Finite Group Databasebeta

← C31×#9505 #21466264

Subgroups — C31×#9505 #21466264

Order 7936.

684 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2973C211yesmaximal, characteristic
2C31×#2409C211yesmaximal, characteristic
3C31×#1784C212yesmaximal, complemented
4C31×#1465C211yesmaximal, characteristic
5C31×#1305C212yesmaximal, complemented
6C23×Q8×C31C2211yescharacteristic
7C2×C31×C22⋊Q8—21no—
8C2×C4×D4×C31C2211yescharacteristic
9C31×#435—24no—
10C31×#432—21no—
11C2×C31×#101—24no—
12C31×#407C2212yes—
13C31×#407C2212yes—
14C2×C31×C22⋊C8—21no—
15C31×#395—21no—
16C31×#393—21no—
17C31×#391—21no—
18C31×#386C412yes—
19C31×#385C412yes—
20C31×#380C2211yescharacteristic
21C31×#324—21no—
22C31×#323—21no—
23C22×Q8×C31—22no—
24C22×Q8×C31—21no—
25C22×Q8×C31—21no—
26C22×Q8×C31—22no—
27C22×Q8×C31—44no—
28C22×D4×C31C2×C411yescharacteristic
29C23×C4×C31C2×C412yes—
30C23×C4×C31D411yescharacteristic
31C23×C4×C31C2311yescharacteristic
32C2×M16×C31—22no—
33C2×M16×C31D414yes—
34C22×C8×C31—22no—
35C31×C22⋊Q8—44no—
36C4×D4×C31—22no—
37C4×D4×C31—22no—
38C2×C4⋊C4×C31—22no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31C2×C411yescharacteristic
43C2×C22⋊C4×C31C2×C412yes—
44C2×C22⋊C4×C31D411yescharacteristic
45C2×C42×C31D411yescharacteristic
46C31×C4⋊C8—22no—
47C31×#101—48no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8—22no—
50C31×C22⋊C8—22no—
51C31×C22⋊C8—22no—
52C31×C22⋊C8—22no—
53C31×#97—22no—
54C31×#95—22no—
55C31×#95—22no—
56C31×#95—21no—
57C24×C31C22⋊C411yescharacteristic
58C24×C31C22×C411yescharacteristic
59C2×Q8×C31—28no—
60C2×Q8×C31—24no—
61C2×Q8×C31—42no—
62C2×Q8×C31—44no—
63C2×Q8×C31—44no—
64C2×Q8×C31—48no—
65C2×Q8×C31—41no—
66C2×Q8×C31—42no—
67C2×Q8×C31—41no—
68C2×D4×C31—22no—
69C2×D4×C31—22no—
70C2×D4×C31—41no—
71C22×C4×C31—22no—
72C22×C4×C31—24no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—22no—
76C22×C4×C31—44no—
77C22×C4×C31—42no—
78C22×C4×C31—42no—
79C22×C4×C31C22⋊C411yescharacteristic
80C22×C4×C31C22⋊C411yescharacteristic
81C22×C4×C31C22⋊C411yescharacteristic
82C22×C4×C31C2×D412yes—
83C22×C4×C31C2×D411yescharacteristic
84C22×C4×C31C2×D411yescharacteristic
85C22×C4×C31#4112yes—
86M16×C31—44no—
87M16×C31—44no—
88C2×C8×C31—24no—
89C2×C8×C31—42no—
90C2×C8×C31—82no—
91C4⋊C4×C31—22no—
92C4⋊C4×C31—42no—
93C4⋊C4×C31—44no—
94C22⋊C4×C31—22no—
95C22⋊C4×C31—22no—
96C22⋊C4×C31—44no—
97C22⋊C4×C31—42no—
98C42×C31—21no—
99C42×C31—21no—
100#9505C3111yesmaximal, characteristic, direct product factor, Sylow
101C23×C31—21no—
102C23×C31—21no—
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31#10112yescomplemented
110C23×C31C2×C22⋊C411yescharacteristic
111C23×C31C4×D412yes—
112Q8×C31—48no—
113Q8×C31—44no—
114Q8×C31—44no—
115D4×C31—44no—
116C2×C4×C31—22no—
117C2×C4×C31—24no—
118C2×C4×C31—24no—
119C2×C4×C31—28no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—41no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—44no—
129C2×C4×C31—42no—
130C2×C4×C31—42no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31—42no—
136C2×C4×C31—81no—
137C2×C4×C31#12011yescharacteristic
138C2×C4×C31#12112yes—
139C2×C4×C31#12311yescharacteristic
140C8×C31—44no—
141C8×C31—82no—
142#2973C2×C3111yescharacteristic
143#2409C2×C3111yescharacteristic
144#1784C2×C3112yescomplemented
145#1465C2×C3111yescharacteristic
146#1305C2×C3112yescomplemented
147C22×C31—21no—
148C22×C31—21no—
149C22×C31—22no—
150C22×C31—22no—
151C22×C31—22no—
152C22×C31—22no—
153C22×C31—42no—
154C22×C31—42no—
155C22×C31—42no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—81no—
159C22×C31#38611yescharacteristic
160C22×C31C2×#10111yescharacteristic
161C22×C31#41311yescharacteristic
162C4×C31—22no—
163C4×C31—21no—
164C4×C31—44no—
165C4×C31—41no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31—82no—
169C4×C31B(4,𝔽2)12yes—
170C23×Q8C22×C3111yescharacteristic
171C2×C22⋊Q8—21no—
172C2×C4×D4C22×C3111yescharacteristic
173#435—24no—
174#432—21no—
175C2×#101—24no—
176#407C22×C3112yes—
177#407C22×C3112yes—
178C2×C22⋊C8—21no—
179#395—21no—
180#393—21no—
181#391—21no—
182#386C4×C3112yes—
183#385C4×C3112yes—
184#380C22×C3111yescharacteristic
185#324—21no—
186#323—21no—
187C2×C31—21no—
188C2×C31—21no—
189C2×C31—42no—
190C2×C31—42no—
191C2×C31#188911yescharacteristic
192C2×C31#190111yescharacteristic
193C2×C31#191511yescharacteristic
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—44no—
199C22×D4C2×C4×C3111yescharacteristic
200C23×C4C2×C4×C3112yes—
201C23×C4D4×C3111yescharacteristic
202C23×C4C23×C3111yescharacteristic
203C2×M16—22no—
204C2×M16D4×C3114yes—
205C22×C8—22no—
206C22⋊Q8—44no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4D4×C3111yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4C2×C4×C3111yescharacteristic
214C2×C22⋊C4C2×C4×C3112yes—
215C2×C22⋊C4D4×C3111yescharacteristic
216C2×C42D4×C3111yescharacteristic
217C4⋊C8—22no—
218#101—48no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224#97—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228C31#950511yescharacteristic, direct product factor, Sylow
229C24C22⋊C4×C3111yescharacteristic
230C24C22×C4×C3111yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊C4×C3111yescharacteristic
252C22×C4C22⋊C4×C3111yescharacteristic
253C22×C4C22⋊C4×C3111yescharacteristic
254C22×C4C2×D4×C3112yes—
255C22×C4C2×D4×C3111yescharacteristic
256C22×C4C2×D4×C3111yescharacteristic
257C22×C4#41×C3112yes—
258M16—44no—
259M16—44no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C31×#10112yescomplemented
281C23C2×C22⋊C4×C3111yescharacteristic
282C23C4×D4×C3112yes—
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4C31×#12011yescharacteristic
309C2×C4C31×#12112yes—
310C2×C4C31×#12311yescharacteristic
311C8—44no—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22C31×#38611yescharacteristic
326C22C2×C31×#10111yescharacteristic
327C22C31×#41311yescharacteristic
328C4—22no—
329C4—21no—
330C4—44no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4C31×B(4,𝔽2)12yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2C31×#188911yescharacteristic
341C2C31×#190111yescharacteristic
342C2C31×#191511yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC24C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1784C2semidirect
5. C31×#1305C2semidirect
100. #9505C31direct (N × H)
109. C23×C31#101semidirect
144. #1784C2×C31semidirect
146. #1305C2×C31semidirect
228. C31#9505direct (N × H)
280. C23C31×#101semidirect

Maximal subgroups

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

subgroupindex
C31×#29732
C31×#24092
C31×#17842
C31×#14652
C31×#13052
#950531