Finite Group Databasebeta

← C71×#9303 #31830175

Subgroups — C71×#9303 #31830175

Order 18176.

488 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3077C211yesmaximal, characteristic
2C71×#2972C211yesmaximal, characteristic
3C71×#1804C211yesmaximal, characteristic
4C71×#1518C211yesmaximal, characteristic, complemented
5C71×#1481C211yesmaximal, characteristic, complemented
6C71×#1301C211yesmaximal, characteristic, complemented
7C71×#1286C211yesmaximal, characteristic, complemented
8#533×C71C2211yescharacteristic
9C2×C4×D4×C71C2211yescharacteristic
10#478×C71—22no—
11#476×C71C412yescomplemented
12#474×C71C412yescomplemented
13C8×D4×C71—22no—
14C8×D4×C71—22no—
15#431×C71C2211yescharacteristic
16#431×C71C2211yescharacteristic
17#428×C71—21no—
18#423×C71C2211yescharacteristic
19C2×C4⋊C8×C71C2211yescharacteristic
20#418×C71—21no—
21C2×#102×C71—21no—
22C2×C22⋊C8×C71—22no—
23#382×C71C2211yescharacteristic
24#340×C71—21no—
25#328×C71—22no—
26Q8⋊C8×C71—22no—
27C22×D4×C71D411yescharacteristic
28C23×C4×C71—21no—
29C2×M16×C71—21no—
30C22×C8×C71—24no—
31C22×C8×C71—21no—
32C4⋊Q8×C71C2×C411yescharacteristic
33C4⋊Q8×C71C2×C411yescharacteristic
34#125×C71—21no—
35C22⋊Q8×C71—21no—
36C22⋊Q8×C71—21no—
37C4×Q8×C71—21no—
38C4×Q8×C71—21no—
39C4×D4×C71—21no—
40C4×D4×C71—21no—
41C4×D4×C71C2×C412yes—
42C4×D4×C71D412yes—
43#117×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C42×C71C2311yescharacteristic
48#107×C71—22no—
49#106×C71—22no—
50C4⋊C8×C71—21no—
51C4⋊C8×C71—21no—
52C4⋊C8×C71—21no—
53C4⋊C8×C71—21no—
54C4⋊C8×C71—21no—
55C4⋊C8×C71C2×C412yes—
56#102×C71—22no—
57#102×C71—22no—
58#102×C71—42no—
59C22⋊C8×C71—24no—
60C22⋊C8×C71—21no—
61C22⋊C8×C71D412yes—
62C4×C8×C71—21no—
63C4×C8×C71D412yes—
64#95×C71—21no—
65#95×C71—21no—
66C24×C71—21no—
67C2×Q8×C71—21no—
68C2×Q8×C71—21no—
69C2×D4×C71—21no—
70C2×D4×C71—21no—
71C2×D4×C71—21no—
72C2×D4×C71C22⋊C412yes—
73C2×D4×C71SD1612yes—
74C2×D4×C71Q1612yes—
75C22×C4×C71—21no—
76C22×C4×C71—22no—
77C22×C4×C71—22no—
78C22×C4×C71—41no—
79C22×C4×C71—82no—
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4111yescharacteristic
83M16×C71—41no—
84M16×C71—41no—
85C2×C8×C71—22no—
86C2×C8×C71—22no—
87C2×C8×C71—22no—
88C2×C8×C71—41no—
89C2×C8×C71—41no—
90C2×C8×C71—41no—
91C2×C8×C71—84no—
92C4⋊C4×C71—21no—
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—21no—
96C4⋊C4×C71—21no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71C22⋊C412yes—
102C22⋊C4×C71—22no—
103C22⋊C4×C71—22no—
104C22⋊C4×C71—41no—
105C42×C71—41no—
106C42×C71C22×C411yescharacteristic
107C42×C71C2×D411yescharacteristic
108C42×C71C2×D411yescharacteristic
109C42×C71#4111yescharacteristic
110C23×C71—22no—
111C23×C71—22no—
112C23×C71—22no—
113C23×C71—81no—
114C23×C71—81no—
115C23×C71#12111yescharacteristic
116Q8×C71—42no—
117Q8×C71—81no—
118D4×C71—22no—
119D4×C71—22no—
120D4×C71—22no—
121D4×C71#10314yes—
122C2×C4×C71—22no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—44no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—82no—
129C2×C4×C71—81no—
130C2×C4×C71—81no—
131C2×C4×C71—81no—
132C2×C4×C71—81no—
133C2×C4×C71C2×C22⋊C411yescharacteristic
134C2×C4×C71C4×D411yescharacteristic
135C2×C4×C71C4×D411yescharacteristic
136C2×C4×C71#12011yescharacteristic
137C2×C4×C71#12111yescharacteristic
138C2×C4×C71#12311yescharacteristic
139C2×C4×C71C2×SD1611yescharacteristic
140C2×C4×C71C2×Q1611yescharacteristic
141C2×C4×C71#13511yescharacteristic
142C2×C4×C71#13511yescharacteristic
143C8×C71—42no—
144C8×C71—42no—
145C8×C71—42no—
146C8×C71—81no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—24no—
150C22×C71—42no—
151C22×C71—42no—
152C22×C71—82no—
153C22×C71—82no—
154C22×C71#38611yescharacteristic
155C22×C71#46511yescharacteristic
156C22×C71#46711yescharacteristic
157C4×C71—21no—
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—42no—
162C4×C71—81no—
163C4×C71—81no—
164C4×C71—161no—
165C4×C71C2×#10311yescharacteristic
166C4×C71#41611yescharacteristic
167C4×C71#46511yescharacteristic
168C4×C71#46711yescharacteristic
169#9303C7111yesmaximal, characteristic, direct product factor, Sylow
170C2×C71—21no—
171C2×C71—21no—
172C2×C71—44no—
173C2×C71#188111yescharacteristic
174C2×C71#189211yescharacteristic
175C2×C71#190411yescharacteristic
176#3077C2×C7111yescharacteristic
177#2972C2×C7111yescharacteristic
178#1804C2×C7111yescharacteristic
179#1518C2×C7111yescharacteristic, complemented
180#1481C2×C7111yescharacteristic, complemented
181#1301C2×C7111yescharacteristic, complemented
182#1286C2×C7111yescharacteristic, complemented
183C71#930311yescharacteristic, direct product factor, Sylow
184#533C22×C7111yescharacteristic
185C2×C4×D4C22×C7111yescharacteristic
186#478—22no—
187#476C4×C7112yescomplemented
188#474C4×C7112yescomplemented
189C8×D4—22no—
190C8×D4—22no—
191#431C22×C7111yescharacteristic
192#431C22×C7111yescharacteristic
193#428—21no—
194#423C22×C7111yescharacteristic
195C2×C4⋊C8C22×C7111yescharacteristic
196#418—21no—
197C2×#102—21no—
198C2×C22⋊C8—22no—
199#382C22×C7111yescharacteristic
200#340—21no—
201#328—22no—
202Q8⋊C8—22no—
203C22×D4D4×C7111yescharacteristic
204C23×C4—21no—
205C2×M16—21no—
206C22×C8—24no—
207C22×C8—21no—
208C4⋊Q8C2×C4×C7111yescharacteristic
209C4⋊Q8C2×C4×C7111yescharacteristic
210#125—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4C2×C4×C7112yes—
218C4×D4D4×C7112yes—
219#117—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4D4×C7111yescharacteristic
222C2×C22⋊C4—21no—
223C2×C42C23×C7111yescharacteristic
224#107—22no—
225#106—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8C2×C4×C7112yes—
232#102—22no—
233#102—22no—
234#102—42no—
235C22⋊C8—24no—
236C22⋊C8—21no—
237C22⋊C8D4×C7112yes—
238C4×C8—21no—
239C4×C8D4×C7112yes—
240#95—21no—
241#95—21no—
242C24—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊C4×C7112yes—
249C2×D4SD16×C7112yes—
250C2×D4Q16×C7112yes—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4C2×D4×C7111yescharacteristic
257C22×C4C2×D4×C7111yescharacteristic
258C22×C4#41×C7111yescharacteristic
259M16—41no—
260M16—41no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—84no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4C22⋊C4×C7112yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—41no—
281C42—41no—
282C42C22×C4×C7111yescharacteristic
283C42C2×D4×C7111yescharacteristic
284C42C2×D4×C7111yescharacteristic
285C42#41×C7111yescharacteristic
286C23—22no—
287C23—22no—
288C23—22no—
289C23—81no—
290C23—81no—
291C23#121×C7111yescharacteristic
292Q8—42no—
293Q8—81no—
294D4—22no—
295D4—22no—
296D4—22no—
297D4#103×C7114yes—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4C2×C22⋊C4×C7111yescharacteristic
310C2×C4C4×D4×C7111yescharacteristic
311C2×C4C4×D4×C7111yescharacteristic
312C2×C4#120×C7111yescharacteristic
313C2×C4#121×C7111yescharacteristic
314C2×C4#123×C7111yescharacteristic
315C2×C4C2×SD16×C7111yescharacteristic
316C2×C4C2×Q16×C7111yescharacteristic
317C2×C4#135×C7111yescharacteristic
318C2×C4#135×C7111yescharacteristic
319C8—42no—
320C8—42no—
321C8—42no—
322C8—81no—
323C22—21no—
324C22—21no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22#386×C7111yescharacteristic
331C22#465×C7111yescharacteristic
332C22#467×C7111yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—42no—
338C4—81no—
339C4—81no—
340C4—161no—
341C4C2×#103×C7111yescharacteristic
342C4#416×C7111yescharacteristic
343C4#465×C7111yescharacteristic
344C4#467×C7111yescharacteristic
345C2—21no—
346C2—21no—
347C2—44no—
348C2C71×#188111yescharacteristic
349C2C71×#189211yescharacteristic
350C2C71×#190411yescharacteristic
subgroupG/H
center, socleC22×C71#386
commutatorC42C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC22×D4×C71D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1518C2semidirect
5. C71×#1481C2semidirect
6. C71×#1301C2semidirect
7. C71×#1286C2semidirect
11. #476×C71C4semidirect
12. #474×C71C4semidirect
169. #9303C71direct (N × H)
179. #1518C2×C71semidirect
180. #1481C2×C71semidirect
181. #1301C2×C71semidirect
182. #1286C2×C71semidirect
183. C71#9303direct (N × H)
187. #476C4×C71semidirect
188. #474C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#30772
C71×#29722
C71×#18042
C71×#15182
C71×#14812
C71×#13012
C71×#12862
#930371