Finite Group Databasebeta

← C31×#7804 #21464563

Subgroups — C31×#7804 #21464563

Order 7936.

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

normal subgroups only

This group has 422 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
1C31×#2804C211yesmaximal, characteristic
2C31×#2096C211yesmaximal, characteristic, complemented
3C31×#2072C211yesmaximal, characteristic, complemented
4C31×#2071C211yesmaximal, characteristic
5C31×#1272C211yesmaximal, characteristic, complemented
6C31×#1272C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic
8C2×C31×#106—21no—
9C2×C31×#103—21no—
10C2×C31×#103—21no—
11C2×C31×#102—21no—
12C31×#400—21no—
13C31×#393—21no—
14C31×#393—21no—
15C31×#391C2211yescharacteristic
16C31×#391C2211yescharacteristic
17C31×#390C2211yescharacteristic
18C31×#386—21no—
19C31×#385—21no—
20C31×#385—21no—
21C31×#384C2211yescharacteristic
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31D411yescharacteristic
30C23×C4×C31—21no—
31C22×C8×C31—21no—
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31D411yescharacteristic
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C42×C31C2311yescharacteristic
47C31×#106—42no—
48C31×#103—42no—
49C31×#103—42no—
50C31×#102—42no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C24×C31—21no—
59C2×Q8×C31—21no—
60C2×Q8×C31—21no—
61C2×Q8×C31—21no—
62C2×Q8×C31—21no—
63C2×Q8×C31—41no—
64C2×Q8×C31—41no—
65C2×Q8×C31—41no—
66C2×Q8×C31—41no—
67C2×D4×C31—21no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31SD1612yes—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—41no—
77C22×C4×C31—81no—
78C22×C4×C31—81no—
79C22×C4×C31—81no—
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31C2×D411yescharacteristic
82C22×C4×C31C2×D411yescharacteristic
83C2×C8×C31—22no—
84C2×C8×C31—41no—
85C2×C8×C31—41no—
86C2×C8×C31—41no—
87C2×C8×C31—41no—
88C2×C8×C31—41no—
89C2×C8×C31—41no—
90C2×C8×C31—41no—
91C2×C8×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—81no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—81no—
107C22⋊C4×C31—81no—
108C22⋊C4×C31—81no—
109C42×C31—41no—
110#7804C3111yesmaximal, characteristic, direct product factor, Sylow
111C23×C31—22no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31#12011yescharacteristic
119Q8×C31—42no—
120Q8×C31—42no—
121Q8×C31—81no—
122Q8×C31—81no—
123D4×C31—42no—
124D4×C31—81no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—84no—
147C2×C4×C31—81no—
148C2×C4×C31—81no—
149C2×C4×C31—81no—
150C2×C4×C31C2×SD1611yescharacteristic
151C2×C4×C31#13511yescharacteristic
152C2×C4×C31#13511yescharacteristic
153C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C31#13711yescharacteristic
155C2×C4×C31#13711yescharacteristic
156C8×C31—42no—
157C8×C31—81no—
158C8×C31—81no—
159#2804C2×C3111yescharacteristic
160#2096C2×C3111yescharacteristic, complemented
161#2072C2×C3111yescharacteristic, complemented
162#2071C2×C3111yescharacteristic
163#1272C2×C3111yescharacteristic, complemented
164#1272C2×C3111yescharacteristic, complemented
165#1270C2×C3111yescharacteristic
166C22×C31—42no—
167C22×C31—42no—
168C22×C31—42no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31—81no—
172C22×C31—81no—
173C22×C31#44811yescharacteristic
174C22×C31#44911yescharacteristic
175C22×C31#45211yescharacteristic
176C22×C31#45311yescharacteristic
177C22×C31#45511yescharacteristic
178C22×C31#45511yescharacteristic
179C22×C31#45811yescharacteristic
180C4×C31—21no—
181C4×C31—21no—
182C4×C31—21no—
183C4×C31—21no—
184C4×C31—21no—
185C4×C31—21no—
186C4×C31—81no—
187C4×C31—81no—
188C4×C31—81no—
189C4×C31—81no—
190C4×C31—81no—
191C4×C31—161no—
192C2×#106—21no—
193C2×#103—21no—
194C2×#103—21no—
195C2×#102—21no—
196#400—21no—
197#393—21no—
198#393—21no—
199#391C22×C3111yescharacteristic
200#391C22×C3111yescharacteristic
201#390C22×C3111yescharacteristic
202#386—21no—
203#385—21no—
204#385—21no—
205#384C22×C3111yescharacteristic
206#340—21no—
207#340—21no—
208#336C22×C3111yescharacteristic
209#336C22×C3111yescharacteristic
210#336C22×C3111yescharacteristic
211C2×C31—82no—
212C2×C31#160811yescharacteristic
213C2×C31#161211yescharacteristic
214C2×C31#161811yescharacteristic
215C2×C31#162211yescharacteristic
216C2×C31#164011yescharacteristic
217C2×C31#164311yescharacteristic
218C2×C31#165911yescharacteristic
219C22×Q8D4×C3111yescharacteristic
220C22×Q8D4×C3111yescharacteristic
221C22×D4D4×C3111yescharacteristic
222C23×C4—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4D4×C3111yescharacteristic
231C2×C4⋊C4D4×C3111yescharacteristic
232C2×C4⋊C4D4×C3111yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42C23×C3111yescharacteristic
239#106—42no—
240#103—42no—
241#103—42no—
242#102—42no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250C31#780411yescharacteristic, direct product factor, Sylow
251C24—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4SD16×C3112yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4C2×D4×C3111yescharacteristic
274C22×C4C2×D4×C3111yescharacteristic
275C22×C4C2×D4×C3111yescharacteristic
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—41no—
303C23—22no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C31×#12011yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—81no—
314Q8—81no—
315D4—42no—
316D4—81no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—84no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4C2×SD16×C3111yescharacteristic
343C2×C4C31×#13511yescharacteristic
344C2×C4C31×#13511yescharacteristic
345C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C31×#13711yescharacteristic
347C2×C4C31×#13711yescharacteristic
348C8—42no—
349C8—81no—
350C8—81no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22C31×#44811yescharacteristic
359C22C31×#44911yescharacteristic
360C22C31×#45211yescharacteristic
361C22C31×#45311yescharacteristic
362C22C31×#45511yescharacteristic
363C22C31×#45511yescharacteristic
364C22C31×#45811yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—161no—
377C2—82no—
378C2C31×#160811yescharacteristic
379C2C31×#161211yescharacteristic
380C2C31×#161811yescharacteristic
381C2C31×#162211yescharacteristic
382C2C31×#164011yescharacteristic
383C2C31×#164311yescharacteristic
384C2C31×#165911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2096C2semidirect
3. C31×#2072C2semidirect
5. C31×#1272C2semidirect
6. C31×#1272C2semidirect
110. #7804C31direct (N × H)
160. #2096C2×C31semidirect
161. #2072C2×C31semidirect
163. #1272C2×C31semidirect
164. #1272C2×C31semidirect
250. C31#7804direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28042
C31×#20962
C31×#20722
C31×#20712
C31×#12722
C31×#12702
#780431