Finite Group Databasebeta

← C31×#32976 #21489735

Subgroups — C31×#32976 #21489735

Order 7936.

1382 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 298 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
1C4×C31×H(5,𝔽2)C211yesmaximal, characteristic
2C31×#2487C213yesmaximal, complemented
3C31×#1867C216yesmaximal, complemented
4C31×#1799C213yesmaximal, complemented
5C31×#1790C212yesmaximal
6C2×C31×H(5,𝔽2)C2211yescharacteristic
7C31×#520—23no—
8C31×#518—23no—
9C31×#518C2213yes—
10C4×#41×C31—23no—
11C2×C4×D4×C31—23no—
12C2×C4×D4×C31C2213yes—
13C22×C4⋊C4×C31C2211yescharacteristic, complemented
14C2×C31×#99C2216yes—
15C31×#400—23no—
16C31×#399—23no—
17C31×#389—23no—
18C31×#386C2216yes—
19C31×#385C2216yescomplemented
20C31×#380—23no—
21C31×#380C2213yescomplemented
22C31×#342C2216yescomplemented
23C31×H(5,𝔽2)—26no—
24C31×H(5,𝔽2)Q814yes—
25C2×#41×C31—23no—
26C22×D4×C31—23no—
27C22×D4×C31C2313yes—
28C23×C4×C31—21no—
29C23×C4×C31D413yes—
30C23×C4×C31C2312yes—
31C23×C4×C31C2311yescharacteristic
32C4×Q8×C31—21no—
33C4×Q8×C31—23no—
34C4×D4×C31—23no—
35C4×D4×C31—26no—
36C4×D4×C31—23no—
37C4×D4×C31—26no—
38C4×D4×C31—26no—
39C4×D4×C31—26no—
40C4×D4×C31—26no—
41C31×#117—26no—
42C31×#117—23no—
43C2×C4⋊C4×C31—23no—
44C2×C4⋊C4×C31—23no—
45C2×C4⋊C4×C31—23no—
46C2×C4⋊C4×C31—23no—
47C2×C4⋊C4×C31—26no—
48C2×C4⋊C4×C31D413yes—
49C2×C22⋊C4×C31—23no—
50C2×C22⋊C4×C31—23no—
51C2×C22⋊C4×C31—26no—
52C2×C22⋊C4×C31D413yes—
53C2×C22⋊C4×C31C2316yes—
54C2×C22⋊C4×C31C2313yes—
55C2×C42×C31—23no—
56C2×C42×C31D413yes—
57C31×#99—212no—
58C31×#95—23no—
59C31×#95—26no—
60C31×#95—26no—
61C24×C31—21no—
62C24×C31C2×D413yes—
63C24×C31C2411yescharacteristic
64#41×C31—212no—
65#41×C31—212no—
66C2×Q8×C31—21no—
67C2×D4×C31—23no—
68C2×D4×C31—212no—
69C2×D4×C31—212no—
70C2×D4×C31—26no—
71C2×D4×C31—26no—
72C2×D4×C31C2×Q816yes—
73C2×D4×C31#4116yes—
74C22×C4×C31—23no—
75C22×C4×C31—23no—
76C22×C4×C31—23no—
77C22×C4×C31—23no—
78C22×C4×C31—23no—
79C22×C4×C31—26no—
80C22×C4×C31—26no—
81C22×C4×C31—42no—
82C22×C4×C31—46no—
83C22×C4×C31—82no—
84C22×C4×C31—82no—
85C22×C4×C31C2×D413yes—
86C22×C4×C31C2×D413yes—
87C22×C4×C31C2×D413yes—
88C22×C4×C31C2×D413yes—
89C22×C4×C31C2×D413yes—
90C4⋊C4×C31—23no—
91C4⋊C4×C31—23no—
92C4⋊C4×C31—26no—
93C4⋊C4×C31—46no—
94C4⋊C4×C31—43no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—86no—
97C22⋊C4×C31—26no—
98C22⋊C4×C31—26no—
99C22⋊C4×C31—26no—
100C22⋊C4×C31—212no—
101C22⋊C4×C31—86no—
102C42×C31—26no—
103C42×C31—23no—
104C42×C31—23no—
105#32976C3111yesmaximal, characteristic, direct product factor, Sylow
106C23×C31—24no—
107C23×C31—23no—
108C23×C31—23no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—43no—
112C23×C31—43no—
113C23×C31—81no—
114C23×C31—81no—
115C23×C31C22⋊Q816yes—
116C23×C31C22×D413yes—
117C23×C31C22×Q811yescharacteristic
118C23×C31C2×#4113yes—
119Q8×C31—24no—
120D4×C31—212no—
121D4×C31—212no—
122D4×C31—212no—
123C2×C4×C31—212no—
124C2×C4×C31—23no—
125C2×C4×C31—23no—
126C2×C4×C31—26no—
127C2×C4×C31—23no—
128C2×C4×C31—23no—
129C2×C4×C31—43no—
130C2×C4×C31—46no—
131C2×C4×C31—43no—
132C2×C4×C31—43no—
133C2×C4×C31—46no—
134C2×C4×C31—46no—
135C2×C4×C31—46no—
136C2×C4×C31—46no—
137C2×C4×C31—42no—
138C2×C4×C31—46no—
139C2×C4×C31—83no—
140C2×C4×C31—83no—
141C2×C4×C31—83no—
142C2×C4×C31#12013yes—
143C2×C4×C31#12011yescharacteristic
144C2×C4×C31C22⋊Q816yes—
145C4×H(5,𝔽2)C2×C3111yescharacteristic
146#2487C2×C3113yescomplemented
147#1867C2×C3116yescomplemented
148#1799C2×C3113yescomplemented
149#1790C2×C3112yes—
150C22×C31—23no—
151C22×C31—23no—
152C22×C31—26no—
153C22×C31—41no—
154C22×C31—43no—
155C22×C31—46no—
156C22×C31—43no—
157C22×C31—43no—
158C22×C31—83no—
159C22×C31—84no—
160C22×C31C2×#12011yescharacteristic
161C22×C31C2×C22⋊Q813yes—
162C22×C31#52513yes—
163C4×C31—26no—
164C4×C31—23no—
165C4×C31—43no—
166C4×C31—43no—
167C4×C31—86no—
168C4×C31—82no—
169C4×C31#45812yes—
170C2×H(5,𝔽2)C22×C3111yescharacteristic
171#520—23no—
172#518—23no—
173#518C22×C3113yes—
174C4×#41—23no—
175C2×C4×D4—23no—
176C2×C4×D4C22×C3113yes—
177C22×C4⋊C4C22×C3111yescharacteristic, complemented
178C2×#99C22×C3116yes—
179#400—23no—
180#399—23no—
181#389—23no—
182#386C22×C3116yes—
183#385C22×C3116yescomplemented
184#380—23no—
185#380C22×C3113yescomplemented
186#342C22×C3116yescomplemented
187C2×C31—23no—
188C2×C31—23no—
189C2×C31—46no—
190C2×C31#284811yescharacteristic
191C2×C31C2×#45811yescharacteristic
192C2×C31#302711yescharacteristic
193H(5,𝔽2)—26no—
194H(5,𝔽2)Q8×C3114yes—
195C2×#41—23no—
196C22×D4—23no—
197C22×D4C23×C3113yes—
198C23×C4—21no—
199C23×C4D4×C3113yes—
200C23×C4C23×C3112yes—
201C23×C4C23×C3111yescharacteristic
202C4×Q8—21no—
203C4×Q8—23no—
204C4×D4—23no—
205C4×D4—26no—
206C4×D4—23no—
207C4×D4—26no—
208C4×D4—26no—
209C4×D4—26no—
210C4×D4—26no—
211#117—26no—
212#117—23no—
213C2×C4⋊C4—23no—
214C2×C4⋊C4—23no—
215C2×C4⋊C4—23no—
216C2×C4⋊C4—23no—
217C2×C4⋊C4—26no—
218C2×C4⋊C4D4×C3113yes—
219C2×C22⋊C4—23no—
220C2×C22⋊C4—23no—
221C2×C22⋊C4—26no—
222C2×C22⋊C4D4×C3113yes—
223C2×C22⋊C4C23×C3116yes—
224C2×C22⋊C4C23×C3113yes—
225C2×C42—23no—
226C2×C42D4×C3113yes—
227#99—212no—
228#95—26no—
229#95—23no—
230#95—26no—
231C31#3297611yescharacteristic, direct product factor, Sylow
232C24—21no—
233C24C2×D4×C3113yes—
234C24C24×C3111yescharacteristic
235#41—212no—
236#41—212no—
237C2×Q8—21no—
238C2×D4—23no—
239C2×D4—212no—
240C2×D4—212no—
241C2×D4—26no—
242C2×D4—26no—
243C2×D4C2×Q8×C3116yes—
244C2×D4#41×C3116yes—
245C22×C4—23no—
246C22×C4—23no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—42no—
253C22×C4—46no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4C2×D4×C3113yes—
257C22×C4C2×D4×C3113yes—
258C22×C4C2×D4×C3113yes—
259C22×C4C2×D4×C3113yes—
260C22×C4C2×D4×C3113yes—
261C4⋊C4—23no—
262C4⋊C4—23no—
263C4⋊C4—26no—
264C4⋊C4—46no—
265C4⋊C4—43no—
266C4⋊C4—41no—
267C4⋊C4—86no—
268C22⋊C4—26no—
269C22⋊C4—26no—
270C22⋊C4—26no—
271C22⋊C4—212no—
272C22⋊C4—86no—
273C42—26no—
274C42—23no—
275C42—23no—
276C23—24no—
277C23—23no—
278C23—23no—
279C23—41no—
280C23—41no—
281C23—43no—
282C23—43no—
283C23—81no—
284C23—81no—
285C23C31×C22⋊Q816yes—
286C23C22×D4×C3113yes—
287C23C22×Q8×C3111yescharacteristic
288C23C2×#41×C3113yes—
289Q8—24no—
290D4—212no—
291D4—212no—
292D4—212no—
293C2×C4—212no—
294C2×C4—23no—
295C2×C4—23no—
296C2×C4—26no—
297C2×C4—23no—
298C2×C4—23no—
299C2×C4—43no—
300C2×C4—46no—
301C2×C4—43no—
302C2×C4—43no—
303C2×C4—46no—
304C2×C4—46no—
305C2×C4—46no—
306C2×C4—46no—
307C2×C4—42no—
308C2×C4—46no—
309C2×C4—83no—
310C2×C4—83no—
311C2×C4—83no—
312C2×C4C31×#12013yes—
313C2×C4C31×#12011yescharacteristic
314C2×C4C31×C22⋊Q816yes—
315C22—23no—
316C22—23no—
317C22—26no—
318C22—41no—
319C22—43no—
320C22—46no—
321C22—43no—
322C22—43no—
323C22—83no—
324C22—84no—
325C22C2×C31×#12011yescharacteristic
326C22C2×C31×C22⋊Q813yes—
327C22C31×#52513yes—
328C4—26no—
329C4—23no—
330C4—43no—
331C4—43no—
332C4—86no—
333C4—82no—
334C4C31×#45812yes—
335C2—23no—
336C2—23no—
337C2—46no—
338C2C31×#284811yescharacteristic
339C2C2×C31×#45811yescharacteristic
340C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC2×C31×H(5,𝔽2)C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2487C2semidirect
3. C31×#1867C2semidirect
4. C31×#1799C2semidirect
13. C22×C4⋊C4×C31C22semidirect
19. C31×#385C22semidirect
21. C31×#380C22semidirect
22. C31×#342C22semidirect
105. #32976C31direct (N × H)
146. #2487C2×C31semidirect
147. #1867C2×C31semidirect
148. #1799C2×C31semidirect
177. C22×C4⋊C4C22×C31semidirect
183. #385C22×C31semidirect
185. #380C22×C31semidirect
186. #342C22×C31semidirect
231. C31#32976direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C31×H(5,𝔽2)2
C31×#24872
C31×#18672
C31×#17992
C31×#17902
#3297631