4⋊C4Dic6812yes—
269C42—41no—
270C42—682no—
271C23#1058756811yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—342no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—1361no—
297C2×C4—1361no—
298C2×C4—1361no—
299C2×C4C2×Dic6811yescharacteristic
300C2×C4#1058759511yescharacteristic
301C2×C4D8×D1711yescharacteristic
302C2×C4#1058760011yescharacteristic
303C2×C4#1058760011yescharacteristic
304C2×C4#1058760511yescharacteristic
305C8—42no—
306C8—42no—
307C8—1361no—
308C22#1176294411yescharacteristic
309C22#1176295711yescharacteristic
310C22#1176296711yescharacteristic
311C22#1176298411yescharacteristic
312C22#1176299311yescharacteristic
313C22#1176299311yescharacteristic
314C22#1176300011yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—81no—
323C4—2721no—
324C4—682no—
325C4—1361no—
326C4—1361no—
327C4—1361no—
328C2#1452778411yescharacteristic
329C2#1452778811yescharacteristic
330C2#1452782811yescharacteristic
331C2#1452787411yescharacteristic
332C2#1452802911yescharacteristic
333C2#1452803611yescharacteristic
334C2#1452811311yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socle, prime radicalC23×C17#120
hypercenter#1271D17
hypocenterC17#7792
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
225. C17#7792semidirect

Maximal subgroups

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

subgroupindex
#145357352
#145303422
#145302572
#145266082
C17×#12712
#779217