Finite Group Databasebeta

← C2×C31×#1662 #21475744

Subgroups — C2×C31×#1662 #21475744

Order 7936.

958 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 238 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×#562C211yesmaximal, characteristic
2C2×C31×#491C211yesmaximal, characteristic, complemented
3C2×C31×#489C212yesmaximal, complemented
4C31×#1662C218yesmaximal, direct product factor
5C2×C31×#330C211yesmaximal, characteristic, complemented
6C2×C31×#329C212yesmaximal, complemented
7C31×#562—22no—
8C31×#562—24no—
9C31×#562C2214yes—
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C31×#125C2211yescharacteristic
12C2×C31×#124C2212yes—
13C2×C31×#123—22no—
14C2×C31×C22⋊Q8—22no—
15C2×C31×#121—22no—
16C31×#491—22no—
17C31×#491C2214yescomplemented
18C31×#489—24no—
19C31×#489C2218yescomplemented
20C2×C31×#103—22no—
21C2×C31×#103—22no—
22C2×C31×#102—22no—
23C2×C31×#97C2212yes—
24C2×C31×#97C2211yescharacteristic
25C31×#330C2214yescomplemented
26C31×#329C2218yescomplemented
27C22×Q8×C31—21no—
28C22×Q8×C31D412yes—
29C22×D4×C31—22no—
30C22×D4×C31D412yes—
31C23×C4×C31—22no—
32C22×C8×C31—22no—
33C22×C8×C31—21no—
34C31×C4⋊Q8—22no—
35C31×C4⋊Q8—21no—
36C31×C4⋊Q8C2312yes—
37C31×#125—22no—
38C31×#125D412yes—
39C31×#125C2312yes—
40C31×#124—22no—
41C31×#124—22no—
42C31×#124D414yes—
43C31×#124C2314yes—
44C31×#123—28no—
45C31×#123—28no—
46C31×C22⋊Q8—28no—
47C31×C22⋊Q8—28no—
48C31×#121—28no—
49C31×#121—28no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—22no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31D412yes—
54C2×C22⋊C4×C31—22no—
55C2×C22⋊C4×C31—22no—
56C2×C22⋊C4×C31—22no—
57C2×C42×C31C2311yescharacteristic
58C31×#103—28no—
59C31×#103—28no—
60C31×#102—28no—
61C31×#97—22no—
62C31×#97—21no—
63C31×#97C2314yes—
64C31×#97C2312yes—
65C24×C31—22no—
66C2×Q8×C31—22no—
67C2×Q8×C31—22no—
68C2×Q8×C31—44no—
69C2×Q8×C31—44no—
70C2×Q8×C31—41no—
71C2×Q8×C31C2×D414yes—
72C2×D4×C31—28no—
73C2×D4×C31—22no—
74C2×D4×C31—44no—
75C2×D4×C31—84no—
76C2×D4×C31C2×D414yes—
77C22×C4×C31—22no—
78C22×C4×C31—22no—
79C22×C4×C31—28no—
80C22×C4×C31—41no—
81C22×C4×C31—84no—
82C22×C4×C31C2×D411yescharacteristic
83C22×C4×C31C2×D412yes—
84C2×C8×C31—24no—
85C2×C8×C31—22no—
86C2×C8×C31—42no—
87C2×C8×C31—44no—
88C4⋊C4×C31—28no—
89C4⋊C4×C31—24no—
90C4⋊C4×C31—22no—
91C4⋊C4×C31—24no—
92C4⋊C4×C31—24no—
93C4⋊C4×C31—42no—
94C4⋊C4×C31C2×D414yes—
95C22⋊C4×C31—28no—
96C22⋊C4×C31—24no—
97C22⋊C4×C31—24no—
98C22⋊C4×C31—24no—
99C22⋊C4×C31—24no—
100C42×C31C2×D411yescharacteristic
101C42×C31C2×D412yes—
102C42×C31C2411yescharacteristic
103C2×#1662C3111yesmaximal, characteristic, direct product factor, Sylow
104C23×C31—24no—
105C23×C31—44no—
106C23×C31—44no—
107C23×C31—44no—
108C23×C31#12011yescharacteristic
109Q8×C31—44no—
110Q8×C31—44no—
111Q8×C31—82no—
112Q8×C31—82no—
113D4×C31—44no—
114D4×C31—82no—
115D4×C31—88no—
116C2×C4×C31—22no—
117C2×C4×C31—24no—
118C2×C4×C31—24no—
119C2×C4×C31—24no—
120C2×C4×C31—44no—
121C2×C4×C31—44no—
122C2×C4×C31—42no—
123C2×C4×C31—88no—
124C2×C4×C31—82no—
125C2×C4×C31#12011yescharacteristic
126C2×C4×C31#12012yes—
127C2×C4×C31C22×D411yescharacteristic
128C2×C4×C31C22×D412yes—
129C8×C31—44no—
130C8×C31—42no—
131C2×#562C2×C3111yescharacteristic
132C2×#491C2×C3111yescharacteristic, complemented
133C2×#489C2×C3112yescomplemented
134#1662C2×C3118yesdirect product factor
135C2×#330C2×C3111yescharacteristic, complemented
136C2×#329C2×C3112yescomplemented
137C22×C31—44no—
138C22×C31—44no—
139C22×C31—44no—
140C22×C31—88no—
141C22×C31#45412yes—
142C22×C31#45514yes—
143C22×C31C2×#12011yescharacteristic
144C4×C31—22no—
145C4×C31—22no—
146C4×C31—21no—
147C4×C31—21no—
148C4×C31—44no—
149C4×C31—44no—
150C4×C31—82no—
151#562—22no—
152#562—24no—
153#562C22×C3114yes—
154C2×C4⋊Q8C22×C3111yescharacteristic
155C2×#125C22×C3111yescharacteristic
156C2×#124C22×C3112yes—
157C2×#123—22no—
158C2×C22⋊Q8—22no—
159C2×#121—22no—
160#491—22no—
161#491C22×C3114yescomplemented
162#489—24no—
163#489C22×C3118yescomplemented
164C2×#103—22no—
165C2×#103—22no—
166C2×#102—22no—
167C2×#97C22×C3112yes—
168C2×#97C22×C3111yescharacteristic
169#330C22×C3114yescomplemented
170#329C22×C3118yescomplemented
171C2×C31—84no—
172C2×C31#166214yesdirect product factor
173C2×C31C2×#45512yes—
174C2×C31C2×#45411yescharacteristic
175C22×Q8—21no—
176C22×Q8D4×C3112yes—
177C22×D4—22no—
178C22×D4D4×C3112yes—
179C23×C4—22no—
180C22×C8—22no—
181C22×C8—21no—
182C4⋊Q8—22no—
183C4⋊Q8—21no—
184C4⋊Q8C23×C3112yes—
185#125—22no—
186#125D4×C3112yes—
187#125C23×C3112yes—
188#124—22no—
189#124—22no—
190#124D4×C3114yes—
191#124C23×C3114yes—
192#123—28no—
193#123—28no—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196#121—28no—
197#121—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4D4×C3112yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42C23×C3111yescharacteristic
206#103—28no—
207#103—28no—
208#102—28no—
209#97—22no—
210#97—21no—
211#97C23×C3112yes—
212#97C23×C3114yes—
213C31C2×#166211yescharacteristic, direct product factor, Sylow
214C24—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—41no—
220C2×Q8C2×D4×C3114yes—
221C2×D4—28no—
222C2×D4—22no—
223C2×D4—44no—
224C2×D4—84no—
225C2×D4C2×D4×C3114yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—41no—
230C22×C4—84no—
231C22×C4C2×D4×C3111yescharacteristic
232C22×C4C2×D4×C3112yes—
233C2×C8—24no—
234C2×C8—22no—
235C2×C8—42no—
236C2×C8—44no—
237C4⋊C4—28no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—42no—
243C4⋊C4C2×D4×C3114yes—
244C22⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C42C2×D4×C3111yescharacteristic
250C42C2×D4×C3112yes—
251C42C24×C3111yescharacteristic
252C23—24no—
253C23—44no—
254C23—44no—
255C23—44no—
256C23C31×#12011yescharacteristic
257Q8—44no—
258Q8—44no—
259Q8—82no—
260Q8—82no—
261D4—44no—
262D4—82no—
263D4—88no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—88no—
272C2×C4—82no—
273C2×C4C31×#12011yescharacteristic
274C2×C4C31×#12012yes—
275C2×C4C22×D4×C3111yescharacteristic
276C2×C4C22×D4×C3112yes—
277C8—44no—
278C8—42no—
279C22—44no—
280C22—44no—
281C22—44no—
282C22—88no—
283C22C31×#45412yes—
284C22C31×#45514yes—
285C22C2×C31×#12011yescharacteristic
286C4—22no—
287C4—22no—
288C4—21no—
289C4—21no—
290C4—44no—
291C4—44no—
292C4—82no—
293C2—84no—
294C2C31×#166214yesdirect product factor
295C2C2×C31×#45512yes—
296C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×C31×#562C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#491C2semidirect
3. C2×C31×#489C2semidirect
4. C31×#1662C2direct (N × H)
5. C2×C31×#330C2semidirect
6. C2×C31×#329C2semidirect
17. C31×#491C22semidirect
19. C31×#489C22semidirect
25. C31×#330C22semidirect
26. C31×#329C22semidirect
103. C2×#1662C31direct (N × H)
132. C2×#491C2×C31semidirect
133. C2×#489C2×C31semidirect
134. #1662C2×C31direct (N × H)
135. C2×#330C2×C31semidirect
136. C2×#329C2×C31semidirect
161. #491C22×C31semidirect
163. #489C22×C31semidirect
169. #330C22×C31semidirect
170. #329C22×C31semidirect
172. C2×C31#1662direct (N × H)
213. C31C2×#1662direct (N × H)
294. C2C31×#1662direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5622
C2×C31×#4912
C2×C31×#4892
C31×#16622
C2×C31×#3302
C2×C31×#3292
C2×#166231