Finite Group Databasebeta

← C31×#7806 #21464565

Subgroups — C31×#7806 #21464565

Order 7936.

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

normal subgroups only

This group has 350 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×#2805C211yesmaximal, characteristic
2C31×#2099C211yesmaximal, characteristic
3C31×#2083C211yesmaximal, characteristic
4C31×#2072C211yesmaximal, characteristic
5C31×#1272C211yesmaximal, characteristic
6C31×#1271C211yesmaximal, characteristic
7C31×#1271C211yesmaximal, characteristic
8C2×C31×#107—21no—
9C2×C31×#103—21no—
10C31×#400—21no—
11C31×#400—21no—
12C31×#395—21no—
13C31×#391C2211yescharacteristic
14C31×#389—21no—
15C31×#389C2211yescharacteristic
16C31×#389C2211yescharacteristic
17C31×#387—21no—
18C31×#387—21no—
19C31×#384C2211yescharacteristic
20C31×#340—21no—
21C31×#340—21no—
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×C8×C31—21no—
29C22×C8×C31—21no—
30C22×C8×C31—21no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
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×C31D411yescharacteristic
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C42×C31—21no—
44C2×C42×C31C2311yescharacteristic
45C31×#107—42no—
46C31×#103—42no—
47C31×#95—21no—
48C31×#95—21no—
49C31×#95—21no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C2×Q8×C31—21no—
57C2×Q8×C31—21no—
58C2×Q8×C31—41no—
59C2×Q8×C31—41no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—41no—
67C22×C4×C31C2×D411yescharacteristic
68C22×C4×C31C2×D411yescharacteristic
69C22×C4×C31C2×D411yescharacteristic
70C2×C8×C31—22no—
71C2×C8×C31—41no—
72C2×C8×C31—41no—
73C2×C8×C31—41no—
74C2×C8×C31—41no—
75C2×C8×C31—41no—
76C2×C8×C31—41no—
77C2×C8×C31—41no—
78C2×C8×C31—41no—
79C4⋊C4×C31—21no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31—21no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—42no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—81no—
95C4⋊C4×C31—81no—
96C4⋊C4×C31Q1612yes—
97C42×C31—41no—
98C42×C31—81no—
99#7806C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31#12011yescharacteristic
101Q8×C31—42no—
102Q8×C31—81no—
103C2×C4×C31—21no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—22no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—41no—
113C2×C4×C31—41no—
114C2×C4×C31—41no—
115C2×C4×C31—41no—
116C2×C4×C31—41no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—41no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—81no—
128C2×C4×C31—81no—
129C2×C4×C31—81no—
130C2×C4×C31C2×Q1611yescharacteristic
131C2×C4×C31#13511yescharacteristic
132C2×C4×C31#13511yescharacteristic
133C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C31#13711yescharacteristic
136C8×C31—42no—
137C8×C31—81no—
138C8×C31—81no—
139#2805C2×C3111yescharacteristic
140#2099C2×C3111yescharacteristic
141#2083C2×C3111yescharacteristic
142#2072C2×C3111yescharacteristic
143#1272C2×C3111yescharacteristic
144#1271C2×C3111yescharacteristic
145#1271C2×C3111yescharacteristic
146C22×C31#44911yescharacteristic
147C22×C31#44911yescharacteristic
148C22×C31#45111yescharacteristic
149C22×C31#45411yescharacteristic
150C22×C31#45511yescharacteristic
151C22×C31#45611yescharacteristic
152C22×C31#45811yescharacteristic
153C4×C31—21no—
154C4×C31—21no—
155C4×C31—21no—
156C4×C31—21no—
157C4×C31—21no—
158C4×C31—21no—
159C4×C31—42no—
160C4×C31—81no—
161C4×C31—81no—
162C4×C31—81no—
163C4×C31—81no—
164C4×C31—81no—
165C4×C31—161no—
166C2×#107—21no—
167C2×#103—21no—
168#400—21no—
169#400—21no—
170#395—21no—
171#391C22×C3111yescharacteristic
172#389—21no—
173#389C22×C3111yescharacteristic
174#389C22×C3111yescharacteristic
175#387—21no—
176#387—21no—
177#384C22×C3111yescharacteristic
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182#336C22×C3111yescharacteristic
183#336C22×C3111yescharacteristic
184#336C22×C3111yescharacteristic
185C2×C31#160911yescharacteristic
186C2×C31#161211yescharacteristic
187C2×C31#161711yescharacteristic
188C2×C31#162811yescharacteristic
189C2×C31#164411yescharacteristic
190C2×C31#164811yescharacteristic
191C2×C31#166411yescharacteristic
192C22×Q8D4×C3111yescharacteristic
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4D4×C3111yescharacteristic
204C2×C4⋊C4D4×C3111yescharacteristic
205C2×C4⋊C4D4×C3111yescharacteristic
206C2×C4⋊C4D4×C3111yescharacteristic
207C2×C4⋊C4D4×C3111yescharacteristic
208C2×C42—21no—
209C2×C42C23×C3111yescharacteristic
210#107—42no—
211#103—42no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221C31#780611yescharacteristic, direct product factor, Sylow
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—41no—
225C2×Q8—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4C2×D4×C3111yescharacteristic
234C22×C4C2×D4×C3111yescharacteristic
235C22×C4C2×D4×C3111yescharacteristic
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4Q16×C3112yes—
263C42—41no—
264C42—81no—
265C23C31×#12011yescharacteristic
266Q8—42no—
267Q8—81no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4C2×Q16×C3111yescharacteristic
296C2×C4C31×#13511yescharacteristic
297C2×C4C31×#13511yescharacteristic
298C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
299C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
300C2×C4C31×#13711yescharacteristic
301C8—42no—
302C8—81no—
303C8—81no—
304C22C31×#44911yescharacteristic
305C22C31×#44911yescharacteristic
306C22C31×#45111yescharacteristic
307C22C31×#45411yescharacteristic
308C22C31×#45511yescharacteristic
309C22C31×#45611yescharacteristic
310C22C31×#45811yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—161no—
324C2C31×#160911yescharacteristic
325C2C31×#161211yescharacteristic
326C2C31×#161711yescharacteristic
327C2C31×#162811yescharacteristic
328C2C31×#164411yescharacteristic
329C2C31×#164811yescharacteristic
330C2C31×#166411yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
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
99. #7806C31direct (N × H)
221. C31#7806direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28052
C31×#20992
C31×#20832
C31×#20722
C31×#12722
C31×#12712
#780631