Finite Group Databasebeta

← C31×#10258 #21467017

Subgroups — C31×#10258 #21467017

Order 7936.

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

normal subgroups only

This group has 526 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, complemented
2C31×#3077C212yesmaximal
3C31×#1954C211yesmaximal, characteristic
4C31×#1493C212yesmaximal, complemented
5C31×#1292C211yesmaximal, characteristic, complemented
6C31×#533C2212yes—
7C2×C4×D4×C31C2211yescharacteristic
8C31×#478—24no—
9C31×#477—22no—
10C31×#476—22no—
11C31×#474—22no—
12C31×#467—22no—
13C31×#462—22no—
14C31×#428—22no—
15C31×#423C2212yes—
16C2×C31×C4⋊C8C2211yescharacteristic
17C31×#418—22no—
18C2×C31×C22⋊C8—22no—
19C2×C4×C8×C31C2211yescharacteristic
20C31×Q8⋊C8—24no—
21C22×D4×C31D411yescharacteristic
22C23×C4×C31—21no—
23C2×Q16×C31—22no—
24C2×M16×C31—22no—
25C22×C8×C31—21no—
26C22×C8×C31D412yes—
27C31×C4⋊Q8D412yes—
28C31×C4⋊Q8D412yes—
29C31×#125—22no—
30C31×C22⋊Q8—22no—
31C31×C22⋊Q8—22no—
32C4×Q8×C31—22no—
33C4×Q8×C31—22no—
34C4×D4×C31—22no—
35C4×D4×C31—22no—
36C31×#117—22no—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C22⋊C4×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×#102—24no—
48C31×#102—24no—
49C31×C22⋊C8—44no—
50C4×C8×C31—22no—
51C24×C31—21no—
52C2×Q8×C31—22no—
53C2×Q8×C31—22no—
54C2×D4×C31—22no—
55C2×D4×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31Q1612yescomplemented
58C22×C4×C31—21no—
59C22×C4×C31—42no—
60C22×C4×C31—42no—
61C22×C4×C31—41no—
62C22×C4×C31—41no—
63C22×C4×C31C2×D411yescharacteristic
64C22×C4×C31C2×D411yescharacteristic
65C22×C4×C31#4111yescharacteristic
66Q16×C31—82no—
67M16×C31—42no—
68M16×C31—42no—
69C2×C8×C31—22no—
70C2×C8×C31—24no—
71C2×C8×C31—22no—
72C2×C8×C31—41no—
73C2×C8×C31—41no—
74C2×C8×C31—42no—
75C4⋊C4×C31—22no—
76C4⋊C4×C31—22no—
77C4⋊C4×C31—22no—
78C4⋊C4×C31—22no—
79C4⋊C4×C31—22no—
80C4⋊C4×C31—42no—
81C4⋊C4×C31—42no—
82C22⋊C4×C31—42no—
83C22⋊C4×C31—42no—
84C42×C31—42no—
85C42×C31C2×D412yes—
86C42×C31C2×D412yes—
87#10258C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—22no—
89C23×C31—42no—
90C23×C31—42no—
91C23×C31—41no—
92C23×C31—41no—
93C23×C31#12111yescharacteristic
94Q8×C31—44no—
95Q8×C31—42no—
96Q8×C31—42no—
97D4×C31—24no—
98D4×C31—42no—
99C2×C4×C31—21no—
100C2×C4×C31—21no—
101C2×C4×C31—41no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—42no—
105C2×C4×C31—41no—
106C2×C4×C31—82no—
107C2×C4×C31—81no—
108C2×C4×C31—81no—
109C2×C4×C31#12012yes—
110C2×C4×C31#12011yescharacteristic
111C2×C4×C31#12112yes—
112C2×C4×C31#12711yescharacteristic
113C2×C4×C31C2×Q1611yescharacteristic
114C2×C4×C31#13511yescharacteristic
115C2×C4×C31#13712yes—
116C8×C31—42no—
117C8×C31—44no—
118C8×C31—42no—
119#3249C2×C3111yescharacteristic, complemented
120#3077C2×C3112yes—
121#1954C2×C3111yescharacteristic
122#1493C2×C3112yescomplemented
123#1292C2×C3111yescharacteristic, complemented
124C22×C31—21no—
125C22×C31—21no—
126C22×C31—44no—
127C22×C31—42no—
128C22×C31—82no—
129C22×C31—81no—
130C22×C31—81no—
131C22×C31#39211yescharacteristic
132C22×C31#46711yescharacteristic
133C22×C31#47011yescharacteristic
134C4×C31—21no—
135C4×C31—22no—
136C4×C31—81no—
137C4×C31—82no—
138C4×C31—82no—
139C4×C31#45112yes—
140C4×C31#45212yes—
141C4×C31#45311yescharacteristic
142C4×C31#45411yescharacteristic
143#533C22×C3112yes—
144C2×C4×D4C22×C3111yescharacteristic
145#478—24no—
146#477—22no—
147#476—22no—
148#474—22no—
149#467—22no—
150#462—22no—
151#428—22no—
152#423C22×C3112yes—
153C2×C4⋊C8C22×C3111yescharacteristic
154#418—22no—
155C2×C22⋊C8—22no—
156C2×C4×C8C22×C3111yescharacteristic
157Q8⋊C8—24no—
158C2×C31—21no—
159C2×C31—21no—
160C2×C31—82no—
161C2×C31#200111yescharacteristic
162C2×C31#200611yescharacteristic
163C2×C31#201711yescharacteristic
164C22×D4D4×C3111yescharacteristic
165C23×C4—21no—
166C2×Q16—22no—
167C2×M16—22no—
168C22×C8—21no—
169C22×C8D4×C3112yes—
170C4⋊Q8D4×C3112yes—
171C4⋊Q8D4×C3112yes—
172#125—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×D4—22no—
178C4×D4—22no—
179#117—22no—
180C2×C4⋊C4D4×C3111yescharacteristic
181C2×C22⋊C4—21no—
182C2×C42C23×C3111yescharacteristic
183#107—24no—
184#107—22no—
185#106—24no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8D4×C3114yes—
189#103—24no—
190#102—24no—
191#102—24no—
192C22⋊C8—44no—
193C4×C8—22no—
194C31#1025811yescharacteristic, direct product factor, Sylow
195C24—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4Q16×C3112yescomplemented
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C2×D4×C3111yescharacteristic
208C22×C4C2×D4×C3111yescharacteristic
209C22×C4#41×C3111yescharacteristic
210Q16—82no—
211M16—42no—
212M16—42no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—42no—
229C42C2×D4×C3112yes—
230C42C2×D4×C3112yes—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23C31×#12111yescharacteristic
237Q8—44no—
238Q8—42no—
239Q8—42no—
240D4—24no—
241D4—42no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4C31×#12012yes—
253C2×C4C31×#12011yescharacteristic
254C2×C4C31×#12112yes—
255C2×C4C31×#12711yescharacteristic
256C2×C4C2×Q16×C3111yescharacteristic
257C2×C4C31×#13511yescharacteristic
258C2×C4C31×#13712yes—
259C8—42no—
260C8—44no—
261C8—42no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—42no—
266C22—82no—
267C22—81no—
268C22—81no—
269C22C31×#39211yescharacteristic
270C22C31×#46711yescharacteristic
271C22C31×#47011yescharacteristic
272C4—21no—
273C4—22no—
274C4—81no—
275C4—82no—
276C4—82no—
277C4C31×#45112yes—
278C4C31×#45212yes—
279C4C31×#45311yescharacteristic
280C4C31×#45411yescharacteristic
281C2—21no—
282C2—21no—
283C2—82no—
284C2C31×#200111yescharacteristic
285C2C31×#200611yescharacteristic
286C2C31×#201711yescharacteristic
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
1. C31×#3249C2semidirect
4. C31×#1493C2semidirect
5. C31×#1292C2semidirect
57. C2×D4×C31Q16semidirect
87. #10258C31direct (N × H)
119. #3249C2×C31semidirect
122. #1493C2×C31semidirect
123. #1292C2×C31semidirect
194. C31#10258direct (N × H)
201. C2×D4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32492
C31×#30772
C31×#19542
C31×#14932
C31×#12922
#1025831