Finite Group Databasebeta

← C31×#13565 #21470324

Subgroups — C31×#13565 #21470324

Order 7936.

496 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 90 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×#455C211yesmaximal, characteristic
2C2×C31×#355C211yesmaximal, characteristic, complemented
3C31×#1533C211yesmaximal, characteristic, complemented
4C31×#1403C214yesmaximal, complemented
5C2×C31×H(5,𝔽2)C411yescharacteristic
6C2×C31×#137—21no—
7C2×C31×#124C2211yescharacteristic
8C31×#455—21no—
9C31×#455—21no—
10C31×#455C412yes—
11C31×#455C2212yes—
12C31×#432—21no—
13C2×C31×#104—21no—
14C2×C31×#101—21no—
15C2×C31×#100C411yescharacteristic
16C31×#355—21no—
17C31×#355C2212yescomplemented
18C31×#329—21no—
19C31×#329C2212yescomplemented
20C31×H(5,𝔽2)—21no—
21C31×H(5,𝔽2)—41no—
22C31×H(5,𝔽2)—42no—
23C31×H(5,𝔽2)C2×C412yes—
24C2×#41×C31—21no—
25C2×#41×C31—41no—
26C22×Q8×C31D411yescharacteristic
27C22×D4×C31—41no—
28C22×D4×C31—41no—
29C22×D4×C31C2×C411yescharacteristic
30C31×#137—22no—
31C31×#137—22no—
32C31×#137—42no—
33C2×Q16×C31—41no—
34C2×SD16×C31—41no—
35C2×M16×C31—21no—
36C2×M16×C31—41no—
37C31×#124—41no—
38C31×#124C2×C411yescharacteristic
39C31×#124D411yescharacteristic
40C31×#124D411yescharacteristic
41C31×#124C2311yescharacteristic
42C2×C22⋊C4×C31—41no—
43C2×C42×C31D411yescharacteristic
44C31×C4⋊C8—41no—
45C31×#104—22no—
46C31×#104—22no—
47C31×#101—22no—
48C31×#101—41no—
49C31×#100—21no—
50C31×#100C2×C412yes—
51C31×C22⋊C8—41no—
52C31×#97—22no—
53C24×C31—21no—
54C24×C31—41no—
55#41×C31—22no—
56#41×C31—22no—
57#41×C31—41no—
58#41×C31—42no—
59#41×C31—44no—
60#41×C31—41no—
61#41×C31—42no—
62C2×Q8×C31—21no—
63C2×Q8×C31—21no—
64C2×Q8×C31—41no—
65C2×Q8×C31—41no—
66C2×Q8×C31C22⋊C411yescharacteristic
67C2×Q8×C31C2×D411yescharacteristic
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—41no—
71C2×D4×C31—41no—
72C2×D4×C31—42no—
73C2×D4×C31—42no—
74C2×D4×C31—42no—
75C2×D4×C31—42no—
76C2×D4×C31—44no—
77C2×D4×C31—42no—
78C2×D4×C31—41no—
79C2×D4×C31—42no—
80C2×D4×C31—42no—
81C2×D4×C31C22⋊C411yescharacteristic
82C2×D4×C31C22×C411yescharacteristic
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—41no—
86C22×C4×C31—41no—
87C22×C4×C31C22⋊C411yescharacteristic
88Q16×C31—42no—
89Q16×C31—42no—
90SD16×C31—42no—
91SD16×C31—42no—
92M16×C31—22no—
93M16×C31—41no—
94M16×C31—42no—
95M16×C31—81no—
96C2×C8×C31—41no—
97C2×C8×C31—42no—
98C2×C8×C31—81no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C42×C31—21no—
104C42×C31C22⋊C411yescharacteristic
105C42×C31C2×D411yescharacteristic
106#13565C3111yesmaximal, characteristic, direct product factor, Sylow
107C23×C31—21no—
108C23×C31—21no—
109C23×C31—42no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—42no—
113C23×C31—41no—
114C23×C31—82no—
115C23×C31—162no—
116C23×C31#9911yescharacteristic
117Q8×C31—22no—
118Q8×C31—41no—
119Q8×C31—41no—
120Q8×C31—41no—
121Q8×C31—81no—
122D4×C31—22no—
123D4×C31—41no—
124D4×C31—41no—
125D4×C31—41no—
126D4×C31—42no—
127D4×C31—44no—
128D4×C31—44no—
129D4×C31—42no—
130D4×C31—42no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—42no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—42no—
144C2×C4×C31—81no—
145C2×C4×C31#9911yescharacteristic
146C2×C4×C31C2×C22⋊C411yescharacteristic
147C8×C31—42no—
148C8×C31—82no—
149C8×C31—82no—
150C2×#455C2×C3111yescharacteristic
151C2×#355C2×C3111yescharacteristic, complemented
152#1533C2×C3111yescharacteristic, complemented
153#1403C2×C3114yescomplemented
154C22×C31—41no—
155C22×C31—42no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—81no—
159C22×C31—81no—
160C22×C31—82no—
161C22×C31—82no—
162C22×C31—161no—
163C22×C31—162no—
164C22×C31—162no—
165C22×C31AΣL(1,𝔽16)11yescharacteristic
166C22×C31AΣL(1,𝔽16)11yescharacteristic
167C22×C31C2×#9911yescharacteristic
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—41no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—42no—
175C2×H(5,𝔽2)C4×C3111yescharacteristic
176C2×#137—21no—
177C2×#124C22×C3111yescharacteristic
178#455—21no—
179#455—21no—
180#455C4×C3112yes—
181#455C22×C3112yes—
182#432—21no—
183C2×#104—21no—
184C2×#101—21no—
185C2×#100C4×C3111yescharacteristic
186#355—21no—
187#355C22×C3112yescomplemented
188#329—21no—
189#329C22×C3112yescomplemented
190C2×C31—21no—
191C2×C31—21no—
192C2×C31—21no—
193C2×C31—81no—
194C2×C31—81no—
195C2×C31—82no—
196C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
197H(5,𝔽2)—21no—
198H(5,𝔽2)—41no—
199H(5,𝔽2)—42no—
200H(5,𝔽2)C2×C4×C3112yes—
201C2×#41—21no—
202C2×#41—41no—
203C22×Q8D4×C3111yescharacteristic
204C22×D4—41no—
205C22×D4—41no—
206C22×D4C2×C4×C3111yescharacteristic
207#137—22no—
208#137—22no—
209#137—42no—
210C2×Q16—41no—
211C2×SD16—41no—
212C2×M16—21no—
213C2×M16—41no—
214#124—41no—
215#124C2×C4×C3111yescharacteristic
216#124D4×C3111yescharacteristic
217#124D4×C3111yescharacteristic
218#124C23×C3111yescharacteristic
219C2×C22⋊C4—41no—
220C2×C42D4×C3111yescharacteristic
221C4⋊C8—41no—
222#104—22no—
223#104—22no—
224#101—22no—
225#101—41no—
226#100—21no—
227#100C2×C4×C3112yes—
228C22⋊C8—41no—
229#97—22no—
230C31#1356511yescharacteristic, direct product factor, Sylow
231C24—21no—
232C24—41no—
233#41—22no—
234#41—22no—
235#41—41no—
236#41—42no—
237#41—44no—
238#41—41no—
239#41—42no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8C22⋊C4×C3111yescharacteristic
245C2×Q8C2×D4×C3111yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4C22⋊C4×C3111yescharacteristic
260C2×D4C22×C4×C3111yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22⋊C4×C3111yescharacteristic
266Q16—42no—
267Q16—42no—
268SD16—42no—
269SD16—42no—
270M16—22no—
271M16—41no—
272M16—42no—
273M16—81no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—81no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42—21no—
282C42C22⋊C4×C3111yescharacteristic
283C42C2×D4×C3111yescharacteristic
284C23—21no—
285C23—21no—
286C23—42no—
287C23—41no—
288C23—41no—
289C23—42no—
290C23—41no—
291C23—82no—
292C23—162no—
293C23C31×#9911yescharacteristic
294Q8—22no—
295Q8—41no—
296Q8—41no—
297Q8—41no—
298Q8—81no—
299D4—22no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—42no—
304D4—44no—
305D4—44no—
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4C31×#9911yescharacteristic
323C2×C4C2×C22⋊C4×C3111yescharacteristic
324C8—42no—
325C8—82no—
326C8—82no—
327C22—41no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—81no—
332C22—81no—
333C22—82no—
334C22—82no—
335C22—161no—
336C22—162no—
337C22—162no—
338C22C31×AΣL(1,𝔽16)11yescharacteristic
339C22C31×AΣL(1,𝔽16)11yescharacteristic
340C22C2×C31×#9911yescharacteristic
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—42no—
348C2—21no—
349C2—21no—
350C2—21no—
351C2—81no—
352C2—81no—
353C2—82no—
354C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
center, socleC2×C31C2×AΣL(1,𝔽16)
commutatorC2×D4C22×C4×C31
Frattini, cosocle#124C23×C31
prime radicalC2×C31×H(5,𝔽2)C4
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C22×C4×C31 ◁ C2×C31×#124 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#355C2semidirect
3. C31×#1533C2semidirect
4. C31×#1403C2semidirect
17. C31×#355C22semidirect
19. C31×#329C22semidirect
106. #13565C31direct (N × H)
151. C2×#355C2×C31semidirect
152. #1533C2×C31semidirect
153. #1403C2×C31semidirect
187. #355C22×C31semidirect
189. #329C22×C31semidirect
230. C31#13565direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4552
C2×C31×#3552
C31×#15332
C31×#14032
#1356531