Finite Group Databasebeta

← C71×#12241 #31833113

Subgroups — C71×#12241 #31833113

Order 18176.

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C71×#2769C211yesmaximal, characteristic
2C71×#2061C211yesmaximal, characteristic
3C71×#1617C211yesmaximal, characteristic
4C71×#1611C211yesmaximal, characteristic
5C71×#1353C211yesmaximal, characteristic
6C71×#1346C211yesmaximal, characteristic
7C71×#1324C211yesmaximal, characteristic
8#544×C71C2211yescharacteristic
9#484×C71—21no—
10#483×C71—21no—
11#428×C71—21no—
12#400×C71—21no—
13#400×C71C2211yescharacteristic
14#398×C71—21no—
15#398×C71C2211yescharacteristic
16#389×C71—21no—
17#387×C71—21no—
18#387×C71—21no—
19#383×C71—21no—
20#382×C71C2211yescharacteristic
21#343×C71C2211yescharacteristic
22#339×C71—21no—
23#328×C71—21no—
24#328×C71—21no—
25#324×C71C2211yescharacteristic
26#324×C71C2211yescharacteristic
27C2×M16×C71—21no—
28C4⋊Q8×C71—41no—
29#125×C71—41no—
30C22⋊Q8×C71—21no—
31C22⋊Q8×C71—21no—
32#117×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71C2311yescharacteristic
41C2×C42×C71—21no—
42C2×C42×C71D411yescharacteristic
43#107×C71—41no—
44#107×C71—41no—
45#106×C71—41no—
46#106×C71—41no—
47#103×C71—41no—
48#103×C71—41no—
49C22⋊C8×C71—21no—
50C22⋊C8×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71D411yescharacteristic
61#95×C71D411yescharacteristic
62C2×Q8×C71—21no—
63C2×Q8×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73M16×C71—22no—
74M16×C71—41no—
75C2×C8×C71—41no—
76C2×C8×C71—41no—
77C2×C8×C71—41no—
78C4⋊C4×C71—21no—
79C4⋊C4×C71—21no—
80C4⋊C4×C71—21no—
81C4⋊C4×C71—21no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—81no—
93C22⋊C4×C71—41no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—42no—
97C42×C71—41no—
98C23×C71#12011yescharacteristic
99Q8×C71—81no—
100Q8×C71—81no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—22no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71#13511yescharacteristic
128C2×C4×C71#13711yescharacteristic
129C8×C71—42no—
130C8×C71—81no—
131C8×C71—81no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71#45211yescharacteristic
135C22×C71#45611yescharacteristic
136C22×C71B(4,𝔽2)11yescharacteristic
137C4×C71—21no—
138C4×C71—21no—
139C4×C71—42no—
140C4×C71—41no—
141C4×C71—41no—
142C4×C71—81no—
143C4×C71—81no—
144C4×C71—81no—
145C4×C71—81no—
146C4×C71—81no—
147C4×C71—81no—
148C4×C71—81no—
149#12241C7111yesmaximal, characteristic, direct product factor, Sylow
150C2×C71—21no—
151C2×C71—21no—
152C2×C71#160311yescharacteristic
153C2×C71#220111yescharacteristic
154C2×C71#220411yescharacteristic
155#2769C2×C7111yescharacteristic
156#2061C2×C7111yescharacteristic
157#1617C2×C7111yescharacteristic
158#1611C2×C7111yescharacteristic
159#1353C2×C7111yescharacteristic
160#1346C2×C7111yescharacteristic
161#1324C2×C7111yescharacteristic
162C71#1224111yescharacteristic, direct product factor, Sylow
163#544C22×C7111yescharacteristic
164#484—21no—
165#483—21no—
166#428—21no—
167#400—21no—
168#400C22×C7111yescharacteristic
169#398—21no—
170#398C22×C7111yescharacteristic
171#389—21no—
172#387—21no—
173#387—21no—
174#383—21no—
175#382C22×C7111yescharacteristic
176#343C22×C7111yescharacteristic
177#339—21no—
178#328—21no—
179#328—21no—
180#324C22×C7111yescharacteristic
181#324C22×C7111yescharacteristic
182C2×M16—21no—
183C4⋊Q8—41no—
184#125—41no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187#117—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4D4×C7111yescharacteristic
193C2×C4⋊C4D4×C7111yescharacteristic
194C2×C4⋊C4D4×C7111yescharacteristic
195C2×C4⋊C4C23×C7111yescharacteristic
196C2×C42—21no—
197C2×C42D4×C7111yescharacteristic
198#107—41no—
199#107—41no—
200#106—41no—
201#106—41no—
202#103—41no—
203#103—41no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95D4×C7111yescharacteristic
216#95D4×C7111yescharacteristic
217C2×Q8—21no—
218C2×Q8—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C2×D4×C7111yescharacteristic
226C22×C4C2×D4×C7111yescharacteristic
227C22×C4C2×D4×C7111yescharacteristic
228M16—22no—
229M16—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—81no—
248C22⋊C4—41no—
249C42—21no—
250C42—21no—
251C42—42no—
252C42—41no—
253C23#120×C7111yescharacteristic
254Q8—81no—
255Q8—81no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4#135×C7111yescharacteristic
283C2×C4#137×C7111yescharacteristic
284C8—42no—
285C8—81no—
286C8—81no—
287C22—21no—
288C22—21no—
289C22#452×C7111yescharacteristic
290C22#456×C7111yescharacteristic
291C22B(4,𝔽2)×C7111yescharacteristic
292C4—21no—
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C2—21no—
305C2—21no—
306C2C71×#160311yescharacteristic
307C2C71×#220111yescharacteristic
308C2C71×#220411yescharacteristic
subgroupG/H
center, socleC22×C71B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C71
prime radicalC23×C71#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C71 ◁ C2×C4⋊C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
149. #12241C71direct (N × H)
162. C71#12241direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#27692
C71×#20612
C71×#16172
C71×#16112
C71×#13532
C71×#13462
C71×#13242
#1224171