Finite Group Databasebeta

← #1325×D71 #31906604

Subgroups — #1325×D71 #31906604

Order 18176.

598 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#384×D71C211yesmaximal, characteristic
2#324×D71C212yesmaximal
3#22586286C211yesmaximal, characteristic, complemented
4#22586285C212yesmaximal, complemented
5C71×#1325C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D71—22no—
7C2×C4⋊C4×D71C411yescharacteristic
8C2×C4⋊C4×D71C2211yescharacteristic
9C2×C42×D71C411yescharacteristic
10C22⋊C8×D71—22no—
11#17985869C411yescharacteristic
12#17985868—22no—
13#17985868C411yescharacteristic
14#17985868C2211yescharacteristic
15#17985725C2212yes—
16#17985724—22no—
17#384×C71C2211yescharacteristic
18#324×C71C2212yes—
19C22×C4×D71—21no—
20C22×C4×D71—21no—
21C22×C4×D71C2×C411yescharacteristic
22C22×C4×D71D412yes—
23C2×C8×D71—42no—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×Dic71⋊C4—21no—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4—22no—
30C2×Dic71⋊C4D412yes—
31C4⋊C4×D71—41no—
32C4⋊C4×D71—41no—
33C4⋊C4×D71—48no—
34C4⋊C4×D71—41no—
35C4⋊C4×D71—41no—
36C2×C4×Dic71—21no—
37C2×C4×Dic71C2×C411yescharacteristic
38C42×D71—22no—
39C42×D71C814yescomplemented
40C22⋊(C71⋊C8)—22no—
41(C2×D71)⋊C8—42no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2311yescharacteristic
45C2×C42×C71C2×C411yescharacteristic
46C22⋊C8×C71—22no—
47C23×D71C22⋊C411yescharacteristic
48C22×Dic71—21no—
49C22×Dic71—21no—
50C22×Dic71C22⋊C411yescharacteristic
51C22×Dic71C22⋊C412yes—
52C2×C4×D71—24no—
53C2×C4×D71—24no—
54C2×C4×D71—24no—
55C2×C4×D71—21no—
56C2×C4×D71—21no—
57C2×C4×D71—22no—
58C2×C4×D71—22no—
59C2×C4×D71—22no—
60C2×C4×D71—44no—
61C2×C4×D71—44no—
62C2×C4×D71—41no—
63C2×C4×D71—41no—
64C2×C4×D71C2×C812yes—
65C2×C4×D71M1612yes—
66C2×C71⋊C8—42no—
67C8×D71—84no—
68C4⋊Dic71—41no—
69C4⋊Dic71—44no—
70C4⋊Dic71—41no—
71Dic71⋊C4—42no—
72Dic71⋊C4—44no—
73Dic71⋊C4—44no—
74Dic71⋊C4—42no—
75C4×Dic71—21no—
76C4×Dic71—24no—
77C4×Dic71C2×C812yescomplemented
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D412yes—
82C2×C8×C71—42no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—44no—
86C42×C71—21no—
87C42×C71C2×C812yescomplemented
88C23×C71C2×C22⋊C411yescharacteristic
89C22×D71—22no—
90C22×D71—22no—
91C22×D71C22⋊C812yes—
92C22×D71#9911yescharacteristic
93C22×D71#9911yescharacteristic
94C22×D71#10112yes—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—22no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—41no—
104C2×C4×C71C22×C811yescharacteristic
105C2×C4×C71C2×M1611yescharacteristic
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71—21no—
110C2×Dic71—21no—
111C2×Dic71—22no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—41no—
115C2×Dic71C22⋊C812yes—
116C4×D71—216no—
117C4×D71—22no—
118C4×D71—22no—
119C4×D71—22no—
120C4×D71—22no—
121C4×D71—44no—
122C4×D71—82no—
123C4×D71—82no—
124C4×D71—82no—
125C8×C71—82no—
126C71⋊C8—82no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71C2×C22⋊C811yescharacteristic
130C22×C71C2×#9911yescharacteristic
131C22×C71C2×#10111yescharacteristic
132C2×D71—21no—
133C2×D71—21no—
134C2×D71—24no—
135C2×D71—24no—
136C2×D71#32412yes—
137C2×D71#32412yes—
138C2×D71#35312yes—
139C2×D71#35612yes—
140C4×C71—24no—
141C4×C71—21no—
142C4×C71—21no—
143C4×C71—42no—
144C4×C71—81no—
145Dic71—24no—
146Dic71—22no—
147Dic71—42no—
148Dic71—81no—
149C2×#1325—711nomaximal, Sylow
150C2×C71—21no—
151C2×C71—21no—
152C2×C71C2×#32411yescharacteristic
153C2×C71C2×#35311yescharacteristic
154C2×C71C2×#35611yescharacteristic
155D71—22no—
156D71#132514yesdirect product factor
157C2×#384—711no—
158C2×#324—712no—
159#1325—711no—
160#1325—712no—
161#1325D7111yescharacteristic, direct product factor
162C71C2×#132511yescharacteristic, complemented, Sylow
163C22×C4⋊C4—711no—
164C22×C4⋊C4—711no—
165C22×C4⋊C4—1422no—
166C22×C42—711no—
167C2×C22⋊C8—1422no—
168#384—711no—
169#384—711no—
170#384—711no—
171#384—1422no—
172#384C2×D7111yescharacteristic
173#324—712no—
174#324—1422no—
175#324C2×D7112yes—
176C23×C4—712no—
177C23×C4—711no—
178C23×C4—1421no—
179C23×C4—1421no—
180C22×C8—2842no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—2848no—
183C2×C4⋊C4—2841no—
184C2×C4⋊C4—2841no—
185C2×C4⋊C4—2841no—
186C2×C4⋊C4—2841no—
187C2×C4⋊C4—711no—
188C2×C4⋊C4—712no—
189C2×C4⋊C4—711no—
190C2×C4⋊C4—1421no—
191C2×C4⋊C4—1422no—
192C2×C4⋊C4—1422no—
193C2×C4⋊C4—1422no—
194C2×C4⋊C4C4×D7111yescharacteristic
195C2×C4⋊C4C22×D7111yescharacteristic
196C2×C42—711no—
197C2×C42—714no—
198C2×C42—1422no—
199C2×C42—1421no—
200C2×C42C4×D7111yescharacteristic
201C22⋊C8—22no—
202C22⋊C8—2842no—
203C22⋊C8—1422no—
204C24—711no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—2844no—
208C22×C4—2841no—
209C22×C4—2841no—
210C22×C4—2844no—
211C22×C4—712no—
212C22×C4—711no—
213C22×C4—712no—
214C22×C4—712no—
215C22×C4—1422no—
216C22×C4—1422no—
217C22×C4—1424no—
218C22×C4—1421no—
219C22×C4—1421no—
220C22×C4—1422no—
221C22×C4—1424no—
222C22×C4—1424no—
223C22×C4—1421no—
224C22×C4—1421no—
225C22×C4C2×C4×D7111yescharacteristic
226C22×C4D4×D7112yes—
227C2×C8—42no—
228C2×C8—2842no—
229C2×C8—5684no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—44no—
233C4⋊C4—2841no—
234C4⋊C4—2844no—
235C4⋊C4—2842no—
236C4⋊C4—2841no—
237C4⋊C4—2844no—
238C4⋊C4—2844no—
239C4⋊C4—2842no—
240C42—21no—
241C42—712no—
242C42—1424no—
243C42—1421no—
244C42C8×D7112yescomplemented
245C23—711no—
246C23—712no—
247C23—711no—
248C23—712no—
249C23—1422no—
250C23—1422no—
251C23C22⋊C4×D7111yescharacteristic
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—2844no—
262C2×C4—2842no—
263C2×C4—2842no—
264C2×C4—2841no—
265C2×C4—5682no—
266C2×C4—5682no—
267C2×C4—5682no—
268C2×C4—712no—
269C2×C4—1422no—
270C2×C4—1422no—
271C2×C4—1422no—
272C2×C4—1422no—
273C2×C4—14216no—
274C2×C4—1422no—
275C2×C4—1422no—
276C2×C4—1422no—
277C2×C4—1422no—
278C2×C4—1421no—
279C2×C4—1421no—
280C2×C4C2×C8×D7111yescharacteristic
281C2×C4M16×D7111yescharacteristic
282C8—82no—
283C8—5682no—
284C22—21no—
285C22—21no—
286C22—712no—
287C22—712no—
288C22—712no—
289C22—712no—
290C22—1424no—
291C22—1424no—
292C22—1421no—
293C22—1421no—
294C22C22⋊C8×D7111yescharacteristic
295C22#99×D7111yescharacteristic
296C22#101×D7111yescharacteristic
297C4—24no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—81no—
302C4—2842no—
303C4—5681no—
304C4—1424no—
305C4—1422no—
306C2—21no—
307C2—21no—
308C2—714no—
309C2—1422no—
310C2#324×D7111yescharacteristic
311C2#353×D7111yescharacteristic
312C2#356×D7111yescharacteristic
subgroupG/H
centerC22#99×D71
commutatorC2×C4×C71C22×C8
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1325C2
socleC22×C71C2×#99
prime radicalC23×D71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1325D71
hypocenterC71C2×#1325
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #22586286C2semidirect
4. #22586285C2semidirect
5. C71×#1325C2semidirect
39. C42×D71C8semidirect
77. C4×Dic71C2×C8semidirect
87. C42×C71C2×C8semidirect
156. D71#1325direct (N × H)
161. #1325D71direct (N × H)
162. C71C2×#1325semidirect
244. C42C8×D71semidirect

Maximal subgroups

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

subgroupindex
#384×D712
#324×D712
#225862862
#225862852
C71×#13252
C2×#132571