Finite Group Databasebeta

← C31×#9970 #21466729

Subgroups — C31×#9970 #21466729

Order 7936.

772 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×C31×#480C212yesmaximal, complemented
2C31×#2656C211yesmaximal, characteristic
3C2×C31×#327C212yesmaximal, complemented
4C31×#1445C211yesmaximal, characteristic, complemented
5C31×#1280C211yesmaximal, characteristic, complemented
6C2×C31×#121—22no—
7C2×C31×#121C2212yes—
8C22×C4⋊C4×C31C2211yescharacteristic
9C31×#480—24no—
10C31×#480—24no—
11C2×C31×#107—22no—
12C2×C31×#102—22no—
13C2×C31×C22⋊C8C2212yes—
14C31×#399—22no—
15C31×#392—22no—
16C2×C31×#95—22no—
17C2×C31×#95C2211yescharacteristic
18C2×C31×#95C2211yescharacteristic
19C31×#327—24no—
20C22×D4×C31—22no—
21C22×D4×C31—22no—
22C22×D4×C31D412yes—
23C23×C4×C31—21no—
24C23×C4×C31—21no—
25C23×C4×C31D412yes—
26C23×C4×C31C2311yescharacteristic
27C22×C8×C31—22no—
28C31×#121—22no—
29C31×#121—22no—
30C31×#121—22no—
31C31×#121—22no—
32C31×#121—44no—
33C31×#121—44no—
34C2×C4⋊C4×C31—22no—
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31—22no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31D412yes—
39C2×C4⋊C4×C31D412yescomplemented
40C2×C22⋊C4×C31—24no—
41C2×C22⋊C4×C31—22no—
42C31×#107—28no—
43C31×#102—28no—
44C31×C22⋊C8—22no—
45C31×C22⋊C8D414yescomplemented
46C31×#95—22no—
47C31×#95—22no—
48C31×#95—44no—
49C24×C31—22no—
50C24×C31C2×D411yescharacteristic
51C2×D4×C31—22no—
52C2×D4×C31—22no—
53C2×D4×C31—44no—
54C2×D4×C31—44no—
55C2×D4×C31—42no—
56C2×D4×C31—42no—
57C2×D4×C31—44no—
58C2×D4×C31—44no—
59C2×D4×C31—82no—
60C2×D4×C31—82no—
61C22×C4×C31—22no—
62C22×C4×C31—22no—
63C22×C4×C31—22no—
64C22×C4×C31—22no—
65C22×C4×C31—22no—
66C22×C4×C31—42no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31—41no—
70C22×C4×C31—42no—
71C22×C4×C31—41no—
72C22×C4×C31—41no—
73C22×C4×C31—41no—
74C22×C4×C31—42no—
75C22×C4×C31—42no—
76C22×C4×C31—42no—
77C22×C4×C31D814yes—
78C22×C4×C31C2×D412yes—
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D412yes—
81C22×C4×C31#4111yescharacteristic
82C2×C8×C31—24no—
83C2×C8×C31—44no—
84C4⋊C4×C31—24no—
85C4⋊C4×C31—22no—
86C4⋊C4×C31—22no—
87C4⋊C4×C31—44no—
88C4⋊C4×C31—42no—
89C22⋊C4×C31—48no—
90C22⋊C4×C31—42no—
91C22⋊C4×C31—42no—
92#9970C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—21no—
94C23×C31—21no—
95C23×C31—22no—
96C23×C31—42no—
97C23×C31—44no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—44no—
101C23×C31#12011yescharacteristic
102C23×C31#12111yescharacteristic
103C23×C31#12711yescharacteristic
104C23×C31C2×D812yes—
105C23×C31AGL(1,ℤ/8)12yes—
106D4×C31—44no—
107D4×C31—82no—
108D4×C31—88no—
109D4×C31—84no—
110D4×C31—82no—
111D4×C31—82no—
112C2×C4×C31—24no—
113C2×C4×C31—24no—
114C2×C4×C31—41no—
115C2×C4×C31—42no—
116C2×C4×C31—41no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—44no—
120C2×C4×C31—44no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—44no—
125C2×C4×C31—44no—
126C2×C4×C31—82no—
127C2×C4×C31—82no—
128C2×C4×C31—81no—
129C2×C4×C31—82no—
130C2×C4×C31—81no—
131C2×C4×C31—81no—
132C2×C4×C31—81no—
133C2×C4×C31#12012yes—
134C2×C4×C31#12112yes—
135C8×C31—44no—
136C2×#480C2×C3112yescomplemented
137#2656C2×C3111yescharacteristic
138C2×#327C2×C3112yescomplemented
139#1445C2×C3111yescharacteristic, complemented
140#1280C2×C3111yescharacteristic, complemented
141C22×C31—22no—
142C22×C31—22no—
143C22×C31—22no—
144C22×C31—22no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—21no—
149C22×C31—84no—
150C22×C31—84no—
151C22×C31—82no—
152C22×C31—82no—
153C22×C31—82no—
154C22×C31#39211yescharacteristic
155C22×C31#44712yes—
156C22×C31#45511yescharacteristic
157C22×C31#45611yescharacteristic
158C22×C31#45812yes—
159C22×C31#45912yes—
160C22×C31#49311yescharacteristic
161C22×C31#49611yescharacteristic
162C4×C31—22no—
163C4×C31—22no—
164C4×C31—44no—
165C4×C31—82no—
166C4×C31—82no—
167C4×C31—82no—
168C2×#121—22no—
169C2×#121C22×C3112yes—
170C22×C4⋊C4C22×C3111yescharacteristic
171#480—24no—
172#480—24no—
173C2×#107—22no—
174C2×#102—22no—
175C2×C22⋊C8C22×C3112yes—
176#399—22no—
177#392—22no—
178C2×#95—22no—
179C2×#95C22×C3111yescharacteristic
180C2×#95C22×C3111yescharacteristic
181#327—24no—
182C2×C31—21no—
183C2×C31—22no—
184C2×C31—21no—
185C2×C31—162no—
186C2×C31#160712yes—
187C2×C31#161012yes—
188C2×C31#200711yescharacteristic
189C2×C31#201111yescharacteristic
190C2×C31#202211yescharacteristic
191C22×D4—22no—
192C22×D4—22no—
193C22×D4D4×C3112yes—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4D4×C3112yes—
197C23×C4C23×C3111yescharacteristic
198C22×C8—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—44no—
204#121—44no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4D4×C3112yes—
210C2×C4⋊C4D4×C3112yescomplemented
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213#107—28no—
214#102—28no—
215C22⋊C8—22no—
216C22⋊C8D4×C3114yescomplemented
217#95—22no—
218#95—22no—
219#95—44no—
220C31#997011yescharacteristic, direct product factor, Sylow
221C24—22no—
222C24C2×D4×C3111yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—82no—
232C2×D4—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4D8×C3114yes—
250C22×C4C2×D4×C3112yes—
251C22×C4C2×D4×C3111yescharacteristic
252C22×C4C2×D4×C3112yes—
253C22×C4#41×C3111yescharacteristic
254C2×C8—24no—
255C2×C8—44no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C22⋊C4—48no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—44no—
271C23—42no—
272C23C31×#12011yescharacteristic
273C23C31×#12111yescharacteristic
274C23C31×#12711yescharacteristic
275C23C2×D8×C3112yes—
276C23C31×AGL(1,ℤ/8)12yes—
277D4—44no—
278D4—82no—
279D4—88no—
280D4—84no—
281D4—82no—
282D4—82no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4C31×#12012yes—
305C2×C4C31×#12112yes—
306C8—44no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—84no—
316C22—84no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22C31×#39211yescharacteristic
321C22C31×#44712yes—
322C22C31×#45511yescharacteristic
323C22C31×#45611yescharacteristic
324C22C31×#45812yes—
325C22C31×#45912yes—
326C22C31×#49311yescharacteristic
327C22C31×#49611yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—82no—
333C4—82no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—162no—
338C2C31×#160712yes—
339C2C31×#161012yes—
340C2C31×#200711yescharacteristic
341C2C31×#201111yescharacteristic
342C2C31×#202211yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#2656C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#480C2semidirect
3. C2×C31×#327C2semidirect
4. C31×#1445C2semidirect
5. C31×#1280C2semidirect
39. C2×C4⋊C4×C31D4semidirect
45. C31×C22⋊C8D4semidirect
92. #9970C31direct (N × H)
136. C2×#480C2×C31semidirect
138. C2×#327C2×C31semidirect
139. #1445C2×C31semidirect
140. #1280C2×C31semidirect
210. C2×C4⋊C4D4×C31semidirect
216. C22⋊C8D4×C31semidirect
220. C31#9970direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4802
C31×#26562
C2×C31×#3272
C31×#14452
C31×#12802
#997031