Finite Group Databasebeta

← C31×#12775 #21469534

Subgroups — C31×#12775 #21469534

Order 7936.

740 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×C31×#563C211yesmaximal, characteristic
2C2×C31×#356C212yesmaximal, complemented
3C2×C31×#355C211yesmaximal, characteristic
4C31×#1888C211yesmaximal, characteristic
5C31×#1886C212yesmaximal, complemented
6C31×#563—24no—
7C31×#563—22no—
8C31×#563C414yescomplemented
9C2×C31×C4⋊Q8C2212yes—
10C2×C31×#126—22no—
11C2×C31×#125—21no—
12C2×C31×#124C2211yescharacteristic
13C2×C31×#123—21no—
14C2×C31×C22⋊Q8—22no—
15C31×#519C2211yescharacteristic
16C31×#419—22no—
17C31×#418—21no—
18C2×C31×#101C2212yes—
19C2×C31×#101C2211yescharacteristic
20C31×#356—24no—
21C31×#355—22no—
22C31×#339—22no—
23C31×#339—21no—
24C22×Q8×C31D412yes—
25C22×Q8×C31C2311yescharacteristic
26C22×D4×C31D411yescharacteristic
27C23×C4×C31—21no—
28C2×M16×C31—21no—
29C2×M16×C31—22no—
30C31×C4⋊Q8—22no—
31C31×C4⋊Q8—22no—
32C31×C4⋊Q8C2×C414yes—
33C31×C4⋊Q8D414yes—
34C31×#126—28no—
35C31×#126—28no—
36C31×#125—24no—
37C31×#125—24no—
38C31×#124—22no—
39C31×#124C2×C412yes—
40C31×#124D412yes—
41C31×#123—24no—
42C31×#123—24no—
43C31×C22⋊Q8—28no—
44C31×C22⋊Q8—28no—
45C4×Q8×C31—42no—
46C31×#117—21no—
47C31×#117—22no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—22no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—22no—
52C2×C4⋊C4×C31—22no—
53C2×C22⋊C4×C31—22no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—22no—
56C2×C42×C31D411yescharacteristic
57C2×C42×C31D412yes—
58C31×#103—44no—
59C31×#102—42no—
60C31×#101—22no—
61C31×#101—22no—
62C31×#101—22no—
63C24×C31—21no—
64C2×Q8×C31—22no—
65C2×Q8×C31—41no—
66C2×Q8×C31—41no—
67C2×Q8×C31—42no—
68C2×Q8×C31—42no—
69C2×Q8×C31C22⋊C414yes—
70C2×Q8×C31C22×C411yescharacteristic
71C2×Q8×C31C2×D412yes—
72C2×Q8×C31C2×D411yescharacteristic
73C2×D4×C31—21no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31C22⋊C412yes—
77C22×C4×C31—21no—
78C22×C4×C31—22no—
79C22×C4×C31—22no—
80C22×C4×C31—24no—
81C22×C4×C31—81no—
82C22×C4×C31—81no—
83C22×C4×C31C2×D412yes—
84C22×C4×C31C2×D411yescharacteristic
85M16×C31—41no—
86M16×C31—42no—
87M16×C31—42no—
88M16×C31—41no—
89C2×C8×C31—41no—
90C2×C8×C31—42no—
91C4⋊C4×C31—24no—
92C4⋊C4×C31—28no—
93C4⋊C4×C31—24no—
94C4⋊C4×C31—24no—
95C4⋊C4×C31—24no—
96C4⋊C4×C31—24no—
97C4⋊C4×C31—24no—
98C4⋊C4×C31—22no—
99C4⋊C4×C31—24no—
100C22⋊C4×C31—28no—
101C22⋊C4×C31—24no—
102C22⋊C4×C31—24no—
103C22⋊C4×C31—24no—
104C22⋊C4×C31—41no—
105C42×C31—21no—
106C42×C31—22no—
107C42×C31—41no—
108C42×C31—42no—
109C42×C31C22⋊C412yes—
110C42×C31C22⋊C414yes—
111#12775C3111yesmaximal, characteristic, direct product factor, Sylow
112C23×C31—22no—
113C23×C31—44no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31#12011yescharacteristic
117Q8×C31—41no—
118Q8×C31—42no—
119Q8×C31—41no—
120Q8×C31—44no—
121Q8×C31—82no—
122D4×C31—42no—
123D4×C31—81no—
124C2×C4×C31—21no—
125C2×C4×C31—22no—
126C2×C4×C31—21no—
127C2×C4×C31—22no—
128C2×C4×C31—22no—
129C2×C4×C31—24no—
130C2×C4×C31—24no—
131C2×C4×C31—42no—
132C2×C4×C31—44no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31—42no—
136C2×C4×C31—84no—
137C2×C4×C31C2×C22⋊C412yes—
138C2×C4×C31C2×C22⋊C411yescharacteristic
139C2×C4×C31#12012yes—
140C2×C4×C31#12011yescharacteristic
141C8×C31—81no—
142C8×C31—82no—
143C2×#563C2×C3111yescharacteristic
144C2×#356C2×C3112yescomplemented
145C2×#355C2×C3111yescharacteristic
146#1888C2×C3111yescharacteristic
147#1886C2×C3112yescomplemented
148C22×C31—22no—
149C22×C31—44no—
150C22×C31—42no—
151C22×C31—82no—
152C22×C31—82no—
153C22×C31#37911yescharacteristic
154C22×C31B(4,𝔽2)11yescharacteristic
155C22×C31#45811yescharacteristic
156C4×C31—21no—
157C4×C31—22no—
158C4×C31—21no—
159C4×C31—22no—
160C4×C31—42no—
161C4×C31—44no—
162C4×C31—44no—
163C4×C31—82no—
164#563—24no—
165#563—22no—
166#563C4×C3114yescomplemented
167C2×C4⋊Q8C22×C3112yes—
168C2×#126—22no—
169C2×#125—21no—
170C2×#124C22×C3111yescharacteristic
171C2×#123—21no—
172C2×C22⋊Q8—22no—
173#519C22×C3111yescharacteristic
174#419—22no—
175#418—21no—
176C2×#101C22×C3112yes—
177C2×#101C22×C3111yescharacteristic
178#356—24no—
179#355—22no—
180#339—22no—
181#339—21no—
182C2×C31—21no—
183C2×C31—21no—
184C2×C31—82no—
185C2×C31#179011yescharacteristic
186C2×C31#220312yes—
187C22×Q8D4×C3112yes—
188C22×Q8C23×C3111yescharacteristic
189C22×D4D4×C3111yescharacteristic
190C23×C4—21no—
191C2×M16—21no—
192C2×M16—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8C2×C4×C3114yes—
196C4⋊Q8D4×C3114yes—
197#126—28no—
198#126—28no—
199#125—24no—
200#125—24no—
201#124—22no—
202#124C2×C4×C3112yes—
203#124D4×C3112yes—
204#123—24no—
205#123—24no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208C4×Q8—42no—
209#117—21no—
210#117—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C42D4×C3111yescharacteristic
220C2×C42D4×C3112yes—
221#103—44no—
222#102—42no—
223#101—22no—
224#101—22no—
225#101—22no—
226C31#1277511yescharacteristic, direct product factor, Sylow
227C24—21no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8C22⋊C4×C3114yes—
234C2×Q8C22×C4×C3111yescharacteristic
235C2×Q8C2×D4×C3112yes—
236C2×Q8C2×D4×C3111yescharacteristic
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊C4×C3112yes—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×D4×C3112yes—
248C22×C4C2×D4×C3111yescharacteristic
249M16—41no—
250M16—42no—
251M16—42no—
252M16—41no—
253C2×C8—41no—
254C2×C8—42no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C22⋊C4—28no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—41no—
269C42—21no—
270C42—22no—
271C42—41no—
272C42—42no—
273C42C22⋊C4×C3112yes—
274C42C22⋊C4×C3114yes—
275C23—22no—
276C23—44no—
277C23—41no—
278C23—41no—
279C23C31×#12011yescharacteristic
280Q8—41no—
281Q8—42no—
282Q8—41no—
283Q8—44no—
284Q8—82no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4C2×C22⋊C4×C3112yes—
301C2×C4C2×C22⋊C4×C3111yescharacteristic
302C2×C4C31×#12012yes—
303C2×C4C31×#12011yescharacteristic
304C8—81no—
305C8—82no—
306C22—22no—
307C22—44no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22C31×#37911yescharacteristic
312C22C31×B(4,𝔽2)11yescharacteristic
313C22C31×#45811yescharacteristic
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—82no—
322C2—21no—
323C2—21no—
324C2—82no—
325C2C31×#179011yescharacteristic
326C2C31×#220312yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#356C2semidirect
5. C31×#1886C2semidirect
8. C31×#563C4semidirect
111. #12775C31direct (N × H)
144. C2×#356C2×C31semidirect
147. #1886C2×C31semidirect
166. #563C4×C31semidirect
226. C31#12775direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5632
C2×C31×#3562
C2×C31×#3552
C31×#18882
C31×#18862
#1277531