Finite Group Databasebeta

← C31×#10174 #21466933

Subgroups — C31×#10174 #21466933

Order 7936.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3226C211yesmaximal, characteristic
2C2×C31×#500C211yesmaximal, characteristic, complemented
3C31×#3152C211yesmaximal, characteristic
4C31×#1966C211yesmaximal, characteristic, complemented
5C31×#1546C211yesmaximal, characteristic, complemented
6C31×#1492C211yesmaximal, characteristic
7C31×#1278C211yesmaximal, characteristic, complemented
8C2×C31×#137—22no—
9C31×#532C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C31×C4⋊Q8C2211yescharacteristic
12C31×#500—22no—
13C31×#500—21no—
14C31×#500—22no—
15C31×#497—22no—
16C31×#494—21no—
17C31×#492—21no—
18C31×#486—21no—
19C31×#480—22no—
20C31×#474—22no—
21C31×#431C2211yescharacteristic
22C2×C31×#107—22no—
23C31×#423C2211yescharacteristic
24C2×C31×#101—22no—
25C4×M16×C31C2211yescharacteristic
26C2×C4×C8×C31C2211yescharacteristic
27C31×#332—22no—
28C31×D4⋊C8—22no—
29C2×#41×C31—21no—
30C22×Q8×C31—21no—
31C22×Q8×C31—21no—
32C31×#137—44no—
33C31×#137—44no—
34C2×Q16×C31—24no—
35C2×SD16×C31—24no—
36C2×M16×C31—21no—
37C2×M16×C31D412yes—
38C22×C8×C31—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8D411yescharacteristic
47C31×C4⋊Q8D412yes—
48C31×#127D411yescharacteristic
49C31×#124—21no—
50C31×#121—21no—
51C31×#121—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31D412yes—
58C2×C42×C31C2311yescharacteristic
59C31×#107—28no—
60C31×#107—24no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8D412yes—
64C31×#102—24no—
65C31×#101—44no—
66C31×C22⋊C8—21no—
67C31×#97—21no—
68C4×C8×C31—21no—
69C4×C8×C31—21no—
70C4×C8×C31—21no—
71C4×C8×C31D412yes—
72#41×C31—42no—
73#41×C31—42no—
74C2×Q8×C31—21no—
75C2×Q8×C31—22no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—41no—
79C2×Q8×C31—42no—
80C2×Q8×C31—42no—
81C2×Q8×C31—42no—
82C2×Q8×C31—42no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—21no—
86C2×D4×C31—41no—
87C22×C4×C31—22no—
88C22×C4×C31—41no—
89C22×C4×C31—41no—
90C22×C4×C31C2×D411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31#4111yescharacteristic
93Q16×C31—48no—
94SD16×C31—48no—
95M16×C31—22no—
96M16×C31—42no—
97M16×C31D814yes—
98C2×C8×C31—24no—
99C2×C8×C31—22no—
100C2×C8×C31—22no—
101C2×C8×C31—22no—
102C2×C8×C31—41no—
103C4⋊C4×C31—22no—
104C4⋊C4×C31—22no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—41no—
110C4⋊C4×C31—41no—
111C4⋊C4×C31—41no—
112C4⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C42×C31C2×D411yescharacteristic
116C42×C31C2×D411yescharacteristic
117C42×C31C2×D411yescharacteristic
118C42×C31C2×D411yescharacteristic
119#10174C3111yesmaximal, characteristic, direct product factor, Sylow
120C23×C31—41no—
121C23×C31#12111yescharacteristic
122Q8×C31—44no—
123Q8×C31—42no—
124Q8×C31—82no—
125Q8×C31—81no—
126Q8×C31—81no—
127Q8×C31—81no—
128Q8×C31—81no—
129D4×C31—42no—
130D4×C31—42no—
131D4×C31—42no—
132D4×C31—82no—
133C2×C4×C31—22no—
134C2×C4×C31—22no—
135C2×C4×C31—41no—
136C2×C4×C31—42no—
137C2×C4×C31—42no—
138C2×C4×C31—44no—
139C2×C4×C31—82no—
140C2×C4×C31—81no—
141C2×C4×C31—81no—
142C2×C4×C31#12011yescharacteristic
143C2×C4×C31#12011yescharacteristic
144C2×C4×C31#12011yescharacteristic
145C2×C4×C31#12111yescharacteristic
146C2×C4×C31#12111yescharacteristic
147C2×C4×C31#12711yescharacteristic
148C2×C4×C31C2×D812yes—
149C2×C4×C31AGL(1,ℤ/8)12yes—
150C8×C31—24no—
151C8×C31—24no—
152C8×C31—42no—
153C8×C31—81no—
154#3226C2×C3111yescharacteristic
155C2×#500C2×C3111yescharacteristic, complemented
156#3152C2×C3111yescharacteristic
157#1966C2×C3111yescharacteristic, complemented
158#1546C2×C3111yescharacteristic, complemented
159#1492C2×C3111yescharacteristic
160#1278C2×C3111yescharacteristic, complemented
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31—81no—
166C22×C31#39211yescharacteristic
167C22×C31#45912yes—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—82no—
171C4×C31—81no—
172C4×C31—82no—
173C4×C31#44712yes—
174C4×C31#45612yes—
175C4×C31#47312yes—
176C4×C31#49311yescharacteristic
177C4×C31#49611yescharacteristic
178C2×#137—22no—
179#532C22×C3111yescharacteristic
180C2×C4⋊Q8C22×C3111yescharacteristic
181C2×C4⋊Q8C22×C3111yescharacteristic
182#500—22no—
183#500—21no—
184#500—22no—
185#497—22no—
186#494—21no—
187#492—21no—
188#486—21no—
189#480—22no—
190#474—22no—
191#431C22×C3111yescharacteristic
192C2×#107—22no—
193#423C22×C3111yescharacteristic
194C2×#101—22no—
195C4×M16C22×C3111yescharacteristic
196C2×C4×C8C22×C3111yescharacteristic
197#332—22no—
198D4⋊C8—22no—
199C2×C31—22no—
200C2×C31—161no—
201C2×C31#200511yescharacteristic
202C2×C31#201111yescharacteristic
203C2×C31#202011yescharacteristic
204C2×#41—21no—
205C22×Q8—21no—
206C22×Q8—21no—
207#137—44no—
208#137—44no—
209C2×Q16—24no—
210C2×SD16—24no—
211C2×M16—21no—
212C2×M16D4×C3112yes—
213C22×C8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8D4×C3112yes—
222C4⋊Q8D4×C3111yescharacteristic
223#127D4×C3111yescharacteristic
224#124—21no—
225#121—21no—
226#121—21no—
227C4×D4—21no—
228C4×D4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4D4×C3112yes—
233C2×C42C23×C3111yescharacteristic
234#107—28no—
235#107—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8D4×C3112yes—
239#102—24no—
240#101—44no—
241C22⋊C8—21no—
242#97—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8D4×C3112yes—
247C31#1017411yescharacteristic, direct product factor, Sylow
248#41—42no—
249#41—42no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×D4×C3111yescharacteristic
267C22×C4C2×D4×C3111yescharacteristic
268C22×C4#41×C3111yescharacteristic
269Q16—48no—
270SD16—48no—
271M16—22no—
272M16—42no—
273M16D8×C3114yes—
274C2×C8—24no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42C2×D4×C3111yescharacteristic
292C42C2×D4×C3111yescharacteristic
293C42C2×D4×C3111yescharacteristic
294C42C2×D4×C3111yescharacteristic
295C23—41no—
296C23C31×#12111yescharacteristic
297Q8—44no—
298Q8—42no—
299Q8—82no—
300Q8—81no—
301Q8—81no—
302Q8—81no—
303Q8—81no—
304D4—42no—
305D4—42no—
306D4—42no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4C31×#12011yescharacteristic
318C2×C4C31×#12011yescharacteristic
319C2×C4C31×#12011yescharacteristic
320C2×C4C31×#12111yescharacteristic
321C2×C4C31×#12111yescharacteristic
322C2×C4C31×#12711yescharacteristic
323C2×C4C2×D8×C3112yes—
324C2×C4C31×AGL(1,ℤ/8)12yes—
325C8—24no—
326C8—24no—
327C8—42no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C31×#39211yescharacteristic
335C22C31×#45912yes—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—81no—
340C4—82no—
341C4C31×#44712yes—
342C4C31×#45612yes—
343C4C31×#47312yes—
344C4C31×#49311yescharacteristic
345C4C31×#49611yescharacteristic
346C2—22no—
347C2—161no—
348C2C31×#200511yescharacteristic
349C2C31×#201111yescharacteristic
350C2C31×#202011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#500C2semidirect
4. C31×#1966C2semidirect
5. C31×#1546C2semidirect
7. C31×#1278C2semidirect
119. #10174C31direct (N × H)
155. C2×#500C2×C31semidirect
157. #1966C2×C31semidirect
158. #1546C2×C31semidirect
160. #1278C2×C31semidirect
247. C31#10174direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32262
C2×C31×#5002
C31×#31522
C31×#19662
C31×#15462
C31×#14922
C31×#12782
#1017431