Finite Group Databasebeta

← C31×#12100 #21468859

Subgroups — C31×#12100 #21468859

Order 7936.

522 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2733C211yesmaximal, characteristic, complemented
2C31×#2007C211yesmaximal, characteristic, complemented
3C31×#1598C212yesmaximal, complemented
4C31×#1345C212yesmaximal, complemented
5C31×#1319C211yesmaximal, characteristic, complemented
6C2×C31×#137—21no—
7C31×#537C2211yescharacteristic
8C2×C31×#124—21no—
9C31×#447—22no—
10C2×C31×#104—21no—
11C31×#394C2212yes—
12C31×#392—22no—
13C31×#391C2211yescharacteristic
14C31×#390—21no—
15C31×#388—22no—
16C31×#343C2211yescharacteristic
17C31×#327—22no—
18C31×#324C2212yes—
19C2×#41×C31—21no—
20C22×Q8×C31D411yescharacteristic
21C22×D4×C31—21no—
22C22×D4×C31—21no—
23C22×D4×C31D412yes—
24C31×#137—42no—
25C31×#137—42no—
26C2×Q16×C31—41no—
27C2×SD16×C31—41no—
28C2×D8×C31—42no—
29C2×M16×C31—21no—
30C31×#124—42no—
31C31×#124—42no—
32C31×C22⋊Q8—41no—
33C31×#121—22no—
34C31×#121—41no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31C2311yescharacteristic
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C42×C31—21no—
42C2×C42×C31D411yescharacteristic
43C31×#104—42no—
44C31×#102—42no—
45C31×C22⋊C8—22no—
46C31×#95—22no—
47C31×#95D412yes—
48C24×C31—22no—
49#41×C31—41no—
50#41×C31—41no—
51#41×C31—41no—
52#41×C31—41no—
53C2×Q8×C31—21no—
54C2×Q8×C31—21no—
55C2×Q8×C31—41no—
56C2×Q8×C31—41no—
57C2×Q8×C31D812yescomplemented
58C2×D4×C31—21no—
59C2×D4×C31—22no—
60C2×D4×C31—42no—
61C2×D4×C31—42no—
62C2×D4×C31—42no—
63C2×D4×C31—41no—
64C2×D4×C31—82no—
65C2×D4×C31—81no—
66C2×D4×C31—81no—
67C2×D4×C31—81no—
68C2×D4×C31—81no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—22no—
72C22×C4×C31—41no—
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D412yes—
75Q16×C31—44no—
76SD16×C31—44no—
77D8×C31—84no—
78M16×C31—22no—
79M16×C31—41no—
80C2×C8×C31—41no—
81C2×C8×C31—42no—
82C4⋊C4×C31—22no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—81no—
85C22⋊C4×C31—44no—
86C22⋊C4×C31—44no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—82no—
91C42×C31—42no—
92C42×C31—41no—
93#12100C3111yesmaximal, characteristic, direct product factor, Sylow
94C23×C31—42no—
95C23×C31—42no—
96C23×C31—42no—
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—41no—
101C23×C31—42no—
102C23×C31#12011yescharacteristic
103Q8×C31—42no—
104Q8×C31—41no—
105Q8×C31—41no—
106Q8×C31—81no—
107D4×C31—41no—
108D4×C31—41no—
109D4×C31—44no—
110D4×C31—82no—
111D4×C31—82no—
112D4×C31—84no—
113D4×C31—84no—
114D4×C31—81no—
115D4×C31—81no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—22no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—42no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—41no—
130C2×C4×C31—81no—
131C2×C4×C31—81no—
132C2×C4×C31C2×D811yescharacteristic
133C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
134C8×C31—42no—
135C8×C31—82no—
136#2733C2×C3111yescharacteristic, complemented
137#2007C2×C3111yescharacteristic, complemented
138#1598C2×C3112yescomplemented
139#1345C2×C3112yescomplemented
140#1319C2×C3111yescharacteristic, complemented
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—82no—
144C22×C31—82no—
145C22×C31—82no—
146C22×C31—82no—
147C22×C31—81no—
148C22×C31—82no—
149C22×C31—82no—
150C22×C31—81no—
151C22×C31—81no—
152C22×C31—82no—
153C22×C31#44711yescharacteristic
154C22×C31#45311yescharacteristic
155C22×C31B(4,𝔽2)11yescharacteristic
156C4×C31—21no—
157C4×C31—21no—
158C4×C31—42no—
159C4×C31—42no—
160C4×C31—42no—
161C4×C31—82no—
162C4×C31—81no—
163C2×#137—21no—
164#537C22×C3111yescharacteristic
165C2×#124—21no—
166#447—22no—
167C2×#104—21no—
168#394C22×C3112yes—
169#392—22no—
170#391C22×C3111yescharacteristic
171#390—21no—
172#388—22no—
173#343C22×C3111yescharacteristic
174#327—22no—
175#324C22×C3112yes—
176C2×C31—21no—
177C2×C31—21no—
178C2×C31—161no—
179C2×C31—162no—
180C2×C31#159511yescharacteristic
181C2×C31#219911yescharacteristic
182C2×C31#220411yescharacteristic
183C2×#41—21no—
184C22×Q8D4×C3111yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4D4×C3112yes—
188#137—42no—
189#137—42no—
190C2×Q16—41no—
191C2×SD16—41no—
192C2×D8—42no—
193C2×M16—21no—
194#124—42no—
195#124—42no—
196C22⋊Q8—41no—
197#121—22no—
198#121—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C23×C3111yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42D4×C3111yescharacteristic
207#104—42no—
208#102—42no—
209C22⋊C8—22no—
210#95—22no—
211#95D4×C3112yes—
212C31#1210011yescharacteristic, direct product factor, Sylow
213C24—22no—
214#41—41no—
215#41—41no—
216#41—41no—
217#41—41no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8D8×C3112yescomplemented
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—82no—
230C2×D4—81no—
231C2×D4—81no—
232C2×D4—81no—
233C2×D4—81no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4C2×D4×C3111yescharacteristic
239C22×C4C2×D4×C3112yes—
240Q16—44no—
241SD16—44no—
242D8—84no—
243M16—22no—
244M16—41no—
245C2×C8—41no—
246C2×C8—42no—
247C4⋊C4—22no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—82no—
256C42—41no—
257C42—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—42no—
266C23C31×#12011yescharacteristic
267Q8—42no—
268Q8—41no—
269Q8—41no—
270Q8—81no—
271D4—41no—
272D4—41no—
273D4—44no—
274D4—82no—
275D4—82no—
276D4—84no—
277D4—84no—
278D4—81no—
279D4—81no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4C2×D8×C3111yescharacteristic
297C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
298C8—42no—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22—82no—
311C22—81no—
312C22C31×#44711yescharacteristic
313C22C31×#45311yescharacteristic
314C22C31×B(4,𝔽2)11yescharacteristic
315C4—21no—
316C4—21no—
317C4—42no—
318C4—42no—
319C4—42no—
320C4—82no—
321C4—81no—
322C2—21no—
323C2—21no—
324C2—162no—
325C2—161no—
326C2C31×#159511yescharacteristic
327C2C31×#219911yescharacteristic
328C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2733C2semidirect
2. C31×#2007C2semidirect
3. C31×#1598C2semidirect
4. C31×#1345C2semidirect
5. C31×#1319C2semidirect
57. C2×Q8×C31D8semidirect
93. #12100C31direct (N × H)
136. #2733C2×C31semidirect
137. #2007C2×C31semidirect
138. #1598C2×C31semidirect
139. #1345C2×C31semidirect
140. #1319C2×C31semidirect
212. C31#12100direct (N × H)
222. C2×Q8D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27332
C31×#20072
C31×#15982
C31×#13452
C31×#13192
#1210031