Finite Group Databasebeta

← C31×#10338 #21467097

Subgroups — C31×#10338 #21467097

Order 7936.

998 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2483C211yesmaximal, characteristic, complemented
2C31×#2405C212yesmaximal
3C31×#1819C212yesmaximal, complemented
4C31×#1781C211yesmaximal, characteristic
5C31×#1304C211yesmaximal, characteristic, complemented
6C23×D4×C31C2211yescharacteristic
7C2×C31×#123—22no—
8C22×C22⋊C4×C31C2212yes—
9C2×C31×#99—22no—
10C31×#397—22no—
11C31×#394—24no—
12C31×#386—22no—
13C31×#385—22no—
14C31×#385—22no—
15C31×#381—22no—
16C31×#379—22no—
17C31×#379C2211yescharacteristic
18C31×#379C2212yes—
19C31×#379C2211yescharacteristic
20C31×#342—24no—
21C25×C31D411yescharacteristic
22C25×C31C2311yescharacteristic
23C22×D4×C31—21no—
24C22×D4×C31—22no—
25C22×D4×C31—22no—
26C22×D4×C31—21no—
27C22×D4×C31—44no—
28C23×C4×C31D412yes—
29C23×C4×C31D412yes—
30C23×C4×C31D411yescharacteristic
31C31×#123—48no—
32C2×C4⋊C4×C31—24no—
33C2×C4⋊C4×C31—22no—
34C2×C22⋊C4×C31—24no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31—22no—
43C2×C22⋊C4×C31—22no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—24no—
47C2×C22⋊C4×C31D412yes—
48C2×C22⋊C4×C31D414yes—
49C31×#99—44no—
50C31×#95—22no—
51C31×#95—24no—
52C31×#95—24no—
53C24×C31—22no—
54C24×C31—21no—
55C24×C31—42no—
56C24×C31—42no—
57C24×C31—44no—
58C24×C31—44no—
59C24×C31C2×D411yescharacteristic
60C24×C31C2×D412yes—
61C24×C31C2×D411yescharacteristic
62C24×C31C2×D412yes—
63C24×C31#4111yescharacteristic
64C2×D4×C31—24no—
65C2×D4×C31—28no—
66C2×D4×C31—44no—
67C2×D4×C31—42no—
68C2×D4×C31—44no—
69C2×D4×C31—42no—
70C2×D4×C31—44no—
71C2×D4×C31—42no—
72C2×D4×C31—41no—
73C2×D4×C31—42no—
74C2×D4×C31—41no—
75C22×C4×C31—22no—
76C22×C4×C31—21no—
77C22×C4×C31—22no—
78C22×C4×C31—22no—
79C22×C4×C31—22no—
80C22×C4×C31—22no—
81C22×C4×C31—24no—
82C22×C4×C31—22no—
83C22×C4×C31—22no—
84C22×C4×C31—22no—
85C22×C4×C31—42no—
86C22×C4×C31—44no—
87C22×C4×C31—44no—
88C22×C4×C31—41no—
89C4⋊C4×C31—48no—
90C4⋊C4×C31—82no—
91C22⋊C4×C31—44no—
92C22⋊C4×C31—44no—
93C22⋊C4×C31—44no—
94C22⋊C4×C31—48no—
95C22⋊C4×C31—42no—
96C22⋊C4×C31—42no—
97C22⋊C4×C31—42no—
98C22⋊C4×C31—42no—
99C22⋊C4×C31—42no—
100C22⋊C4×C31—44no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—44no—
103C22⋊C4×C31—84no—
104#10338C3111yesmaximal, characteristic, direct product factor, Sylow
105C23×C31—24no—
106C23×C31—24no—
107C23×C31—24no—
108C23×C31—22no—
109C23×C31—24no—
110C23×C31—24no—
111C23×C31—22no—
112C23×C31—41no—
113C23×C31—42no—
114C23×C31—42no—
115C23×C31—41no—
116C23×C31—42no—
117C23×C31—42no—
118C23×C31—42no—
119C23×C31—42no—
120C23×C31—41no—
121C23×C31—44no—
122C23×C31—41no—
123C23×C31—44no—
124C23×C31—82no—
125C23×C31—88no—
126C23×C31—88no—
127C23×C31#12011yescharacteristic
128C23×C31#12012yes—
129C23×C31#12111yescharacteristic
130C23×C31#12112yes—
131C23×C31#12711yescharacteristic
132D4×C31—48no—
133D4×C31—44no—
134D4×C31—44no—
135C2×C4×C31—24no—
136C2×C4×C31—22no—
137C2×C4×C31—42no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—44no—
141C2×C4×C31—44no—
142C2×C4×C31—44no—
143C2×C4×C31—44no—
144C2×C4×C31—44no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—44no—
149C2×C4×C31—44no—
150C2×C4×C31—44no—
151C2×C4×C31—82no—
152C2×C4×C31—82no—
153C2×C4×C31—81no—
154C2×C4×C31—82no—
155#2483C2×C3111yescharacteristic, complemented
156#2405C2×C3112yes—
157#1819C2×C3112yescomplemented
158#1781C2×C3111yescharacteristic
159#1304C2×C3111yescharacteristic, complemented
160C22×C31—22no—
161C22×C31—22no—
162C22×C31—22no—
163C22×C31—21no—
164C22×C31—22no—
165C22×C31—21no—
166C22×C31—22no—
167C22×C31—22no—
168C22×C31—22no—
169C22×C31—22no—
170C22×C31—48no—
171C22×C31—44no—
172C22×C31—48no—
173C22×C31—42no—
174C22×C31—84no—
175C22×C31—82no—
176C22×C31—84no—
177C22×C31—82no—
178C22×C31—82no—
179C22×C31—82no—
180C22×C31—82no—
181C22×C31#39211yescharacteristic
182C22×C31B(4,𝔽2)12yes—
183C22×C31#45814yes—
184C4×C31—42no—
185C4×C31—84no—
186C4×C31—82no—
187C4×C31—82no—
188C4×C31—82no—
189C4×C31—161no—
190C23×D4C22×C3111yescharacteristic
191C2×#123—22no—
192C22×C22⋊C4C22×C3112yes—
193C2×#99—22no—
194#397—22no—
195#394—24no—
196#386—22no—
197#385—22no—
198#385—22no—
199#381—22no—
200#379—22no—
201#379C22×C3111yescharacteristic
202#379C22×C3112yes—
203#379C22×C3111yescharacteristic
204#342—24no—
205C2×C31—24no—
206C2×C31—22no—
207C2×C31—41no—
208C2×C31—42no—
209C2×C31—41no—
210C2×C31—82no—
211C2×C31#202111yescharacteristic
212C2×C31#202212yes—
213C25D4×C3111yescharacteristic
214C25C23×C3111yescharacteristic
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—44no—
220C23×C4D4×C3112yes—
221C23×C4D4×C3112yes—
222C23×C4D4×C3111yescharacteristic
223#123—48no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4D4×C3114yes—
240C2×C22⋊C4D4×C3112yes—
241#99—44no—
242#95—22no—
243#95—24no—
244#95—24no—
245C31#1033811yescharacteristic, direct product factor, Sylow
246C24—22no—
247C24—21no—
248C24—42no—
249C24—42no—
250C24—44no—
251C24—44no—
252C24C2×D4×C3111yescharacteristic
253C24C2×D4×C3112yes—
254C24C2×D4×C3111yescharacteristic
255C24C2×D4×C3112yes—
256C24#41×C3111yescharacteristic
257C2×D4—24no—
258C2×D4—28no—
259C2×D4—44no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—41no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—41no—
282C4⋊C4—48no—
283C4⋊C4—82no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—48no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—84no—
297C23—24no—
298C23—24no—
299C23—24no—
300C23—22no—
301C23—24no—
302C23—24no—
303C23—22no—
304C23—41no—
305C23—42no—
306C23—42no—
307C23—41no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—44no—
314C23—44no—
315C23—41no—
316C23—82no—
317C23—88no—
318C23—88no—
319C23C31×#12011yescharacteristic
320C23C31×#12012yes—
321C23C31×#12111yescharacteristic
322C23C31×#12112yes—
323C23C31×#12711yescharacteristic
324D4—48no—
325D4—44no—
326D4—44no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—82no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—44no—
358C22—48no—
359C22—48no—
360C22—42no—
361C22—84no—
362C22—82no—
363C22—84no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22C31×#39211yescharacteristic
369C22C31×B(4,𝔽2)12yes—
370C22C31×#45814yes—
371C4—42no—
372C4—84no—
373C4—82no—
374C4—82no—
375C4—82no—
376C4—161no—
377C2—24no—
378C2—22no—
379C2—41no—
380C2—41no—
381C2—42no—
382C2—82no—
383C2C31×#202111yescharacteristic
384C2C31×#202212yes—
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC25C23×C31
prime radicalC23×D4×C31C22
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2483C2semidirect
3. C31×#1819C2semidirect
5. C31×#1304C2semidirect
104. #10338C31direct (N × H)
155. #2483C2×C31semidirect
157. #1819C2×C31semidirect
159. #1304C2×C31semidirect
245. C31#10338direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24832
C31×#24052
C31×#18192
C31×#17812
C31×#13042
#1033831