Finite Group Databasebeta

← C31×#10255 #21467014

Subgroups — C31×#10255 #21467014

Order 7936.

582 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3245C211yesmaximal, characteristic
2C31×#3037C212yesmaximal, complemented
3C31×#1984C211yesmaximal, characteristic, complemented
4C31×#1492C212yesmaximal, complemented
5C31×#1292C211yesmaximal, characteristic, complemented
6C2×C31×#137—22no—
7C31×#532C2212yes—
8C2×C4×Q8×C31C2211yescharacteristic
9C31×#474—22no—
10C31×#466—22no—
11C31×#464—24no—
12C31×#462—22no—
13C31×#460—22no—
14C31×#459—22no—
15C31×#423C2212yes—
16C2×C31×C4⋊C8C2211yescharacteristic
17C31×#417—22no—
18C2×C4×C8×C31C2211yescharacteristic
19C31×#336—22no—
20C31×D4⋊C8—24no—
21C2×#41×C31—22no—
22C22×Q8×C31D411yescharacteristic
23C31×#137—44no—
24C31×#137—44no—
25C2×Q16×C31—24no—
26C2×SD16×C31—24no—
27C2×D8×C31—22no—
28C2×M16×C31—22no—
29C22×C8×C31—21no—
30C22×C8×C31D412yes—
31C31×C4⋊Q8D412yes—
32C31×#127D412yes—
33C31×#124—22no—
34C31×#121—22no—
35C31×#121—22no—
36C4×Q8×C31—22no—
37C4×Q8×C31—22no—
38C4×D4×C31—22no—
39C4×D4×C31—22no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C42×C31—21no—
43C2×C42×C31C2311yescharacteristic
44C31×#107—22no—
45C31×C4⋊C8—22no—
46C31×C4⋊C8—22no—
47C31×C4⋊C8D414yes—
48C31×#103—24no—
49C31×#102—24no—
50C31×#102—24no—
51C4×C8×C31—22no—
52#41×C31—42no—
53#41×C31—42no—
54#41×C31—42no—
55#41×C31—42no—
56C2×Q8×C31—21no—
57C2×Q8×C31—22no—
58C2×Q8×C31—22no—
59C2×Q8×C31—22no—
60C2×Q8×C31D812yescomplemented
61C2×D4×C31—22no—
62C2×D4×C31—22no—
63C2×D4×C31—22no—
64C2×D4×C31—42no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—42no—
68C22×C4×C31C2×D411yescharacteristic
69C22×C4×C31C2×D411yescharacteristic
70C22×C4×C31#4111yescharacteristic
71Q16×C31—48no—
72SD16×C31—48no—
73D8×C31—82no—
74M16×C31—42no—
75M16×C31—42no—
76C2×C8×C31—22no—
77C2×C8×C31—22no—
78C2×C8×C31—24no—
79C2×C8×C31—41no—
80C2×C8×C31—41no—
81C2×C8×C31—42no—
82C4⋊C4×C31—22no—
83C4⋊C4×C31—22no—
84C4⋊C4×C31—22no—
85C4⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—42no—
88C42×C31—41no—
89C42×C31—41no—
90C42×C31C2×D412yes—
91C42×C31C2×D412yes—
92#10255C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—42no—
94C23×C31#12111yescharacteristic
95Q8×C31—24no—
96Q8×C31—42no—
97Q8×C31—42no—
98Q8×C31—42no—
99D4×C31—44no—
100D4×C31—42no—
101D4×C31—42no—
102D4×C31—42no—
103D4×C31—42no—
104D4×C31—82no—
105D4×C31—82no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—42no—
113C2×C4×C31—41no—
114C2×C4×C31—42no—
115C2×C4×C31—82no—
116C2×C4×C31—82no—
117C2×C4×C31#12012yes—
118C2×C4×C31#12011yescharacteristic
119C2×C4×C31#12112yes—
120C2×C4×C31#12711yescharacteristic
121C2×C4×C31C2×D811yescharacteristic
122C2×C4×C31#13511yescharacteristic
123C2×C4×C31AGL(1,ℤ/8)12yes—
124C8×C31—42no—
125C8×C31—44no—
126C8×C31—42no—
127#3245C2×C3111yescharacteristic
128#3037C2×C3112yescomplemented
129#1984C2×C3111yescharacteristic, complemented
130#1492C2×C3112yescomplemented
131#1292C2×C3111yescharacteristic, complemented
132C22×C31—21no—
133C22×C31—21no—
134C22×C31—82no—
135C22×C31—82no—
136C22×C31—82no—
137C22×C31#39211yescharacteristic
138C22×C31#46611yescharacteristic
139C22×C31#46911yescharacteristic
140C4×C31—21no—
141C4×C31—22no—
142C4×C31—42no—
143C4×C31—82no—
144C4×C31—81no—
145C4×C31#44712yes—
146C4×C31#44912yes—
147C4×C31#45511yescharacteristic
148C4×C31#45611yescharacteristic
149C2×#137—22no—
150#532C22×C3112yes—
151C2×C4×Q8C22×C3111yescharacteristic
152#474—22no—
153#466—22no—
154#464—24no—
155#462—22no—
156#460—22no—
157#459—22no—
158#423C22×C3112yes—
159C2×C4⋊C8C22×C3111yescharacteristic
160#417—22no—
161C2×C4×C8C22×C3111yescharacteristic
162#336—22no—
163D4⋊C8—24no—
164C2×C31—21no—
165C2×C31—21no—
166C2×C31—162no—
167C2×C31#199911yescharacteristic
168C2×C31#200711yescharacteristic
169C2×C31#201711yescharacteristic
170C2×#41—22no—
171C22×Q8D4×C3111yescharacteristic
172#137—44no—
173#137—44no—
174C2×Q16—24no—
175C2×SD16—24no—
176C2×D8—22no—
177C2×M16—22no—
178C22×C8—21no—
179C22×C8D4×C3112yes—
180C4⋊Q8D4×C3112yes—
181#127D4×C3112yes—
182#124—22no—
183#121—22no—
184#121—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×D4—22no—
188C4×D4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4D4×C3111yescharacteristic
191C2×C42—21no—
192C2×C42C23×C3111yescharacteristic
193#107—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8D4×C3114yes—
197#103—24no—
198#102—24no—
199#102—24no—
200C4×C8—22no—
201C31#1025511yescharacteristic, direct product factor, Sylow
202#41—42no—
203#41—42no—
204#41—42no—
205#41—42no—
206C2×Q8—21no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8D8×C3112yescomplemented
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—42no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4C2×D4×C3111yescharacteristic
219C22×C4C2×D4×C3111yescharacteristic
220C22×C4#41×C3111yescharacteristic
221Q16—48no—
222SD16—48no—
223D8—82no—
224M16—42no—
225M16—42no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—24no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C42—41no—
239C42—41no—
240C42C2×D4×C3112yes—
241C42C2×D4×C3112yes—
242C23—42no—
243C23C31×#12111yescharacteristic
244Q8—24no—
245Q8—42no—
246Q8—42no—
247Q8—42no—
248D4—44no—
249D4—42no—
250D4—42no—
251D4—42no—
252D4—42no—
253D4—82no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4C31×#12012yes—
267C2×C4C31×#12011yescharacteristic
268C2×C4C31×#12112yes—
269C2×C4C31×#12711yescharacteristic
270C2×C4C2×D8×C3111yescharacteristic
271C2×C4C31×#13511yescharacteristic
272C2×C4C31×AGL(1,ℤ/8)12yes—
273C8—42no—
274C8—44no—
275C8—42no—
276C22—21no—
277C22—21no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22C31×#39211yescharacteristic
282C22C31×#46611yescharacteristic
283C22C31×#46911yescharacteristic
284C4—21no—
285C4—22no—
286C4—42no—
287C4—82no—
288C4—81no—
289C4C31×#44712yes—
290C4C31×#44912yes—
291C4C31×#45511yescharacteristic
292C4C31×#45611yescharacteristic
293C2—21no—
294C2—21no—
295C2—162no—
296C2C31×#199911yescharacteristic
297C2C31×#200711yescharacteristic
298C2C31×#201711yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3245C2
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×#3037C2semidirect
3. C31×#1984C2semidirect
4. C31×#1492C2semidirect
5. C31×#1292C2semidirect
60. C2×Q8×C31D8semidirect
92. #10255C31direct (N × H)
128. #3037C2×C31semidirect
129. #1984C2×C31semidirect
130. #1492C2×C31semidirect
131. #1292C2×C31semidirect
201. C31#10255direct (N × H)
210. C2×Q8D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32452
C31×#30372
C31×#19842
C31×#14922
C31×#12922
#1025531