Finite Group Databasebeta

← C31×#10010 #21466769

Subgroups — C31×#10010 #21466769

Order 7936.

534 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3215C211yesmaximal, characteristic
2C31×#3113C211yesmaximal, characteristic, complemented
3C31×#2592C211yesmaximal, characteristic
4C31×#1986C211yesmaximal, characteristic, complemented
5C31×#1527C211yesmaximal, characteristic
6C31×#1491C211yesmaximal, characteristic, complemented
7C31×#1285C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C2×C4×Q8×C31C2211yescharacteristic
11C31×#497—21no—
12C31×#489—21no—
13C31×#483—21no—
14C31×#475—21no—
15C31×#464—22no—
16C31×#462—21no—
17C31×#451—21no—
18C31×#451—21no—
19C31×#450—21no—
20C31×#441—22no—
21C31×#432C2211yescharacteristic
22C31×#432C2211yescharacteristic
23C4×M16×C31C2211yescharacteristic
24C31×#397—21no—
25C31×#393—21no—
26C31×#391—21no—
27C31×#390—21no—
28C31×#388—22no—
29C4×C4⋊C4×C31C2211yescharacteristic
30C31×#339—22no—
31C31×#329—22no—
32C31×Q8⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C22×Q8×C31D411yescharacteristic
35C22×D4×C31D411yescharacteristic
36C2×Q16×C31—22no—
37C2×Q16×C31—22no—
38C2×SD16×C31—22no—
39C2×M16×C31—21no—
40C31×C4⋊Q8D411yescharacteristic
41C31×C4⋊Q8D411yescharacteristic
42C31×#125—21no—
43C31×#124—21no—
44C31×#124—21no—
45C31×#124—21no—
46C31×#124—21no—
47C31×C22⋊Q8—21no—
48C31×C22⋊Q8—21no—
49C4×Q8×C31—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×Q8×C31—21no—
54C4×Q8×C31—21no—
55C31×#117—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—22no—
63C2×C42×C31—21no—
64C2×C42×C31D412yes—
65C2×C42×C31C2311yescharacteristic
66C31×#106—22no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8D412yes—
69C31×#103—22no—
70C31×#103—22no—
71C31×#103—22no—
72C31×#102—22no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×#97—21no—
76C31×#97—21no—
77C31×#97D412yes—
78C4×C8×C31—21no—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—21no—
82C24×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—21no—
89C2×Q8×C31—42no—
90C2×Q8×C31—42no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—42no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—22no—
97C22×C4×C31—21no—
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101Q16×C31—44no—
102Q16×C31—82no—
103SD16×C31—82no—
104M16×C31—22no—
105M16×C31—41no—
106C2×C8×C31—22no—
107C2×C8×C31—22no—
108C2×C8×C31—41no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—42no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C22⋊C4×C31—44no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—82no—
127C42×C31—42no—
128C42×C31—41no—
129C42×C31—41no—
130C42×C31—41no—
131C42×C31C2×D411yescharacteristic
132C42×C31C2×D411yescharacteristic
133C42×C31C2×D411yescharacteristic
134C42×C31#4111yescharacteristic
135#10010C3111yesmaximal, characteristic, direct product factor, Sylow
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31#12011yescharacteristic
142Q8×C31—22no—
143Q8×C31—22no—
144Q8×C31—41no—
145Q8×C31—41no—
146Q8×C31—42no—
147Q8×C31—42no—
148Q8×C31—42no—
149Q8×C31—81no—
150D4×C31—42no—
151D4×C31—81no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—22no—
155C2×C4×C31—22no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—42no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31#12011yescharacteristic
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12111yescharacteristic
172C2×C4×C31#12111yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12711yescharacteristic
175C8×C31—42no—
176C8×C31—42no—
177C8×C31—42no—
178#3215C2×C3111yescharacteristic
179#3113C2×C3111yescharacteristic, complemented
180#2592C2×C3111yescharacteristic
181#1986C2×C3111yescharacteristic, complemented
182#1527C2×C3111yescharacteristic
183#1491C2×C3111yescharacteristic, complemented
184#1285C2×C3111yescharacteristic, complemented
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—82no—
188C22×C31—82no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31#39211yescharacteristic
193C22×C31#45511yescharacteristic
194C22×C31#45611yescharacteristic
195C4×C31—21no—
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—21no—
200C4×C31—42no—
201C4×C31—82no—
202C4×C31—81no—
203C4×C31—81no—
204C4×C31—81no—
205C4×C31—81no—
206C4×C31#45512yes—
207#533C22×C3111yescharacteristic
208C2×#124C22×C3111yescharacteristic
209C2×C4×Q8C22×C3111yescharacteristic
210#497—21no—
211#489—21no—
212#483—21no—
213#475—21no—
214#464—22no—
215#462—21no—
216#451—21no—
217#451—21no—
218#450—21no—
219#441—22no—
220#432C22×C3111yescharacteristic
221#432C22×C3111yescharacteristic
222C4×M16C22×C3111yescharacteristic
223#397—21no—
224#393—21no—
225#391—21no—
226#390—21no—
227#388—22no—
228C4×C4⋊C4C22×C3111yescharacteristic
229#339—22no—
230#329—22no—
231Q8⋊C8—22no—
232C2×C31—21no—
233C2×C31—21no—
234C2×C31—161no—
235C2×C31#200411yescharacteristic
236C2×C31#200711yescharacteristic
237C2×C31#201011yescharacteristic
238C22×Q8D4×C3111yescharacteristic
239C22×Q8D4×C3111yescharacteristic
240C22×D4D4×C3111yescharacteristic
241C2×Q16—22no—
242C2×Q16—22no—
243C2×SD16—22no—
244C2×M16—21no—
245C4⋊Q8D4×C3111yescharacteristic
246C4⋊Q8D4×C3111yescharacteristic
247#125—21no—
248#124—21no—
249#124—21no—
250#124—21no—
251#124—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260#117—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4D4×C3111yescharacteristic
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C42—21no—
269C2×C42D4×C3112yes—
270C2×C42C23×C3111yescharacteristic
271#106—22no—
272C4⋊C8—21no—
273C4⋊C8D4×C3112yes—
274#103—22no—
275#103—22no—
276#103—22no—
277#102—22no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280#97—21no—
281#97—21no—
282#97D4×C3112yes—
283C4×C8—21no—
284#95—22no—
285#95—22no—
286#95—21no—
287C31#1001011yescharacteristic, direct product factor, Sylow
288C24—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4C2×D4×C3111yescharacteristic
305C22×C4C2×D4×C3111yescharacteristic
306C22×C4C2×D4×C3111yescharacteristic
307Q16—44no—
308Q16—82no—
309SD16—82no—
310M16—22no—
311M16—41no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—41no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—44no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—82no—
333C42—42no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42C2×D4×C3111yescharacteristic
338C42C2×D4×C3111yescharacteristic
339C42C2×D4×C3111yescharacteristic
340C42#41×C3111yescharacteristic
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C31×#12011yescharacteristic
347Q8—22no—
348Q8—22no—
349Q8—41no—
350Q8—41no—
351Q8—42no—
352Q8—42no—
353Q8—42no—
354Q8—81no—
355D4—42no—
356D4—81no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4C31×#12011yescharacteristic
375C2×C4C31×#12011yescharacteristic
376C2×C4C31×#12111yescharacteristic
377C2×C4C31×#12111yescharacteristic
378C2×C4C31×#12111yescharacteristic
379C2×C4C31×#12711yescharacteristic
380C8—42no—
381C8—42no—
382C8—42no—
383C22—21no—
384C22—21no—
385C22—82no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C31×#39211yescharacteristic
391C22C31×#45511yescharacteristic
392C22C31×#45611yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4C31×#45512yes—
405C2—21no—
406C2—21no—
407C2—161no—
408C2C31×#200411yescharacteristic
409C2C31×#200711yescharacteristic
410C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
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. C31×#3113C2semidirect
4. C31×#1986C2semidirect
6. C31×#1491C2semidirect
7. C31×#1285C2semidirect
135. #10010C31direct (N × H)
179. #3113C2×C31semidirect
181. #1986C2×C31semidirect
183. #1491C2×C31semidirect
184. #1285C2×C31semidirect
287. C31#10010direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32152
C31×#31132
C31×#25922
C31×#19862
C31×#15272
C31×#14912
C31×#12852
#1001031