Finite Group Databasebeta

← C31×#12253 #21469012

Subgroups — C31×#12253 #21469012

Order 7936.

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

normal subgroups only

This group has 398 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×#2763C211yesmaximal, characteristic
2C31×#2062C211yesmaximal, characteristic
3C31×#1602C211yesmaximal, characteristic
4C31×#1602C211yesmaximal, characteristic
5C31×#1351C211yesmaximal, characteristic
6C31×#1351C211yesmaximal, characteristic
7C31×#1326C211yesmaximal, characteristic
8C31×#543C2211yescharacteristic
9C31×#451—21no—
10C31×#451—21no—
11C31×#400—21no—
12C31×#395—21no—
13C31×#393C2211yescharacteristic
14C31×#393C2211yescharacteristic
15C31×#391—21no—
16C31×#388—21no—
17C31×#385—21no—
18C31×#385—21no—
19C31×#385C2211yescharacteristic
20C31×#382—21no—
21C31×#342—21no—
22C31×#342C2211yescharacteristic
23C31×#328—21no—
24C31×#328—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C23×C4×C31D411yescharacteristic
30C2×Q16×C31—41no—
31C2×Q16×C31—41no—
32C31×C22⋊Q8—21no—
33C31×C22⋊Q8—21no—
34C31×C22⋊Q8—41no—
35C31×C22⋊Q8—41no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—41no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31D411yescharacteristic
46C2×C22⋊C4×C31C2311yescharacteristic
47C2×C42×C31—21no—
48C31×#103—41no—
49C31×#103—41no—
50C31×C22⋊C8—21no—
51C31×C22⋊C8—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—41no—
60C31×#95D411yescharacteristic
61C31×#95D411yescharacteristic
62C24×C31C2×D411yescharacteristic
63C2×Q8×C31—21no—
64C2×Q8×C31—21no—
65C2×Q8×C31—42no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31—81no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—41no—
78C22×C4×C31—41no—
79C22×C4×C31—81no—
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31C2×D411yescharacteristic
82Q16×C31—82no—
83Q16×C31—82no—
84C2×C8×C31—41no—
85C2×C8×C31—41no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—81no—
91C4⋊C4×C31—81no—
92C4⋊C4×C31—81no—
93C4⋊C4×C31—81no—
94C4⋊C4×C31—82no—
95C22⋊C4×C31—21no—
96C22⋊C4×C31—22no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—42no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—81no—
103C22⋊C4×C31Q1612yescomplemented
104C42×C31—42no—
105#12253C3111yesmaximal, characteristic, direct product factor, Sylow
106C23×C31—21no—
107C23×C31—21no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31#12011yescharacteristic
111C23×C31C2×Q1611yescharacteristic
112C23×C31#13711yescharacteristic
113Q8×C31—42no—
114Q8×C31—42no—
115Q8×C31—81no—
116Q8×C31—81no—
117Q8×C31—81no—
118Q8×C31—81no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—42no—
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—42no—
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—81no—
146C2×C4×C31—81no—
147C2×C4×C31—81no—
148C2×C4×C31—81no—
149C2×C4×C31—81no—
150C8×C31—81no—
151C8×C31—81no—
152#2763C2×C3111yescharacteristic
153#2062C2×C3111yescharacteristic
154#1602C2×C3111yescharacteristic
155#1602C2×C3111yescharacteristic
156#1351C2×C3111yescharacteristic
157#1351C2×C3111yescharacteristic
158#1326C2×C3111yescharacteristic
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31—41no—
167C22×C31—81no—
168C22×C31#45111yescharacteristic
169C22×C31#45611yescharacteristic
170C22×C31B(4,𝔽2)11yescharacteristic
171C4×C31—42no—
172C4×C31—41no—
173C4×C31—42no—
174C4×C31—41no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—82no—
178C4×C31—81no—
179C4×C31—81no—
180C4×C31—81no—
181C4×C31—81no—
182#543C22×C3111yescharacteristic
183#451—21no—
184#451—21no—
185#400—21no—
186#395—21no—
187#393C22×C3111yescharacteristic
188#393C22×C3111yescharacteristic
189#391—21no—
190#388—21no—
191#385—21no—
192#385—21no—
193#385C22×C3111yescharacteristic
194#382—21no—
195#342—21no—
196#342C22×C3111yescharacteristic
197#328—21no—
198#328—21no—
199#323C22×C3111yescharacteristic
200#323C22×C3111yescharacteristic
201C2×C31—21no—
202C2×C31—21no—
203C2×C31—41no—
204C2×C31—41no—
205C2×C31#160511yescharacteristic
206C2×C31#219711yescharacteristic
207C2×C31#219911yescharacteristic
208C22×Q8D4×C3111yescharacteristic
209C22×Q8D4×C3111yescharacteristic
210C23×C4D4×C3111yescharacteristic
211C2×Q16—41no—
212C2×Q16—41no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4D4×C3111yescharacteristic
227C2×C22⋊C4C23×C3111yescharacteristic
228C2×C42—21no—
229#103—41no—
230#103—41no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—41no—
241#95D4×C3111yescharacteristic
242#95D4×C3111yescharacteristic
243C31#1225311yescharacteristic, direct product factor, Sylow
244C24C2×D4×C3111yescharacteristic
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—81no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4C2×D4×C3111yescharacteristic
263C22×C4C2×D4×C3111yescharacteristic
264Q16—82no—
265Q16—82no—
266C2×C8—41no—
267C2×C8—41no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C4⋊C4—82no—
277C22⋊C4—21no—
278C22⋊C4—22no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—81no—
285C22⋊C4Q16×C3112yescomplemented
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23C31×#12011yescharacteristic
292C23C2×Q16×C3111yescharacteristic
293C23C31×#13711yescharacteristic
294Q8—42no—
295Q8—42no—
296Q8—81no—
297Q8—81no—
298Q8—81no—
299Q8—81no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C8—81no—
332C8—81no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22C31×#45111yescharacteristic
343C22C31×#45611yescharacteristic
344C22C31×B(4,𝔽2)11yescharacteristic
345C4—42no—
346C4—41no—
347C4—42no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—82no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C2—21no—
357C2—21no—
358C2—41no—
359C2—41no—
360C2C31×#160511yescharacteristic
361C2C31×#219711yescharacteristic
362C2C31×#219911yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC24×C31C2×D4
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
103. C22⋊C4×C31Q16semidirect
105. #12253C31direct (N × H)
243. C31#12253direct (N × H)
285. C22⋊C4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27632
C31×#20622
C31×#16022
C31×#13512
C31×#13262
#1225331