Finite Group Databasebeta

← C31×#13362 #21470121

Subgroups — C31×#13362 #21470121

Order 7936.

670 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C31×#3202C211yesmaximal, characteristic, complemented
2C31×#3022C212yesmaximal
3C31×#2123C212yesmaximal, complemented
4C31×#1794C211yesmaximal, characteristic
5C31×#1392C211yesmaximal, characteristic, complemented
6C31×#585C2211yescharacteristic
7C2×C31×#137—22no—
8C31×#536C2212yes—
9C31×#485—22no—
10C31×#470—21no—
11C31×#467—21no—
12C31×#456—24no—
13C31×#455—24no—
14C31×#451—22no—
15C31×#421—22no—
16C31×#421—21no—
17C31×#416—21no—
18C31×#413C2212yes—
19C31×#410C2211yescharacteristic
20C31×#408C2211yescharacteristic
21C31×#352—24no—
22C31×2−1+4—41no—
23C31×2−1+4D412yes—
24C31×H(5,𝔽2)—22no—
25C31×H(5,𝔽2)—21no—
26C31×H(5,𝔽2)—41no—
27C2×#41×C31—21no—
28C2×#41×C31—22no—
29C2×#41×C31—42no—
30C2×#41×C31C2311yescharacteristic
31C22×Q8×C31D412yes—
32C31×#137—28no—
33C31×#137—44no—
34C2×Q16×C31—21no—
35C2×Q16×C31—21no—
36C2×Q16×C31—42no—
37C2×SD16×C31—42no—
38C2×M16×C31—22no—
39C2×M16×C31D411yescharacteristic
40C22×C8×C31D411yescharacteristic
41C31×C4⋊Q8—22no—
42C31×#124—22no—
43C31×#123—22no—
44C31×#123—22no—
45C31×C22⋊Q8—22no—
46C31×C22⋊Q8—22no—
47C31×#117—21no—
48C31×#117D412yes—
49C31×#107—21no—
50C31×#106—21no—
51C31×#104—28no—
52C31×#104—42no—
53C31×#103—22no—
54C31×#103—22no—
55C31×#103—41no—
56C31×#102—41no—
57C31×#101—22no—
58C31×#100D414yes—
59C31×#99—22no—
60C31×C22⋊C8—22no—
61#41×C31—22no—
62#41×C31—24no—
63#41×C31—24no—
64#41×C31—22no—
65#41×C31—41no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—42no—
69#41×C31—42no—
70#41×C31—44no—
71#41×C31—42no—
72#41×C31—41no—
73C2×Q8×C31—21no—
74C2×Q8×C31—22no—
75C2×Q8×C31—22no—
76C2×Q8×C31—22no—
77C2×Q8×C31—42no—
78C2×Q8×C31—42no—
79C2×Q8×C31—42no—
80C2×Q8×C31C2×D411yescharacteristic
81C2×D4×C31—21no—
82C2×D4×C31—22no—
83C2×D4×C31—44no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31—41no—
87C2×D4×C31C2×D412yes—
88C2×D4×C31#4111yescharacteristic
89C22×C4×C31—22no—
90C22×C4×C31—41no—
91C22×C4×C31—42no—
92C22×C4×C31C2×D412yes—
93C22×C4×C31C2×D411yescharacteristic
94Q16×C31—48no—
95Q16×C31—81no—
96Q16×C31—81no—
97SD16×C31—48no—
98M16×C31—24no—
99M16×C31—42no—
100M16×C31—41no—
101C2×C8×C31—21no—
102C2×C8×C31—21no—
103C2×C8×C31—42no—
104C2×C8×C31—41no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—42no—
108C4⋊C4×C31—42no—
109C22⋊C4×C31—22no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—41no—
113C42×C31—22no—
114C42×C31—41no—
115#13362C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—42no—
117C23×C31—41no—
118C23×C31#12112yes—
119C23×C31#12711yescharacteristic
120Q8×C31—24no—
121Q8×C31—22no—
122Q8×C31—41no—
123Q8×C31—44no—
124Q8×C31—41no—
125Q8×C31—42no—
126Q8×C31—42no—
127D4×C31—24no—
128D4×C31—22no—
129D4×C31—42no—
130D4×C31—41no—
131D4×C31—44no—
132D4×C31—42no—
133D4×C31—42no—
134D4×C31—41no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—22no—
138C2×C4×C31—22no—
139C2×C4×C31—44no—
140C2×C4×C31—42no—
141C2×C4×C31—42no—
142C2×C4×C31—41no—
143C2×C4×C31—42no—
144C2×C4×C31—42no—
145C2×C4×C31—42no—
146C2×C4×C31—82no—
147C2×C4×C31—81no—
148C2×C4×C31#12012yes—
149C2×C4×C31#12011yescharacteristic
150C2×C4×C31#12111yescharacteristic
151C8×C31—44no—
152C8×C31—41no—
153C8×C31—41no—
154#3202C2×C3111yescharacteristic, complemented
155#3022C2×C3112yes—
156#2123C2×C3112yescomplemented
157#1794C2×C3111yescharacteristic
158#1392C2×C3111yescharacteristic, complemented
159C22×C31—22no—
160C22×C31—21no—
161C22×C31—42no—
162C22×C31—41no—
163C22×C31—42no—
164C22×C31—161no—
165C22×C31—162no—
166C22×C31#39211yescharacteristic
167C4×C31—22no—
168C4×C31—21no—
169C4×C31—42no—
170C4×C31—82no—
171C4×C31—82no—
172C4×C31—161no—
173C4×C31B(4,𝔽2)11yescharacteristic
174C4×C31B(4,𝔽2)11yescharacteristic
175#585C22×C3111yescharacteristic
176C2×#137—22no—
177#536C22×C3112yes—
178#485—22no—
179#470—21no—
180#467—21no—
181#456—24no—
182#455—24no—
183#451—22no—
184#421—22no—
185#421—21no—
186#416—21no—
187#413C22×C3112yes—
188#410C22×C3111yescharacteristic
189#408C22×C3111yescharacteristic
190#352—24no—
191C2×C31—21no—
192C2×C31—41no—
193C2×C31—42no—
194C2×C31—82no—
195C2×C31#202111yescharacteristic
1962−1+4—41no—
1972−1+4D4×C3112yes—
198H(5,𝔽2)—22no—
199H(5,𝔽2)—21no—
200H(5,𝔽2)—41no—
201C2×#41—22no—
202C2×#41—21no—
203C2×#41—42no—
204C2×#41C23×C3111yescharacteristic
205C22×Q8D4×C3112yes—
206#137—28no—
207#137—44no—
208C2×Q16—21no—
209C2×Q16—21no—
210C2×Q16—42no—
211C2×SD16—42no—
212C2×M16—22no—
213C2×M16D4×C3111yescharacteristic
214C22×C8D4×C3111yescharacteristic
215C4⋊Q8—22no—
216#124—22no—
217#123—22no—
218#123—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221#117—21no—
222#117D4×C3112yes—
223#107—21no—
224#106—21no—
225#104—28no—
226#104—42no—
227#103—22no—
228#103—22no—
229#103—41no—
230#102—41no—
231#101—22no—
232#100D4×C3114yes—
233#99—22no—
234C22⋊C8—22no—
235C31#1336211yescharacteristic, direct product factor, Sylow
236#41—22no—
237#41—24no—
238#41—24no—
239#41—22no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—42no—
244#41—42no—
245#41—44no—
246#41—42no—
247#41—41no—
248C2×Q8—21no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8C2×D4×C3111yescharacteristic
256C2×D4—21no—
257C2×D4—22no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4C2×D4×C3112yes—
263C2×D4#41×C3111yescharacteristic
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4C2×D4×C3112yes—
268C22×C4C2×D4×C3111yescharacteristic
269Q16—48no—
270Q16—81no—
271Q16—81no—
272SD16—48no—
273M16—24no—
274M16—42no—
275M16—41no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—42no—
279C2×C8—41no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C22⋊C4—22no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C42—22no—
289C42—41no—
290C23—42no—
291C23—41no—
292C23C31×#12112yes—
293C23C31×#12711yescharacteristic
294Q8—24no—
295Q8—22no—
296Q8—41no—
297Q8—44no—
298Q8—41no—
299Q8—42no—
300Q8—42no—
301D4—24no—
302D4—22no—
303D4—42no—
304D4—41no—
305D4—44no—
306D4—42no—
307D4—42no—
308D4—41no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4C31×#12012yes—
323C2×C4C31×#12011yescharacteristic
324C2×C4C31×#12111yescharacteristic
325C8—44no—
326C8—41no—
327C8—41no—
328C22—22no—
329C22—21no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—161no—
334C22—162no—
335C22C31×#39211yescharacteristic
336C4—22no—
337C4—21no—
338C4—42no—
339C4—82no—
340C4—82no—
341C4—161no—
342C4C31×B(4,𝔽2)11yescharacteristic
343C4C31×B(4,𝔽2)11yescharacteristic
344C2—21no—
345C2—41no—
346C2—42no—
347C2—82no—
348C2C31×#202111yescharacteristic
subgroupG/H
center, socleC2×C31#2021
commutator, Frattini, cosocleC2×#41C23×C31
prime radicalC31×#585C22
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3202C2semidirect
3. C31×#2123C2semidirect
5. C31×#1392C2semidirect
115. #13362C31direct (N × H)
154. #3202C2×C31semidirect
156. #2123C2×C31semidirect
158. #1392C2×C31semidirect
235. C31#13362direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32022
C31×#30222
C31×#21232
C31×#17942
C31×#13922
#1336231