Finite Group Databasebeta

← C31×#10172 #21466931

Subgroups — C31×#10172 #21466931

Order 7936.

562 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3227C211yesmaximal, characteristic
2C2×C31×#498C211yesmaximal, characteristic, complemented
3C31×#3152C211yesmaximal, characteristic
4C31×#1997C211yesmaximal, 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×#498—22no—
13C31×#498—21no—
14C31×#498—22no—
15C31×#497—22no—
16C31×#494—21no—
17C31×#492—21no—
18C31×#486—21no—
19C31×#481—22no—
20C31×#476—22no—
21C31×#431C2211yescharacteristic
22C2×C31×#106—22no—
23C31×#423C2211yescharacteristic
24C4×M16×C31C2211yescharacteristic
25C2×C4×C8×C31C2211yescharacteristic
26C31×#343—22no—
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×#106—28no—
60C31×#106—24no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8D412yes—
64C31×#102—24no—
65C31×C22⋊C8—21no—
66C31×#97—21no—
67C4×C8×C31—21no—
68C4×C8×C31—21no—
69C4×C8×C31—21no—
70C4×C8×C31D412yes—
71#41×C31—42no—
72#41×C31—42no—
73C2×Q8×C31—21no—
74C2×Q8×C31—22no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—42no—
79C2×Q8×C31—42no—
80C2×Q8×C31—82no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—41no—
85C22×C4×C31—22no—
86C22×C4×C31—41no—
87C22×C4×C31—41no—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31#4111yescharacteristic
91Q16×C31—48no—
92SD16×C31—48no—
93M16×C31—22no—
94M16×C31—42no—
95M16×C31SD1614yes—
96C2×C8×C31—24no—
97C2×C8×C31—22no—
98C2×C8×C31—22no—
99C2×C8×C31—22no—
100C2×C8×C31—41no—
101C4⋊C4×C31—22no—
102C4⋊C4×C31—22no—
103C4⋊C4×C31—22no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—41no—
108C4⋊C4×C31—41no—
109C4⋊C4×C31—41no—
110C4⋊C4×C31—41no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C42×C31C2×D411yescharacteristic
114C42×C31C2×D411yescharacteristic
115C42×C31C2×D411yescharacteristic
116C42×C31C2×D411yescharacteristic
117#10172C3111yesmaximal, characteristic, direct product factor, Sylow
118C23×C31—41no—
119C23×C31#12111yescharacteristic
120Q8×C31—44no—
121Q8×C31—42no—
122Q8×C31—82no—
123Q8×C31—82no—
124Q8×C31—81no—
125Q8×C31—81no—
126D4×C31—42no—
127D4×C31—42no—
128D4×C31—42no—
129D4×C31—82no—
130C2×C4×C31—22no—
131C2×C4×C31—22no—
132C2×C4×C31—41no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31—82no—
136C2×C4×C31—81no—
137C2×C4×C31—81no—
138C2×C4×C31—82no—
139C2×C4×C31#12011yescharacteristic
140C2×C4×C31#12011yescharacteristic
141C2×C4×C31#12011yescharacteristic
142C2×C4×C31#12111yescharacteristic
143C2×C4×C31#12111yescharacteristic
144C2×C4×C31#12711yescharacteristic
145C2×C4×C31C2×SD1612yes—
146C2×C4×C31AGL(1,ℤ/8)12yes—
147C8×C31—24no—
148C8×C31—24no—
149C8×C31—42no—
150C8×C31—81no—
151#3227C2×C3111yescharacteristic
152C2×#498C2×C3111yescharacteristic, complemented
153#3152C2×C3111yescharacteristic
154#1997C2×C3111yescharacteristic, complemented
155#1546C2×C3111yescharacteristic, complemented
156#1492C2×C3111yescharacteristic
157#1278C2×C3111yescharacteristic, complemented
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—81no—
161C22×C31—81no—
162C22×C31—81no—
163C22×C31#39211yescharacteristic
164C22×C31#46012yes—
165C4×C31—21no—
166C4×C31—21no—
167C4×C31—81no—
168C4×C31—82no—
169C4×C31—82no—
170C4×C31#45012yes—
171C4×C31#45612yes—
172C4×C31#47112yes—
173C4×C31#49211yescharacteristic
174C4×C31#49611yescharacteristic
175C2×#137—22no—
176#532C22×C3111yescharacteristic
177C2×C4⋊Q8C22×C3111yescharacteristic
178C2×C4⋊Q8C22×C3111yescharacteristic
179#498—22no—
180#498—21no—
181#498—22no—
182#497—22no—
183#494—21no—
184#492—21no—
185#486—21no—
186#481—22no—
187#476—22no—
188#431C22×C3111yescharacteristic
189C2×#106—22no—
190#423C22×C3111yescharacteristic
191C4×M16C22×C3111yescharacteristic
192C2×C4×C8C22×C3111yescharacteristic
193#343—22no—
194#332—22no—
195D4⋊C8—22no—
196C2×C31—22no—
197C2×C31—161no—
198C2×C31#200511yescharacteristic
199C2×C31#201411yescharacteristic
200C2×C31#201911yescharacteristic
201C2×#41—21no—
202C22×Q8—21no—
203C22×Q8—21no—
204#137—44no—
205#137—44no—
206C2×Q16—24no—
207C2×SD16—24no—
208C2×M16—21no—
209C2×M16D4×C3112yes—
210C22×C8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8D4×C3112yes—
219C4⋊Q8D4×C3111yescharacteristic
220#127D4×C3111yescharacteristic
221#124—21no—
222#121—21no—
223#121—21no—
224C4×D4—21no—
225C4×D4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C3112yes—
230C2×C42C23×C3111yescharacteristic
231#106—28no—
232#106—24no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8D4×C3112yes—
236#102—24no—
237C22⋊C8—21no—
238#97—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8D4×C3112yes—
243C31#1017211yescharacteristic, direct product factor, Sylow
244#41—42no—
245#41—42no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—82no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4C2×D4×C3111yescharacteristic
262C22×C4C2×D4×C3111yescharacteristic
263C22×C4#41×C3111yescharacteristic
264Q16—48no—
265SD16—48no—
266M16—22no—
267M16—42no—
268M16SD16×C3114yes—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C42C2×D4×C3111yescharacteristic
287C42C2×D4×C3111yescharacteristic
288C42C2×D4×C3111yescharacteristic
289C42C2×D4×C3111yescharacteristic
290C23—41no—
291C23C31×#12111yescharacteristic
292Q8—44no—
293Q8—42no—
294Q8—82no—
295Q8—82no—
296Q8—81no—
297Q8—81no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4C31×#12011yescharacteristic
312C2×C4C31×#12011yescharacteristic
313C2×C4C31×#12011yescharacteristic
314C2×C4C31×#12111yescharacteristic
315C2×C4C31×#12111yescharacteristic
316C2×C4C31×#12711yescharacteristic
317C2×C4C2×SD16×C3112yes—
318C2×C4C31×AGL(1,ℤ/8)12yes—
319C8—24no—
320C8—24no—
321C8—42no—
322C8—81no—
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C31×#39211yescharacteristic
329C22C31×#46012yes—
330C4—21no—
331C4—21no—
332C4—81no—
333C4—82no—
334C4—82no—
335C4C31×#45012yes—
336C4C31×#45612yes—
337C4C31×#47112yes—
338C4C31×#49211yescharacteristic
339C4C31×#49611yescharacteristic
340C2—22no—
341C2—161no—
342C2C31×#200511yescharacteristic
343C2C31×#201411yescharacteristic
344C2C31×#201911yescharacteristic
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×#498C2semidirect
4. C31×#1997C2semidirect
5. C31×#1546C2semidirect
7. C31×#1278C2semidirect
117. #10172C31direct (N × H)
152. C2×#498C2×C31semidirect
154. #1997C2×C31semidirect
155. #1546C2×C31semidirect
157. #1278C2×C31semidirect
243. C31#10172direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32272
C2×C31×#4982
C31×#31522
C31×#19972
C31×#15462
C31×#14922
C31×#12782
#1017231