Finite Group Databasebeta

← C31×#13359 #21470118

Subgroups — C31×#13359 #21470118

Order 7936.

704 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 98 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×#456C212yesmaximal
2C31×#2734C211yesmaximal, characteristic, complemented
3C2×C31×#356C212yesmaximal, complemented
4C31×#1793C211yesmaximal, characteristic
5C31×#1394C211yesmaximal, characteristic, complemented
6C2×C31×2−1+4C2211yescharacteristic
7C2×C31×#137—22no—
8C2×C31×C4⋊Q8C2212yes—
9C31×#456—24no—
10C31×#456—24no—
11C2×C31×#104—22no—
12C2×C31×#104—21no—
13C31×#417—21no—
14C2×C31×#101C2211yescharacteristic
15C2×C31×#101C2212yes—
16C31×#398—21no—
17C31×#395—22no—
18C31×#391C2211yescharacteristic
19C31×#389—21no—
20C31×#387—22no—
21C31×#356—24no—
22C31×2−1+4—22no—
23C31×2−1+4—21no—
24C31×2−1+4—42no—
25C31×2−1+4D412yes—
26C2×#41×C31—21no—
27C2×#41×C31—22no—
28C2×#41×C31—41no—
29C22×Q8×C31—41no—
30C22×Q8×C31D412yes—
31C22×Q8×C31C2311yescharacteristic
32C31×#137—28no—
33C31×#137—44no—
34C2×Q16×C31—42no—
35C2×SD16×C31—42no—
36C2×M16×C31—22no—
37C2×M16×C31—21no—
38C31×C4⋊Q8—22no—
39C31×C4⋊Q8—22no—
40C31×C4⋊Q8—22no—
41C31×C4⋊Q8—22no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—22no—
44C2×C4⋊C4×C31—22no—
45C2×C4⋊C4×C31—21no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31D411yescharacteristic
48C2×C42×C31D412yes—
49C2×C42×C31D411yescharacteristic
50C31×#104—28no—
51C31×#104—42no—
52C31×#103—41no—
53C31×#102—41no—
54C31×#101—22no—
55C31×#101—22no—
56C31×#101D414yes—
57C31×#95—22no—
58C31×#95—22no—
59C31×#95—21no—
60C31×#95—22no—
61#41×C31—22no—
62#41×C31—24no—
63#41×C31—24no—
64#41×C31—22no—
65#41×C31—41no—
66#41×C31—42no—
67#41×C31—42no—
68#41×C31—44no—
69#41×C31—42no—
70#41×C31—42no—
71#41×C31—41no—
72C2×Q8×C31—21no—
73C2×Q8×C31—22no—
74C2×Q8×C31—22no—
75C2×Q8×C31—22no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—42no—
79C2×Q8×C31—42no—
80C2×Q8×C31—42no—
81C2×Q8×C31—44no—
82C2×Q8×C31—42no—
83C2×Q8×C31C2×D411yescharacteristic
84C2×Q8×C31C2×D412yes—
85C2×Q8×C31#4111yescharacteristic
86C2×D4×C31—21no—
87C2×D4×C31—22no—
88C2×D4×C31—41no—
89C22×C4×C31—21no—
90C22×C4×C31—22no—
91C22×C4×C31—21no—
92C22×C4×C31—22no—
93C22×C4×C31—41no—
94C22×C4×C31—42no—
95C22×C4×C31C2×D412yes—
96C22×C4×C31C2×D411yescharacteristic
97Q16×C31—48no—
98SD16×C31—48no—
99M16×C31—24no—
100M16×C31—42no—
101M16×C31—41no—
102M16×C31—41no—
103C2×C8×C31—42no—
104C2×C8×C31—41no—
105C4⋊C4×C31—24no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—42no—
108C4⋊C4×C31—42no—
109C4⋊C4×C31—42no—
110C4⋊C4×C31—42no—
111C4⋊C4×C31—81no—
112C4⋊C4×C31—81no—
113C42×C31—22no—
114C42×C31—22no—
115C42×C31—41no—
116#13359C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—41no—
118C23×C31#12011yescharacteristic
119Q8×C31—24no—
120Q8×C31—22no—
121Q8×C31—41no—
122Q8×C31—42no—
123Q8×C31—41no—
124Q8×C31—41no—
125Q8×C31—42no—
126Q8×C31—44no—
127Q8×C31—82no—
128D4×C31—24no—
129D4×C31—22no—
130D4×C31—41no—
131D4×C31—42no—
132D4×C31—42no—
133D4×C31—42no—
134C2×C4×C31—21no—
135C2×C4×C31—22no—
136C2×C4×C31—21no—
137C2×C4×C31—22no—
138C2×C4×C31—22no—
139C2×C4×C31—24no—
140C2×C4×C31—44no—
141C2×C4×C31—42no—
142C2×C4×C31—42no—
143C2×C4×C31—44no—
144C2×C4×C31—41no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—42no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31#12012yes—
155C2×C4×C31#12112yes—
156C2×C4×C31#12111yescharacteristic
157C2×C4×C31#12711yescharacteristic
158C8×C31—44no—
159C8×C31—81no—
160C2×#456C2×C3112yes—
161#2734C2×C3111yescharacteristic, complemented
162C2×#356C2×C3112yescomplemented
163#1793C2×C3111yescharacteristic
164#1394C2×C3111yescharacteristic, complemented
165C22×C31—21no—
166C22×C31—21no—
167C22×C31—42no—
168C22×C31—81no—
169C22×C31—81no—
170C22×C31#39211yescharacteristic
171C22×C31B(4,𝔽2)11yescharacteristic
172C22×C31B(4,𝔽2)11yescharacteristic
173C4×C31—22no—
174C4×C31—21no—
175C4×C31—21no—
176C4×C31—22no—
177C4×C31—44no—
178C4×C31—42no—
179C4×C31—42no—
180C4×C31—81no—
181C4×C31—82no—
182C2×2−1+4C22×C3111yescharacteristic
183C2×#137—22no—
184C2×C4⋊Q8C22×C3112yes—
185#456—24no—
186#456—24no—
187C2×#104—22no—
188C2×#104—21no—
189#417—21no—
190C2×#101C22×C3111yescharacteristic
191C2×#101C22×C3112yes—
192#398—21no—
193#395—22no—
194#391C22×C3111yescharacteristic
195#389—21no—
196#387—22no—
197#356—24no—
198C2×C31—21no—
199C2×C31—21no—
200C2×C31—82no—
201C2×C31#202111yescharacteristic
202C2×C31#220311yescharacteristic
203C2×C31#220511yescharacteristic
2042−1+4—22no—
2052−1+4—21no—
2062−1+4—42no—
2072−1+4D4×C3112yes—
208C2×#41—22no—
209C2×#41—21no—
210C2×#41—41no—
211C22×Q8—41no—
212C22×Q8D4×C3112yes—
213C22×Q8C23×C3111yescharacteristic
214#137—28no—
215#137—44no—
216C2×Q16—42no—
217C2×SD16—42no—
218C2×M16—22no—
219C2×M16—21no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C3111yescharacteristic
230C2×C42D4×C3112yes—
231C2×C42D4×C3111yescharacteristic
232#104—28no—
233#104—42no—
234#103—41no—
235#102—41no—
236#101—22no—
237#101—22no—
238#101D4×C3114yes—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—22no—
243C31#1335911yescharacteristic, direct product factor, Sylow
244#41—22no—
245#41—24no—
246#41—24no—
247#41—22no—
248#41—41no—
249#41—42no—
250#41—42no—
251#41—44no—
252#41—42no—
253#41—42no—
254#41—41no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—42no—
266C2×Q8C2×D4×C3111yescharacteristic
267C2×Q8C2×D4×C3112yes—
268C2×Q8#41×C3111yescharacteristic
269C2×D4—21no—
270C2×D4—22no—
271C2×D4—41no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4C2×D4×C3112yes—
279C22×C4C2×D4×C3111yescharacteristic
280Q16—48no—
281SD16—48no—
282M16—24no—
283M16—42no—
284M16—41no—
285M16—41no—
286C2×C8—42no—
287C2×C8—41no—
288C4⋊C4—24no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C42—22no—
297C42—22no—
298C42—41no—
299C23—41no—
300C23C31×#12011yescharacteristic
301Q8—24no—
302Q8—22no—
303Q8—41no—
304Q8—42no—
305Q8—41no—
306Q8—41no—
307Q8—42no—
308Q8—44no—
309Q8—82no—
310D4—24no—
311D4—22no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—42no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4C31×#12012yes—
337C2×C4C31×#12112yes—
338C2×C4C31×#12111yescharacteristic
339C2×C4C31×#12711yescharacteristic
340C8—44no—
341C8—81no—
342C22—21no—
343C22—21no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22C31×#39211yescharacteristic
348C22C31×B(4,𝔽2)11yescharacteristic
349C22C31×B(4,𝔽2)11yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—44no—
355C4—42no—
356C4—42no—
357C4—81no—
358C4—82no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2C31×#202111yescharacteristic
363C2C31×#220311yescharacteristic
364C2C31×#220511yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C31
prime radicalC2×C31×2−1+4C22
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2734C2semidirect
3. C2×C31×#356C2semidirect
5. C31×#1394C2semidirect
116. #13359C31direct (N × H)
161. #2734C2×C31semidirect
162. C2×#356C2×C31semidirect
164. #1394C2×C31semidirect
243. C31#13359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4562
C31×#27342
C2×C31×#3562
C31×#17932
C31×#13942
#1335931