Finite Group Databasebeta

← C2×#14539320 #17903954

Subgroups — C2×#14539320 #17903954

Order 4352.

1854 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763440C212yesmaximal
2#14539320C2116yesmaximal, direct product factor
3C2×#11763217C214yesmaximal, complemented
4C2×#11763213C212yesmaximal, complemented
5C2×#11763210C212yesmaximal, complemented
6C2×#11762849C212yesmaximal
7C2×#11762847C211yesmaximal, characteristic
8C2×#11762764C211yesmaximal, characteristic
9C22×C17×#107C211yesmaximal, characteristic
10C22×#63223C2211yescharacteristic
11#11763440—28no—
12#11763440C22116yes—
13C2×#10587574—22no—
14C2×#10587574C2214yes—
15C2×#10587576—22no—
16C2×#10587576C2214yes—
17C2×#10587610—24no—
18C2×#10587612C2212yes—
19C2×#10587612C2211yescharacteristic
20#11763217C22132yescomplemented
21#11763213C22116yescomplemented
22#11763210C22116yescomplemented
23#11762849C22116yes—
24#11762847C2218yes—
25C2×#10587489C2211yescharacteristic
26C2×#10587489C2211yescharacteristic
27C2×#10587478C2214yes—
28C2×#10587488C2212yes—
29C2×#10587479C2214yes—
30#11762764C2218yes—
31C2×#10587486C2212yes—
32C2×C8×Dic17C2212yes—
33C23×C8×C17C2211yescharacteristic
34C22×C4⋊C4×C17C2212yes—
35C2×C17×#107C2212yes—
36C2×C17×#107C2212yes—
37C2×C17×#107C2218yes—
38C23×C4×C17—22no—
39C23×C4×C17C2311yescharacteristic
40C22×Dic34—24no—
41C22×C8×C17Q818yescomplemented
42C22×C8×C17C2311yescharacteristic
43C22×C8×C17C2311yescharacteristic
44C22×C8×C17C2314yes—
45C2×C4⋊C4×C17—22no—
46C2×C4⋊C4×C17—24no—
47C2×C4⋊C4×C17C2314yes—
48C2×C4⋊C4×C17C2318yes—
49C2×C22⋊Dic17—22no—
50C2×C22⋊Dic17D412yes—
51#10587612—24no—
52#10587612D414yes—
53#10587612C2318yes—
54#10587612C2314yes—
55C2×C4⋊Dic17—24no—
56C2×C4⋊Dic17D411yescharacteristic
57C2×C4⋊Dic17D411yescharacteristic
58C2×C4⋊Dic17C2312yes—
59C2×C4⋊Dic17C2312yes—
60#10587610—232no—
61C2×#63225—24no—
62C2×#63225—24no—
63C2×C4×Dic17—22no—
64C2×C4×Dic17C2312yes—
65C2×#63223Q818yescomplemented
66C2×#63223C2314yes—
67C22×#3622C2312yes—
68#10587576—216no—
69#10587576—28no—
70#10587576C23116yes—
71#10587574—216no—
72#10587574—28no—
73#10587574C23116yes—
74C17×#107C2318yes—
75C17×#107C2318yes—
76#10587489C2314yes—
77#10587489C2314yes—
78#10587488C2318yes—
79#10587486C2318yes—
80C8×Dic17C2318yes—
81#10587479C23116yes—
82#10587478C23116yes—
83C24×C17C2×D411yescharacteristic
84C22×C4×C17—24no—
85C22×C4×C17—28no—
86C22×C4×C17—84no—
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D412yes—
89C22×C4×C17C2×Q814yes—
90C22×C4×C17C2×Q814yes—
91C22×C4×C17C2411yescharacteristic
92C22×C4×C17C2412yes—
93C22×Dic17—22no—
94C22×Dic17—22no—
95C22×Dic17C2×D412yes—
96C22×Dic17C2×D412yes—
97C2×Dic34—216no—
98C2×Dic34—88no—
99C2×C8×C17C2×Q814yes—
100C2×C8×C17C2×Q814yes—
101C2×C8×C17C2×Q8116yescomplemented
102C2×C8×C17C2412yes—
103C2×C8×C17C2412yes—
104C4⋊C4×C17—28no—
105C4⋊C4×C17—24no—
106C4⋊C4×C17C2418yes—
107C22⋊Dic17—28no—
108C22⋊Dic17C2×D418yes—
109C4⋊Dic17—22no—
110C4⋊Dic17—22no—
111C4⋊Dic17—216no—
112C4⋊Dic17C2×D414yes—
113C4⋊Dic17C2×D414yes—
114C4⋊Dic17C2414yes—
115C4⋊Dic17C2414yes—
116#63225—216no—
117#63225—216no—
118C4×Dic17—28no—
119C4×Dic17C2×D414yes—
120C4×Dic17C2414yes—
121#63223C2×Q8116yescomplemented
122C2×#3622C2×Q818yes—
123C2×#3622C2414yes—
124C2×#3164—171nomaximal, Sylow
125C23×C17—22no—
126C23×C17—22no—
127C23×C17C4⋊Q814yes—
128C23×C17C22×D412yes—
129C23×C17C22×D411yescharacteristic
130C2×C4×C17—28no—
131C2×C4×C17—48no—
132C2×C4×C17—88no—
133C2×C4×C17C4⋊Q818yes—
134C2×C4×C17C4⋊Q814yes—
135C2×C4×C17C22×D412yes—
136C2×C4×C17C22×D411yescharacteristic
137C2×C4×C17C22×Q812yes—
138C2×C4×C17C22×Q814yes—
139C2×C4×C17C22×Q812yes—
140C2×C4×C17C22×Q814yes—
141C2×C4×C17C2511yescharacteristic
142C2×Dic17—24no—
143C2×Dic17—24no—
144C2×Dic17—24no—
145C2×Dic17—24no—
146C2×Dic17—44no—
147C2×Dic17—44no—
148C2×Dic17C22×D414yes—
149C2×Dic17C22×D414yes—
150Dic34—816no—
151C8×C17C22×Q814yes—
152C8×C17C22×Q814yes—
153#3622C22×Q818yes—
154C2×#544—172no—
155#3164—1716no—
156C2×#501—174no—
157C2×#500—172no—
158C2×#499—172no—
159C2×#428—172no—
160C2×#427—171no—
161C22×#107D1711yescharacteristic
162C2×#405—171no—
163C22×C17—28no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17C2×C4⋊Q814yes—
167C22×C17C2×C4⋊Q812yes—
168C22×C17#57612yes—
169C22×C17#57812yes—
170C22×C17C23×D411yescharacteristic
171C4×C17—48no—
172C4×C17C2×C4⋊Q814yes—
173C4×C17C2×C4⋊Q812yes—
174C4×C17C23×Q811yescharacteristic
175C4×C17C23×Q811yescharacteristic
176Dic17—24no—
177Dic17—24no—
178Dic17—44no—
179Dic17—44no—
180C22×M16—171no—
181C23×C8C2×D1711yescharacteristic
182#544—1716no—
183#544—348no—
184C2×C4⋊Q8—174no—
185C2×C4⋊Q8—342no—
186C2×#125—174no—
187C2×#125—342no—
188C2×C22⋊Q8—344no—
189C2×#117—171no—
190C2×#117—172no—
191C22×C4⋊C4C2×D1712yes—
192#501—1732no—
193#500—1716no—
194#499—1716no—
195#428—1716no—
196#427—178no—
197C2×#107—171no—
198C2×#107—171no—
199C2×#107—174no—
200C2×#107C2×D1712yes—
201C2×#107C2×D1712yes—
202C2×#107C2×D1718yes—
203C2×#106—172no—
204C2×#106—174no—
205#405—178no—
206C2×#97—172no—
207C2×C4×C8—172no—
208C2×C17—24no—
209C2×C17#316414yescomplemented
210C2×C17C22×C4⋊Q811yescharacteristic
211C2×C17C2×#57611yescharacteristic
212C2×C17C2×#57811yescharacteristic
213C22×Q8—344no—
214C23×C4—22no—
215C23×C4C22×D1711yescharacteristic
216C2×M16—178no—
217C2×M16—174no—
218C22×C8—172no—
219C22×C8Dic3418yescomplemented
220C22×C8C22×D1711yescharacteristic
221C22×C8C22×D1711yescharacteristic
222C22×C8C22×D1714yes—
223C4⋊Q8—1716no—
224C4⋊Q8—348no—
225C4⋊Q8—3416no—
226#125—1716no—
227#125—348no—
228#125—3416no—
229C22⋊Q8—3432no—
230#117—174no—
231#117—174no—
232#117—178no—
233#117—344no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—171no—
237C2×C4⋊C4—171no—
238C2×C4⋊C4—172no—
239C2×C4⋊C4—172no—
240C2×C4⋊C4—344no—
241C2×C4⋊C4—344no—
242C2×C4⋊C4—344no—
243C2×C4⋊C4C22×D1714yes—
244C2×C4⋊C4C22×D1718yes—
245C2×C22⋊C4—172no—
246C2×C22⋊C4—342no—
247C2×C42—172no—
248C2×C42—342no—
249#107—174no—
250#107—174no—
251#107—1716no—
252#107C22×D1718yes—
253#107C22×D1718yes—
254#106—178no—
255#106—1716no—
256#97—178no—
257C4×C8—178no—
258C17C2×#316411yescharacteristic, complemented, Sylow
259C24D4×D1711yescharacteristic
260C2×Q8—3416no—
261C2×Q8—1368no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—84no—
265C22×C4—172no—
266C22×C4—172no—
267C22×C4—342no—
268C22×C4—342no—
269C22×C4C2×Dic3414yes—
270C22×C4D4×D1712yes—
271C22×C4D4×D1711yescharacteristic
272C22×C4Q8×D1714yes—
273C22×C4C23×D1712yes—
274C22×C4C23×D1711yescharacteristic
275M16—1716no—
276C2×C8—178no—
277C2×C8—174no—
278C2×C8C2×Dic3414yes—
279C2×C8C2×Dic3414yes—
280C2×C8C2×Dic34116yescomplemented
281C2×C8C23×D1712yes—
282C2×C8C23×D1712yes—
283C4⋊C4—24no—
284C4⋊C4—28no—
285C4⋊C4—174no—
286C4⋊C4—174no—
287C4⋊C4—174no—
288C4⋊C4—174no—
289C4⋊C4—3416no—
290C4⋊C4—3416no—
291C4⋊C4—3416no—
292C4⋊C4—342no—
293C4⋊C4—342no—
294C4⋊C4C23×D1718yes—
295C22⋊C4—178no—
296C22⋊C4—348no—
297C42—174no—
298C42—174no—
299C42—348no—
300C23—22no—
301C23—22no—
302C23#1058757414yes—
303C23C2×D4×D1712yes—
304C23C2×D4×D1711yescharacteristic
305Q8—13616no—
306C2×C4—28no—
307C2×C4—48no—
308C2×C4—88no—
309C2×C4—174no—
310C2×C4—174no—
311C2×C4—344no—
312C2×C4—344no—
313C2×C4—344no—
314C2×C4—344no—
315C2×C4—684no—
316C2×C4—684no—
317C2×C4#1058757418yes—
318C2×C4#1058757414yes—
319C2×C4C22×Dic3412yes—
320C2×C4C22×Dic3414yes—
321C2×C4C2×D4×D1711yescharacteristic
322C2×C4C2×D4×D1712yes—
323C2×C4C2×Q8×D1712yes—
324C2×C4C2×Q8×D1714yes—
325C2×C4C24×D1711yescharacteristic
326C8—178no—
327C8C22×Dic3414yes—
328C8C22×Dic3414yes—
329C22—28no—
330C22—22no—
331C22—22no—
332C22C2×#1058757414yes—
333C22C2×#1058757412yes—
334C22#1176361012yes—
335C22#1176362212yes—
336C22C22×D4×D1711yescharacteristic
337C4—48no—
338C4—344no—
339C4—344no—
340C4—684no—
341C4—684no—
342C4C2×#1058757412yes—
343C4C2×#1058757414yes—
344C4C22×Q8×D1711yescharacteristic
345C4C23×Dic3411yescharacteristic
346C2—24no—
347C2#1453932014yesdirect product factor
348C2C22×#1058757411yescharacteristic
349C2C2×#1176361011yescharacteristic
350C2C2×#1176362211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC2×C4×C17C25
FrattiniC2×C4C24×D17
FittingC22×C17×#107C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC22×#107D17
hypocenterC17C2×#3164
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. #14539320C2direct (N × H)
3. C2×#11763217C2semidirect
4. C2×#11763213C2semidirect
5. C2×#11763210C2semidirect
20. #11763217C22semidirect
21. #11763213C22semidirect
22. #11763210C22semidirect
41. C22×C8×C17Q8semidirect
65. C2×#63223Q8semidirect
101. C2×C8×C17C2×Q8semidirect
121. #63223C2×Q8semidirect
209. C2×C17#3164semidirect
219. C22×C8Dic34semidirect
258. C17C2×#3164semidirect
280. C2×C8C2×Dic34semidirect
347. C2#14539320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117634402
#145393202
C2×#117632172
C2×#117632132
C2×#117632102
C2×#117628492
C2×#117628472
C2×#117627642
C22×C17×#1072
C2×#316417