Finite Group Databasebeta

← C31×#12701 #21469460

Subgroups — C31×#12701 #21469460

Order 7936.

1070 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1070 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×#532C211yesmaximal, characteristic
2C2×C31×#356C212yesmaximal, complemented
3C2×C31×#355C212yesmaximal, complemented
4C31×#1785C212yesmaximal, complemented
5C22×#41×C31C411yescharacteristic
6C22×#41×C31C2211yescharacteristic
7C31×#532—24no—
8C31×#532—24no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#127C411yescharacteristic
11C2×C31×#124C411yescharacteristic
12C2×C31×#124C2211yescharacteristic
13C2×C31×#121—22no—
14C2×C4×D4×C31—22no—
15C22×C42×C31C411yescharacteristic
16C2×C31×#101C2214yes—
17C2×C31×#100—22no—
18C31×#407—22no—
19C31×#356—24no—
20C31×#355—24no—
21C2×#41×C31—22no—
22C2×#41×C31—22no—
23C2×#41×C31—22no—
24C2×#41×C31—22no—
25C2×#41×C31—42no—
26C2×#41×C31—42no—
27C2×#41×C31—42no—
28C2×#41×C31—42no—
29C22×Q8×C31C2×C411yescharacteristic
30C22×Q8×C31C2311yescharacteristic
31C22×D4×C31—22no—
32C22×D4×C31—21no—
33C22×D4×C31C2×C411yescharacteristic
34C22×D4×C31C2×C411yescharacteristic
35C22×D4×C31D412yes—
36C23×C4×C31—21no—
37C23×C4×C31—21no—
38C23×C4×C31C2×C411yescharacteristic
39C23×C4×C31D412yes—
40C2×M16×C31—24no—
41C31×C4⋊Q8—22no—
42C31×C4⋊Q8D414yes—
43C31×#127—22no—
44C31×#127—22no—
45C31×#124—22no—
46C31×#124—22no—
47C31×#124—22no—
48C31×#124D414yes—
49C31×#121—216no—
50C4×D4×C31—216no—
51C2×C4⋊C4×C31—22no—
52C2×C22⋊C4×C31—22no—
53C2×C22⋊C4×C31—22no—
54C2×C42×C31—21no—
55C2×C42×C31—22no—
56C2×C42×C31—22no—
57C2×C42×C31C2×C411yescharacteristic
58C2×C42×C31C2×C411yescharacteristic
59C31×#101—24no—
60C31×#101—24no—
61C31×#100—44no—
62C31×C22⋊C8—44no—
63C24×C31—21no—
64C24×C31C22⋊C412yes—
65C24×C31C22⋊C411yescharacteristic
66#41×C31—48no—
67#41×C31—48no—
68#41×C31—48no—
69#41×C31—48no—
70C2×Q8×C31—21no—
71C2×Q8×C31—41no—
72C2×Q8×C31—41no—
73C2×Q8×C31—41no—
74C2×Q8×C31—41no—
75C2×Q8×C31C22⋊C412yes—
76C2×Q8×C31C2×D412yes—
77C2×Q8×C31C2×D412yes—
78C2×D4×C31—28no—
79C2×D4×C31—24no—
80C2×D4×C31—24no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—24no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—42no—
88C2×D4×C31—48no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—42no—
92C2×D4×C31—42no—
93C2×D4×C31C22⋊C412yes—
94C22×C4×C31—22no—
95C22×C4×C31—21no—
96C22×C4×C31—22no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31—24no—
100C22×C4×C31—24no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—24no—
104C22×C4×C31—24no—
105C22×C4×C31—42no—
106C22×C4×C31—44no—
107C22×C4×C31—42no—
108C22×C4×C31—42no—
109C22×C4×C31C22⋊C411yescharacteristic
110C22×C4×C31C22×C411yescharacteristic
111C22×C4×C31C2×D412yes—
112M16×C31—44no—
113M16×C31—44no—
114C2×C8×C31—44no—
115C4⋊C4×C31—28no—
116C22⋊C4×C31—28no—
117C22⋊C4×C31—28no—
118C42×C31—24no—
119C42×C31—21no—
120C42×C31—21no—
121C42×C31C22⋊C412yes—
122C42×C31C22⋊C412yes—
123#12701C3111yesmaximal, characteristic, direct product factor, Sylow
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—22no—
127C23×C31—22no—
128C23×C31—22no—
129C23×C31—22no—
130C23×C31—42no—
131C23×C31—44no—
132C23×C31—44no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31#9912yes—
136C23×C31C2×C22⋊C411yescharacteristic
137Q8×C31—42no—
138Q8×C31—42no—
139Q8×C31—42no—
140Q8×C31—42no—
141D4×C31—48no—
142D4×C31—42no—
143D4×C31—416no—
144D4×C31—42no—
145D4×C31—42no—
146D4×C31—42no—
147D4×C31—48no—
148C2×C4×C31—24no—
149C2×C4×C31—24no—
150C2×C4×C31—24no—
151C2×C4×C31—21no—
152C2×C4×C31—22no—
153C2×C4×C31—22no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—22no—
157C2×C4×C31—21no—
158C2×C4×C31—22no—
159C2×C4×C31—22no—
160C2×C4×C31—48no—
161C2×C4×C31—48no—
162C2×C4×C31—48no—
163C2×C4×C31—42no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31#9912yes—
167C2×C4×C31C2×C22⋊C411yescharacteristic
168C2×C4×C31C2×C22⋊C411yescharacteristic
169C2×C4×C31#12014yes—
170C8×C31—84no—
171C2×#532C2×C3111yescharacteristic
172C2×#356C2×C3112yescomplemented
173C2×#355C2×C3112yescomplemented
174#1785C2×C3112yescomplemented
175C22×C31—22no—
176C22×C31—22no—
177C22×C31—42no—
178C22×C31—42no—
179C22×C31—44no—
180C22×C31—44no—
181C22×C31—42no—
182C22×C31—44no—
183C22×C31—42no—
184C22×C31—42no—
185C22×C31—82no—
186C22×C31—82no—
187C22×C31—82no—
188C22×C31#37911yescharacteristic
189C22×C31C2×#9911yescharacteristic
190C22×C31C2×#9911yescharacteristic
191C4×C31—24no—
192C4×C31—24no—
193C4×C31—22no—
194C4×C31—21no—
195C4×C31—21no—
196C4×C31—42no—
197C4×C31—42no—
198C22×#41C4×C3111yescharacteristic
199C22×#41C22×C3111yescharacteristic
200#532—24no—
201#532—24no—
202C2×C4⋊Q8C22×C3111yescharacteristic
203C2×#127C4×C3111yescharacteristic
204C2×#124C4×C3111yescharacteristic
205C2×#124C22×C3111yescharacteristic
206C2×#121—22no—
207C2×C4×D4—22no—
208C22×C42C4×C3111yescharacteristic
209C2×#101C22×C3114yes—
210C2×#100—22no—
211#407—22no—
212#356—24no—
213#355—24no—
214C2×C31—21no—
215C2×C31—21no—
216C2×C31—42no—
217C2×C31—82no—
218C2×C31—82no—
219C2×C31#178111yescharacteristic
220C2×C31#213212yes—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—42no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41—42no—
229C22×Q8C2×C4×C3111yescharacteristic
230C22×Q8C23×C3111yescharacteristic
231C22×D4—22no—
232C22×D4—21no—
233C22×D4C2×C4×C3111yescharacteristic
234C22×D4C2×C4×C3111yescharacteristic
235C22×D4D4×C3112yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×C4×C3111yescharacteristic
239C23×C4D4×C3112yes—
240C2×M16—24no—
241C4⋊Q8—22no—
242C4⋊Q8D4×C3114yes—
243#127—22no—
244#127—22no—
245#124—22no—
246#124—22no—
247#124—22no—
248#124D4×C3114yes—
249#121—216no—
250C4×D4—216no—
251C2×C4⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42C2×C4×C3111yescharacteristic
258C2×C42C2×C4×C3111yescharacteristic
259#101—24no—
260#101—24no—
261#100—44no—
262C22⋊C8—44no—
263C31#1270111yescharacteristic, direct product factor, Sylow
264C24—21no—
265C24C22⋊C4×C3112yes—
266C24C22⋊C4×C3111yescharacteristic
267#41—48no—
268#41—48no—
269#41—48no—
270#41—48no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8C22⋊C4×C3112yes—
277C2×Q8C2×D4×C3112yes—
278C2×Q8C2×D4×C3112yes—
279C2×D4—28no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—24no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—48no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4C22⋊C4×C3112yes—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4C22⋊C4×C3111yescharacteristic
311C22×C4C22×C4×C3111yescharacteristic
312C22×C4C2×D4×C3112yes—
313M16—44no—
314M16—44no—
315C2×C8—44no—
316C4⋊C4—28no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C42—24no—
320C42—21no—
321C42—21no—
322C42C22⋊C4×C3112yes—
323C42C22⋊C4×C3112yes—
324C23—21no—
325C23—21no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—42no—
331C23—44no—
332C23—44no—
333C23—41no—
334C23—41no—
335C23C31×#9912yes—
336C23C2×C22⋊C4×C3111yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—42no—
340Q8—42no—
341D4—48no—
342D4—42no—
343D4—416no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—48no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4C31×#9912yes—
367C2×C4C2×C22⋊C4×C3111yescharacteristic
368C2×C4C2×C22⋊C4×C3111yescharacteristic
369C2×C4C31×#12014yes—
370C8—84no—
371C22—22no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—44no—
376C22—44no—
377C22—42no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22C31×#37911yescharacteristic
385C22C2×C31×#9911yescharacteristic
386C22C2×C31×#9911yescharacteristic
387C4—24no—
388C4—24no—
389C4—22no—
390C4—21no—
391C4—21no—
392C4—42no—
393C4—42no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—82no—
398C2—82no—
399C2C31×#178111yescharacteristic
400C2C31×#213212yes—
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC2×C31×#532C2
LCSG ▷ C22×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
2. C2×C31×#356C2semidirect
3. C2×C31×#355C2semidirect
4. C31×#1785C2semidirect
123. #12701C31direct (N × H)
172. C2×#356C2×C31semidirect
173. C2×#355C2×C31semidirect
174. #1785C2×C31semidirect
263. C31#12701direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5322
C2×C31×#3562
C2×C31×#3552
C31×#17852
#1270131