Finite Group Databasebeta

← C71×#13388 #31834260

Subgroups — C71×#13388 #31834260

Order 18176.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3059C211yesmaximal, characteristic
2C71×#2159C211yesmaximal, characteristic, complemented
3C2×#362×C71C211yesmaximal, characteristic, complemented
4C2×#358×C71C211yesmaximal, characteristic, complemented
5C2×#350×C71C211yesmaximal, characteristic
6C71×#1816C211yesmaximal, characteristic
7C71×#1382C211yesmaximal, characteristic, complemented
8C22×Q16×C71C2211yescharacteristic
9C2×#131×C71C2211yescharacteristic
10#536×C71—21no—
11C2×M32×C71C2211yescharacteristic
12C22×C16×C71C2211yescharacteristic
13#470×C71—22no—
14#467×C71—21no—
15#467×C71—21no—
16C2×#108×C71C2211yescharacteristic
17#427×C71C2211yescharacteristic
18C2×#107×C71C2211yescharacteristic
19#419×C71—21no—
20#417×C71—21no—
21#367×C71—21no—
22#366×C71—21no—
23#362×C71—22no—
24#358×C71—22no—
25#350×C71—22no—
26#343×C71—21no—
27C2×#41×C71D411yescharacteristic
28C22×Q8×C71—21no—
29C2×Q16×C71—21no—
30C2×Q16×C71—21no—
31C2×Q16×C71—21no—
32C2×Q16×C71—21no—
33C2×Q16×C71—21no—
34C2×Q16×C71D412yes—
35#131×C71—21no—
36#131×C71—21no—
37#131×C71—21no—
38#131×C71—21no—
39C2×M16×C71—21no—
40C2×M16×C71—21no—
41C2×M16×C71D411yescharacteristic
42C22×C8×C71—21no—
43C22×C8×C71C2311yescharacteristic
44C4⋊Q8×C71—41no—
45#124×C71—41no—
46#123×C71—42no—
47C22⋊Q8×C71—22no—
48C22⋊Q8×C71—22no—
49#117×C71—21no—
50C2×C4⋊C4×C71—21no—
51M32×C71—21no—
52M32×C71D412yes—
53C2×C16×C71—21no—
54C2×C16×C71—21no—
55C2×C16×C71—21no—
56C2×C16×C71—21no—
57#108×C71—21no—
58#108×C71—21no—
59#107×C71—21no—
60#107×C71—21no—
61#107×C71D411yescharacteristic
62#107×C71D411yescharacteristic
63#106×C71—21no—
64#103×C71—22no—
65#103×C71—22no—
66#103×C71—41no—
67#102×C71—41no—
68#41×C71—21no—
69#41×C71—21no—
70#41×C71—21no—
71#41×C71—21no—
72C2×Q8×C71—22no—
73C2×Q8×C71—41no—
74C2×Q8×C71—41no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C2×Q8×C71D811yescharacteristic
79C2×D4×C71—21no—
80C2×D4×C71D811yescharacteristic
81C22×C4×C71—21no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71C2×D411yescharacteristic
85Q16×C71—22no—
86Q16×C71—22no—
87Q16×C71—41no—
88Q16×C71—41no—
89M16×C71—21no—
90M16×C71—21no—
91M16×C71—41no—
92M16×C71—41no—
93M16×C71—41no—
94M16×C71—41no—
95C2×C8×C71—21no—
96C2×C8×C71—41no—
97C2×C8×C71—41no—
98C2×C8×C71—41no—
99C2×C8×C71C2×D411yescharacteristic
100C2×C8×C71C2×D411yescharacteristic
101C2×C8×C71C2×D411yescharacteristic
102C2×C8×C71#4111yescharacteristic
103C2×C8×C71#4111yescharacteristic
104C2×C8×C71#4111yescharacteristic
105C4⋊C4×C71—21no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71—42no—
108C4⋊C4×C71—42no—
109C4⋊C4×C71—41no—
110C4⋊C4×C71—41no—
111C22⋊C4×C71—42no—
112C22⋊C4×C71—42no—
113C22⋊C4×C71—41no—
114C42×C71—41no—
115C16×C71—22no—
116C16×C71—41no—
117C23×C71—21no—
118C23×C71AGL(1,ℤ/8)11yescharacteristic
119Q8×C71—21no—
120Q8×C71—21no—
121Q8×C71—42no—
122Q8×C71—42no—
123Q8×C71—81no—
124Q8×C71—81no—
125D4×C71—21no—
126D4×C71—21no—
127D4×C71—41no—
128D4×C71—41no—
129C2×C4×C71—21no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—42no—
135C2×C4×C71—81no—
136C2×C4×C71—81no—
137C2×C4×C71—81no—
138C2×C4×C71—161no—
139C2×C4×C71#12011yescharacteristic
140C2×C4×C71#12111yescharacteristic
141C2×C4×C71#12311yescharacteristic
142C2×C4×C71C2×D811yescharacteristic
143C2×C4×C71#13511yescharacteristic
144C2×C4×C71#13711yescharacteristic
145C8×C71—41no—
146C8×C71—81no—
147C8×C71#12111yescharacteristic
148C8×C71#12111yescharacteristic
149C8×C71#12311yescharacteristic
150C8×C71#12411yescharacteristic
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71#44711yescharacteristic
157C22×C71#46811yescharacteristic
158C22×C71#48211yescharacteristic
159C4×C71—41no—
160C4×C71—81no—
161C4×C71—81no—
162C4×C71—82no—
163C4×C71—161no—
164C4×C71#39411yescharacteristic
165C4×C71#45211yescharacteristic
166C4×C71#46611yescharacteristic
167C4×C71#48011yescharacteristic
168#13388C7111yesmaximal, characteristic, direct product factor, Sylow
169C2×C71—21no—
170C2×C71—21no—
171C2×C71—81no—
172C2×C71#203311yescharacteristic
173C2×C71#219311yescharacteristic
174C2×C71#219511yescharacteristic
175#3059C2×C7111yescharacteristic
176#2159C2×C7111yescharacteristic, complemented
177C2×#362C2×C7111yescharacteristic, complemented
178C2×#358C2×C7111yescharacteristic, complemented
179C2×#350C2×C7111yescharacteristic
180#1816C2×C7111yescharacteristic
181#1382C2×C7111yescharacteristic, complemented
182C71#1338811yescharacteristic, direct product factor, Sylow
183C22×Q16C22×C7111yescharacteristic
184C2×#131C22×C7111yescharacteristic
185#536—21no—
186C2×M32C22×C7111yescharacteristic
187C22×C16C22×C7111yescharacteristic
188#470—22no—
189#467—21no—
190#467—21no—
191C2×#108C22×C7111yescharacteristic
192#427C22×C7111yescharacteristic
193C2×#107C22×C7111yescharacteristic
194#419—21no—
195#417—21no—
196#367—21no—
197#366—21no—
198#362—22no—
199#358—22no—
200#350—22no—
201#343—21no—
202C2×#41D4×C7111yescharacteristic
203C22×Q8—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—21no—
207C2×Q16—21no—
208C2×Q16—21no—
209C2×Q16D4×C7112yes—
210#131—21no—
211#131—21no—
212#131—21no—
213#131—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16D4×C7111yescharacteristic
217C22×C8—21no—
218C22×C8C23×C7111yescharacteristic
219C4⋊Q8—41no—
220#124—41no—
221#123—42no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224#117—21no—
225C2×C4⋊C4—21no—
226M32—21no—
227M32D4×C7112yes—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232#108—21no—
233#108—21no—
234#107—21no—
235#107—21no—
236#107D4×C7111yescharacteristic
237#107D4×C7111yescharacteristic
238#106—21no—
239#103—22no—
240#103—22no—
241#103—41no—
242#102—41no—
243#41—21no—
244#41—21no—
245#41—21no—
246#41—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8D8×C7111yescharacteristic
254C2×D4—21no—
255C2×D4D8×C7111yescharacteristic
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C2×D4×C7111yescharacteristic
260Q16—22no—
261Q16—22no—
262Q16—41no—
263Q16—41no—
264M16—21no—
265M16—21no—
266M16—41no—
267M16—41no—
268M16—41no—
269M16—41no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8C2×D4×C7111yescharacteristic
275C2×C8C2×D4×C7111yescharacteristic
276C2×C8C2×D4×C7111yescharacteristic
277C2×C8#41×C7111yescharacteristic
278C2×C8#41×C7111yescharacteristic
279C2×C8#41×C7111yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C42—41no—
290C16—22no—
291C16—41no—
292C23—21no—
293C23AGL(1,ℤ/8)×C7111yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—42no—
298Q8—81no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—161no—
314C2×C4#120×C7111yescharacteristic
315C2×C4#121×C7111yescharacteristic
316C2×C4#123×C7111yescharacteristic
317C2×C4C2×D8×C7111yescharacteristic
318C2×C4#135×C7111yescharacteristic
319C2×C4#137×C7111yescharacteristic
320C8—41no—
321C8—81no—
322C8#121×C7111yescharacteristic
323C8#121×C7111yescharacteristic
324C8#123×C7111yescharacteristic
325C8#124×C7111yescharacteristic
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22#447×C7111yescharacteristic
332C22#468×C7111yescharacteristic
333C22#482×C7111yescharacteristic
334C4—41no—
335C4—82no—
336C4—81no—
337C4—81no—
338C4—161no—
339C4#394×C7111yescharacteristic
340C4#452×C7111yescharacteristic
341C4#466×C7111yescharacteristic
342C4#480×C7111yescharacteristic
343C2—21no—
344C2—21no—
345C2—81no—
346C2C71×#203311yescharacteristic
347C2C71×#219311yescharacteristic
348C2C71×#219511yescharacteristic
subgroupG/H
center, socleC22×C71#447
commutator, Frattini, cosocleC22×C8C23×C71
prime radicalC2×#41×C71D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×#131×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2159C2semidirect
3. C2×#362×C71C2semidirect
4. C2×#358×C71C2semidirect
7. C71×#1382C2semidirect
168. #13388C71direct (N × H)
176. #2159C2×C71semidirect
177. C2×#362C2×C71semidirect
178. C2×#358C2×C71semidirect
181. #1382C2×C71semidirect
182. C71#13388direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#30592
C71×#21592
C2×#362×C712
C2×#358×C712
C2×#350×C712
C71×#18162
C71×#13822
#1338871