Finite Group Databasebeta

← C31×#10321 #21467080

Subgroups — C31×#10321 #21467080

Order 7936.

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

normal subgroups only

This group has 726 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×#3186C211yesmaximal, characteristic
2C31×#2854C212yesmaximal, complemented
3C31×#1817C211yesmaximal, characteristic, complemented
4C31×#1519C212yesmaximal, complemented
5C31×#1289C211yesmaximal, characteristic, complemented
6C2×C31×#123—22no—
7C2×C31×#121C2212yes—
8C22×C42×C31C2211yescharacteristic
9C31×#480—22no—
10C31×#466—22no—
11C31×#447—22no—
12C2×C31×C22⋊C8C2211yescharacteristic
13C2×C31×C22⋊C8C2211yescharacteristic
14C31×#390—22no—
15C31×#389—22no—
16C31×#388—24no—
17C31×#386—22no—
18C31×#380C2212yes—
19C31×#336—22no—
20C31×#327—24no—
21C22×D4×C31—22no—
22C22×D4×C31D412yes—
23C23×C4×C31D411yescharacteristic
24C23×C4×C31D411yescharacteristic
25C23×C4×C31C2311yescharacteristic
26C2×D8×C31—22no—
27C22×C8×C31—21no—
28C22×C8×C31D412yes—
29C31×#123—44no—
30C31×#123—44no—
31C31×#121—22no—
32C31×#121—22no—
33C31×#121—22no—
34C31×#121—22no—
35C2×C4⋊C4×C31—24no—
36C2×C4⋊C4×C31—22no—
37C2×C4⋊C4×C31D412yes—
38C2×C22⋊C4×C31—24no—
39C2×C22⋊C4×C31—24no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C42×C31—21no—
43C2×C42×C31—22no—
44C2×C42×C31—22no—
45C2×C42×C31—21no—
46C31×#107—22no—
47C31×#102—24no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8—22no—
50C31×#95—24no—
51C31×#95D414yes—
52C24×C31—22no—
53C24×C31C2×D411yescharacteristic
54C2×D4×C31—22no—
55C2×D4×C31—22no—
56C2×D4×C31—44no—
57C2×D4×C31—42no—
58C2×D4×C31—42no—
59C2×D4×C31—42no—
60C2×D4×C31—42no—
61C2×D4×C31—42no—
62C2×D4×C31—42no—
63C22×C4×C31—22no—
64C22×C4×C31—22no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—22no—
69C22×C4×C31—22no—
70C22×C4×C31—24no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31D812yescomplemented
74C22×C4×C31C2×D412yes—
75C22×C4×C31C2×D411yescharacteristic
76C22×C4×C31C2×D412yes—
77C22×C4×C31#4111yescharacteristic
78D8×C31—82no—
79C2×C8×C31—22no—
80C2×C8×C31—42no—
81C2×C8×C31—42no—
82C2×C8×C31—41no—
83C4⋊C4×C31—22no—
84C4⋊C4×C31—22no—
85C4⋊C4×C31—44no—
86C4⋊C4×C31—48no—
87C22⋊C4×C31—48no—
88C22⋊C4×C31—48no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—42no—
91C22⋊C4×C31—42no—
92C22⋊C4×C31—42no—
93C42×C31—41no—
94C42×C31—42no—
95C42×C31—41no—
96#10321C3111yesmaximal, characteristic, direct product factor, Sylow
97C23×C31—41no—
98C23×C31—41no—
99C23×C31—44no—
100C23×C31—44no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—42no—
104C23×C31#12012yes—
105C23×C31#12111yescharacteristic
106C23×C31C2×D811yescharacteristic
107C23×C31#13511yescharacteristic
108C23×C31AGL(1,ℤ/8)12yes—
109D4×C31—44no—
110D4×C31—82no—
111D4×C31—82no—
112D4×C31—82no—
113D4×C31—82no—
114D4×C31—82no—
115C2×C4×C31—24no—
116C2×C4×C31—22no—
117C2×C4×C31—24no—
118C2×C4×C31—22no—
119C2×C4×C31—24no—
120C2×C4×C31—22no—
121C2×C4×C31—42no—
122C2×C4×C31—41no—
123C2×C4×C31—42no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—42no—
128C2×C4×C31—41no—
129C2×C4×C31—44no—
130C2×C4×C31—44no—
131C2×C4×C31—42no—
132C2×C4×C31—44no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31#12011yescharacteristic
136C2×C4×C31#12112yes—
137C2×C4×C31#12711yescharacteristic
138C8×C31—42no—
139C8×C31—81no—
140#3186C2×C3111yescharacteristic
141#2854C2×C3112yescomplemented
142#1817C2×C3111yescharacteristic, complemented
143#1519C2×C3112yescomplemented
144#1289C2×C3111yescharacteristic, complemented
145C22×C31—22no—
146C22×C31—21no—
147C22×C31—22no—
148C22×C31—21no—
149C22×C31—41no—
150C22×C31—41no—
151C22×C31—42no—
152C22×C31—84no—
153C22×C31—84no—
154C22×C31—82no—
155C22×C31—82no—
156C22×C31—82no—
157C22×C31#39211yescharacteristic
158C22×C31#44712yes—
159C22×C31#44912yes—
160C22×C31#46611yescharacteristic
161C22×C31#46911yescharacteristic
162C4×C31—21no—
163C4×C31—22no—
164C4×C31—42no—
165C4×C31—42no—
166C4×C31—84no—
167C4×C31—82no—
168C4×C31#45812yes—
169C2×#123—22no—
170C2×#121C22×C3112yes—
171C22×C42C22×C3111yescharacteristic
172#480—22no—
173#466—22no—
174#447—22no—
175C2×C22⋊C8C22×C3111yescharacteristic
176C2×C22⋊C8C22×C3111yescharacteristic
177#390—22no—
178#389—22no—
179#388—24no—
180#386—22no—
181#380C22×C3112yes—
182#336—22no—
183#327—24no—
184C2×C31—22no—
185C2×C31—22no—
186C2×C31—21no—
187C2×C31—21no—
188C2×C31—162no—
189C2×C31#199911yescharacteristic
190C2×C31#201011yescharacteristic
191C2×C31#202211yescharacteristic
192C22×D4—22no—
193C22×D4D4×C3112yes—
194C23×C4D4×C3111yescharacteristic
195C23×C4D4×C3111yescharacteristic
196C23×C4C23×C3111yescharacteristic
197C2×D8—22no—
198C22×C8—21no—
199C22×C8D4×C3112yes—
200#123—44no—
201#123—44no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4D4×C3112yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217#107—22no—
218#102—24no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221#95—24no—
222#95D4×C3114yes—
223C31#1032111yescharacteristic, direct product factor, Sylow
224C24—22no—
225C24C2×D4×C3111yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—44no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4D8×C3112yescomplemented
246C22×C4C2×D4×C3112yes—
247C22×C4C2×D4×C3111yescharacteristic
248C22×C4C2×D4×C3112yes—
249C22×C4#41×C3111yescharacteristic
250D8—82no—
251C2×C8—22no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—41no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—44no—
258C4⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C42—41no—
266C42—42no—
267C42—41no—
268C23—41no—
269C23—41no—
270C23—44no—
271C23—44no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23C31×#12012yes—
276C23C31×#12111yescharacteristic
277C23C2×D8×C3111yescharacteristic
278C23C31×#13511yescharacteristic
279C23C31×AGL(1,ℤ/8)12yes—
280D4—44no—
281D4—82no—
282D4—82no—
283D4—82no—
284D4—82no—
285D4—82no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4C31×#12011yescharacteristic
307C2×C4C31×#12112yes—
308C2×C4C31×#12711yescharacteristic
309C8—42no—
310C8—81no—
311C22—22no—
312C22—21no—
313C22—22no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—84no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22C31×#39211yescharacteristic
324C22C31×#44712yes—
325C22C31×#44912yes—
326C22C31×#46611yescharacteristic
327C22C31×#46911yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—84no—
333C4—82no—
334C4C31×#45812yes—
335C2—22no—
336C2—22no—
337C2—21no—
338C2—21no—
339C2—162no—
340C2C31×#199911yescharacteristic
341C2C31×#201011yescharacteristic
342C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2854C2semidirect
3. C31×#1817C2semidirect
4. C31×#1519C2semidirect
5. C31×#1289C2semidirect
73. C22×C4×C31D8semidirect
96. #10321C31direct (N × H)
141. #2854C2×C31semidirect
142. #1817C2×C31semidirect
143. #1519C2×C31semidirect
144. #1289C2×C31semidirect
223. C31#10321direct (N × H)
245. C22×C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31862
C31×#28542
C31×#18172
C31×#15192
C31×#12892
#1032131