Finite Group Databasebeta

← C71×#13249 #31834121

Subgroups — C71×#13249 #31834121

Order 18176.

444 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#509×C71C211yesmaximal, characteristic
2C2×#365×C71C211yesmaximal, characteristic, complemented
3C2×#361×C71C211yesmaximal, characteristic, complemented
4C2×#360×C71C211yesmaximal, characteristic, complemented
5C2×#347×C71C211yesmaximal, characteristic, complemented
6C71×#1975C211yesmaximal, characteristic, complemented
7C71×#1972C211yesmaximal, characteristic, complemented
8C2×#135×C71C2211yescharacteristic
9C22×D8×C71C2211yescharacteristic
10#509×C71—22no—
11#509×C71—22no—
12C2×SD32×C71C412yescomplemented
13C2×D16×C71C412yescomplemented
14C2×M32×C71C2211yescharacteristic
15C2×M32×C71C2211yescharacteristic
16C2×#108×C71C2211yescharacteristic
17C2×#106×C71C2211yescharacteristic
18C2×#104×C71—21no—
19C2×#102×C71—21no—
20#408×C71—21no—
21C2×#97×C71C2211yescharacteristic
22#390×C71—21no—
23#365×C71—22no—
24#361×C71—22no—
25#360×C71—22no—
26#347×C71—22no—
27C2×#41×C71—21no—
28C22×D4×C71—21no—
29#135×C71—21no—
30#135×C71—21no—
31#135×C71—21no—
32#135×C71—21no—
33C2×Q16×C71C2×C411yescharacteristic
34C2×SD16×C71—21no—
35C2×D8×C71—21no—
36C2×D8×C71—21no—
37C2×D8×C71—21no—
38C2×D8×C71—21no—
39C2×D8×C71—21no—
40C2×D8×C71C2×C411yescharacteristic
41C2×D8×C71C2×C412yes—
42C2×M16×C71—21no—
43C22×C8×C71D411yescharacteristic
44C22×C8×C71C2311yescharacteristic
45C2×C4⋊C4×C71—21no—
46C2×C22⋊C4×C71—42no—
47C2×C42×C71D411yescharacteristic
48SD32×C71—24no—
49D16×C71—24no—
50M32×C71—21no—
51M32×C71D412yes—
52M32×C71D412yes—
53M32×C71D412yes—
54C2×C16×C71—21no—
55C2×C16×C71C2×C412yes—
56#108×C71—21no—
57#108×C71—21no—
58#106×C71—21no—
59#106×C71—21no—
60#104×C71—42no—
61#102×C71—42no—
62C22⋊C8×C71—42no—
63#97×C71—21no—
64#97×C71—21no—
65C24×C71—41no—
66#41×C71—41no—
67#41×C71—41no—
68#41×C71—41no—
69#41×C71—41no—
70C2×Q8×C71—21no—
71C2×D4×C71—22no—
72C2×D4×C71—21no—
73C2×D4×C71—41no—
74C2×D4×C71—41no—
75C2×D4×C71—41no—
76C2×D4×C71—41no—
77C2×D4×C71—41no—
78C2×D4×C71—41no—
79C22×C4×C71—21no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71C2×D411yescharacteristic
83Q16×C71—21no—
84Q16×C71—21no—
85SD16×C71—41no—
86SD16×C71—41no—
87D8×C71—21no—
88D8×C71—22no—
89D8×C71—22no—
90D8×C71—21no—
91D8×C71—41no—
92D8×C71—41no—
93M16×C71—41no—
94M16×C71—41no—
95C2×C8×C71—21no—
96C2×C8×C71—21no—
97C2×C8×C71—21no—
98C2×C8×C71—41no—
99C2×C8×C71C22×C411yescharacteristic
100C2×C8×C71C2×D411yescharacteristic
101C2×C8×C71C2×D411yescharacteristic
102C2×C8×C71C2×D411yescharacteristic
103C2×C8×C71#4111yescharacteristic
104C2×C8×C71#4111yescharacteristic
105C4⋊C4×C71—41no—
106C4⋊C4×C71—41no—
107C22⋊C4×C71—84no—
108C42×C71—21no—
109C42×C71—21no—
110C16×C71—22no—
111C16×C71C22⋊C414yescomplemented
112C23×C71—41no—
113C23×C71—42no—
114C23×C71—82no—
115C23×C71—82no—
116C23×C71—81no—
117C23×C71—81no—
118C23×C71AGL(1,ℤ/8)11yescharacteristic
119Q8×C71—41no—
120Q8×C71—41no—
121D4×C71—42no—
122D4×C71—41no—
123D4×C71—42no—
124D4×C71—41no—
125D4×C71—81no—
126D4×C71—81no—
127D4×C71—81no—
128D4×C71—81no—
129C2×C4×C71—42no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—81no—
133C2×C4×C71—81no—
134C2×C4×C71—82no—
135C2×C4×C71—81no—
136C2×C4×C71C4×D411yescharacteristic
137C2×C4×C71#12111yescharacteristic
138C2×C4×C71#12711yescharacteristic
139C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
142C8×C71—41no—
143C8×C71—81no—
144C8×C71C2×C22⋊C411yescharacteristic
145C8×C71C4×D411yescharacteristic
146C8×C71#12111yescharacteristic
147C8×C71#12411yescharacteristic
148C22×C71—22no—
149C22×C71—82no—
150C22×C71—84no—
151C22×C71—81no—
152C22×C71—82no—
153C22×C71—162no—
154C22×C71—161no—
155C22×C71—161no—
156C22×C71#44211yescharacteristic
157C22×C71#46911yescharacteristic
158C22×C71#49611yescharacteristic
159C4×C71—81no—
160C4×C71—81no—
161C4×C71—161no—
162C4×C71#39011yescharacteristic
163C4×C71#44211yescharacteristic
164C4×C71#46911yescharacteristic
165C4×C71#49611yescharacteristic
166#13249C7111yesmaximal, characteristic, direct product factor, Sylow
167C2×C71—21no—
168C2×C71—21no—
169C2×C71—161no—
170C2×C71—162no—
171C2×C71#197211yescharacteristic
172C2×C71AGL(1,ℤ/16)12yes—
173C2×#509C2×C7111yescharacteristic
174C2×#365C2×C7111yescharacteristic, complemented
175C2×#361C2×C7111yescharacteristic, complemented
176C2×#360C2×C7111yescharacteristic, complemented
177C2×#347C2×C7111yescharacteristic, complemented
178#1975C2×C7111yescharacteristic, complemented
179#1972C2×C7111yescharacteristic, complemented
180C71#1324911yescharacteristic, direct product factor, Sylow
181C2×#135C22×C7111yescharacteristic
182C22×D8C22×C7111yescharacteristic
183#509—22no—
184#509—22no—
185C2×SD32C4×C7112yescomplemented
186C2×D16C4×C7112yescomplemented
187C2×M32C22×C7111yescharacteristic
188C2×M32C22×C7111yescharacteristic
189C2×#108C22×C7111yescharacteristic
190C2×#106C22×C7111yescharacteristic
191C2×#104—21no—
192C2×#102—21no—
193#408—21no—
194C2×#97C22×C7111yescharacteristic
195#390—21no—
196#365—22no—
197#361—22no—
198#360—22no—
199#347—22no—
200C2×#41—21no—
201C22×D4—21no—
202#135—21no—
203#135—21no—
204#135—21no—
205#135—21no—
206C2×Q16C2×C4×C7111yescharacteristic
207C2×SD16—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8C2×C4×C7112yes—
214C2×D8C2×C4×C7111yescharacteristic
215C2×M16—21no—
216C22×C8D4×C7111yescharacteristic
217C22×C8C23×C7111yescharacteristic
218C2×C4⋊C4—21no—
219C2×C22⋊C4—42no—
220C2×C42D4×C7111yescharacteristic
221SD32—24no—
222D16—24no—
223M32—21no—
224M32D4×C7112yes—
225M32D4×C7112yes—
226M32D4×C7112yes—
227C2×C16—21no—
228C2×C16C2×C4×C7112yes—
229#108—21no—
230#108—21no—
231#106—21no—
232#106—21no—
233#104—42no—
234#102—42no—
235C22⋊C8—42no—
236#97—21no—
237#97—21no—
238C24—41no—
239#41—41no—
240#41—41no—
241#41—41no—
242#41—41no—
243C2×Q8—21no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×D4×C7111yescharacteristic
256Q16—21no—
257Q16—21no—
258SD16—41no—
259SD16—41no—
260D8—21no—
261D8—22no—
262D8—22no—
263D8—21no—
264D8—41no—
265D8—41no—
266M16—41no—
267M16—41no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8C22×C4×C7111yescharacteristic
273C2×C8C2×D4×C7111yescharacteristic
274C2×C8C2×D4×C7111yescharacteristic
275C2×C8C2×D4×C7111yescharacteristic
276C2×C8#41×C7111yescharacteristic
277C2×C8#41×C7111yescharacteristic
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—84no—
281C42—21no—
282C42—21no—
283C16—22no—
284C16C22⋊C4×C7114yescomplemented
285C23—41no—
286C23—42no—
287C23—82no—
288C23—82no—
289C23—81no—
290C23—81no—
291C23AGL(1,ℤ/8)×C7111yescharacteristic
292Q8—41no—
293Q8—41no—
294D4—42no—
295D4—41no—
296D4—42no—
297D4—41no—
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4C4×D4×C7111yescharacteristic
310C2×C4#121×C7111yescharacteristic
311C2×C4#127×C7111yescharacteristic
312C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
313C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
314C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
315C8—41no—
316C8—81no—
317C8C2×C22⋊C4×C7111yescharacteristic
318C8C4×D4×C7111yescharacteristic
319C8#121×C7111yescharacteristic
320C8#124×C7111yescharacteristic
321C22—22no—
322C22—84no—
323C22—82no—
324C22—81no—
325C22—82no—
326C22—162no—
327C22—161no—
328C22—161no—
329C22#442×C7111yescharacteristic
330C22#469×C7111yescharacteristic
331C22#496×C7111yescharacteristic
332C4—81no—
333C4—81no—
334C4—161no—
335C4#390×C7111yescharacteristic
336C4#442×C7111yescharacteristic
337C4#469×C7111yescharacteristic
338C4#496×C7111yescharacteristic
339C2—21no—
340C2—21no—
341C2—162no—
342C2—161no—
343C2C71×#197211yescharacteristic
344C2C71×AGL(1,ℤ/16)12yes—
subgroupG/H
center, socleC22×C71#442
commutatorC2×C8C22×C4×C71
Frattini, cosocleC22×C8C23×C71
prime radicalC2×#509×C71C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×#97×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#365×C71C2semidirect
3. C2×#361×C71C2semidirect
4. C2×#360×C71C2semidirect
5. C2×#347×C71C2semidirect
6. C71×#1975C2semidirect
7. C71×#1972C2semidirect
12. C2×SD32×C71C4semidirect
13. C2×D16×C71C4semidirect
111. C16×C71C22⋊C4semidirect
166. #13249C71direct (N × H)
174. C2×#365C2×C71semidirect
175. C2×#361C2×C71semidirect
176. C2×#360C2×C71semidirect
177. C2×#347C2×C71semidirect
178. #1975C2×C71semidirect
179. #1972C2×C71semidirect
180. C71#13249direct (N × H)
185. C2×SD32C4×C71semidirect
186. C2×D16C4×C71semidirect
284. C16C22⋊C4×C71semidirect

Maximal subgroups

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

subgroupindex
C2×#509×C712
C2×#365×C712
C2×#361×C712
C2×#360×C712
C2×#347×C712
C71×#19752
C71×#19722
#1324971