Finite Group Databasebeta

← #31885566

Subgroups — #31885566

Order 18176.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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#22590883C211yesmaximal, characteristic
2#22588455C212yesmaximal, complemented
3#22586166C211yesmaximal, characteristic, complemented
4#22586160C212yesmaximal
5C71×#1272C211yesmaximal, characteristic, complemented
6C2×#14556117C411yescharacteristic
7C2×Q8×Dic71C411yescharacteristic
8C2×#14556090C411yescharacteristic
9C2×Dic71⋊C8—22no—
10#17985901C2211yescharacteristic
11#17985891—22no—
12#17985864C411yescharacteristic
13#17985846—22no—
14C42×Dic71C2211yescharacteristic
15#17985754C2212yes—
16#17985753—22no—
17#391×C71C2211yescharacteristic
18#336×C71C2212yes—
19C22×D284C2×C411yescharacteristic
20C22×C4×D71—21no—
21C22×C71⋊C8—22no—
22#14556117—22no—
23#14556117C814yescomplemented
24Q8×Dic71—42no—
25#14556090—42no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×(C2×D71)⋊C4—22no—
29C2×(C2×D71)⋊C4—21no—
30C2×(C2×D71)⋊C4—22no—
31C2×C4×Dic71—21no—
32C2×C4×Dic71C2×C411yescharacteristic
33C2×C4×Dic71C2×C411yescharacteristic
34C2×C4×Dic71D412yes—
35C2×C4×Dic71D412yes—
36Dic71⋊C8—44no—
37C71⋊#95—22no—
38C22×Q8×C71C2×C411yescharacteristic
39C22×C8×C71—22no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C42×C71C2311yescharacteristic
42#95×C71—22no—
43C23×D71—21no—
44C2×D284—21no—
45C2×D284—41no—
46C2×D284—41no—
47C2×D284C2×C812yes—
48C22×Dic71—21no—
49C22×Dic71C22⋊C411yescharacteristic
50C22×Dic71C22⋊C412yes—
51C22×Dic71C22⋊C411yescharacteristic
52C2×C4×D71—44no—
53C2×C4×D71—42no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—42no—
58C4⋊Dic71—41no—
59(C2×D71)⋊C4—28no—
60(C2×D71)⋊C4—42no—
61(C2×D71)⋊C4—42no—
62(C2×D71)⋊C4—42no—
63C4×Dic71—22no—
64C4×Dic71—22no—
65C4×Dic71—22no—
66C4×Dic71—42no—
67C4×Dic71—42no—
68C4×Dic71C2×C812yes—
69C4×Dic71M1612yes—
70C2×Q8×C71—21no—
71C2×Q8×C71—41no—
72C2×Q8×C71—41no—
73C2×Q8×C71C2×C812yes—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D412yes—
78C2×C8×C71—42no—
79C2×C8×C71—42no—
80C2×C8×C71—42no—
81C4⋊C4×C71—41no—
82C42×C71—41no—
83C23×C71C2×C22⋊C411yescharacteristic
84C22×D71—22no—
85C22×D71—41no—
86C22×D71—41no—
87C22×D71—42no—
88C22×D71—42no—
89Q8×C71—42no—
90Q8×C71—42no—
91C2×C4×C71—22no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71C22×C811yescharacteristic
101C2×C4×C71C2×M1611yescharacteristic
102C2×C4×C71AGL(1,ℤ/8)12yes—
103C2×C4×C71#13712yes—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—21no—
108C2×Dic71—22no—
109C2×Dic71—41no—
110C2×Dic71—42no—
111C2×Dic71C22⋊C814yes—
112C2×Dic71#10012yes—
113D284—42no—
114D284—42no—
115C4×D71—84no—
116C8×C71—82no—
117C71⋊C8—82no—
118C22×C71C2×C22⋊C811yescharacteristic
119C22×C71#41011yescharacteristic
120C22×C71C2×#10011yescharacteristic
121C22×C71#41712yes—
122C22×C71#42112yes—
123C2×D71—42no—
124C2×D71—44no—
125C2×D71—81no—
126C2×D71—82no—
127C2×D71—81no—
128C4×C71—21no—
129C4×C71—21no—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—42no—
133C4×C71—81no—
134Dic71—24no—
135Dic71—22no—
136Dic71—41no—
137Dic71—81no—
138#7601—711nomaximal, Sylow
139C2×C71#146011yescharacteristic
140C2×C71#147712yes—
141C2×C71#151212yes—
142C2×C71#153311yescharacteristic
143C2×C71#156011yescharacteristic
144D71—82no—
145C4×#124—711no—
146#1831—712no—
147#1272—711no—
148#1272D7111yescharacteristic, complemented
149#1270—712no—
150C71#760111yescharacteristic, complemented, Sylow
151C2×#124—711no—
152C2×C4×Q8—711no—
153C2×C4×D4—711no—
154C2×C4⋊C8—1422no—
155#391C2×D7111yescharacteristic
156#390—711no—
157#388—1422no—
158#382—711no—
159C4×C22⋊C4—1422no—
160C43—711no—
161#336—712no—
162#336—1422no—
163#336C2×D7112yes—
164C22×Q8C4×D7111yescharacteristic
165C22×D4—711no—
166C23×C4—1421no—
167C22×C8—22no—
168C22×C8—1422no—
169#124—714no—
170#124—1422no—
171C4×Q8—2842no—
172C4×D4—2842no—
173C2×C4⋊C4—711no—
174C2×C4⋊C4—1421no—
175C2×C4⋊C4C4×D7111yescharacteristic
176C2×C22⋊C4—1421no—
177C2×C22⋊C4—1422no—
178C2×C22⋊C4—1422no—
179C2×C42—712no—
180C2×C42—711no—
181C2×C42—711no—
182C2×C42—712no—
183C2×C42—1421no—
184C2×C42C22×D7111yescharacteristic
185C4⋊C8—2844no—
186#95—22no—
187#95—1422no—
188C24—1421no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C8×D7112yes—
193C2×D4—2841no—
194C2×D4—2841no—
195C2×D4—712no—
196C2×D4—1421no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—2842no—
200C22×C4—2844no—
201C22×C4—712no—
202C22×C4—711no—
203C22×C4—711no—
204C22×C4—1421no—
205C22×C4C2×C4×D7111yescharacteristic
206C22×C4D4×D7112yes—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—2842no—
211C2×C8—2842no—
212C2×C8—2842no—
213C4⋊C4—41no—
214C4⋊C4—2841no—
215C4⋊C4—2842no—
216C22⋊C4—2842no—
217C22⋊C4—2842no—
218C22⋊C4—2842no—
219C22⋊C4—1428no—
220C42—41no—
221C42—2842no—
222C42—2842no—
223C42—712no—
224C42—712no—
225C42—1422no—
226C42—1422no—
227C42—1422no—
228C23—2841no—
229C23—2841no—
230C23—2842no—
231C23—2842no—
232C23—1422no—
233C23C22⋊C4×D7111yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—2842no—
237D4—2842no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—2841no—
248C2×C4—2842no—
249C2×C4—5684no—
250C2×C4—714no—
251C2×C4—712no—
252C2×C4—1421no—
253C2×C4—1422no—
254C2×C4—1422no—
255C2×C4—1422no—
256C2×C4—1422no—
257C2×C4C2×C8×D7111yescharacteristic
258C2×C4M16×D7111yescharacteristic
259C2×C4#1455615512yes—
260C2×C4#1455616012yes—
261C8—82no—
262C8—5682no—
263C22—2842no—
264C22—2844no—
265C22—5681no—
266C22—5681no—
267C22—5682no—
268C22C22⋊C8×D7111yescharacteristic
269C22#1798597111yescharacteristic
270C22#100×D7111yescharacteristic
271C22#1798599912yes—
272C22#1798601412yes—
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—81no—
279C4—2841no—
280C4—5681no—
281C4—1422no—
282C4—1424no—
283C2—5682no—
284C2#2258665911yescharacteristic
285C2#2258672612yes—
286C2#2258687712yes—
287C2#2258696211yescharacteristic
288C2#2258707411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC71×#1272C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1272D71
hypocenterC71#7601
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #22588455C2semidirect
3. #22586166C2semidirect
5. C71×#1272C2semidirect
23. #14556117C8semidirect
148. #1272D71semidirect
150. C71#7601semidirect

Maximal subgroups

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

subgroupindex
#225908832
#225884552
#225861662
#225861602
C71×#12722
#760171