Finite Group Databasebeta

← C2×#2135×C131 #45719252

Subgroups — C2×#2135×C131 #45719252

Order 33536.

1198 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536×C131C211yesmaximal, characteristic
2C2×#413×C131C212yesmaximal
3#2135×C131C218yesmaximal, direct product factor
4C2×#356×C131C214yesmaximal, complemented
5C22×#41×C131C2211yescharacteristic
6C23×Q8×C131C411yescharacteristic
7C22×M16×C131—22no—
8#536×C131—24no—
9#536×C131C414yes—
10#536×C131C2214yes—
11C2×C4⋊Q8×C131C2212yes—
12C2×#124×C131C412yes—
13C2×#123×C131—22no—
14C2×C22⋊Q8×C131—22no—
15C2×#117×C131C411yescharacteristic
16#413×C131—24no—
17#413×C131C2218yes—
18C2×#101×C131C2214yes—
19C2×#100×C131—22no—
20#356×C131C22116yescomplemented
21C2×#41×C131—42no—
22C2×#41×C131—42no—
23C2×#41×C131C2×C412yes—
24C2×#41×C131D414yes—
25C2×#41×C131C2312yes—
26C22×Q8×C131—22no—
27C22×Q8×C131—21no—
28C22×Q8×C131—42no—
29C22×Q8×C131—42no—
30C22×Q8×C131C2×C412yes—
31C22×Q8×C131C2×C414yes—
32C22×Q8×C131C2311yescharacteristic
33C22×D4×C131C2×C411yescharacteristic
34C22×D4×C131D412yes—
35C23×C4×C131—21no—
36C23×C4×C131C2×C411yescharacteristic
37C23×C4×C131D412yes—
38C2×M16×C131—24no—
39C2×M16×C131—28no—
40C2×M16×C131—42no—
41C2×M16×C131—44no—
42C22×C8×C131—42no—
43C4⋊Q8×C131—24no—
44C4⋊Q8×C131C2×C414yes—
45C4⋊Q8×C131C2314yes—
46#124×C131—24no—
47#124×C131C2×C418yes—
48#123×C131—28no—
49#123×C131—28no—
50C22⋊Q8×C131—28no—
51C22⋊Q8×C131—28no—
52#117×C131—22no—
53#117×C131C2×C414yes—
54C2×C4⋊C4×C131—21no—
55C2×C4⋊C4×C131—24no—
56C2×C22⋊C4×C131—21no—
57C2×C22⋊C4×C131—24no—
58C2×C42×C131C2×C412yes—
59#101×C131—24no—
60#101×C131C2318yes—
61#100×C131—28no—
62C24×C131C22⋊C412yes—
63C24×C131C22⋊C411yescharacteristic
64#41×C131—44no—
65#41×C131—44no—
66#41×C131—44no—
67#41×C131—44no—
68C2×Q8×C131—24no—
69C2×Q8×C131—22no—
70C2×Q8×C131—41no—
71C2×Q8×C131—41no—
72C2×Q8×C131—48no—
73C2×Q8×C131—48no—
74C2×Q8×C131—42no—
75C2×Q8×C131—42no—
76C2×Q8×C131—42no—
77C2×Q8×C131—42no—
78C2×Q8×C131C22×C414yes—
79C2×Q8×C131C22×C411yescharacteristic
80C2×Q8×C131C2×D412yes—
81C2×Q8×C131C2411yescharacteristic
82C2×D4×C131—41no—
83C2×D4×C131—41no—
84C2×D4×C131—42no—
85C2×D4×C131—42no—
86C2×D4×C131C22⋊C414yes—
87C2×D4×C131C22⋊C412yes—
88C2×D4×C131C22×C412yes—
89C2×D4×C131C2×D414yes—
90C22×C4×C131—22no—
91C22×C4×C131—24no—
92C22×C4×C131—22no—
93C22×C4×C131—22no—
94C22×C4×C131—22no—
95C22×C4×C131—22no—
96C22×C4×C131—22no—
97C22×C4×C131—24no—
98C22×C4×C131—42no—
99C22×C4×C131—42no—
100C22×C4×C131C22⋊C412yes—
101C22×C4×C131C22⋊C414yes—
102C22×C4×C131C22⋊C411yescharacteristic
103C22×C4×C131C22×C411yescharacteristic
104C22×C4×C131C22×C412yes—
105C22×C4×C131C2×D412yes—
106C22×C4×C131C2×D414yes—
107M16×C131—28no—
108M16×C131—44no—
109M16×C131—48no—
110C2×C8×C131—44no—
111C2×C8×C131—48no—
112C4⋊C4×C131—24no—
113C4⋊C4×C131—28no—
114C4⋊C4×C131—28no—
115C22⋊C4×C131—24no—
116C22⋊C4×C131—216no—
117C42×C131—22no—
118C42×C131C22×C414yes—
119C23×C131—21no—
120C23×C131—21no—
121C23×C131—22no—
122C23×C131—22no—
123C23×C131—42no—
124C23×C131—44no—
125C23×C131C2×C22⋊C411yescharacteristic
126C23×C131C2×C22⋊C414yes—
127C23×C131C2×C22⋊C412yes—
128Q8×C131—41no—
129Q8×C131—42no—
130Q8×C131—41no—
131Q8×C131—44no—
132Q8×C131—44no—
133Q8×C131—48no—
134D4×C131—44no—
135D4×C131—42no—
136D4×C131—42no—
137D4×C131—44no—
138C2×C4×C131—22no—
139C2×C4×C131—21no—
140C2×C4×C131—22no—
141C2×C4×C131—21no—
142C2×C4×C131—24no—
143C2×C4×C131—24no—
144C2×C4×C131—24no—
145C2×C4×C131—24no—
146C2×C4×C131—24no—
147C2×C4×C131—24no—
148C2×C4×C131—24no—
149C2×C4×C131—24no—
150C2×C4×C131—44no—
151C2×C4×C131—48no—
152C2×C4×C131—42no—
153C2×C4×C131—42no—
154C2×C4×C131#9912yes—
155C2×C4×C131#9912yes—
156C2×C4×C131C2×C22⋊C411yescharacteristic
157C2×C4×C131C2×C22⋊C412yes—
158C2×C4×C131C2×C22⋊C412yes—
159C2×C4×C131C23×C411yescharacteristic
160C2×C4×C131C22×D412yes—
161C8×C131—48no—
162C22×C131—22no—
163C22×C131—22no—
164C22×C131—24no—
165C22×C131—42no—
166C22×C131—44no—
167C22×C131—42no—
168C22×C131—42no—
169C22×C131—44no—
170C22×C131—82no—
171C22×C131C2×#9911yescharacteristic
172C22×C131C2×#9911yescharacteristic
173C22×C131C22×C22⋊C411yescharacteristic
174C4×C131—22no—
175C4×C131—22no—
176C4×C131—21no—
177C4×C131—21no—
178C4×C131—44no—
179C4×C131—44no—
180C4×C131C2×#9914yes—
181C2×C131—21no—
182C2×C131—21no—
183C2×C131—42no—
184C2×C131—44no—
185C2×C131#213512yesdirect product factor
186C2×C131C22×#9911yescharacteristic
187C2×#2135C13111yesmaximal, characteristic, direct product factor, Sylow
188C131C2×#213511yescharacteristic, direct product factor, Sylow
189C2×#536C2×C13111yescharacteristic
190C2×#413C2×C13112yes—
191#2135C2×C13118yesdirect product factor
192C2×#356C2×C13114yescomplemented
193C22×#41C22×C13111yescharacteristic
194C23×Q8C4×C13111yescharacteristic
195C22×M16—22no—
196#536—24no—
197#536C4×C13114yes—
198#536C22×C13114yes—
199C2×C4⋊Q8C22×C13112yes—
200C2×#124C4×C13112yes—
201C2×#123—22no—
202C2×C22⋊Q8—22no—
203C2×#117C4×C13111yescharacteristic
204#413—24no—
205#413C22×C13118yes—
206C2×#101C22×C13114yes—
207C2×#100—22no—
208#356C22×C131116yescomplemented
209C2×#41—42no—
210C2×#41—42no—
211C2×#41C2×C4×C13112yes—
212C2×#41D4×C13114yes—
213C2×#41C23×C13112yes—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8C2×C4×C13112yes—
219C22×Q8C2×C4×C13114yes—
220C22×Q8C23×C13111yescharacteristic
221C22×D4C2×C4×C13111yescharacteristic
222C22×D4D4×C13112yes—
223C23×C4—21no—
224C23×C4C2×C4×C13111yescharacteristic
225C23×C4D4×C13112yes—
226C2×M16—24no—
227C2×M16—28no—
228C2×M16—42no—
229C2×M16—44no—
230C22×C8—42no—
231C4⋊Q8—24no—
232C4⋊Q8C2×C4×C13114yes—
233C4⋊Q8C23×C13114yes—
234#124—24no—
235#124C2×C4×C13118yes—
236#123—28no—
237#123—28no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240#117—22no—
241#117C2×C4×C13114yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—24no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C42C2×C4×C13112yes—
247#101—24no—
248#101C23×C13118yes—
249#100—28no—
250C24C22⋊C4×C13112yes—
251C24C22⋊C4×C13111yescharacteristic
252#41—44no—
253#41—44no—
254#41—44no—
255#41—44no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—48no—
261C2×Q8—48no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8C22×C4×C13114yes—
267C2×Q8C22×C4×C13111yescharacteristic
268C2×Q8C2×D4×C13112yes—
269C2×Q8C24×C13111yescharacteristic
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4C22⋊C4×C13114yes—
275C2×D4C22⋊C4×C13112yes—
276C2×D4C22×C4×C13112yes—
277C2×D4C2×D4×C13114yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊C4×C13112yes—
289C22×C4C22⋊C4×C13114yes—
290C22×C4C22⋊C4×C13111yescharacteristic
291C22×C4C22×C4×C13111yescharacteristic
292C22×C4C22×C4×C13112yes—
293C22×C4C2×D4×C13112yes—
294C22×C4C2×D4×C13114yes—
295M16—28no—
296M16—44no—
297M16—48no—
298C2×C8—44no—
299C2×C8—48no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C22⋊C4—24no—
304C22⋊C4—216no—
305C42—22no—
306C42C22×C4×C13114yes—
307C23—21no—
308C23—21no—
309C23—22no—
310C23—22no—
311C23—42no—
312C23—44no—
313C23C2×C22⋊C4×C13111yescharacteristic
314C23C2×C22⋊C4×C13114yes—
315C23C2×C22⋊C4×C13112yes—
316Q8—41no—
317Q8—42no—
318Q8—41no—
319Q8—44no—
320Q8—44no—
321Q8—48no—
322D4—42no—
323D4—44no—
324D4—42no—
325D4—44no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4#99×C13112yes—
343C2×C4#99×C13112yes—
344C2×C4C2×C22⋊C4×C13111yescharacteristic
345C2×C4C2×C22⋊C4×C13112yes—
346C2×C4C2×C22⋊C4×C13112yes—
347C2×C4C23×C4×C13111yescharacteristic
348C2×C4C22×D4×C13112yes—
349C8—48no—
350C22—22no—
351C22—22no—
352C22—24no—
353C22—42no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—44no—
358C22—82no—
359C22C2×#99×C13111yescharacteristic
360C22C2×#99×C13111yescharacteristic
361C22C22×C22⋊C4×C13111yescharacteristic
362C4—22no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—44no—
367C4—44no—
368C4C2×#99×C13114yes—
369C2—21no—
370C2—21no—
371C2—42no—
372C2—44no—
373C2#2135×C13112yesdirect product factor
374C2C22×#99×C13111yescharacteristic
subgroupG/H
center, socleC22×C131C2×#99
commutatorC2×C4C23×C4×C131
Frattini, cosocleC2×Q8C24×C131
prime radicalC22×#41×C131C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C22×C4×C131 ◁ C22×#41×C131 ◁ G

Complements

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

NH ≅ G/Ntype
3. #2135×C131C2direct (N × H)
4. C2×#356×C131C2semidirect
20. #356×C131C22semidirect
185. C2×C131#2135direct (N × H)
187. C2×#2135C131direct (N × H)
188. C131C2×#2135direct (N × H)
191. #2135C2×C131direct (N × H)
192. C2×#356C2×C131semidirect
208. #356C22×C131semidirect
373. C2#2135×C131direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#536×C1312
C2×#413×C1312
#2135×C1312
C2×#356×C1312
C2×#2135131