Finite Group Databasebeta

← C31×#7798 #21464557

Subgroups — C31×#7798 #21464557

Order 7936.

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

normal subgroups only

This group has 374 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×#2726C211yesmaximal, characteristic
2C31×#2099C211yesmaximal, characteristic
3C31×#2084C211yesmaximal, characteristic
4C31×#2083C211yesmaximal, characteristic
5C31×#1271C211yesmaximal, characteristic
6C31×#1271C211yesmaximal, characteristic
7C31×#1271C211yesmaximal, characteristic
8C2×C31×#125—21no—
9C2×C31×#107—21no—
10C2×C31×#106—21no—
11C31×#398—21no—
12C31×#398—21no—
13C31×#398—21no—
14C31×#389—21no—
15C31×#389C2211yescharacteristic
16C31×#389C2211yescharacteristic
17C31×#389C2211yescharacteristic
18C31×#387—21no—
19C31×#384C2211yescharacteristic
20C31×#340—21no—
21C31×#340—21no—
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×C8×C31—21no—
28C22×C8×C31—21no—
29C22×C8×C31—21no—
30C31×#125—42no—
31C31×#125—42no—
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×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C4⋊C4×C31D411yescharacteristic
45C2×C4⋊C4×C31D411yescharacteristic
46C2×C4⋊C4×C31D411yescharacteristic
47C2×C4⋊C4×C31D411yescharacteristic
48C2×C42×C31—21no—
49C2×C42×C31C2311yescharacteristic
50C31×#107—42no—
51C31×#106—42no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C22×C4×C31—21no—
59C22×C4×C31—21no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—41no—
65C22×C4×C31C2×D411yescharacteristic
66C22×C4×C31C2×D411yescharacteristic
67C22×C4×C31C2×D411yescharacteristic
68C2×C8×C31—22no—
69C2×C8×C31—22no—
70C2×C8×C31—41no—
71C2×C8×C31—41no—
72C2×C8×C31—41no—
73C2×C8×C31—41no—
74C2×C8×C31—41no—
75C2×C8×C31—41no—
76C2×C8×C31—41no—
77C4⋊C4×C31—21no—
78C4⋊C4×C31—21no—
79C4⋊C4×C31—21no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31—42no—
82C4⋊C4×C31—42no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—42no—
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—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—81no—
99C4⋊C4×C31D812yes—
100C4⋊C4×C31SD1612yes—
101C42×C31—41no—
102C42×C31—42no—
103#7798C3111yesmaximal, characteristic, direct product factor, Sylow
104C23×C31#12011yescharacteristic
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—22no—
112C2×C4×C31—22no—
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—41no—
128C2×C4×C31—41no—
129C2×C4×C31—81no—
130C2×C4×C31—81no—
131C2×C4×C31—81no—
132C2×C4×C31C2×D811yescharacteristic
133C2×C4×C31C2×SD1611yescharacteristic
134C2×C4×C31#13511yescharacteristic
135C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C31#13711yescharacteristic
137C2×C4×C31#13711yescharacteristic
138C8×C31—42no—
139C8×C31—42no—
140C8×C31—81no—
141#2726C2×C3111yescharacteristic
142#2099C2×C3111yescharacteristic
143#2084C2×C3111yescharacteristic
144#2083C2×C3111yescharacteristic
145#1271C2×C3111yescharacteristic
146#1271C2×C3111yescharacteristic
147#1271C2×C3111yescharacteristic
148C22×C31#44711yescharacteristic
149C22×C31#44811yescharacteristic
150C22×C31#45211yescharacteristic
151C22×C31#45411yescharacteristic
152C22×C31#45611yescharacteristic
153C22×C31#45611yescharacteristic
154C22×C31#45811yescharacteristic
155C4×C31—21no—
156C4×C31—21no—
157C4×C31—21no—
158C4×C31—21no—
159C4×C31—21no—
160C4×C31—21no—
161C4×C31—42no—
162C4×C31—42no—
163C4×C31—81no—
164C4×C31—81no—
165C4×C31—81no—
166C4×C31—81no—
167C4×C31—161no—
168C2×#125—21no—
169C2×#107—21no—
170C2×#106—21no—
171#398—21no—
172#398—21no—
173#398—21no—
174#389—21no—
175#389C22×C3111yescharacteristic
176#389C22×C3111yescharacteristic
177#389C22×C3111yescharacteristic
178#387—21no—
179#384C22×C3111yescharacteristic
180#340—21no—
181#340—21no—
182#340—21no—
183#340—21no—
184#336C22×C3111yescharacteristic
185#336C22×C3111yescharacteristic
186#336C22×C3111yescharacteristic
187C2×C31#161011yescharacteristic
188C2×C31#161111yescharacteristic
189C2×C31#161511yescharacteristic
190C2×C31#162711yescharacteristic
191C2×C31#164711yescharacteristic
192C2×C31#165311yescharacteristic
193C2×C31#166311yescharacteristic
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197#125—42no—
198#125—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4D4×C3111yescharacteristic
210C2×C4⋊C4D4×C3111yescharacteristic
211C2×C4⋊C4D4×C3111yescharacteristic
212C2×C4⋊C4D4×C3111yescharacteristic
213C2×C4⋊C4D4×C3111yescharacteristic
214C2×C4⋊C4D4×C3111yescharacteristic
215C2×C42—21no—
216C2×C42C23×C3111yescharacteristic
217#107—42no—
218#106—42no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225C31#779811yescharacteristic, direct product factor, Sylow
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—22no—
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—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4D8×C3112yes—
268C4⋊C4SD16×C3112yes—
269C42—41no—
270C42—42no—
271C23C31×#12011yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
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—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4C2×D8×C3111yescharacteristic
300C2×C4C2×SD16×C3111yescharacteristic
301C2×C4C31×#13511yescharacteristic
302C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
303C2×C4C31×#13711yescharacteristic
304C2×C4C31×#13711yescharacteristic
305C8—42no—
306C8—42no—
307C8—81no—
308C22C31×#44711yescharacteristic
309C22C31×#44811yescharacteristic
310C22C31×#45211yescharacteristic
311C22C31×#45411yescharacteristic
312C22C31×#45611yescharacteristic
313C22C31×#45611yescharacteristic
314C22C31×#45811yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—161no—
328C2C31×#161011yescharacteristic
329C2C31×#161111yescharacteristic
330C2C31×#161511yescharacteristic
331C2C31×#162711yescharacteristic
332C2C31×#164711yescharacteristic
333C2C31×#165311yescharacteristic
334C2C31×#166311yescharacteristic
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
103. #7798C31direct (N × H)
225. C31#7798direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27262
C31×#20992
C31×#20842
C31×#20832
C31×#12712
#779831