Finite Group Databasebeta

← C31×#12702 #21469461

Subgroups — C31×#12702 #21469461

Order 7936.

962 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C31×#536C211yesmaximal, characteristic
2C2×C31×#356C212yesmaximal, complemented
3C2×C31×#355C212yesmaximal, complemented
4C31×#1784C212yesmaximal, complemented
5C22×#41×C31C411yescharacteristic
6C23×Q8×C31C2211yescharacteristic
7C31×#536—24no—
8C31×#536—24no—
9C2×C31×C4⋊Q8C411yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C31×#124C411yescharacteristic
12C2×C31×#124C2211yescharacteristic
13C2×C31×#123—22no—
14C2×C31×C22⋊Q8—22no—
15C2×C31×#117C411yescharacteristic
16C2×C31×#101—22no—
17C2×C31×#101C2214yes—
18C31×#407—22no—
19C31×#356—24no—
20C31×#355—24no—
21C2×#41×C31—22no—
22C2×#41×C31—22no—
23C2×#41×C31—42no—
24C2×#41×C31—42no—
25C22×Q8×C31—22no—
26C22×Q8×C31—22no—
27C22×Q8×C31—42no—
28C22×Q8×C31—42no—
29C22×Q8×C31C2×C411yescharacteristic
30C22×Q8×C31C2×C411yescharacteristic
31C22×Q8×C31D412yes—
32C22×Q8×C31C2311yescharacteristic
33C22×D4×C31—21no—
34C22×D4×C31C2×C411yescharacteristic
35C23×C4×C31—21no—
36C23×C4×C31C2×C411yescharacteristic
37C23×C4×C31D412yes—
38C2×M16×C31—24no—
39C31×C4⋊Q8—22no—
40C31×C4⋊Q8—22no—
41C31×C4⋊Q8—22no—
42C31×C4⋊Q8D414yes—
43C31×#124—22no—
44C31×#124—22no—
45C31×#124—22no—
46C31×#124D414yes—
47C31×#123—216no—
48C31×C22⋊Q8—216no—
49C31×#117—22no—
50C31×#117—22no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—22no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—22no—
56C2×C22⋊C4×C31—22no—
57C2×C42×C31C2×C411yescharacteristic
58C2×C42×C31C2×C411yescharacteristic
59C31×#101—24no—
60C31×#101—24no—
61C31×#101—44no—
62C31×C22⋊C8—44no—
63C24×C31—21no—
64C24×C31C22⋊C411yescharacteristic
65#41×C31—48no—
66#41×C31—48no—
67C2×Q8×C31—24no—
68C2×Q8×C31—21no—
69C2×Q8×C31—21no—
70C2×Q8×C31—21no—
71C2×Q8×C31—24no—
72C2×Q8×C31—41no—
73C2×Q8×C31—41no—
74C2×Q8×C31—42no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—48no—
78C2×Q8×C31—48no—
79C2×Q8×C31—41no—
80C2×Q8×C31—41no—
81C2×Q8×C31C22⋊C412yes—
82C2×Q8×C31C2×D412yes—
83C2×Q8×C31C2×D412yes—
84C2×D4×C31—24no—
85C2×D4×C31—21no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—42no—
89C2×D4×C31—42no—
90C2×D4×C31C22⋊C412yes—
91C22×C4×C31—22no—
92C22×C4×C31—22no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—24no—
96C22×C4×C31—24no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—24no—
101C22×C4×C31—42no—
102C22×C4×C31—42no—
103C22×C4×C31—42no—
104C22×C4×C31C22⋊C412yes—
105C22×C4×C31C22⋊C411yescharacteristic
106C22×C4×C31C22×C411yescharacteristic
107C22×C4×C31C2×D412yes—
108M16×C31—44no—
109M16×C31—44no—
110C2×C8×C31—44no—
111C4⋊C4×C31—24no—
112C4⋊C4×C31—28no—
113C4⋊C4×C31—28no—
114C22⋊C4×C31—24no—
115C22⋊C4×C31—28no—
116C22⋊C4×C31—28no—
117C42×C31—21no—
118C42×C31—21no—
119C42×C31C22⋊C412yes—
120C42×C31C22⋊C412yes—
121#12702C3111yesmaximal, characteristic, direct product factor, Sylow
122C23×C31—21no—
123C23×C31—21no—
124C23×C31—22no—
125C23×C31—42no—
126C23×C31—44no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31#9912yes—
130C23×C31C2×C22⋊C411yescharacteristic
131Q8×C31—42no—
132Q8×C31—42no—
133Q8×C31—48no—
134Q8×C31—42no—
135Q8×C31—42no—
136Q8×C31—42no—
137Q8×C31—42no—
138D4×C31—42no—
139D4×C31—42no—
140D4×C31—48no—
141C2×C4×C31—24no—
142C2×C4×C31—24no—
143C2×C4×C31—21no—
144C2×C4×C31—22no—
145C2×C4×C31—21no—
146C2×C4×C31—22no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—22no—
150C2×C4×C31—22no—
151C2×C4×C31—22no—
152C2×C4×C31—22no—
153C2×C4×C31—22no—
154C2×C4×C31—22no—
155C2×C4×C31—48no—
156C2×C4×C31—48no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31#9912yes—
162C2×C4×C31C2×C22⋊C411yescharacteristic
163C2×C4×C31C2×C22⋊C411yescharacteristic
164C2×C4×C31#12014yes—
165C8×C31—84no—
166C2×#536C2×C3111yescharacteristic
167C2×#356C2×C3112yescomplemented
168C2×#355C2×C3112yescomplemented
169#1784C2×C3112yescomplemented
170C22×C31—22no—
171C22×C31—24no—
172C22×C31—22no—
173C22×C31—42no—
174C22×C31—42no—
175C22×C31—44no—
176C22×C31—42no—
177C22×C31—82no—
178C22×C31—82no—
179C22×C31#37911yescharacteristic
180C22×C31C2×#9911yescharacteristic
181C22×C31C2×#9911yescharacteristic
182C4×C31—24no—
183C4×C31—22no—
184C4×C31—21no—
185C4×C31—21no—
186C4×C31—42no—
187C4×C31—42no—
188C4×C31—42no—
189C4×C31—42no—
190C22×#41C4×C3111yescharacteristic
191C23×Q8C22×C3111yescharacteristic
192#536—24no—
193#536—24no—
194C2×C4⋊Q8C4×C3111yescharacteristic
195C2×C4⋊Q8C22×C3111yescharacteristic
196C2×#124C4×C3111yescharacteristic
197C2×#124C22×C3111yescharacteristic
198C2×#123—22no—
199C2×C22⋊Q8—22no—
200C2×#117C4×C3111yescharacteristic
201C2×#101—22no—
202C2×#101C22×C3114yes—
203#407—22no—
204#356—24no—
205#355—24no—
206C2×C31—21no—
207C2×C31—21no—
208C2×C31—42no—
209C2×C31—82no—
210C2×C31#178111yescharacteristic
211C2×C31#213312yes—
212C2×#41—22no—
213C2×#41—22no—
214C2×#41—42no—
215C2×#41—42no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8C2×C4×C3111yescharacteristic
221C22×Q8C2×C4×C3111yescharacteristic
222C22×Q8D4×C3112yes—
223C22×Q8C23×C3111yescharacteristic
224C22×D4—21no—
225C22×D4C2×C4×C3111yescharacteristic
226C23×C4—21no—
227C23×C4C2×C4×C3111yescharacteristic
228C23×C4D4×C3112yes—
229C2×M16—24no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8D4×C3114yes—
234#124—22no—
235#124—22no—
236#124—22no—
237#124D4×C3114yes—
238#123—216no—
239C22⋊Q8—216no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42C2×C4×C3111yescharacteristic
249C2×C42C2×C4×C3111yescharacteristic
250#101—24no—
251#101—24no—
252#101—44no—
253C22⋊C8—44no—
254C31#1270211yescharacteristic, direct product factor, Sylow
255C24—21no—
256C24C22⋊C4×C3111yescharacteristic
257#41—48no—
258#41—48no—
259C2×Q8—24no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—24no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8C22⋊C4×C3112yes—
274C2×Q8C2×D4×C3112yes—
275C2×Q8C2×D4×C3112yes—
276C2×D4—24no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4C22⋊C4×C3112yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C22⋊C4×C3112yes—
297C22×C4C22⋊C4×C3111yescharacteristic
298C22×C4C22×C4×C3111yescharacteristic
299C22×C4C2×D4×C3112yes—
300M16—44no—
301M16—44no—
302C2×C8—44no—
303C4⋊C4—24no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C22⋊C4—24no—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C42—21no—
310C42—21no—
311C42C22⋊C4×C3112yes—
312C42C22⋊C4×C3112yes—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—42no—
317C23—44no—
318C23—41no—
319C23—41no—
320C23C31×#9912yes—
321C23C2×C22⋊C4×C3111yescharacteristic
322Q8—42no—
323Q8—42no—
324Q8—48no—
325Q8—42no—
326Q8—42no—
327Q8—42no—
328Q8—42no—
329D4—42no—
330D4—42no—
331D4—48no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4C31×#9912yes—
353C2×C4C2×C22⋊C4×C3111yescharacteristic
354C2×C4C2×C22⋊C4×C3111yescharacteristic
355C2×C4C31×#12014yes—
356C8—84no—
357C22—22no—
358C22—24no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—44no—
363C22—42no—
364C22—82no—
365C22—82no—
366C22C31×#37911yescharacteristic
367C22C2×C31×#9911yescharacteristic
368C22C2×C31×#9911yescharacteristic
369C4—24no—
370C4—22no—
371C4—21no—
372C4—21no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—82no—
381C2C31×#178111yescharacteristic
382C2C31×#213312yes—
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC22×#41×C31C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C31 ◁ C23×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#356C2semidirect
3. C2×C31×#355C2semidirect
4. C31×#1784C2semidirect
121. #12702C31direct (N × H)
167. C2×#356C2×C31semidirect
168. C2×#355C2×C31semidirect
169. #1784C2×C31semidirect
254. C31#12702direct (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×#3562
C2×C31×#3552
C31×#17842
#1270231