Finite Group Databasebeta

← C2×#364×D71 #32150446

Subgroups — C2×#364×D71 #32150446

Order 18176.

782 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D71C212yesmaximal
2C2×C4×C8×D71C211yesmaximal, characteristic
3C2×#17985820C211yesmaximal, characteristic, complemented
4C2×#17985819C212yesmaximal, complemented
5#364×D71C218yesmaximal, direct product factor
6C2×#364×C71C211yesmaximal, characteristic, complemented
7C22×C8×D71C411yescharacteristic
8C22×C8×D71C2211yescharacteristic
9C2×C42×D71C411yescharacteristic
10C2×#14556068C2212yes—
11C2×#14556067—22no—
12M32×D71—24no—
13M32×D71C2218yes—
14C2×C16×D71—22no—
15C2×C4×C71⋊C8C411yescharacteristic
16C2×C8×Dic71C411yescharacteristic
17C2×C8×Dic71C2211yescharacteristic
18C4×C8×D71C414yes—
19C4×C8×D71C2214yes—
20#17985820C2214yescomplemented
21#17985819C2218yescomplemented
22C2×M32×C71C2212yes—
23C2×C4×C8×C71C2211yescharacteristic
24#364×C71C2214yescomplemented
25C22×C4×D71—21no—
26C22×C4×D71C2×C411yescharacteristic
27C22×C71⋊C8C2×C411yescharacteristic
28C22×C71⋊C8C2×C411yescharacteristic
29C2×C8×D71C814yes—
30C2×C8×D71C812yes—
31C2×C8×D71C812yes—
32C2×C8×D71C2×C412yes—
33C2×C8×D71C2×C414yes—
34C2×C8×D71D411yescharacteristic
35C2×C8×D71D412yes—
36C2×C8×D71D411yescharacteristic
37C2×C8×D71Q812yes—
38C2×C8×D71Q811yescharacteristic
39C2×C8×D71Q811yescharacteristic
40C2×C8×D71C2312yes—
41C2×C4×Dic71—21no—
42C2×C4×Dic71C2×C411yescharacteristic
43C42×D71—22no—
44C42×D71C2×C414yes—
45#14556068—22no—
46#14556068C2314yes—
47#14556067—28no—
48C2×C71⋊C16—22no—
49C16×D71—28no—
50C4×C71⋊C8C2×C414yes—
51C8×Dic71C2×C412yes—
52C8×Dic71C2×C414yes—
53C8×Dic71C2312yes—
54C22×C8×C71C2×C411yescharacteristic
55C22×C8×C71C2311yescharacteristic
56C2×C42×C71C2×C411yescharacteristic
57M32×C71—22no—
58M32×C71C2314yes—
59C2×C16×C71—22no—
60C4×C8×C71C2×C412yes—
61C4×C8×C71C2312yes—
62C23×D71M1611yescharacteristic
63C22×Dic71—21no—
64C22×Dic71M1611yescharacteristic
65C2×C4×D71—28no—
66C2×C4×D71—24no—
67C2×C4×D71C4⋊C412yes—
68C2×C4×D71C4⋊C411yescharacteristic
69C2×C4×D71C4⋊C411yescharacteristic
70C2×C4×D71C2×C811yescharacteristic
71C2×C4×D71C2×C812yes—
72C2×C4×D71C2×C811yescharacteristic
73C2×C4×D71M1612yes—
74C2×C4×D71C22×C412yes—
75C2×C71⋊C8C4⋊C411yescharacteristic
76C2×C71⋊C8C4⋊C412yes—
77C2×C71⋊C8C4⋊C411yescharacteristic
78C2×C71⋊C8C2×C812yes—
79C2×C71⋊C8C2×C812yes—
80C2×C71⋊C8C22×C412yes—
81C2×C71⋊C8C22×C412yes—
82C8×D71C4⋊C412yes—
83C8×D71C4⋊C412yes—
84C8×D71C4⋊C412yes—
85C8×D71C4⋊C412yes—
86C8×D71C2×C818yes—
87C8×D71C2×C818yes—
88C8×D71C2×D412yes—
89C8×D71C2×D412yes—
90C8×D71C2×Q812yes—
91C8×D71C2×Q812yes—
92C4×Dic71—21no—
93C4×Dic71—24no—
94C4×Dic71C22×C412yes—
95C71⋊C16—24no—
96C22×C4×C71—21no—
97C22×C4×C71C22×C411yescharacteristic
98C2×C8×C71C2×C812yes—
99C2×C8×C71C2×C812yes—
100C2×C8×C71C22×C411yescharacteristic
101C2×C8×C71C22×C412yes—
102C2×C8×C71C2×D411yescharacteristic
103C2×C8×C71C2×D411yescharacteristic
104C2×C8×C71C2×Q811yescharacteristic
105C2×C8×C71C2×Q811yescharacteristic
106C2×C8×C71C2411yescharacteristic
107C42×C71—21no—
108C42×C71C22×C412yes—
109C16×C71—24no—
110C23×C71C2×M1611yescharacteristic
111C22×D71—24no—
112C22×D71C4⋊C811yescharacteristic
113C22×D71C4⋊C812yes—
114C22×D71C4⋊C811yescharacteristic
115C22×D71C2×M1612yes—
116C2×C4×C71—24no—
117C2×C4×C71—22no—
118C2×C4×C71C2×C4⋊C411yescharacteristic
119C2×C4×C71C2×C4⋊C411yescharacteristic
120C2×C4×C71C22×C811yescharacteristic
121C2×C4×C71C22×C811yescharacteristic
122C2×C4×C71C2×M1611yescharacteristic
123C2×C4×C71C23×C411yescharacteristic
124C2×Dic71—24no—
125C2×Dic71—22no—
126C2×Dic71C4⋊C812yes—
127C2×Dic71C4⋊C811yescharacteristic
128C2×Dic71C4⋊C811yescharacteristic
129C2×Dic71C2×M1612yes—
130C4×D71—216no—
131C4×D71C4⋊C812yes—
132C4×D71C4⋊C812yes—
133C4×D71C4⋊C812yes—
134C4×D71C4⋊C812yes—
135C4×D71C2×C4⋊C412yes—
136C4×D71C2×C4⋊C412yes—
137C4×D71C22×C812yes—
138C4×D71C22×C812yes—
139C8×C71C2×C4⋊C411yescharacteristic
140C8×C71C2×C4⋊C411yescharacteristic
141C8×C71C22×C814yes—
142C8×C71C22×D411yescharacteristic
143C8×C71C22×Q811yescharacteristic
144C71⋊C8C2×C4⋊C412yes—
145C71⋊C8C2×C4⋊C412yes—
146C71⋊C8C22×C814yes—
147C22×C71—22no—
148C22×C71C2×C4⋊C811yescharacteristic
149C22×C71C2×C4⋊C811yescharacteristic
150C22×C71C22×M1611yescharacteristic
151C2×D71—22no—
152C2×D71—24no—
153C2×D71—24no—
154C2×D71#36414yesdirect product factor
155C2×D71C2×C4⋊C812yes—
156C2×D71C2×C4⋊C812yes—
157C4×C71—24no—
158C4×C71C2×C4⋊C811yescharacteristic
159C4×C71C2×C4⋊C811yescharacteristic
160C4×C71C22×C4⋊C411yescharacteristic
161C4×C71C23×C811yescharacteristic
162Dic71—24no—
163Dic71C2×C4⋊C812yes—
164Dic71C2×C4⋊C812yes—
165C22×#364—711nomaximal, Sylow
166C2×C71—21no—
167C2×C71—21no—
168C2×C71C2×#36412yescomplemented
169C2×C71C22×C4⋊C811yescharacteristic
170D71—22no—
171D71C2×#36414yesdirect product factor
172C22×M32—712no—
173C22×C4×C8—711no—
174C2×#364—712no—
175C2×#364—711no—
176C2×#364—718no—
177C2×#364D7111yescharacteristic, direct product factor
178C71C22×#36411yescharacteristic, complemented, Sylow
179C23×C8—711no—
180C23×C8—711no—
181C22×C42—711no—
182C2×M32—712no—
183C2×M32—718no—
184C2×M32—1422no—
185C2×M32—1424no—
186C2×M32C2×D7112yes—
187C22×C16—1422no—
188C2×C4×C8—711no—
189C2×C4×C8—711no—
190C2×C4×C8—711no—
191C2×C4×C8—714no—
192C2×C4×C8—714no—
193C2×C4×C8C2×D7111yescharacteristic
194#364—718no—
195#364—714no—
196#364C2×D7114yesdirect product factor
197C23×C4—711no—
198C23×C4—1421no—
199C22×C8—711no—
200C22×C8—711no—
201C22×C8—712no—
202C22×C8—714no—
203C22×C8—712no—
204C22×C8—712no—
205C22×C8—714no—
206C22×C8—712no—
207C22×C8—711no—
208C22×C8—711no—
209C22×C8—712no—
210C22×C8—711no—
211C22×C8—711no—
212C22×C8—712no—
213C22×C8C4×D7111yescharacteristic
214C22×C8C22×D7111yescharacteristic
215C2×C42—711no—
216C2×C42—714no—
217C2×C42—1422no—
218C2×C42—1421no—
219C2×C42C4×D7111yescharacteristic
220M32—22no—
221M32—714no—
222M32—1428no—
223M32—1422no—
224M32C22×D7114yes—
225C2×C16—22no—
226C2×C16—1428no—
227C2×C16—1422no—
228C4×C8—712no—
229C4×C8—714no—
230C4×C8—712no—
231C4×C8—714no—
232C4×C8C4×D7112yes—
233C4×C8C22×D7112yes—
234C24—711no—
235C22×C4—21no—
236C22×C4—711no—
237C22×C4—712no—
238C22×C4—712no—
239C22×C4—711no—
240C22×C4—711no—
241C22×C4—711no—
242C22×C4—712no—
243C22×C4—711no—
244C22×C4—712no—
245C22×C4—1424no—
246C22×C4—1428no—
247C22×C4—1421no—
248C22×C4C2×C4×D7111yescharacteristic
249C2×C8—712no—
250C2×C8—712no—
251C2×C8—712no—
252C2×C8—711no—
253C2×C8—711no—
254C2×C8—712no—
255C2×C8—712no—
256C2×C8—712no—
257C2×C8—712no—
258C2×C8—712no—
259C2×C8—712no—
260C2×C8—712no—
261C2×C8—712no—
262C2×C8—712no—
263C2×C8—712no—
264C2×C8—718no—
265C2×C8—718no—
266C2×C8C8×D7112yes—
267C2×C8C8×D7112yes—
268C2×C8C2×C4×D7111yescharacteristic
269C2×C8C2×C4×D7112yes—
270C2×C8D4×D7111yescharacteristic
271C2×C8D4×D7111yescharacteristic
272C2×C8Q8×D7111yescharacteristic
273C2×C8Q8×D7111yescharacteristic
274C2×C8C23×D7111yescharacteristic
275C42—21no—
276C42—712no—
277C42—1424no—
278C42—1421no—
279C42C2×C4×D7112yes—
280C16—24no—
281C16—1424no—
282C23—711no—
283C23—712no—
284C23—712no—
285C23—711no—
286C23—1424no—
287C23M16×D7111yescharacteristic
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—712no—
291C2×C4—712no—
292C2×C4—711no—
293C2×C4—711no—
294C2×C4—712no—
295C2×C4—712no—
296C2×C4—712no—
297C2×C4—712no—
298C2×C4—712no—
299C2×C4—712no—
300C2×C4—712no—
301C2×C4—712no—
302C2×C4—14216no—
303C2×C4—1424no—
304C2×C4—1422no—
305C2×C4C4⋊C4×D7111yescharacteristic
306C2×C4C4⋊C4×D7111yescharacteristic
307C2×C4C2×C8×D7111yescharacteristic
308C2×C4C2×C8×D7111yescharacteristic
309C2×C4M16×D7111yescharacteristic
310C2×C4C22×C4×D7111yescharacteristic
311C8—712no—
312C8—712no—
313C8—714no—
314C8C4⋊C4×D7111yescharacteristic
315C8C4⋊C4×D7111yescharacteristic
316C8C2×C8×D7114yes—
317C8C2×D4×D7111yescharacteristic
318C8C2×Q8×D7111yescharacteristic
319C22—22no—
320C22—712no—
321C22—714no—
322C22—712no—
323C22—1424no—
324C22—1424no—
325C22—1422no—
326C22C4⋊C8×D7111yescharacteristic
327C22C4⋊C8×D7111yescharacteristic
328C22C2×M16×D7111yescharacteristic
329C4—24no—
330C4—712no—
331C4—712no—
332C4—1424no—
333C4C4⋊C8×D7111yescharacteristic
334C4C4⋊C8×D7111yescharacteristic
335C4C2×C4⋊C4×D7111yescharacteristic
336C4C22×C8×D7111yescharacteristic
337C2—21no—
338C2—21no—
339C2—714no—
340C2—1422no—
341C2#364×D7112yesdirect product factor
342C2C2×C4⋊C8×D7111yescharacteristic
subgroupG/H
centerC2×C8D4×D71
commutatorC4×C71C23×C8
FrattiniC2×C8C23×D71
FittingC2×#364×C71C2
socleC22×C71C2×C4⋊C8
prime radicalC23×D71M16
cosocleC2×C8×C71C24
hypercenterC2×#364D71
hypocenterC71C22×#364
LCSG ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#364

Complements

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

NH ≅ G/Ntype
3. C2×#17985820C2semidirect
4. C2×#17985819C2semidirect
5. #364×D71C2direct (N × H)
6. C2×#364×C71C2semidirect
20. #17985820C22semidirect
21. #17985819C22semidirect
24. #364×C71C22semidirect
154. C2×D71#364direct (N × H)
168. C2×C71C2×#364semidirect
171. D71C2×#364direct (N × H)
177. C2×#364D71direct (N × H)
178. C71C22×#364semidirect
196. #364C2×D71direct (N × H)
341. C2#364×D71direct (N × H)

Maximal subgroups

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

subgroupindex
C2×M32×D712
C2×C4×C8×D712
C2×#179858202
C2×#179858192
#364×D712
C2×#364×C712
C22×#36471