Finite Group Databasebeta

← C31×#10024 #21466783

Subgroups — C31×#10024 #21466783

Order 7936.

512 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C31×#3140C211yesmaximal, characteristic
2C31×#3113C211yesmaximal, characteristic, complemented
3C31×#2632C211yesmaximal, characteristic
4C31×#1533C211yesmaximal, characteristic
5C31×#1515C211yesmaximal, characteristic, complemented
6C31×#1491C211yesmaximal, characteristic, complemented
7C31×#1288C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C2×C4×Q8×C31C2211yescharacteristic
11C31×#489—21no—
12C31×#489—21no—
13C31×#488—22no—
14C31×#483—21no—
15C31×#475—21no—
16C31×#462—21no—
17C31×#451—21no—
18C31×#441—22no—
19C31×#432C2211yescharacteristic
20C31×#432C2211yescharacteristic
21C31×#419—21no—
22C31×#418—21no—
23C4×M16×C31C2211yescharacteristic
24C31×#399—21no—
25C31×#395—21no—
26C31×#394—21no—
27C31×#393—21no—
28C31×#388—22no—
29C31×#383C2211yescharacteristic
30C31×#329—22no—
31C31×#328—22no—
32C31×Q8⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C22×Q8×C31D411yescharacteristic
35C22×D4×C31D411yescharacteristic
36C2×Q16×C31—22no—
37C2×M16×C31—21no—
38C31×C4⋊Q8D411yescharacteristic
39C31×C4⋊Q8D411yescharacteristic
40C31×#125—21no—
41C31×#124—21no—
42C31×#124—21no—
43C31×#124—21no—
44C31×#124—21no—
45C31×C22⋊Q8—21no—
46C31×C22⋊Q8—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31—21no—
49C4×Q8×C31—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C31×#117—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C22⋊C4×C31—22no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—22no—
61C2×C42×C31—21no—
62C2×C42×C31C2311yescharacteristic
63C31×#106—22no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8—21no—
66C31×#103—22no—
67C31×#103—22no—
68C31×#103—22no—
69C31×#103—22no—
70C31×#102—22no—
71C31×#102—22no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×#97—21no—
75C31×#97—21no—
76C31×#97D412yes—
77C4×C8×C31D412yes—
78C31×#95—22no—
79C31×#95—21no—
80C31×#95—21no—
81C31×#95D412yes—
82C24×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—21no—
89C2×Q8×C31—42no—
90C2×Q8×C31—42no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—42no—
94C22×C4×C31—21no—
95C22×C4×C31—22no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101Q16×C31—44no—
102M16×C31—42no—
103C2×C8×C31—22no—
104C2×C8×C31—22no—
105C2×C8×C31—41no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—41no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—82no—
117C22⋊C4×C31—44no—
118C22⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—82no—
124C42×C31—41no—
125C42×C31—41no—
126C42×C31—41no—
127C42×C31C2×D411yescharacteristic
128C42×C31C2×D411yescharacteristic
129C42×C31C2×D411yescharacteristic
130C42×C31#4111yescharacteristic
131#10024C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31—42no—
133C23×C31—42no—
134C23×C31—41no—
135C23×C31—42no—
136C23×C31#12011yescharacteristic
137Q8×C31—22no—
138Q8×C31—22no—
139Q8×C31—41no—
140Q8×C31—41no—
141Q8×C31—42no—
142Q8×C31—42no—
143Q8×C31—81no—
144Q8×C31—81no—
145D4×C31—42no—
146D4×C31—81no—
147C2×C4×C31—22no—
148C2×C4×C31—22no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31—42no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31#12011yescharacteristic
163C2×C4×C31#12011yescharacteristic
164C2×C4×C31#12111yescharacteristic
165C2×C4×C31#12111yescharacteristic
166C2×C4×C31#12111yescharacteristic
167C2×C4×C31#12711yescharacteristic
168C8×C31—42no—
169C8×C31—42no—
170C8×C31—81no—
171#3140C2×C3111yescharacteristic
172#3113C2×C3111yescharacteristic, complemented
173#2632C2×C3111yescharacteristic
174#1533C2×C3111yescharacteristic
175#1515C2×C3111yescharacteristic, complemented
176#1491C2×C3111yescharacteristic, complemented
177#1288C2×C3111yescharacteristic, complemented
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—84no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31#39211yescharacteristic
185C22×C31#45512yes—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—21no—
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—42no—
193C4×C31—82no—
194C4×C31—81no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—81no—
198#533C22×C3111yescharacteristic
199C2×#124C22×C3111yescharacteristic
200C2×C4×Q8C22×C3111yescharacteristic
201#489—21no—
202#489—21no—
203#488—22no—
204#483—21no—
205#475—21no—
206#462—21no—
207#451—21no—
208#441—22no—
209#432C22×C3111yescharacteristic
210#432C22×C3111yescharacteristic
211#419—21no—
212#418—21no—
213C4×M16C22×C3111yescharacteristic
214#399—21no—
215#395—21no—
216#394—21no—
217#393—21no—
218#388—22no—
219#383C22×C3111yescharacteristic
220#329—22no—
221#328—22no—
222Q8⋊C8—22no—
223C2×C31—22no—
224C2×C31—161no—
225C2×C31#200411yescharacteristic
226C2×C31#200911yescharacteristic
227C2×C31#201011yescharacteristic
228C22×Q8D4×C3111yescharacteristic
229C22×Q8D4×C3111yescharacteristic
230C22×D4D4×C3111yescharacteristic
231C2×Q16—22no—
232C2×M16—21no—
233C4⋊Q8D4×C3111yescharacteristic
234C4⋊Q8D4×C3111yescharacteristic
235#125—21no—
236#124—21no—
237#124—21no—
238#124—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248#117—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4D4×C3111yescharacteristic
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C42—21no—
257C2×C42C23×C3111yescharacteristic
258#106—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261#103—22no—
262#103—22no—
263#103—22no—
264#103—22no—
265#102—22no—
266#102—22no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269#97—21no—
270#97—21no—
271#97D4×C3112yes—
272C4×C8D4×C3112yes—
273#95—22no—
274#95—21no—
275#95—21no—
276#95D4×C3112yes—
277C31#1002411yescharacteristic, direct product factor, Sylow
278C24—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4C2×D4×C3111yescharacteristic
295C22×C4C2×D4×C3111yescharacteristic
296C22×C4C2×D4×C3111yescharacteristic
297Q16—44no—
298M16—42no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C22⋊C4—44no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—82no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42C2×D4×C3111yescharacteristic
324C42C2×D4×C3111yescharacteristic
325C42C2×D4×C3111yescharacteristic
326C42#41×C3111yescharacteristic
327C23—42no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23C31×#12011yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340D4—42no—
341D4—81no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4C31×#12011yescharacteristic
358C2×C4C31×#12011yescharacteristic
359C2×C4C31×#12111yescharacteristic
360C2×C4C31×#12111yescharacteristic
361C2×C4C31×#12111yescharacteristic
362C2×C4C31×#12711yescharacteristic
363C8—42no—
364C8—42no—
365C8—81no—
366C22—21no—
367C22—21no—
368C22—84no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C31×#39211yescharacteristic
373C22C31×#45512yes—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C2—22no—
387C2—161no—
388C2C31×#200411yescharacteristic
389C2C31×#200911yescharacteristic
390C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3113C2semidirect
5. C31×#1515C2semidirect
6. C31×#1491C2semidirect
7. C31×#1288C2semidirect
131. #10024C31direct (N × H)
172. #3113C2×C31semidirect
175. #1515C2×C31semidirect
176. #1491C2×C31semidirect
177. #1288C2×C31semidirect
277. C31#10024direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31402
C31×#31132
C31×#26322
C31×#15332
C31×#15152
C31×#14912
C31×#12882
#1002431