Finite Group Databasebeta

← C31×#12729 #21469488

Subgroups — C31×#12729 #21469488

Order 7936.

690 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3510C211yesmaximal, characteristic
2C31×#2133C212yesmaximal, complemented
3C2×C31×#355C212yesmaximal, complemented
4C31×#1830C212yesmaximal, complemented
5C31×#559—22no—
6C31×#556—22no—
7C31×#551—24no—
8C31×#542—22no—
9C31×#536C2211yescharacteristic
10C31×#535C411yescharacteristic
11C31×#533C411yescharacteristic
12C31×#529—22no—
13C2×C31×#124C2211yescharacteristic
14C31×#519C2211yescharacteristic
15C31×#518C411yescharacteristic
16C4×#41×C31C411yescharacteristic
17C2×C31×#101C2214yes—
18C31×#356—22no—
19C31×#355—22no—
20C31×#355—22no—
21C31×#355—22no—
22C31×#344—24no—
23C2×#41×C31C2×C411yescharacteristic
24C22×Q8×C31C2311yescharacteristic
25C22×D4×C31C2×C411yescharacteristic
26C2×M16×C31—24no—
27C31×C4⋊Q8—21no—
28C31×C4⋊Q8D412yes—
29C31×#127—21no—
30C31×#126—24no—
31C31×#126—24no—
32C31×#125—21no—
33C31×#124—24no—
34C31×#124—21no—
35C31×#124—22no—
36C31×#124D412yes—
37C31×#124D412yes—
38C31×#124D412yes—
39C31×#123—24no—
40C31×#123—24no—
41C31×#123—22no—
42C31×C22⋊Q8—24no—
43C31×C22⋊Q8—22no—
44C31×C22⋊Q8—22no—
45C31×#121—22no—
46C31×#120—22no—
47C4×Q8×C31—21no—
48C4×Q8×C31—22no—
49C4×Q8×C31—22no—
50C4×Q8×C31—22no—
51C4×D4×C31—24no—
52C4×D4×C31—21no—
53C4×D4×C31—22no—
54C4×D4×C31—22no—
55C4×D4×C31—22no—
56C31×#117—24no—
57C31×#117—21no—
58C31×#117C2×C411yescharacteristic
59C31×#117C2×C411yescharacteristic
60C31×#117C2×C411yescharacteristic
61C31×#117D414yes—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—22no—
64C2×C42×C31—21no—
65C2×C42×C31C2×C411yescharacteristic
66C31×#101—24no—
67C31×#101—24no—
68C24×C31—21no—
69#41×C31—42no—
70C2×Q8×C31—22no—
71C2×Q8×C31—42no—
72C2×Q8×C31C22⋊C411yescharacteristic
73C2×Q8×C31C2×D412yes—
74C2×Q8×C31C2×D412yes—
75C2×D4×C31—21no—
76C2×D4×C31—24no—
77C2×D4×C31—21no—
78C2×D4×C31—42no—
79C2×D4×C31C22⋊C412yes—
80C2×D4×C31C22⋊C411yescharacteristic
81C22×C4×C31—22no—
82C22×C4×C31—24no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31C22×C411yescharacteristic
86C22×C4×C31C2×D412yes—
87M16×C31—44no—
88M16×C31—44no—
89C2×C8×C31—44no—
90C4⋊C4×C31—22no—
91C4⋊C4×C31—22no—
92C4⋊C4×C31—22no—
93C4⋊C4×C31—22no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—24no—
98C4⋊C4×C31—24no—
99C22⋊C4×C31—24no—
100C22⋊C4×C31—22no—
101C22⋊C4×C31—24no—
102C22⋊C4×C31—21no—
103C22⋊C4×C31—24no—
104C22⋊C4×C31—24no—
105C22⋊C4×C31—24no—
106C22⋊C4×C31C22⋊C412yes—
107C22⋊C4×C31C22⋊C412yes—
108C42×C31—22no—
109C42×C31—21no—
110C42×C31—21no—
111C42×C31—21no—
112C42×C31—24no—
113C42×C31C22⋊C411yescharacteristic
114C42×C31C22⋊C411yescharacteristic
115C42×C31C22⋊C412yes—
116#12729C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—22no—
118C23×C31—21no—
119C23×C31—42no—
120C23×C31—82no—
121C23×C31C2×C22⋊C411yescharacteristic
122Q8×C31—42no—
123Q8×C31—42no—
124Q8×C31—41no—
125Q8×C31—82no—
126D4×C31—42no—
127D4×C31—41no—
128D4×C31—42no—
129D4×C31—42no—
130D4×C31—84no—
131C2×C4×C31—24no—
132C2×C4×C31—22no—
133C2×C4×C31—22no—
134C2×C4×C31—21no—
135C2×C4×C31—22no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—22no—
139C2×C4×C31—42no—
140C2×C4×C31—82no—
141C2×C4×C31—84no—
142C2×C4×C31—81no—
143C2×C4×C31C2×C22⋊C411yescharacteristic
144C2×C4×C31C2×C22⋊C411yescharacteristic
145C2×C4×C31#12014yes—
146C8×C31—84no—
147#3510C2×C3111yescharacteristic
148#2133C2×C3112yescomplemented
149C2×#355C2×C3112yescomplemented
150#1830C2×C3112yescomplemented
151C22×C31—41no—
152C22×C31—42no—
153C22×C31—44no—
154C22×C31—41no—
155C22×C31—42no—
156C22×C31—162no—
157C22×C31#37911yescharacteristic
158C22×C31#41012yes—
159C4×C31—24no—
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—42no—
163C4×C31—41no—
164C4×C31—42no—
165C4×C31—41no—
166C4×C31—42no—
167C4×C31—41no—
168#559—22no—
169#556—22no—
170#551—24no—
171#542—22no—
172#536C22×C3111yescharacteristic
173#535C4×C3111yescharacteristic
174#533C4×C3111yescharacteristic
175#529—22no—
176C2×#124C22×C3111yescharacteristic
177#519C22×C3111yescharacteristic
178#518C4×C3111yescharacteristic
179C4×#41C4×C3111yescharacteristic
180C2×#101C22×C3114yes—
181#356—22no—
182#355—22no—
183#355—22no—
184#355—22no—
185#344—24no—
186C2×C31—21no—
187C2×C31—22no—
188C2×C31—81no—
189C2×C31—82no—
190C2×C31#178211yescharacteristic
191C2×#41C2×C4×C3111yescharacteristic
192C22×Q8C23×C3111yescharacteristic
193C22×D4C2×C4×C3111yescharacteristic
194C2×M16—24no—
195C4⋊Q8—21no—
196C4⋊Q8D4×C3112yes—
197#127—21no—
198#126—24no—
199#126—24no—
200#125—21no—
201#124—24no—
202#124—21no—
203#124—22no—
204#124D4×C3112yes—
205#124D4×C3112yes—
206#124D4×C3112yes—
207#123—24no—
208#123—24no—
209#123—22no—
210C22⋊Q8—24no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213#121—22no—
214#120—22no—
215C4×Q8—21no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×D4—24no—
220C4×D4—21no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224#117—24no—
225#117—21no—
226#117C2×C4×C3111yescharacteristic
227#117C2×C4×C3111yescharacteristic
228#117C2×C4×C3111yescharacteristic
229#117D4×C3114yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—21no—
233C2×C42C2×C4×C3111yescharacteristic
234#101—24no—
235#101—24no—
236C31#1272911yescharacteristic, direct product factor, Sylow
237C24—21no—
238#41—42no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8C22⋊C4×C3111yescharacteristic
242C2×Q8C2×D4×C3112yes—
243C2×Q8C2×D4×C3112yes—
244C2×D4—21no—
245C2×D4—24no—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C22⋊C4×C3112yes—
249C2×D4C22⋊C4×C3111yescharacteristic
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22×C4×C3111yescharacteristic
255C22×C4C2×D4×C3112yes—
256M16—44no—
257M16—44no—
258C2×C8—44no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—22no—
270C22⋊C4—24no—
271C22⋊C4—21no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4C22⋊C4×C3112yes—
276C22⋊C4C22⋊C4×C3112yes—
277C42—22no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—24no—
282C42C22⋊C4×C3111yescharacteristic
283C42C22⋊C4×C3111yescharacteristic
284C42C22⋊C4×C3112yes—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—82no—
289C23C2×C22⋊C4×C3111yescharacteristic
290Q8—42no—
291Q8—42no—
292Q8—41no—
293Q8—82no—
294D4—42no—
295D4—41no—
296D4—42no—
297D4—42no—
298D4—84no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—81no—
311C2×C4C2×C22⋊C4×C3111yescharacteristic
312C2×C4C2×C22⋊C4×C3111yescharacteristic
313C2×C4C31×#12014yes—
314C8—84no—
315C22—41no—
316C22—42no—
317C22—44no—
318C22—41no—
319C22—42no—
320C22—162no—
321C22C31×#37911yescharacteristic
322C22C31×#41012yes—
323C4—24no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C2—21no—
333C2—22no—
334C2—82no—
335C2—81no—
336C2C31×#178211yescharacteristic
subgroupG/H
center, socleC2×C31#1782
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC31×#535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C31×#519 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2133C2semidirect
3. C2×C31×#355C2semidirect
4. C31×#1830C2semidirect
116. #12729C31direct (N × H)
148. #2133C2×C31semidirect
149. C2×#355C2×C31semidirect
150. #1830C2×C31semidirect
236. C31#12729direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35102
C31×#21332
C2×C31×#3552
C31×#18302
#1272931