Finite Group Databasebeta

← #31888373

Subgroups — #31888373

Order 18176.

528 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 129 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#22590879C211yesmaximal, characteristic
2#22588088C212yesmaximal, complemented
3C2×#17985738C212yesmaximal
4#22586221C211yesmaximal, characteristic, complemented
5C71×#1295C211yesmaximal, characteristic, complemented
6C2×#14556116C2211yescharacteristic
7C2×#14556093C411yescharacteristic
8C2×#14556090C411yescharacteristic
9C2×#14556039C2212yes—
10#17985904C411yescharacteristic
11#17985900C411yescharacteristic
12#17985888—22no—
13#17985846—22no—
14C42×Dic71C2211yescharacteristic
15#17985753—22no—
16#17985753—22no—
17#17985738C418yescomplemented
18C2×#97×C71C2212yes—
19#382×C71C2211yescharacteristic
20C22×Dic142C2×C411yescharacteristic
21C22×D284C2×C411yescharacteristic
22C22×C4×D71—21no—
23C22×C71⋊C8—22no—
24#14556116—21no—
25#14556116—21no—
26#14556116C2×C414yes—
27#14556093—42no—
28#14556090—42no—
29C2×Dic71⋊C4—21no—
30C2×(C2×D71)⋊C4—22no—
31C2×(C2×D71)⋊C4—21no—
32C2×(C2×D71)⋊C4—22no—
33C2×C4×Dic71—21no—
34C2×C4×Dic71C2×C411yescharacteristic
35C2×C4×Dic71C2×C411yescharacteristic
36C2×C4×Dic71D412yes—
37C2×C4×Dic71D412yes—
38#14556039—22no—
39#14556039C2×C414yescomplemented
40C71⋊#95—22no—
41C22×C8×C71—22no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C42×C71C2311yescharacteristic
45#97×C71—22no—
46#97×C71C2×C414yescomplemented
47#95×C71—22no—
48C23×D71—21no—
49C2×Dic142—21no—
50C2×Dic142—41no—
51C2×Dic142—41no—
52C2×Dic142C4212yes—
53C2×D284—21no—
54C2×D284—41no—
55C2×D284—41no—
56C2×D284C4212yes—
57C22×Dic71—21no—
58C22×Dic71C22⋊C412yes—
59C22×Dic71C22⋊C411yescharacteristic
60C22×Dic71C22⋊C411yescharacteristic
61C2×C4×D71—44no—
62C2×C4×D71—42no—
63C2×C71⋊C8—24no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66Dic71⋊C4—42no—
67(C2×D71)⋊C4—24no—
68(C2×D71)⋊C4—24no—
69(C2×D71)⋊C4—44no—
70(C2×D71)⋊C4—42no—
71C4×Dic71—22no—
72C4×Dic71—41no—
73C4×Dic71—42no—
74C4×Dic71—42no—
75C4×Dic71—42no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D412yes—
80C2×C8×C71—24no—
81C2×C8×C71—42no—
82C2×C8×C71—42no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C42×C71C22×C412yes—
86C42×C71#4111yescharacteristic
87C42×C71#4111yescharacteristic
88C23×C71C2×C22⋊C411yescharacteristic
89C22×D71—22no—
90C22×D71—41no—
91C22×D71—41no—
92C22×D71—42no—
93C22×D71—42no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71C2×C4211yescharacteristic
103C2×C4×C71#11711yescharacteristic
104C2×C4×C71C4×D412yes—
105C2×C4×C71C4×D412yes—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—21no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—41no—
114C2×Dic71—41no—
115C2×Dic71#10012yes—
116D284—42no—
117D284—42no—
118C4×D71—84no—
119Dic142—42no—
120Dic142—42no—
121C8×C71—44no—
122C71⋊C8—44no—
123C22×C71C4×C22⋊C411yescharacteristic
124C22×C71C2×#10011yescharacteristic
125C22×C71#41311yescharacteristic
126C22×C71#42112yes—
127C22×C71#42112yes—
128C2×D71—42no—
129C2×D71—44no—
130C2×D71—81no—
131C2×D71—81no—
132C2×D71—82no—
133C4×C71—21no—
134C4×C71—21no—
135C4×C71—22no—
136C4×C71—22no—
137C4×C71—81no—
138C4×C71—81no—
139Dic71—22no—
140Dic71—41no—
141Dic71—42no—
142Dic71—42no—
143#8224—711nomaximal, Sylow
144C2×C71#153311yescharacteristic
145C2×C71#153512yes—
146C2×C71#153611yescharacteristic
147C2×C71C4×#10011yescharacteristic
148C2×C71#175912yes—
149D71—82no—
150C4×#124—711no—
151#1745—712no—
152C2×#329—712no—
153#1295—711no—
154#1295D7111yescharacteristic, complemented
155C71#822411yescharacteristic, complemented, Sylow
156C2×#124—711no—
157C2×C4×Q8—711no—
158C2×C4×D4—711no—
159C2×#97—712no—
160C2×#97C2×D7112yes—
161#391—711no—
162#390—711no—
163#388—1422no—
164#382C2×D7111yescharacteristic
165C4×C22⋊C4—1422no—
166C43—711no—
167#336—1422no—
168#336—1422no—
169#329—718no—
170C22×Q8—711no—
171C22×D4—711no—
172C23×C4—1421no—
173C22×C8—22no—
174C22×C8—1422no—
175#124—714no—
176#124—1421no—
177#124—1421no—
178C4×Q8—2842no—
179C4×D4—2842no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4C4×D7111yescharacteristic
182C2×C4⋊C4C4×D7111yescharacteristic
183C2×C22⋊C4—1421no—
184C2×C22⋊C4—1422no—
185C2×C22⋊C4—1422no—
186C2×C42—711no—
187C2×C42—712no—
188C2×C42—712no—
189C2×C42—711no—
190C2×C42—1421no—
191C2×C42C22×D7111yescharacteristic
192#97—22no—
193#97—714no—
194#97—1422no—
195#97C4×D7114yescomplemented
196#95—22no—
197#95—1422no—
198C24—1421no—
199C2×Q8—2841no—
200C2×Q8—2841no—
201C2×Q8—712no—
202C2×Q8—1421no—
203C2×D4—2841no—
204C2×D4—2841no—
205C2×D4—712no—
206C2×D4—1421no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—2842no—
210C22×C4—2844no—
211C22×C4—711no—
212C22×C4—712no—
213C22×C4—711no—
214C22×C4—1421no—
215C22×C4C2×C4×D7111yescharacteristic
216C22×C4D4×D7112yes—
217C2×C8—24no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—2842no—
221C2×C8—2842no—
222C2×C8—1424no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—2842no—
226C22⋊C4—2844no—
227C22⋊C4—2842no—
228C22⋊C4—1424no—
229C22⋊C4—1424no—
230C42—2841no—
231C42—2842no—
232C42—2842no—
233C42—2842no—
234C42—1422no—
235C42C2×C4×D7112yes—
236C42#1176586411yescharacteristic
237C42#1176586611yescharacteristic
238C23—2841no—
239C23—2841no—
240C23—2842no—
241C23—2842no—
242C23—1422no—
243C23C22⋊C4×D7111yescharacteristic
244Q8—2842no—
245Q8—2842no—
246D4—2842no—
247D4—2842no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—2841no—
257C2×C4—2841no—
258C2×C4—5684no—
259C2×C4—712no—
260C2×C4—1422no—
261C2×C4—1421no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4C42×D7111yescharacteristic
268C2×C4#1455608611yescharacteristic
269C2×C4(C22⋊D71)⋊C412yes—
270C2×C4#1455609012yes—
271C8—44no—
272C8—2844no—
273C22—2842no—
274C22—2844no—
275C22—5681no—
276C22—5682no—
277C22—5681no—
278C22#1798584611yescharacteristic
279C22#100×D7111yescharacteristic
280C22#1798598211yescharacteristic
281C22#1798601412yes—
282C22#1798601412yes—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—22no—
287C4—81no—
288C4—81no—
289C4—2842no—
290C4—2841no—
291C4—2842no—
292C4—1422no—
293C2—5682no—
294C2#2258696211yescharacteristic
295C2#2258697012yes—
296C2#2258697311yescharacteristic
297C2#2258813111yescharacteristic
298C2#2258814912yes—
subgroupG/H
centerC23C22⋊C4×D71
commutatorC2×C4×C71C2×C42
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1295D71
hypocenterC71#8224
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #22588088C2semidirect
4. #22586221C2semidirect
5. C71×#1295C2semidirect
17. #17985738C4semidirect
39. #14556039C2×C4semidirect
46. #97×C71C2×C4semidirect
154. #1295D71semidirect
155. C71#8224semidirect
195. #97C4×D71semidirect

Maximal subgroups

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

subgroupindex
#225908792
#225880882
C2×#179857382
#225862212
C71×#12952
#822471