Finite Group Databasebeta

← C2×C31×#1656 #21475739

Subgroups — C2×C31×#1656 #21475739

Order 7936.

1138 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1138 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, complemented
2C2×C31×#496C212yesmaximal, complemented
3C2×C31×#490C211yesmaximal, characteristic, complemented
4C31×#1656C218yesmaximal, direct product factor
5C2×C31×#330C211yesmaximal, characteristic, complemented
6C2×C31×#329C212yesmaximal, complemented
7C22×SD16×C31—22no—
8C22×D8×C31—22no—
9C31×#562—22no—
10C31×#562—24no—
11C31×#562C2214yescomplemented
12C2×C31×#127C2211yescharacteristic
13C2×C31×#125C2211yescharacteristic
14C2×C31×#124C2212yes—
15C2×C31×#123—22no—
16C2×C31×C22⋊Q8—22no—
17C2×C31×#121—22no—
18C31×#496—24no—
19C31×#496C2218yescomplemented
20C31×#490—22no—
21C31×#490C2214yescomplemented
22C2×C31×#102—22no—
23C2×C31×#97C2211yescharacteristic
24C2×C31×#97C2212yes—
25C31×#330C2214yescomplemented
26C31×#329C2218yescomplemented
27C22×Q8×C31D412yes—
28C22×D4×C31—22no—
29C22×D4×C31—21no—
30C22×D4×C31—22no—
31C22×D4×C31D412yes—
32C23×C4×C31—22no—
33C2×SD16×C31—28no—
34C2×SD16×C31—48no—
35C2×D8×C31—28no—
36C2×D8×C31—48no—
37C22×C8×C31—22no—
38C22×C8×C31—21no—
39C31×#127—21no—
40C31×#127—22no—
41C31×#127C2312yes—
42C31×#125—22no—
43C31×#125D412yes—
44C31×#125C2312yes—
45C31×#124—22no—
46C31×#124—22no—
47C31×#124D414yes—
48C31×#124C2314yes—
49C31×#123—28no—
50C31×#123—28no—
51C31×C22⋊Q8—28no—
52C31×C22⋊Q8—28no—
53C31×#121—28no—
54C31×#121—28no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31D412yes—
58C2×C22⋊C4×C31—22no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—22no—
61C2×C42×C31C2311yescharacteristic
62C31×#102—28no—
63C31×#97—21no—
64C31×#97—22no—
65C31×#97C2312yes—
66C31×#97C2314yes—
67C24×C31—22no—
68C24×C31—41no—
69C2×Q8×C31—22no—
70C2×Q8×C31—44no—
71C2×Q8×C31C2×D414yes—
72C2×D4×C31—28no—
73C2×D4×C31—22no—
74C2×D4×C31—22no—
75C2×D4×C31—24no—
76C2×D4×C31—44no—
77C2×D4×C31—44no—
78C2×D4×C31—44no—
79C2×D4×C31—44no—
80C2×D4×C31—41no—
81C2×D4×C31—42no—
82C2×D4×C31—84no—
83C2×D4×C31C2×D414yes—
84C22×C4×C31—22no—
85C22×C4×C31—22no—
86C22×C4×C31—28no—
87C22×C4×C31—84no—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D412yes—
90SD16×C31—416no—
91D8×C31—416no—
92C2×C8×C31—28no—
93C2×C8×C31—24no—
94C2×C8×C31—22no—
95C2×C8×C31—42no—
96C4⋊C4×C31—28no—
97C4⋊C4×C31—24no—
98C4⋊C4×C31—22no—
99C4⋊C4×C31—24no—
100C4⋊C4×C31C2×D414yes—
101C22⋊C4×C31—28no—
102C22⋊C4×C31—24no—
103C22⋊C4×C31—24no—
104C22⋊C4×C31—24no—
105C22⋊C4×C31—24no—
106C42×C31C2×D411yescharacteristic
107C42×C31C2×D412yes—
108C42×C31C2411yescharacteristic
109C2×#1656C3111yesmaximal, characteristic, direct product factor, Sylow
110C23×C31—24no—
111C23×C31—44no—
112C23×C31—44no—
113C23×C31—44no—
114C23×C31—42no—
115C23×C31—84no—
116C23×C31—82no—
117C23×C31#12011yescharacteristic
118Q8×C31—44no—
119Q8×C31—82no—
120D4×C31—44no—
121D4×C31—44no—
122D4×C31—44no—
123D4×C31—44no—
124D4×C31—82no—
125D4×C31—84no—
126D4×C31—82no—
127D4×C31—88no—
128C2×C4×C31—24no—
129C2×C4×C31—22no—
130C2×C4×C31—24no—
131C2×C4×C31—24no—
132C2×C4×C31—44no—
133C2×C4×C31—44no—
134C2×C4×C31—88no—
135C2×C4×C31#12011yescharacteristic
136C2×C4×C31#12012yes—
137C2×C4×C31C22×D411yescharacteristic
138C2×C4×C31C22×D412yes—
139C8×C31—28no—
140C8×C31—42no—
141C2×#562C2×C3111yescharacteristic, complemented
142C2×#496C2×C3112yescomplemented
143C2×#490C2×C3111yescharacteristic, complemented
144#1656C2×C3118yesdirect product factor
145C2×#330C2×C3111yescharacteristic, complemented
146C2×#329C2×C3112yescomplemented
147C22×C31—44no—
148C22×C31—44no—
149C22×C31—44no—
150C22×C31—88no—
151C22×C31—84no—
152C22×C31—82no—
153C22×C31—164no—
154C22×C31#45314yes—
155C22×C31#45412yes—
156C22×C31C2×#12011yescharacteristic
157C4×C31—22no—
158C4×C31—22no—
159C4×C31—21no—
160C4×C31—21no—
161C4×C31—44no—
162C4×C31—44no—
163C22×SD16—22no—
164C22×D8—22no—
165#562—22no—
166#562—24no—
167#562C22×C3114yescomplemented
168C2×#127C22×C3111yescharacteristic
169C2×#125C22×C3111yescharacteristic
170C2×#124C22×C3112yes—
171C2×#123—22no—
172C2×C22⋊Q8—22no—
173C2×#121—22no—
174#496—24no—
175#496C22×C3118yescomplemented
176#490—22no—
177#490C22×C3114yescomplemented
178C2×#102—22no—
179C2×#97C22×C3111yescharacteristic
180C2×#97C22×C3112yes—
181#330C22×C3114yescomplemented
182#329C22×C3118yescomplemented
183C2×C31—84no—
184C2×C31—162no—
185C2×C31#165614yesdirect product factor
186C2×C31C2×#45312yes—
187C2×C31C2×#45411yescharacteristic
188C22×Q8D4×C3112yes—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—21no—
192C22×D4D4×C3112yes—
193C23×C4—22no—
194C2×SD16—28no—
195C2×SD16—48no—
196C2×D8—28no—
197C2×D8—48no—
198C22×C8—22no—
199C22×C8—21no—
200#127—21no—
201#127—22no—
202#127C23×C3112yes—
203#125—22no—
204#125D4×C3112yes—
205#125C23×C3112yes—
206#124—22no—
207#124—22no—
208#124D4×C3114yes—
209#124C23×C3114yes—
210#123—28no—
211#123—28no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214#121—28no—
215#121—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4D4×C3112yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42C23×C3111yescharacteristic
223#102—28no—
224#97—21no—
225#97—22no—
226#97C23×C3112yes—
227#97C23×C3114yes—
228C31C2×#165611yescharacteristic, direct product factor, Sylow
229C24—22no—
230C24—41no—
231C2×Q8—22no—
232C2×Q8—44no—
233C2×Q8C2×D4×C3114yes—
234C2×D4—28no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—24no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—84no—
245C2×D4C2×D4×C3114yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—28no—
249C22×C4—84no—
250C22×C4C2×D4×C3111yescharacteristic
251C22×C4C2×D4×C3112yes—
252SD16—416no—
253D8—416no—
254C2×C8—28no—
255C2×C8—24no—
256C2×C8—22no—
257C2×C8—42no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4C2×D4×C3114yes—
263C22⋊C4—28no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C42C2×D4×C3111yescharacteristic
269C42C2×D4×C3112yes—
270C42C24×C3111yescharacteristic
271C23—24no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23—42no—
276C23—84no—
277C23—82no—
278C23C31×#12011yescharacteristic
279Q8—44no—
280Q8—82no—
281D4—44no—
282D4—44no—
283D4—44no—
284D4—44no—
285D4—82no—
286D4—84no—
287D4—82no—
288D4—88no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—88no—
296C2×C4C31×#12011yescharacteristic
297C2×C4C31×#12012yes—
298C2×C4C22×D4×C3111yescharacteristic
299C2×C4C22×D4×C3112yes—
300C8—28no—
301C8—42no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—88no—
306C22—84no—
307C22—82no—
308C22—164no—
309C22C31×#45314yes—
310C22C31×#45412yes—
311C22C2×C31×#12011yescharacteristic
312C4—22no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—44no—
317C4—44no—
318C2—84no—
319C2—162no—
320C2C31×#165614yesdirect product factor
321C2C2×C31×#45312yes—
322C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC42C24×C31
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
1. C2×C31×#562C2semidirect
2. C2×C31×#496C2semidirect
3. C2×C31×#490C2semidirect
4. C31×#1656C2direct (N × H)
5. C2×C31×#330C2semidirect
6. C2×C31×#329C2semidirect
11. C31×#562C22semidirect
19. C31×#496C22semidirect
21. C31×#490C22semidirect
25. C31×#330C22semidirect
26. C31×#329C22semidirect
109. C2×#1656C31direct (N × H)
141. C2×#562C2×C31semidirect
142. C2×#496C2×C31semidirect
143. C2×#490C2×C31semidirect
144. #1656C2×C31direct (N × H)
145. C2×#330C2×C31semidirect
146. C2×#329C2×C31semidirect
167. #562C22×C31semidirect
175. #496C22×C31semidirect
177. #490C22×C31semidirect
181. #330C22×C31semidirect
182. #329C22×C31semidirect
185. C2×C31#1656direct (N × H)
228. C31C2×#1656direct (N × H)
320. C2C31×#1656direct (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×#4962
C2×C31×#4902
C31×#16562
C2×C31×#3302
C2×C31×#3292
C2×#165631