Finite Group Databasebeta

← #31885679

Subgroups — #31885679

Order 18176.

594 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#17985897C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic
3#22586164C214yesmaximal, complemented
4C71×#1749C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic
6C22×C4⋊Dic71C411yescharacteristic
7C22×Dic71⋊C4—21no—
8C2×C22⋊(C71⋊C8)—21no—
9C2×#14556039—21no—
10#17985897—21no—
11#17985897—21no—
12#17985897C412yes—
13#17985897C2212yes—
14C2×C71⋊#95—21no—
15#17985754C2212yes—
16C22×C42×C71C2211yescharacteristic
17C2×C22⋊C8×C71—21no—
18C2×#97×C71—21no—
19#336×C71C2212yes—
20C23×Dic71—21no—
21C23×Dic71—21no—
22C22×C71⋊C8—22no—
23C2×C4⋊Dic71—24no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—42no—
27C2×C4⋊Dic71C2×C412yes—
28C2×C4⋊Dic71C2×C412yes—
29C2×Dic71⋊C4—22no—
30C2×Dic71⋊C4—22no—
31C2×Dic71⋊C4—44no—
32C22⋊(C71⋊C8)—24no—
33#14556039—42no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C22×C8×C71—22no—
40C2×C42×C71—42no—
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71C2311yescharacteristic
45C22⋊C8×C71—24no—
46#97×C71—42no—
47C22×Dic71—24no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50C22×Dic71—42no—
51C22×Dic71—42no—
52C22×Dic71—42no—
53C22×Dic71—42no—
54C22×Dic71—42no—
55C2×C71⋊C8—24no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—24no—
59C4⋊Dic71—22no—
60C4⋊Dic71—42no—
61C4⋊Dic71—42no—
62C4⋊Dic71M1614yes—
63Dic71⋊C4—44no—
64Dic71⋊C4—44no—
65C24×C71C22⋊C411yescharacteristic
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—41no—
74C22×C4×C71—41no—
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C2×C8×C71—24no—
82C2×C8×C71—42no—
83C2×C8×C71—42no—
84C42×C71—22no—
85C42×C71—22no—
86C42×C71—41no—
87C42×C71—41no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71#10112yes—
93C23×C71#10412yes—
94C23×C71#10412yes—
95C23×C71C2×C22⋊C411yescharacteristic
96C2×C4×C71—28no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—44no—
112C2×C4×C71—44no—
113C2×C4×C71#9912yes—
114C2×C4×C71C2×M1612yes—
115C2×C4×C71AGL(1,ℤ/8)12yes—
116C2×C4×C71#13712yes—
117C2×Dic71—24no—
118C2×Dic71—48no—
119C2×Dic71—42no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124C2×Dic71—84no—
125C8×C71—44no—
126C71⋊C8—44no—
127C22×C71—22no—
128C22×C71—22no—
129C22×C71—22no—
130C22×C71—22no—
131C22×C71—22no—
132C22×C71—22no—
133C22×C71#33014yes—
134C22×C71#40711yescharacteristic
135C22×C71C2×#9911yescharacteristic
136C22×C71C2×#10111yescharacteristic
137C22×C71#41811yescharacteristic
138C22×C71#41911yescharacteristic
139C22×C71C2×#10411yescharacteristic
140C22×C71C2×#10411yescharacteristic
141C4×C71—22no—
142C4×C71—22no—
143C4×C71—22no—
144C4×C71—22no—
145C4×C71—41no—
146C4×C71—41no—
147Dic71—44no—
148Dic71—82no—
149#7627—711nomaximal, Sylow
150C2×C71—24no—
151C2×C71#146311yescharacteristic
152C2×C71#149011yescharacteristic
153C2×C71#149111yescharacteristic
154C2×C71#150711yescharacteristic
155C2×C71#150811yescharacteristic
156C2×C71C2×#33011yescharacteristic
157C2×C71#155611yescharacteristic
158C2×#389—711no—
159#1749—711no—
160#1749D7111yescharacteristic
161#1271—714no—
162C71#762711yescharacteristic, complemented, Sylow
163C22×C4⋊C4—711no—
164C22×C4⋊C4—711no—
165C22×C4⋊C4—1421no—
166C22×C42C2×D7111yescharacteristic
167C2×C22⋊C8—21no—
168C2×C22⋊C8—1421no—
169C2×#97—21no—
170C2×#97—1421no—
171#389—712no—
172#389—712no—
173#389—1421no—
174#389—1421no—
175C2×#95—1421no—
176#336—712no—
177#336C2×D7112yes—
178C23×C4—1421no—
179C23×C4—1421no—
180C23×C4C4×D7111yescharacteristic
181C23×C4D28411yescharacteristic
182C23×C4C22⋊D7111yescharacteristic
183C22×C8—22no—
184C22×C8—1422no—
185C2×C4⋊C4—2844no—
186C2×C4⋊C4—2842no—
187C2×C4⋊C4—712no—
188C2×C4⋊C4—712no—
189C2×C4⋊C4—1421no—
190C2×C4⋊C4—1422no—
191C2×C4⋊C4—1422no—
192C2×C4⋊C4—1424no—
193C2×C4⋊C4—1421no—
194C2×C42—42no—
195C2×C42C4×D7111yescharacteristic
196C2×C42D28411yescharacteristic
197C2×C42C22⋊D7111yescharacteristic
198C2×C42C22×D7111yescharacteristic
199C22⋊C8—24no—
200C22⋊C8—1424no—
201#97—42no—
202#97—2842no—
203#95—1422no—
204#95—1422no—
205C24(C2×D71)⋊C411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—2842no—
216C22×C4—2842no—
217C22×C4—2842no—
218C22×C4—2842no—
219C22×C4—2842no—
220C22×C4—1422no—
221C22×C4—1424no—
222C22×C4—1422no—
223C22×C4(C2×D71)⋊C411yescharacteristic
224C22×C4(C2×D71)⋊C411yescharacteristic
225C22×C4(C2×D71)⋊C411yescharacteristic
226C22×C4C2×C4×D7111yescharacteristic
227C22×C4C2×C22⋊D7111yescharacteristic
228C22×C4C2×D28411yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—2842no—
233C2×C8—2842no—
234C2×C8—1424no—
235C4⋊C4—2842no—
236C4⋊C4—2844no—
237C4⋊C4—2842no—
238C4⋊C4—2844no—
239C4⋊C4—714no—
240C4⋊C4—1424no—
241C4⋊C4—1422no—
242C42—22no—
243C42—22no—
244C42—41no—
245C42—41no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1455604612yes—
251C23#1455605412yes—
252C23#1455605612yes—
253C23C2×(C2×D71)⋊C411yescharacteristic
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—2848no—
272C2×C4—2842no—
273C2×C4—2842no—
274C2×C4—2842no—
275C2×C4—2842no—
276C2×C4—2842no—
277C2×C4—5684no—
278C2×C4—1424no—
279C2×C4#1455604212yes—
280C2×C4M16×D7112yes—
281C2×C4#1455615312yes—
282C2×C4#1455616112yes—
283C8—44no—
284C8—2844no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22#1798574014yes—
292C22#1798596211yescharacteristic
293C22C2×#1455604211yescharacteristic
294C22C2×#1455604611yescharacteristic
295C22#1798600311yescharacteristic
296C22#1798600911yescharacteristic
297C22C2×#1455605411yescharacteristic
298C22C2×#1455605611yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—2844no—
306C4—5682no—
307C2—24no—
308C2#2258666911yescharacteristic
309C2#2258678211yescharacteristic
310C2#2258678811yescharacteristic
311C2#2258685511yescharacteristic
312C2#2258686111yescharacteristic
313C2C2×#1798574011yescharacteristic
314C2#2258705811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1749C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1749D71
hypocenterC71#7627
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #22586164C2semidirect
162. C71#7627semidirect

Maximal subgroups

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

subgroupindex
C2×#179858972
#225881022
#225861642
C71×#17492
#762771