Finite Group Databasebeta

← #17002776

Subgroups — #17002776

Order 4352.

1172 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537895C211yesmaximal, characteristic, complemented
2#14537352C211yesmaximal, characteristic, complemented
3#14527461C213yesmaximal, complemented
4#14527456C213yesmaximal, complemented
5#14527433C213yesmaximal, complemented
6C2×#11762555C213yesmaximal, complemented
7C17×#3499C211yesmaximal, characteristic
8C2×#10587634—21no—
9Q8×#3622—21no—
10D4×#3622—23no—
11#11762862C2213yes—
12#11762836—23no—
13#11762833C2213yes—
14C2×C4⋊#3622C2213yes—
15C2×C4⋊#3622C2213yes—
16#11762555C2213yes—
17#11762555C2216yes—
18#11762555C2213yes—
19#11762553C2211yescharacteristic
20#11762553C2213yes—
21C2×C17×2−1+4—21no—
22D4×Q8×C17—21no—
23D4×Q8×C17C412yes—
24C17×#547—23no—
25C17×#547C416yes—
26C17×#536—23no—
27C17×#532C2213yes—
28C2×C17×C4⋊Q8C2213yes—
29C4×#41×C17C2211yescharacteristic
30C17×2−1+4—44no—
31C17×2−1+4—44no—
32C2×#41×C17—26no—
33C2×#41×C17—23no—
34C2×#41×C17D411yescharacteristic
35C22×Q8×C17—22no—
36C22×Q8×C17—23no—
37C17×C4⋊Q8—23no—
38C17×C4⋊Q8—23no—
39C17×C4⋊Q8C2×C416yes—
40C17×C4⋊Q8C2313yes—
41C17×#127C2311yescharacteristic
42C17×#124—23no—
43C17×#123—212no—
44C17×C22⋊Q8—26no—
45C17×C22⋊Q8—26no—
46C17×#121—23no—
47C17×#121C2×C416yes—
48C4×Q8×C17C2×C411yescharacteristic
49C4×Q8×C17D411yescharacteristic
50C4×D4×C17—23no—
51C4×D4×C17C2×C416yes—
52C4×D4×C17C2×C413yes—
53C4×D4×C17D413yes—
54C17×#117D413yes—
55C2×C4⋊C4×C17—23no—
56C2×C4⋊C4×C17C2×C416yes—
57C2×C42×C17C2313yes—
58#10587634—44no—
59C2×#63223—23no—
60C22×#3622—23no—
61C22×#3622—23no—
62C22⋊#3622—23no—
63C4⋊#3622—23no—
64C4⋊#3622—23no—
65C4⋊#3622—23no—
66C4⋊#3622C2311yescharacteristic
67C4⋊#3622C2313yes—
68C4⋊#3622C2313yes—
69C4⋊#3622C2311yescharacteristic
70C4×#3622—23no—
71#41×C17—424no—
72#41×C17—46no—
73#41×C17—46no—
74#41×C17D814yes—
75#41×C17SD1614yes—
76C2×Q8×C17—26no—
77C2×Q8×C17—23no—
78C2×Q8×C17—48no—
79C2×Q8×C17—46no—
80C2×Q8×C17—46no—
81C2×Q8×C17C22⋊C411yescharacteristic
82C2×D4×C17—26no—
83C2×D4×C17—21no—
84C2×D4×C17—21no—
85C2×D4×C17C22⋊C413yes—
86C2×D4×C17C22×C412yes—
87C22×C4×C17—26no—
88C22×C4×C17—26no—
89C22×C4×C17C22×C413yes—
90C22×C4×C17C2×D413yes—
91C22×C4×C17C2×D413yes—
92C4⋊C4×C17—26no—
93C4⋊C4×C17—26no—
94C4⋊C4×C17—23no—
95C4⋊C4×C17C22⋊C413yes—
96C4⋊C4×C17C22⋊C413yes—
97C4⋊C4×C17C22×C416yes—
98C22⋊C4×C17—26no—
99C22⋊C4×C17—26no—
100C22⋊C4×C17C22⋊C416yes—
101C42×C17C22×C413yes—
102C42×C17C2×D413yes—
103C42×C17C2×D413yes—
104C42×C17C2411yescharacteristic
105#63223—46no—
106C2×#3622—21no—
107C2×#3622—23no—
108C2×#3622—21no—
109C2×#3622—23no—
110C2×#3622—46no—
111C2×#3622—83no—
112#17721—171nomaximal, Sylow
113C23×C17—22no—
114C23×C17C2×C22⋊C413yes—
115Q8×C17—43no—
116Q8×C17—43no—
117Q8×C17—412no—
118Q8×C17#10214yes—
119D4×C17—42no—
120D4×C17—412no—
121D4×C17—41no—
122D4×C17—41no—
123D4×C17—41no—
124D4×C17—41no—
125D4×C17#102112yes—
126C2×C4×C17—23no—
127C2×C4×C17—23no—
128C2×C4×C17—26no—
129C2×C4×C17—43no—
130C2×C4×C17—412no—
131C2×C4×C17—412no—
132C2×C4×C17C2×C22⋊C413yes—
133C2×C4×C17C2×C22⋊C413yes—
134C2×C4×C17C2×C22⋊C413yes—
135C2×C4×C17C2×D816yes—
136C2×C4×C17C2×SD1616yes—
137C2×C4×C17C23×C411yescharacteristic
138C2×C4×C17C22×D411yescharacteristic
139C2×C4×C17C22×D411yescharacteristic
140#3622—41no—
141#3622—43no—
142#3622—43no—
143#3622—41no—
144#3499D1711yescharacteristic
145#2988—171no—
146#2906—171no—
147#1547—173no—
148#1546—173no—
149#1540—173no—
150C2×#332—173no—
151C22×C17—23no—
152C22×C17—23no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17C2×#10216yes—
157C22×C17C22×C22⋊C411yescharacteristic
158C4×C17—21no—
159C4×C17—23no—
160C4×C17—23no—
161C4×C17—46no—
162C4×C17C2×#10216yes—
163C4×C17C22×D811yescharacteristic
164C4×C17C22×SD1611yescharacteristic
165C4×C17#57711yescharacteristic
166C4×C17#57811yescharacteristic
167C2×2−1+4—21no—
168C2×#131—341no—
169D4×Q8—21no—
170D4×Q8Dic1712yes—
171#547—23no—
172#547Dic1716yes—
173#536—23no—
174#532C2×D1713yes—
175C2×C4⋊Q8C2×D1713yes—
176C4×#41C2×D1711yescharacteristic
177C8×Q8—341no—
178C8×D4—343no—
179#431—173no—
180#424—343no—
181#423—173no—
182C2×C4⋊C8—173no—
183C2×C4⋊C8—173no—
184#332—173no—
185#332—176no—
186#332—173no—
187#331—171no—
188#331—173no—
189C2×C17—26no—
190C2×C17—82no—
191C2×C17#288911yescharacteristic
192C2×C17C22×#10211yescharacteristic
193C2×C17#289811yescharacteristic
1942−1+4—44no—
1952−1+4—44no—
196C2×#41—26no—
197C2×#41—23no—
198C2×#41C22⋊D1711yescharacteristic
199C22×Q8—22no—
200C22×Q8—23no—
201#131—684no—
202C2×M16—343no—
203C22×C8—343no—
204C22×C8—343no—
205C4⋊Q8—23no—
206C4⋊Q8—23no—
207C4⋊Q8C2×Dic1716yes—
208C4⋊Q8C22×D1713yes—
209#127C22×D1711yescharacteristic
210#124—23no—
211#123—212no—
212C22⋊Q8—26no—
213C22⋊Q8—26no—
214#121—23no—
215#121C2×Dic1716yes—
216C4×Q8C2×Dic1711yescharacteristic
217C4×Q8C22⋊D1711yescharacteristic
218C4×D4—23no—
219C4×D4C2×Dic1716yes—
220C4×D4C2×Dic1713yes—
221C4×D4C22⋊D1713yes—
222#117C22⋊D1713yes—
223C2×C4⋊C4—23no—
224C2×C4⋊C4C2×Dic1716yes—
225C2×C42C22×D1713yes—
226C4⋊C8—171no—
227C4⋊C8—173no—
228C4⋊C8—173no—
229C4⋊C8—171no—
230C4⋊C8—343no—
231C4⋊C8—343no—
232C4⋊C8—343no—
233C22⋊C8—343no—
234C4×C8—343no—
235C17#1772111yescharacteristic, complemented, Sylow
236#41—424no—
237#41—46no—
238#41—46no—
239#41#6322814yes—
240#41#6322914yes—
241C2×Q8—23no—
242C2×Q8—26no—
243C2×Q8—48no—
244C2×Q8—46no—
245C2×Q8—46no—
246C2×Q8C22⋊Dic1711yescharacteristic
247C2×D4—26no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4C22⋊Dic1713yes—
251C2×D4C22×Dic1712yes—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4C22×Dic1713yes—
255C22×C4C2×C22⋊D1713yes—
256C22×C4C2×C22⋊D1713yes—
257M16—686no—
258C2×C8—341no—
259C2×C8—343no—
260C2×C8—341no—
261C2×C8—343no—
262C2×C8—686no—
263C2×C8—1363no—
264C4⋊C4—23no—
265C4⋊C4—26no—
266C4⋊C4—26no—
267C4⋊C4C22⋊Dic1713yes—
268C4⋊C4C22⋊Dic1713yes—
269C4⋊C4C22×Dic1716yes—
270C22⋊C4—26no—
271C22⋊C4—26no—
272C22⋊C4C22⋊Dic1716yes—
273C42C22×Dic1713yes—
274C42C2×C22⋊D1713yes—
275C42C2×C22⋊D1713yes—
276C42C23×D1711yescharacteristic
277C23—22no—
278C23C2×C22⋊Dic1713yes—
279Q8—412no—
280Q8—43no—
281Q8—43no—
282Q8#1058750314yes—
283D4—42no—
284D4—412no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4#10587503112yes—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—26no—
293C2×C4—43no—
294C2×C4—412no—
295C2×C4—412no—
296C2×C4C2×#6322816yes—
297C2×C4C2×#6322916yes—
298C2×C4C2×C22⋊Dic1713yes—
299C2×C4C2×C22⋊Dic1713yes—
300C2×C4C2×C22⋊Dic1713yes—
301C2×C4C23×Dic1711yescharacteristic
302C2×C4C22×C22⋊D1711yescharacteristic
303C2×C4C22×C22⋊D1711yescharacteristic
304C8—681no—
305C8—683no—
306C8—683no—
307C8—681no—
308C22—23no—
309C22—23no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22C2×#1058750316yes—
314C22C22×C22⋊Dic1711yescharacteristic
315C4—21no—
316C4—23no—
317C4—23no—
318C4—46no—
319C4C2×#1058750316yes—
320C4C22×#6322811yescharacteristic
321C4C22×#6322911yescharacteristic
322C4#1176361711yescharacteristic
323C4#1176362411yescharacteristic
324C2—26no—
325C2—82no—
326C2#1453726711yescharacteristic
327C2C22×#1058750311yescharacteristic
328C2#1453731711yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
Fitting, prime radicalC17×#3499C2
socleC22×C17C22×C22⋊C4
cosocleC42×C17C24
hypercenter#3499D17
hypocenterC17#17721
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #14537895C2semidirect
2. #14537352C2semidirect
3. #14527461C2semidirect
4. #14527456C2semidirect
5. #14527433C2semidirect
6. C2×#11762555C2semidirect
235. C17#17721semidirect

Maximal subgroups

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

subgroupindex
#145378952
#145373522
#145274612
#145274562
#145274332
C2×#117625552
C17×#34992
#1772117