Finite Group Databasebeta

← C31×#9967 #21466726

Subgroups — C31×#9967 #21466726

Order 7936.

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

normal subgroups only

This group has 804 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×#451C212yesmaximal
2C31×#2659C211yesmaximal, characteristic
3C2×C31×#328C212yesmaximal
4C31×#1446C211yesmaximal, characteristic
5C31×#1280C211yesmaximal, characteristic
6C23×Q8×C31C2211yescharacteristic
7C22×Q16×C31—22no—
8C2×C31×C22⋊Q8C2212yes—
9C31×#451—24no—
10C31×#451—24no—
11C2×C31×#103—22no—
12C2×C31×C22⋊C8C2212yes—
13C31×#398—22no—
14C31×#395—22no—
15C31×#380—22no—
16C2×C31×#95—22no—
17C2×C31×#95C2211yescharacteristic
18C2×C31×#95C2211yescharacteristic
19C31×#328—24no—
20C22×Q8×C31—21no—
21C22×Q8×C31—22no—
22C22×Q8×C31—22no—
23C22×Q8×C31—44no—
24C22×Q8×C31D412yes—
25C22×Q8×C31D412yes—
26C23×C4×C31—21no—
27C23×C4×C31—21no—
28C23×C4×C31D412yes—
29C23×C4×C31C2311yescharacteristic
30C2×Q16×C31—28no—
31C2×Q16×C31—48no—
32C22×C8×C31—22no—
33C31×C22⋊Q8—22no—
34C31×C22⋊Q8—22no—
35C31×C22⋊Q8—22no—
36C31×C22⋊Q8—22no—
37C2×C4⋊C4×C31—22no—
38C2×C4⋊C4×C31—22no—
39C2×C4⋊C4×C31—22no—
40C2×C4⋊C4×C31D412yes—
41C2×C22⋊C4×C31—22no—
42C31×#103—28no—
43C31×C22⋊C8—22no—
44C31×C22⋊C8D414yes—
45C31×#95—24no—
46C31×#95—22no—
47C31×#95—22no—
48C31×#95—44no—
49C24×C31C2×D411yescharacteristic
50C2×Q8×C31—28no—
51C2×Q8×C31—24no—
52C2×Q8×C31—22no—
53C2×Q8×C31—22no—
54C2×Q8×C31—44no—
55C2×Q8×C31—44no—
56C2×Q8×C31—44no—
57C2×Q8×C31—44no—
58C2×Q8×C31—48no—
59C2×Q8×C31—42no—
60C22×C4×C31—24no—
61C22×C4×C31—22no—
62C22×C4×C31—22no—
63C22×C4×C31—22no—
64C22×C4×C31—22no—
65C22×C4×C31—22no—
66C22×C4×C31—22no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31—41no—
70C22×C4×C31—41no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31—42no—
74C22×C4×C31—41no—
75C22×C4×C31—41no—
76C22×C4×C31—42no—
77C22×C4×C31Q1614yescomplemented
78C22×C4×C31C2×D412yes—
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D412yes—
81C22×C4×C31#4111yescharacteristic
82Q16×C31—416no—
83C2×C8×C31—24no—
84C2×C8×C31—44no—
85C4⋊C4×C31—22no—
86C4⋊C4×C31—22no—
87C4⋊C4×C31—42no—
88C4⋊C4×C31—42no—
89C4⋊C4×C31—82no—
90C4⋊C4×C31—82no—
91C22⋊C4×C31—42no—
92C22⋊C4×C31—42no—
93#9967C3111yesmaximal, characteristic, direct product factor, Sylow
94C23×C31—21no—
95C23×C31—21no—
96C23×C31—22no—
97C23×C31#12011yescharacteristic
98C23×C31#12111yescharacteristic
99C23×C31#12711yescharacteristic
100C23×C31C2×Q1612yes—
101C23×C31#13712yes—
102Q8×C31—28no—
103Q8×C31—44no—
104Q8×C31—48no—
105Q8×C31—44no—
106Q8×C31—82no—
107C2×C4×C31—24no—
108C2×C4×C31—24no—
109C2×C4×C31—24no—
110C2×C4×C31—41no—
111C2×C4×C31—42no—
112C2×C4×C31—41no—
113C2×C4×C31—42no—
114C2×C4×C31—44no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—44no—
119C2×C4×C31—44no—
120C2×C4×C31—44no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—42no—
126C2×C4×C31—44no—
127C2×C4×C31—82no—
128C2×C4×C31—82no—
129C2×C4×C31—81no—
130C2×C4×C31—81no—
131C2×C4×C31#12012yes—
132C2×C4×C31#12112yes—
133C8×C31—44no—
134C2×#451C2×C3112yes—
135#2659C2×C3111yescharacteristic
136C2×#328C2×C3112yes—
137#1446C2×C3111yescharacteristic
138#1280C2×C3111yescharacteristic
139C22×C31—22no—
140C22×C31—22no—
141C22×C31—22no—
142C22×C31—22no—
143C22×C31—21no—
144C22×C31—21no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31#39211yescharacteristic
148C22×C31#45112yes—
149C22×C31#45511yescharacteristic
150C22×C31#45611yescharacteristic
151C22×C31B(4,𝔽2)12yes—
152C22×C31#46212yes—
153C22×C31#49411yescharacteristic
154C22×C31#49711yescharacteristic
155C4×C31—22no—
156C4×C31—22no—
157C4×C31—44no—
158C4×C31—44no—
159C4×C31—82no—
160C4×C31—82no—
161C4×C31—82no—
162C23×Q8C22×C3111yescharacteristic
163C22×Q16—22no—
164C2×C22⋊Q8C22×C3112yes—
165#451—24no—
166#451—24no—
167C2×#103—22no—
168C2×C22⋊C8C22×C3112yes—
169#398—22no—
170#395—22no—
171#380—22no—
172C2×#95—22no—
173C2×#95C22×C3111yescharacteristic
174C2×#95C22×C3111yescharacteristic
175#328—24no—
176C2×C31—21no—
177C2×C31—22no—
178C2×C31—21no—
179C2×C31#160212yes—
180C2×C31#160512yes—
181C2×C31#200711yescharacteristic
182C2×C31#201611yescharacteristic
183C2×C31#202111yescharacteristic
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—44no—
188C22×Q8D4×C3112yes—
189C22×Q8D4×C3112yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4D4×C3112yes—
193C23×C4C23×C3111yescharacteristic
194C2×Q16—28no—
195C2×Q16—48no—
196C22×C8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4D4×C3112yes—
205C2×C22⋊C4—22no—
206#103—28no—
207C22⋊C8—22no—
208C22⋊C8D4×C3114yes—
209#95—24no—
210#95—22no—
211#95—22no—
212#95—44no—
213C31#996711yescharacteristic, direct product factor, Sylow
214C24C2×D4×C3111yescharacteristic
215C2×Q8—28no—
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4Q16×C3114yescomplemented
243C22×C4C2×D4×C3112yes—
244C22×C4C2×D4×C3111yescharacteristic
245C22×C4C2×D4×C3112yes—
246C22×C4#41×C3111yescharacteristic
247Q16—416no—
248C2×C8—24no—
249C2×C8—44no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23C31×#12011yescharacteristic
262C23C31×#12111yescharacteristic
263C23C31×#12711yescharacteristic
264C23C2×Q16×C3112yes—
265C23C31×#13712yes—
266Q8—28no—
267Q8—44no—
268Q8—48no—
269Q8—44no—
270Q8—82no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4C31×#12012yes—
296C2×C4C31×#12112yes—
297C8—44no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22C31×#39211yescharacteristic
307C22C31×#45112yes—
308C22C31×#45511yescharacteristic
309C22C31×#45611yescharacteristic
310C22C31×B(4,𝔽2)12yes—
311C22C31×#46212yes—
312C22C31×#49411yescharacteristic
313C22C31×#49711yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—82no—
319C4—82no—
320C4—82no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2C31×#160212yes—
325C2C31×#160512yes—
326C2C31×#200711yescharacteristic
327C2C31×#201611yescharacteristic
328C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
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
77. C22×C4×C31Q16semidirect
93. #9967C31direct (N × H)
213. C31#9967direct (N × H)
242. C22×C4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4512
C31×#26592
C2×C31×#3282
C31×#14462
C31×#12802
#996731