Finite Group Databasebeta

← C31×#10259 #21467018

Subgroups — C31×#10259 #21467018

Order 7936.

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

normal subgroups only

This group has 474 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×#3249C211yesmaximal, characteristic
2C31×#3078C212yesmaximal
3C31×#1984C211yesmaximal, characteristic
4C31×#1493C212yesmaximal
5C31×#1292C211yesmaximal, characteristic
6C31×#533C2212yes—
7C2×C4×Q8×C31C2211yescharacteristic
8C31×#479—24no—
9C31×#477—22no—
10C31×#477—22no—
11C31×#475—22no—
12C31×#467—22no—
13C31×#462—22no—
14C31×#428—22no—
15C31×#423C2212yes—
16C2×C31×C4⋊C8C2211yescharacteristic
17C31×#419—22no—
18C2×C4×C8×C31C2211yescharacteristic
19C31×#336—22no—
20C31×Q8⋊C8—24no—
21C22×Q8×C31D411yescharacteristic
22C2×Q16×C31—22no—
23C2×M16×C31—22no—
24C22×C8×C31—21no—
25C22×C8×C31D412yes—
26C31×C4⋊Q8D412yes—
27C31×C4⋊Q8D412yes—
28C31×#125—22no—
29C31×C22⋊Q8—22no—
30C31×C22⋊Q8—22no—
31C4×Q8×C31—22no—
32C4×Q8×C31—22no—
33C4×Q8×C31—22no—
34C4×Q8×C31—22no—
35C31×#117—22no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C42×C31—21no—
39C2×C42×C31C2311yescharacteristic
40C31×#107—24no—
41C31×#107—22no—
42C31×#106—24no—
43C31×C4⋊C8—22no—
44C31×C4⋊C8—22no—
45C31×C4⋊C8D414yes—
46C31×#103—24no—
47C31×#103—24no—
48C31×#103—24no—
49C4×C8×C31—22no—
50C2×Q8×C31—21no—
51C2×Q8×C31—22no—
52C2×Q8×C31—22no—
53C2×Q8×C31—22no—
54C2×Q8×C31—22no—
55C2×Q8×C31Q1612yescomplemented
56C22×C4×C31—21no—
57C22×C4×C31—21no—
58C22×C4×C31C2×D411yescharacteristic
59C22×C4×C31C2×D411yescharacteristic
60C22×C4×C31#4111yescharacteristic
61Q16×C31—82no—
62M16×C31—42no—
63M16×C31—42no—
64C2×C8×C31—22no—
65C2×C8×C31—22no—
66C2×C8×C31—24no—
67C2×C8×C31—41no—
68C2×C8×C31—41no—
69C2×C8×C31—42no—
70C4⋊C4×C31—22no—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—22no—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—22no—
75C4⋊C4×C31—42no—
76C4⋊C4×C31—42no—
77C4⋊C4×C31—42no—
78C22⋊C4×C31—42no—
79C42×C31—41no—
80C42×C31—42no—
81C42×C31—41no—
82C42×C31C2×D412yes—
83C42×C31C2×D412yes—
84#10259C3111yesmaximal, characteristic, direct product factor, Sylow
85C23×C31#12111yescharacteristic
86Q8×C31—24no—
87Q8×C31—44no—
88Q8×C31—42no—
89Q8×C31—42no—
90Q8×C31—42no—
91C2×C4×C31—21no—
92C2×C4×C31—21no—
93C2×C4×C31—22no—
94C2×C4×C31—41no—
95C2×C4×C31—41no—
96C2×C4×C31—41no—
97C2×C4×C31—42no—
98C2×C4×C31—41no—
99C2×C4×C31—42no—
100C2×C4×C31—42no—
101C2×C4×C31#12012yes—
102C2×C4×C31#12011yescharacteristic
103C2×C4×C31#12112yes—
104C2×C4×C31#12711yescharacteristic
105C2×C4×C31C2×Q1611yescharacteristic
106C2×C4×C31#13511yescharacteristic
107C2×C4×C31#13712yes—
108C8×C31—42no—
109C8×C31—44no—
110C8×C31—42no—
111#3249C2×C3111yescharacteristic
112#3078C2×C3112yes—
113#1984C2×C3111yescharacteristic
114#1493C2×C3112yes—
115#1292C2×C3111yescharacteristic
116C22×C31—21no—
117C22×C31—21no—
118C22×C31#39211yescharacteristic
119C22×C31#46711yescharacteristic
120C22×C31#47011yescharacteristic
121C4×C31—21no—
122C4×C31—22no—
123C4×C31—42no—
124C4×C31—82no—
125C4×C31—81no—
126C4×C31—82no—
127C4×C31#45112yes—
128C4×C31#45212yes—
129C4×C31#45511yescharacteristic
130C4×C31#45611yescharacteristic
131#533C22×C3112yes—
132C2×C4×Q8C22×C3111yescharacteristic
133#479—24no—
134#477—22no—
135#477—22no—
136#475—22no—
137#467—22no—
138#462—22no—
139#428—22no—
140#423C22×C3112yes—
141C2×C4⋊C8C22×C3111yescharacteristic
142#419—22no—
143C2×C4×C8C22×C3111yescharacteristic
144#336—22no—
145Q8⋊C8—24no—
146C2×C31—21no—
147C2×C31—21no—
148C2×C31#200111yescharacteristic
149C2×C31#200711yescharacteristic
150C2×C31#201711yescharacteristic
151C22×Q8D4×C3111yescharacteristic
152C2×Q16—22no—
153C2×M16—22no—
154C22×C8—21no—
155C22×C8D4×C3112yes—
156C4⋊Q8D4×C3112yes—
157C4⋊Q8D4×C3112yes—
158#125—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165#117—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4D4×C3111yescharacteristic
168C2×C42—21no—
169C2×C42C23×C3111yescharacteristic
170#107—24no—
171#107—22no—
172#106—24no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8D4×C3114yes—
176#103—24no—
177#103—24no—
178#103—24no—
179C4×C8—22no—
180C31#1025911yescharacteristic, direct product factor, Sylow
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8Q16×C3112yescomplemented
187C22×C4—21no—
188C22×C4—21no—
189C22×C4C2×D4×C3111yescharacteristic
190C22×C4C2×D4×C3111yescharacteristic
191C22×C4#41×C3111yescharacteristic
192Q16—82no—
193M16—42no—
194M16—42no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C22⋊C4—42no—
210C42—41no—
211C42—42no—
212C42—41no—
213C42C2×D4×C3112yes—
214C42C2×D4×C3112yes—
215C23C31×#12111yescharacteristic
216Q8—24no—
217Q8—44no—
218Q8—42no—
219Q8—42no—
220Q8—42no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4C31×#12012yes—
232C2×C4C31×#12011yescharacteristic
233C2×C4C31×#12112yes—
234C2×C4C31×#12711yescharacteristic
235C2×C4C2×Q16×C3111yescharacteristic
236C2×C4C31×#13511yescharacteristic
237C2×C4C31×#13712yes—
238C8—42no—
239C8—44no—
240C8—42no—
241C22—21no—
242C22—21no—
243C22C31×#39211yescharacteristic
244C22C31×#46711yescharacteristic
245C22C31×#47011yescharacteristic
246C4—21no—
247C4—22no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4C31×#45112yes—
253C4C31×#45212yes—
254C4C31×#45511yescharacteristic
255C4C31×#45611yescharacteristic
256C2—21no—
257C2—21no—
258C2C31×#200111yescharacteristic
259C2C31×#200711yescharacteristic
260C2C31×#201711yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC23×C31#121
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
55. C2×Q8×C31Q16semidirect
84. #10259C31direct (N × H)
180. C31#10259direct (N × H)
186. C2×Q8Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32492
C31×#30782
C31×#19842
C31×#14932
C31×#12922
#1025931