Finite Group Databasebeta

← C31×#12728 #21469487

Subgroups — C31×#12728 #21469487

Order 7936.

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

normal subgroups only

This group has 714 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×#2127C212yesmaximal
3C2×C31×#354C212yesmaximal, complemented
4C31×#1828C212yesmaximal, complemented
5C31×#559—22no—
6C31×#556—22no—
7C31×#551—24no—
8C31×#542—22no—
9C31×#536C411yescharacteristic, complemented
10C31×#535C2211yescharacteristic
11C31×#533C411yescharacteristic, complemented
12C31×#529—22no—
13C2×C31×#124C2211yescharacteristic
14C31×#519C411yescharacteristic, complemented
15C31×#518C2211yescharacteristic
16C4×#41×C31C411yescharacteristic, complemented
17C2×C31×#99C2214yes—
18C31×#354—22no—
19C31×#354—22no—
20C31×#354—22no—
21C31×#353—22no—
22C31×#342—24no—
23C2×#41×C31C2×C411yescharacteristic
24C22×Q8×C31C2×C411yescharacteristic
25C22×D4×C31C2311yescharacteristic
26C31×C4⋊Q8—21no—
27C31×C4⋊Q8—21no—
28C31×#127D412yes—
29C31×#126—24no—
30C31×#126—24no—
31C31×#125—21no—
32C31×#124—24no—
33C31×#124—21no—
34C31×#124—22no—
35C31×#124D412yes—
36C31×#124D412yes—
37C31×#124D412yes—
38C31×#123—24no—
39C31×#123—24no—
40C31×#123—22no—
41C31×C22⋊Q8—24no—
42C31×C22⋊Q8—22no—
43C31×C22⋊Q8—22no—
44C31×#121—22no—
45C31×#120—22no—
46C4×Q8×C31—21no—
47C4×Q8×C31—22no—
48C4×Q8×C31—22no—
49C4×Q8×C31—22no—
50C4×D4×C31—24no—
51C4×D4×C31—21no—
52C4×D4×C31—22no—
53C4×D4×C31—22no—
54C4×D4×C31—22no—
55C31×#117—24no—
56C31×#117—22no—
57C31×#117—21no—
58C31×#117C2×C411yescharacteristic
59C31×#117C2×C411yescharacteristic
60C31×#117C2×C411yescharacteristic
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31D414yes—
64C2×C42×C31—21no—
65C2×C42×C31C2×C411yescharacteristic
66C31×#99—24no—
67C31×#99—24no—
68C24×C31C2×D412yes—
69#41×C31—42no—
70C2×Q8×C31—22no—
71C2×Q8×C31—21no—
72C2×Q8×C31—42no—
73C2×Q8×C31C22⋊C412yes—
74C2×Q8×C31C22⋊C411yescharacteristic
75C2×D4×C31—21no—
76C2×D4×C31—24no—
77C2×D4×C31—42no—
78C2×D4×C31C22⋊C411yescharacteristic
79C2×D4×C31C2×D412yes—
80C2×D4×C31C2×D412yes—
81C22×C4×C31—21no—
82C22×C4×C31—22no—
83C22×C4×C31—21no—
84C22×C4×C31—24no—
85C22×C4×C31—21no—
86C22×C4×C31—44no—
87C22×C4×C31C22×C411yescharacteristic
88C4⋊C4×C31—22no—
89C4⋊C4×C31—22no—
90C4⋊C4×C31—24no—
91C4⋊C4×C31—24no—
92C4⋊C4×C31—22no—
93C4⋊C4×C31—22no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31—21no—
97C22⋊C4×C31—22no—
98C22⋊C4×C31—24no—
99C22⋊C4×C31—21no—
100C22⋊C4×C31—24no—
101C22⋊C4×C31—24no—
102C22⋊C4×C31—24no—
103C22⋊C4×C31—24no—
104C22⋊C4×C31—44no—
105C22⋊C4×C31—44no—
106C22⋊C4×C31C22⋊C412yes—
107C22⋊C4×C31C22⋊C412yes—
108C42×C31—22no—
109C42×C31—24no—
110C42×C31—21no—
111C42×C31—21no—
112C42×C31—21no—
113C42×C31C22⋊C411yescharacteristic
114C42×C31C22⋊C411yescharacteristic
115C42×C31C22⋊C412yes—
116#12728C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—24no—
118C23×C31—21no—
119C23×C31—82no—
120C23×C31C2×C22⋊C411yescharacteristic
121C23×C31#12014yes—
122Q8×C31—42no—
123Q8×C31—41no—
124Q8×C31—42no—
125Q8×C31—82no—
126D4×C31—42no—
127D4×C31—42no—
128D4×C31—41no—
129D4×C31—42no—
130D4×C31—84no—
131C2×C4×C31—22no—
132C2×C4×C31—22no—
133C2×C4×C31—21no—
134C2×C4×C31—22no—
135C2×C4×C31—22no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—22no—
139C2×C4×C31—42no—
140C2×C4×C31—42no—
141C2×C4×C31—82no—
142C2×C4×C31—84no—
143C2×C4×C31—81no—
144C2×C4×C31—84no—
145C2×C4×C31—164no—
146C2×C4×C31C2×C22⋊C411yescharacteristic
147C2×C4×C31C2×C22⋊C411yescharacteristic
148#3510C2×C3111yescharacteristic
149#2127C2×C3112yes—
150C2×#354C2×C3112yescomplemented
151#1828C2×C3112yescomplemented
152C22×C31—24no—
153C22×C31—41no—
154C22×C31—44no—
155C22×C31—42no—
156C22×C31—41no—
157C22×C31—84no—
158C22×C31#37911yescharacteristic
159C22×C31#41012yes—
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—42no—
163C4×C31—41no—
164C4×C31—42no—
165C4×C31—41no—
166C4×C31—42no—
167C4×C31—41no—
168C4×C31—42no—
169C4×C31—164no—
170#559—22no—
171#556—22no—
172#551—24no—
173#542—22no—
174#536C4×C3111yescharacteristic, complemented
175#535C22×C3111yescharacteristic
176#533C4×C3111yescharacteristic, complemented
177#529—22no—
178C2×#124C22×C3111yescharacteristic
179#519C4×C3111yescharacteristic, complemented
180#518C22×C3111yescharacteristic
181C4×#41C4×C3111yescharacteristic, complemented
182C2×#99C22×C3114yes—
183#354—22no—
184#354—22no—
185#354—22no—
186#353—22no—
187#342—24no—
188C2×C31—21no—
189C2×C31—22no—
190C2×C31—44no—
191C2×C31—81no—
192C2×C31#178211yescharacteristic
193C2×#41C2×C4×C3111yescharacteristic
194C22×Q8C2×C4×C3111yescharacteristic
195C22×D4C23×C3111yescharacteristic
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198#127D4×C3112yes—
199#126—24no—
200#126—24no—
201#125—21no—
202#124—24no—
203#124—21no—
204#124—22no—
205#124D4×C3112yes—
206#124D4×C3112yes—
207#124D4×C3112yes—
208#123—24no—
209#123—24no—
210#123—22no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214#121—22no—
215#120—22no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×D4—24no—
221C4×D4—21no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225#117—24no—
226#117—22no—
227#117—21no—
228#117C2×C4×C3111yescharacteristic
229#117C2×C4×C3111yescharacteristic
230#117C2×C4×C3111yescharacteristic
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4D4×C3114yes—
234C2×C42—21no—
235C2×C42C2×C4×C3111yescharacteristic
236#99—24no—
237#99—24no—
238C31#1272811yescharacteristic, direct product factor, Sylow
239C24C2×D4×C3112yes—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊C4×C3112yes—
245C2×Q8C22⋊C4×C3111yescharacteristic
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—42no—
249C2×D4C22⋊C4×C3111yescharacteristic
250C2×D4C2×D4×C3112yes—
251C2×D4C2×D4×C3112yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—44no—
258C22×C4C22×C4×C3111yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—21no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4C22⋊C4×C3112yes—
278C22⋊C4C22⋊C4×C3112yes—
279C42—22no—
280C42—24no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42C22⋊C4×C3111yescharacteristic
285C42C22⋊C4×C3111yescharacteristic
286C42C22⋊C4×C3112yes—
287C23—24no—
288C23—21no—
289C23—82no—
290C23C2×C22⋊C4×C3111yescharacteristic
291C23C31×#12014yes—
292Q8—42no—
293Q8—41no—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—84no—
315C2×C4—164no—
316C2×C4C2×C22⋊C4×C3111yescharacteristic
317C2×C4C2×C22⋊C4×C3111yescharacteristic
318C22—24no—
319C22—41no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—84no—
324C22C31×#37911yescharacteristic
325C22C31×#41012yes—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—164no—
336C2—21no—
337C2—22no—
338C2—44no—
339C2—81no—
340C2C31×#178211yescharacteristic
subgroupG/H
center, socleC2×C31#1782
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC31×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C31×#518 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#354C2semidirect
4. C31×#1828C2semidirect
9. C31×#536C4semidirect
11. C31×#533C4semidirect
14. C31×#519C4semidirect
16. C4×#41×C31C4semidirect
116. #12728C31direct (N × H)
150. C2×#354C2×C31semidirect
151. #1828C2×C31semidirect
174. #536C4×C31semidirect
176. #533C4×C31semidirect
179. #519C4×C31semidirect
181. C4×#41C4×C31semidirect
238. C31#12728direct (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×#21272
C2×C31×#3542
C31×#18282
#1272831