Finite Group Databasebeta

← C2×C31×#2135 #21490127

Subgroups — C2×C31×#2135 #21490127

Order 7936.

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×C31×#536C211yesmaximal, characteristic
2C2×C31×#413C212yesmaximal
3C31×#2135C218yesmaximal, direct product factor
4C2×C31×#356C214yesmaximal, complemented
5C22×#41×C31C2211yescharacteristic
6C23×Q8×C31C411yescharacteristic
7C22×M16×C31—22no—
8C31×#536—24no—
9C31×#536C414yes—
10C31×#536C2214yes—
11C2×C31×C4⋊Q8C2212yes—
12C2×C31×#124C412yes—
13C2×C31×#123—22no—
14C2×C31×C22⋊Q8—22no—
15C2×C31×#117C411yescharacteristic
16C31×#413—24no—
17C31×#413C2218yes—
18C2×C31×#101C2214yes—
19C2×C31×#100—22no—
20C31×#356C22116yescomplemented
21C2×#41×C31—42no—
22C2×#41×C31—42no—
23C2×#41×C31C2×C412yes—
24C2×#41×C31D414yes—
25C2×#41×C31C2312yes—
26C22×Q8×C31—22no—
27C22×Q8×C31—21no—
28C22×Q8×C31—42no—
29C22×Q8×C31—42no—
30C22×Q8×C31C2×C412yes—
31C22×Q8×C31C2×C414yes—
32C22×Q8×C31C2311yescharacteristic
33C22×D4×C31C2×C411yescharacteristic
34C22×D4×C31D412yes—
35C23×C4×C31—21no—
36C23×C4×C31C2×C411yescharacteristic
37C23×C4×C31D412yes—
38C2×M16×C31—24no—
39C2×M16×C31—28no—
40C2×M16×C31—42no—
41C2×M16×C31—44no—
42C22×C8×C31—42no—
43C31×C4⋊Q8—24no—
44C31×C4⋊Q8C2×C414yes—
45C31×C4⋊Q8C2314yes—
46C31×#124—24no—
47C31×#124C2×C418yes—
48C31×#123—28no—
49C31×#123—28no—
50C31×C22⋊Q8—28no—
51C31×C22⋊Q8—28no—
52C31×#117—22no—
53C31×#117C2×C414yes—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—24no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31—24no—
58C2×C42×C31C2×C412yes—
59C31×#101—24no—
60C31×#101C2318yes—
61C31×#100—28no—
62C24×C31C22⋊C412yes—
63C24×C31C22⋊C411yescharacteristic
64#41×C31—44no—
65#41×C31—44no—
66#41×C31—44no—
67#41×C31—44no—
68C2×Q8×C31—24no—
69C2×Q8×C31—22no—
70C2×Q8×C31—41no—
71C2×Q8×C31—41no—
72C2×Q8×C31—48no—
73C2×Q8×C31—48no—
74C2×Q8×C31—42no—
75C2×Q8×C31—42no—
76C2×Q8×C31—42no—
77C2×Q8×C31—42no—
78C2×Q8×C31C22×C414yes—
79C2×Q8×C31C22×C411yescharacteristic
80C2×Q8×C31C2×D412yes—
81C2×Q8×C31C2411yescharacteristic
82C2×D4×C31—41no—
83C2×D4×C31—41no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31C22⋊C414yes—
87C2×D4×C31C22⋊C412yes—
88C2×D4×C31C22×C412yes—
89C2×D4×C31C2×D414yes—
90C22×C4×C31—22no—
91C22×C4×C31—24no—
92C22×C4×C31—22no—
93C22×C4×C31—22no—
94C22×C4×C31—22no—
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—24no—
98C22×C4×C31—42no—
99C22×C4×C31—42no—
100C22×C4×C31C22⋊C412yes—
101C22×C4×C31C22⋊C414yes—
102C22×C4×C31C22⋊C411yescharacteristic
103C22×C4×C31C22×C411yescharacteristic
104C22×C4×C31C22×C412yes—
105C22×C4×C31C2×D412yes—
106C22×C4×C31C2×D414yes—
107M16×C31—28no—
108M16×C31—44no—
109M16×C31—48no—
110C2×C8×C31—44no—
111C2×C8×C31—48no—
112C4⋊C4×C31—24no—
113C4⋊C4×C31—28no—
114C4⋊C4×C31—28no—
115C22⋊C4×C31—24no—
116C22⋊C4×C31—216no—
117C42×C31—22no—
118C42×C31C22×C414yes—
119C2×#2135C3111yesmaximal, characteristic, direct product factor, Sylow
120C23×C31—21no—
121C23×C31—21no—
122C23×C31—22no—
123C23×C31—22no—
124C23×C31—42no—
125C23×C31—44no—
126C23×C31C2×C22⋊C411yescharacteristic
127C23×C31C2×C22⋊C414yes—
128C23×C31C2×C22⋊C412yes—
129Q8×C31—41no—
130Q8×C31—42no—
131Q8×C31—41no—
132Q8×C31—44no—
133Q8×C31—44no—
134Q8×C31—48no—
135D4×C31—44no—
136D4×C31—42no—
137D4×C31—42no—
138D4×C31—44no—
139C2×C4×C31—22no—
140C2×C4×C31—21no—
141C2×C4×C31—22no—
142C2×C4×C31—21no—
143C2×C4×C31—24no—
144C2×C4×C31—24no—
145C2×C4×C31—24no—
146C2×C4×C31—24no—
147C2×C4×C31—24no—
148C2×C4×C31—24no—
149C2×C4×C31—24no—
150C2×C4×C31—24no—
151C2×C4×C31—44no—
152C2×C4×C31—48no—
153C2×C4×C31—42no—
154C2×C4×C31—42no—
155C2×C4×C31#9912yes—
156C2×C4×C31#9912yes—
157C2×C4×C31C2×C22⋊C411yescharacteristic
158C2×C4×C31C2×C22⋊C412yes—
159C2×C4×C31C2×C22⋊C412yes—
160C2×C4×C31C23×C411yescharacteristic
161C2×C4×C31C22×D412yes—
162C8×C31—48no—
163C2×#536C2×C3111yescharacteristic
164C2×#413C2×C3112yes—
165#2135C2×C3118yesdirect product factor
166C2×#356C2×C3114yescomplemented
167C22×C31—22no—
168C22×C31—22no—
169C22×C31—24no—
170C22×C31—42no—
171C22×C31—44no—
172C22×C31—42no—
173C22×C31—42no—
174C22×C31—44no—
175C22×C31—82no—
176C22×C31C2×#9911yescharacteristic
177C22×C31C2×#9911yescharacteristic
178C22×C31C22×C22⋊C411yescharacteristic
179C4×C31—22no—
180C4×C31—22no—
181C4×C31—21no—
182C4×C31—21no—
183C4×C31—44no—
184C4×C31—44no—
185C4×C31C2×#9914yes—
186C22×#41C22×C3111yescharacteristic
187C23×Q8C4×C3111yescharacteristic
188C22×M16—22no—
189#536—24no—
190#536C4×C3114yes—
191#536C22×C3114yes—
192C2×C4⋊Q8C22×C3112yes—
193C2×#124C4×C3112yes—
194C2×#123—22no—
195C2×C22⋊Q8—22no—
196C2×#117C4×C3111yescharacteristic
197#413—24no—
198#413C22×C3118yes—
199C2×#101C22×C3114yes—
200C2×#100—22no—
201#356C22×C31116yescomplemented
202C2×C31—21no—
203C2×C31—21no—
204C2×C31—42no—
205C2×C31—44no—
206C2×C31#213512yesdirect product factor
207C2×C31C22×#9911yescharacteristic
208C2×#41—42no—
209C2×#41—42no—
210C2×#41C2×C4×C3112yes—
211C2×#41D4×C3114yes—
212C2×#41C23×C3112yes—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8C2×C4×C3112yes—
218C22×Q8C2×C4×C3114yes—
219C22×Q8C23×C3111yescharacteristic
220C22×D4C2×C4×C3111yescharacteristic
221C22×D4D4×C3112yes—
222C23×C4—21no—
223C23×C4C2×C4×C3111yescharacteristic
224C23×C4D4×C3112yes—
225C2×M16—24no—
226C2×M16—28no—
227C2×M16—42no—
228C2×M16—44no—
229C22×C8—42no—
230C4⋊Q8—24no—
231C4⋊Q8C2×C4×C3114yes—
232C4⋊Q8C23×C3114yes—
233#124—24no—
234#124C2×C4×C3118yes—
235#123—28no—
236#123—28no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239#117—22no—
240#117C2×C4×C3114yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—24no—
245C2×C42C2×C4×C3112yes—
246#101—24no—
247#101C23×C3118yes—
248#100—28no—
249C31C2×#213511yescharacteristic, direct product factor, Sylow
250C24C22⋊C4×C3112yes—
251C24C22⋊C4×C3111yescharacteristic
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×C3114yes—
267C2×Q8C22×C4×C3111yescharacteristic
268C2×Q8C2×D4×C3112yes—
269C2×Q8C24×C3111yescharacteristic
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4C22⋊C4×C3114yes—
275C2×D4C22⋊C4×C3112yes—
276C2×D4C22×C4×C3112yes—
277C2×D4C2×D4×C3114yes—
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×C3112yes—
289C22×C4C22⋊C4×C3114yes—
290C22×C4C22⋊C4×C3111yescharacteristic
291C22×C4C22×C4×C3111yescharacteristic
292C22×C4C22×C4×C3112yes—
293C22×C4C2×D4×C3112yes—
294C22×C4C2×D4×C3114yes—
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×C3114yes—
307C23—21no—
308C23—21no—
309C23—22no—
310C23—22no—
311C23—42no—
312C23—44no—
313C23C2×C22⋊C4×C3111yescharacteristic
314C23C2×C22⋊C4×C3114yes—
315C23C2×C22⋊C4×C3112yes—
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—44no—
339C2×C4—48no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4C31×#9912yes—
343C2×C4C31×#9912yes—
344C2×C4C2×C22⋊C4×C3111yescharacteristic
345C2×C4C2×C22⋊C4×C3112yes—
346C2×C4C2×C22⋊C4×C3112yes—
347C2×C4C23×C4×C3111yescharacteristic
348C2×C4C22×D4×C3112yes—
349C8—48no—
350C22—22no—
351C22—22no—
352C22—24no—
353C22—42no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—44no—
358C22—82no—
359C22C2×C31×#9911yescharacteristic
360C22C2×C31×#9911yescharacteristic
361C22C22×C22⋊C4×C3111yescharacteristic
362C4—22no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—44no—
367C4—44no—
368C4C2×C31×#9914yes—
369C2—21no—
370C2—21no—
371C2—42no—
372C2—44no—
373C2C31×#213512yesdirect product factor
374C2C22×C31×#9911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC2×C4C23×C4×C31
Frattini, cosocleC2×Q8C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2135C2direct (N × H)
4. C2×C31×#356C2semidirect
20. C31×#356C22semidirect
119. C2×#2135C31direct (N × H)
165. #2135C2×C31direct (N × H)
166. C2×#356C2×C31semidirect
201. #356C22×C31semidirect
206. C2×C31#2135direct (N × H)
249. C31C2×#2135direct (N × H)
373. C2C31×#2135direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5362
C2×C31×#4132
C31×#21352
C2×C31×#3562
C2×#213531