Finite Group Databasebeta

← #31913760

Subgroups — #31913760

Order 18176.

442 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D71C211yesmaximal, characteristic
2#22589950C211yesmaximal, characteristic
3#22588375C211yesmaximal, characteristic
4#22586450C211yesmaximal, characteristic, complemented
5#22586449C211yesmaximal, characteristic, complemented
6#22586448C211yesmaximal, characteristic, complemented
7C71×#1387C211yesmaximal, characteristic, complemented
8C22×C8×D71C2211yescharacteristic
9C2×C4⋊C4×D71—21no—
10C2×#14556068C2211yescharacteristic
11C2×#14556064C2211yescharacteristic
12C2×#14556062C2211yescharacteristic
13C2×#14556061—21no—
14#107×D71—21no—
15#107×D71—21no—
16#107×D71C412yes—
17#107×D71C412yes—
18#17985962—21no—
19(C2×D71)⋊C16—21no—
20(C2×D71)⋊C16—21no—
21#17985763—21no—
22C2×M32×C71C2211yescharacteristic
23C2×#108×C71C2211yescharacteristic
24C2×#107×C71C2211yescharacteristic
25C22×C4×D71—41no—
26C22×C4×D71D411yescharacteristic
27C2×#11765849—21no—
28C22×C71⋊C8D411yescharacteristic
29C2×C8×D71—21no—
30C2×C8×D71—21no—
31C2×C8×D71—21no—
32C2×C8×D71—21no—
33C2×C8×D71C2×C411yescharacteristic
34C2×C8×D71C2×C411yescharacteristic
35C2×C8×D71D411yescharacteristic
36C2×C8×D71D411yescharacteristic
37C2×C4⋊Dic71—21no—
38C2×Dic71⋊C4—41no—
39C4⋊C4×D71—22no—
40C4⋊C4×D71—22no—
41C4⋊C4×D71—41no—
42C4⋊C4×D71—41no—
43#14556068—21no—
44#14556068—21no—
45C2×C71⋊C16—21no—
46#14556064—21no—
47#14556064—21no—
48#14556062—21no—
49#14556062C2×C412yes—
50#14556061—22no—
51#14556061—22no—
52(C2×D71)⋊C8—41no—
53(C2×D71)⋊C8—41no—
54C2×M16×C71—21no—
55C22×C8×C71C2311yescharacteristic
56C2×C4⋊C4×C71—21no—
57M32×C71—21no—
58M32×C71—21no—
59C2×C16×C71—21no—
60#108×C71—21no—
61#108×C71—21no—
62#107×C71—21no—
63#107×C71C2×C412yes—
64C23×D71SD1611yescharacteristic
65C22×Dic71—41no—
66C22×Dic71SD1611yescharacteristic
67C2×C4×D71—21no—
68C2×C4×D71—21no—
69C2×C4×D71—21no—
70C2×C4×D71—21no—
71C2×C4×D71—42no—
72C2×C4×D71—42no—
73C2×C4×D71—84no—
74C2×C4×D71C22⋊C411yescharacteristic
75C2×C4×D71C22⋊C411yescharacteristic
76C2×C4×D71D811yescharacteristic
77C2×C4×D71D811yescharacteristic
78#11765849—41no—
79#11765849—41no—
80C2×C71⋊C8—21no—
81C2×C71⋊C8—21no—
82C2×C71⋊C8—41no—
83C2×C71⋊C8C22⋊C411yescharacteristic
84C2×C71⋊C8C22⋊C411yescharacteristic
85C8×D71—21no—
86C8×D71—21no—
87C8×D71—21no—
88C8×D71—21no—
89C8×D71—21no—
90C8×D71—21no—
91C8×D71—21no—
92C8×D71—21no—
93C4⋊Dic71—22no—
94C4⋊Dic71—41no—
95Dic71⋊C4—42no—
96Dic71⋊C4—42no—
97C71⋊C16—41no—
98C22×C4×C71—41no—
99C22×C4×C71C2×D411yescharacteristic
100M16×C71—41no—
101M16×C71—41no—
102C2×C8×C71—41no—
103C2×C8×C71C22×C411yescharacteristic
104C2×C8×C71C2×D411yescharacteristic
105C2×C8×C71#4111yescharacteristic
106C2×C8×C71#4111yescharacteristic
107C2×C8×C71#4111yescharacteristic
108C2×C8×C71#4111yescharacteristic
109C4⋊C4×C71—22no—
110C4⋊C4×C71—41no—
111C16×C71—41no—
112C23×C71C2×SD1611yescharacteristic
113C22×D71—21no—
114C22×D71—21no—
115C22×D71—22no—
116C22×D71—22no—
117C22×D71#10211yescharacteristic
118C22×D71#10211yescharacteristic
119C2×C4×C71—42no—
120C2×C4×C71—82no—
121C2×C4×C71C2×C22⋊C411yescharacteristic
122C2×C4×C71#12311yescharacteristic
123C2×C4×C71#12311yescharacteristic
124C2×C4×C71C2×D811yescharacteristic
125C2×C4×C71#13711yescharacteristic
126C2×C4×C71#13711yescharacteristic
127C2×Dic71—21no—
128C2×Dic71—21no—
129C2×Dic71—42no—
130C2×Dic71—82no—
131C2×Dic71#10211yescharacteristic
132C2×Dic71#10211yescharacteristic
133C4×D71—21no—
134C4×D71—21no—
135C4×D71—21no—
136C4×D71—21no—
137C4×D71—21no—
138C4×D71—21no—
139C4×D71—21no—
140C4×D71—21no—
141C4×D71—84no—
142C4×D71—84no—
143C8×C71—81no—
144C8×C71#11711yescharacteristic
145C8×C71#11711yescharacteristic
146C8×C71#12311yescharacteristic
147C8×C71#12311yescharacteristic
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—81no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71C2×#10211yescharacteristic
154C22×C71#48311yescharacteristic
155C22×C71#48311yescharacteristic
156C2×D71—24no—
157C2×D71—24no—
158C2×D71—21no—
159C2×D71—21no—
160C2×D71—21no—
161C2×D71—21no—
162C2×D71#36112yes—
163C2×D71#36212yes—
164C4×C71—82no—
165C4×C71#38111yescharacteristic
166C4×C71#41911yescharacteristic
167C4×C71#48011yescharacteristic
168C4×C71#48011yescharacteristic
169Dic71—21no—
170Dic71—21no—
171Dic71—82no—
172#12871—711nomaximal, Sylow
173C2×C71—21no—
174C2×C71—21no—
175C2×C71#182311yescharacteristic
176C2×C71C2×#36111yescharacteristic
177C2×C71C2×#36211yescharacteristic
178D71—22no—
179D71—22no—
180C22×#107—711no—
181#2112—711no—
182#1810—711no—
183#1387—711no—
184#1387—711no—
185#1387—711no—
186#1387D7111yescharacteristic, complemented
187C71#1287111yescharacteristic, complemented, Sylow
188C23×C8—711no—
189C22×C4⋊C4—1421no—
190C2×M32—711no—
191C2×M32C2×D7111yescharacteristic
192C2×#108—711no—
193C2×#108C2×D7111yescharacteristic
194C2×#107—711no—
195C2×#107—712no—
196C2×#107—712no—
197C2×#107—1421no—
198C2×#107—1421no—
199C2×#107—1421no—
200C2×#107C2×D7111yescharacteristic
201#407—1421no—
202C22⋊C16—1421no—
203C22⋊C16—1421no—
204#341—1421no—
205C23×C4—2841no—
206C23×C4—711no—
207C2×M16—21no—
208C2×M16—1421no—
209C22×C8—711no—
210C22×C8—711no—
211C22×C8—711no—
212C22×C8—711no—
213C22×C8—711no—
214C22×C8—1421no—
215C22×C8—1421no—
216C22×C8—1421no—
217C22×C8—1421no—
218C22×C8C22×D7111yescharacteristic
219C2×C4⋊C4—21no—
220C2×C4⋊C4—2841no—
221C2×C4⋊C4—2841no—
222C2×C4⋊C4—2841no—
223C2×C4⋊C4—1422no—
224C2×C4⋊C4—1422no—
225C2×C4⋊C4—1421no—
226M32—21no—
227M32—21no—
228M32—1421no—
229M32—1421no—
230C2×C16—21no—
231C2×C16—1421no—
232#108—21no—
233#108—21no—
234#108—1421no—
235#108—1421no—
236#107—21no—
237#107—712no—
238#107—1422no—
239#107—1422no—
240#107—1421no—
241#107C4×D7112yes—
242C22⋊C8—2841no—
243C22⋊C8—2841no—
244C24—711no—
245C22×C4—41no—
246C22×C4—2842no—
247C22×C4—2842no—
248C22×C4—2841no—
249C22×C4—5684no—
250C22×C4—711no—
251C22×C4—711no—
252C22×C4—711no—
253C22×C4—711no—
254C22×C4—711no—
255C22×C4—1421no—
256C22×C4—1421no—
257C22×C4—1421no—
258C22×C4—1421no—
259C22×C4D4×D7111yescharacteristic
260M16—41no—
261M16—41no—
262M16—2841no—
263M16—2841no—
264C2×C8—41no—
265C2×C8—2841no—
266C2×C8—711no—
267C2×C8—711no—
268C2×C8—1421no—
269C2×C8—1421no—
270C2×C8—1421no—
271C2×C8—1421no—
272C2×C8—1421no—
273C2×C8—1421no—
274C2×C8—1421no—
275C2×C8—1421no—
276C2×C8—1421no—
277C2×C8—1421no—
278C2×C8C2×C4×D7111yescharacteristic
279C2×C8D4×D7111yescharacteristic
280C2×C8#1176586411yescharacteristic
281C2×C8#1176586511yescharacteristic
282C2×C8#1176586511yescharacteristic
283C2×C8#1176586611yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—2842no—
287C4⋊C4—2842no—
288C4⋊C4—2841no—
289C4⋊C4—1422no—
290C16—41no—
291C16—2841no—
292C23—711no—
293C23—711no—
294C23—1422no—
295C23—1422no—
296C23—1421no—
297C23—1421no—
298C23SD16×D7111yescharacteristic
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—2842no—
302C2×C4—5684no—
303C2×C4—5684no—
304C2×C4—5682no—
305C2×C4—711no—
306C2×C4—711no—
307C2×C4—1421no—
308C2×C4—1421no—
309C2×C4—1421no—
310C2×C4—1421no—
311C2×C4—1421no—
312C2×C4—1421no—
313C2×C4—1421no—
314C2×C4—1421no—
315C2×C4—1421no—
316C2×C4—1421no—
317C2×C4C22⋊C4×D7111yescharacteristic
318C2×C4#1455611011yescharacteristic
319C2×C4#1455611311yescharacteristic
320C2×C4D8×D7111yescharacteristic
321C2×C4#1455615711yescharacteristic
322C2×C4#1455616011yescharacteristic
323C8—81no—
324C8—5681no—
325C8—1421no—
326C8—1421no—
327C8#1455608511yescharacteristic
328C8#1455608611yescharacteristic
329C8#1455611011yescharacteristic
330C8#1455611311yescharacteristic
331C22—21no—
332C22—21no—
333C22—712no—
334C22—712no—
335C22—1424no—
336C22—1424no—
337C22—1421no—
338C22—1421no—
339C22—1421no—
340C22—1421no—
341C22#102×D7111yescharacteristic
342C22#1798632811yescharacteristic
343C22#1798633111yescharacteristic
344C4—82no—
345C4—5682no—
346C4—1421no—
347C4—1421no—
348C4#1798586111yescharacteristic
349C4#1798600611yescharacteristic
350C4#1798631111yescharacteristic
351C4#1798631411yescharacteristic
352C2—21no—
353C2—21no—
354C2—1422no—
355C2—1422no—
356C2#2258842511yescharacteristic
357C2#361×D7111yescharacteristic
358C2#362×D7111yescharacteristic
subgroupG/H
centerC22#102×D71
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC71×#1387C2
socleC22×C71C2×#102
prime radicalC23×D71SD16
cosocleC22×C8×C71C23
hypercenter#1387D71
hypocenterC71#12871
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1387

Complements

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

NH ≅ G/Ntype
4. #22586450C2semidirect
5. #22586449C2semidirect
6. #22586448C2semidirect
7. C71×#1387C2semidirect
186. #1387D71semidirect
187. C71#12871semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D712
#225899502
#225883752
#225864502
#225864492
#225864482
C71×#13872
#1287171