Finite Group Databasebeta

← C31×#12179 #21468938

Subgroups — C31×#12179 #21468938

Order 7936.

412 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2807C211yesmaximal, characteristic, complemented
2C31×#2044C211yesmaximal, characteristic
3C31×#1613C211yesmaximal, characteristic
4C31×#1613C211yesmaximal, characteristic
5C31×#1352C211yesmaximal, characteristic, complemented
6C31×#1352C211yesmaximal, characteristic, complemented
7C31×#1326C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C31×#481—21no—
10C31×#481—21no—
11C2×C31×#99—21no—
12C31×#400—21no—
13C31×#399C2211yescharacteristic
14C31×#399C2211yescharacteristic
15C31×#398—21no—
16C31×#389—21no—
17C31×#389—21no—
18C31×#388—21no—
19C31×#386—21no—
20C31×#385—21no—
21C31×#385C2211yescharacteristic
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×D4×C31—21no—
28C23×C4×C31D411yescharacteristic
29C31×#123—41no—
30C31×#121—21no—
31C31×#121—21no—
32C31×#120—41no—
33C2×C4⋊C4×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×C31—21no—
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—41no—
45C2×C22⋊C4×C31D411yescharacteristic
46C2×C22⋊C4×C31C2311yescharacteristic
47C2×C42×C31—21no—
48C31×#106—41no—
49C31×#106—41no—
50C31×#102—41no—
51C31×#102—41no—
52C31×#99—42no—
53C31×C22⋊C8—21no—
54C31×C22⋊C8—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—41no—
61C31×#95D411yescharacteristic
62C31×#95D411yescharacteristic
63C24×C31—21no—
64C24×C31C2×D411yescharacteristic
65C2×D4×C31—21no—
66C2×D4×C31—21no—
67C2×D4×C31—41no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31—41no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31—81no—
83C22×C4×C31C2×D411yescharacteristic
84C22×C4×C31C2×D411yescharacteristic
85C2×C8×C31—41no—
86C2×C8×C31—41no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—21no—
89C4⋊C4×C31—42no—
90C4⋊C4×C31—42no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—81no—
94C4⋊C4×C31—81no—
95C4⋊C4×C31—81no—
96C4⋊C4×C31—81no—
97C22⋊C4×C31—21no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—42no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—81no—
105C22⋊C4×C31—82no—
106C22⋊C4×C31SD1612yes—
107C42×C31—81no—
108#12179C3111yesmaximal, characteristic, direct product factor, Sylow
109C23×C31—21no—
110C23×C31—21no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—81no—
117C23×C31—81no—
118C23×C31#12011yescharacteristic
119C23×C31C2×SD1611yescharacteristic
120C23×C31#13711yescharacteristic
121D4×C31—81no—
122D4×C31—81no—
123D4×C31—81no—
124D4×C31—81no—
125D4×C31—81no—
126D4×C31—81no—
127C2×C4×C31—22no—
128C2×C4×C31—22no—
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—41no—
147C2×C4×C31—42no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—81no—
154C2×C4×C31—81no—
155C2×C4×C31—81no—
156C2×C4×C31—81no—
157C8×C31—81no—
158C8×C31—81no—
159#2807C2×C3111yescharacteristic, complemented
160#2044C2×C3111yescharacteristic
161#1613C2×C3111yescharacteristic
162#1613C2×C3111yescharacteristic
163#1352C2×C3111yescharacteristic, complemented
164#1352C2×C3111yescharacteristic, complemented
165#1326C2×C3111yescharacteristic, complemented
166C22×C31—21no—
167C22×C31—21no—
168C22×C31—21no—
169C22×C31—21no—
170C22×C31—41no—
171C22×C31—41no—
172C22×C31—41no—
173C22×C31—41no—
174C22×C31—81no—
175C22×C31—82no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—161no—
180C22×C31#44811yescharacteristic
181C22×C31#45411yescharacteristic
182C22×C31B(4,𝔽2)11yescharacteristic
183C4×C31—42no—
184C4×C31—42no—
185C4×C31—41no—
186C4×C31—41no—
187C4×C31—81no—
188C4×C31—81no—
189C4×C31—81no—
190C4×C31—81no—
191C4×C31—81no—
192C4×C31—161no—
193#534C22×C3111yescharacteristic
194#481—21no—
195#481—21no—
196C2×#99—21no—
197#400—21no—
198#399C22×C3111yescharacteristic
199#399C22×C3111yescharacteristic
200#398—21no—
201#389—21no—
202#389—21no—
203#388—21no—
204#386—21no—
205#385—21no—
206#385C22×C3111yescharacteristic
207#342C22×C3111yescharacteristic
208#327—21no—
209#327—21no—
210#323C22×C3111yescharacteristic
211#323C22×C3111yescharacteristic
212C2×C31—21no—
213C2×C31—21no—
214C2×C31—41no—
215C2×C31—41no—
216C2×C31—161no—
217C2×C31#159911yescharacteristic
218C2×C31#219811yescharacteristic
219C2×C31#220011yescharacteristic
220C22×D4—21no—
221C23×C4D4×C3111yescharacteristic
222#123—41no—
223#121—21no—
224#121—21no—
225#120—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4D4×C3111yescharacteristic
233C2×C4⋊C4D4×C3111yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4D4×C3111yescharacteristic
239C2×C22⋊C4C23×C3111yescharacteristic
240C2×C42—21no—
241#106—41no—
242#106—41no—
243#102—41no—
244#102—41no—
245#99—42no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—41no—
254#95D4×C3111yescharacteristic
255#95D4×C3111yescharacteristic
256C31#1217911yescharacteristic, direct product factor, Sylow
257C24—21no—
258C24C2×D4×C3111yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—81no—
277C22×C4C2×D4×C3111yescharacteristic
278C22×C4C2×D4×C3111yescharacteristic
279C2×C8—41no—
280C2×C8—41no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C22⋊C4—21no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—82no—
300C22⋊C4SD16×C3112yes—
301C42—81no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—81no—
310C23—81no—
311C23C31×#12011yescharacteristic
312C23C2×SD16×C3111yescharacteristic
313C23C31×#13711yescharacteristic
314D4—81no—
315D4—81no—
316D4—81no—
317D4—81no—
318D4—81no—
319D4—81no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
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—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C8—81no—
351C8—81no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—82no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—161no—
366C22C31×#44811yescharacteristic
367C22C31×#45411yescharacteristic
368C22C31×B(4,𝔽2)11yescharacteristic
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—161no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2C31×#159911yescharacteristic
385C2C31×#219811yescharacteristic
386C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2807C2semidirect
5. C31×#1352C2semidirect
6. C31×#1352C2semidirect
7. C31×#1326C2semidirect
108. #12179C31direct (N × H)
159. #2807C2×C31semidirect
163. #1352C2×C31semidirect
164. #1352C2×C31semidirect
165. #1326C2×C31semidirect
256. C31#12179direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28072
C31×#20442
C31×#16132
C31×#13522
C31×#13262
#1217931