Finite Group Databasebeta

← #31892491

Subgroups — #31892491

Order 18176.

616 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 147 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×#17985938C211yesmaximal, characteristic
2#22588428C212yesmaximal
3C2×#17985725C211yesmaximal, characteristic
4#22586218C212yesmaximal
5C71×#2316C211yesmaximal, characteristic
6C22×C4⋊Dic71C2211yescharacteristic
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985938—22no—
9#17985938C414yescomplemented
10C2×C71⋊#95—21no—
11C2×C71⋊#95—21no—
12C2×C71⋊#95C2211yescharacteristic
13#17985761—24no—
14#17985725—22no—
15C2×#117×C71—21no—
16C22×C4⋊C4×C71C2211yescharacteristic
17#389×C71—24no—
18#380×C71—21no—
19#380×C71C2212yes—
20C23×Dic71—21no—
21C23×Dic71—21no—
22C22×C71⋊C8—22no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—22no—
26C2×C4⋊Dic71C2×C412yes—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C71⋊#95—24no—
30C71⋊#95—24no—
31C71⋊#95—21no—
32C71⋊#95C2×C412yes—
33C23×C4×C71D411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C23×C4×C71C2311yescharacteristic
36#117×C71—42no—
37#117×C71—42no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71C2×C412yes—
43C2×C4⋊C4×C71D412yes—
44C2×C4⋊C4×C71D414yes—
45C2×C22⋊C4×C71—22no—
46C2×C42×C71—22no—
47#95×C71—22no—
48#95×C71—22no—
49C22×Dic71—22no—
50C22×Dic71—22no—
51C22×Dic71—42no—
52C22×Dic71—42no—
53C22×Dic71—44no—
54C22×Dic71—41no—
55C22×Dic71—41no—
56C22×Dic71—41no—
57C22×Dic71—41no—
58C2×C71⋊C8—44no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—24no—
61C4⋊Dic71—42no—
62C24×C71C2×D411yescharacteristic
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—24no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71C22⋊C412yes—
73C22×C4×C71C22⋊C412yes—
74C22×C4×C71SD1612yes—
75C22×C4×C71Q1612yes—
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D412yes—
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71#4112yes—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—24no—
82C4⋊C4×C71—44no—
83C4⋊C4×C71—44no—
84C4⋊C4×C71—82no—
85C22⋊C4×C71—42no—
86C22⋊C4×C71—42no—
87C42×C71—44no—
88C23×C71—22no—
89C23×C71—22no—
90C23×C71C2×C22⋊C411yescharacteristic
91C23×C71#12011yescharacteristic
92C23×C71#12311yescharacteristic
93C23×C71C2×SD1611yescharacteristic
94C23×C71C2×Q1611yescharacteristic
95C23×C71AGL(1,ℤ/8)11yescharacteristic
96C23×C71#13711yescharacteristic
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—24no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—44no—
111C2×C4×C71—44no—
112C2×C4×C71—44no—
113C2×C4×C71—81no—
114C2×C4×C71#10314yes—
115C2×C4×C71C4×D412yes—
116C2×C4×C71#12112yes—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—44no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—82no—
123C2×Dic71—82no—
124C2×Dic71—82no—
125C2×Dic71—82no—
126C71⋊C8—82no—
127C22×C71—22no—
128C22×C71—22no—
129C22×C71—22no—
130C22×C71—22no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71#38611yescharacteristic
136C22×C71C2×#10311yescharacteristic
137C22×C71#41711yescharacteristic
138C22×C71#45011yescharacteristic
139C22×C71#45111yescharacteristic
140C22×C71#45412yes—
141C22×C71#45812yes—
142C22×C71#48311yescharacteristic
143C22×C71#48411yescharacteristic
144C4×C71—24no—
145C4×C71—22no—
146C4×C71—22no—
147C4×C71—42no—
148C4×C71—84no—
149Dic71—82no—
150Dic71—82no—
151#9023—711nomaximal, Sylow
152C2×C71—21no—
153C2×C71—21no—
154C2×C71—21no—
155C2×C71—21no—
156C2×C71#161612yes—
157C2×C71#161712yes—
158C2×C71#187211yescharacteristic
159C2×C71#188811yescharacteristic
160C2×C71#189011yescharacteristic
161C2×#400—711no—
162#2316D7111yescharacteristic
163#1824—712no—
164C2×#324—711no—
165#1294—712no—
166C71#902311yescharacteristic, complemented, Sylow
167C2×#117—21no—
168C22×C4⋊C4—711no—
169C22×C4⋊C4C2×D7111yescharacteristic
170C2×C22⋊C8—712no—
171#400—714no—
172#400—1422no—
173#389—24no—
174#380—21no—
175#380C2×D7112yes—
176C2×#95—711no—
177C2×#95—1421no—
178C2×#95—1421no—
179#340—1424no—
180#324—1422no—
181C23×C4—1421no—
182C23×C4—1421no—
183C23×C4C22⋊D7111yescharacteristic
184C23×C4C22⋊D7111yescharacteristic
185C23×C4C22×D7111yescharacteristic
186C22×C8—1422no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—712no—
194C2×C4⋊C4—1421no—
195C2×C4⋊C4—1422no—
196C2×C4⋊C4—1422no—
197C2×C4⋊C4C4×D7112yes—
198C2×C4⋊C4D28412yes—
199C2×C4⋊C4C22⋊D7114yes—
200C2×C22⋊C4—22no—
201C2×C42—22no—
202C22⋊C8—1422no—
203C22⋊C8—1422no—
204#95—22no—
205#95—22no—
206#95—712no—
207#95—1424no—
208#95—1424no—
209#95—1421no—
210C24C2×C22⋊D7111yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—2842no—
221C22×C4—2842no—
222C22×C4—2844no—
223C22×C4—2841no—
224C22×C4—2841no—
225C22×C4—2841no—
226C22×C4—2841no—
227C22×C4—1422no—
228C22×C4—1422no—
229C22×C4(C2×D71)⋊C412yes—
230C22×C4(C2×D71)⋊C412yes—
231C22×C4#1176585312yes—
232C22×C4#1176585512yes—
233C22×C4C2×C4×D7111yescharacteristic
234C22×C4C2×C22⋊D7112yes—
235C22×C4C2×D28411yescharacteristic
236C22×C4#1176586512yes—
237C2×C8—2844no—
238C2×C8—2842no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—82no—
244C4⋊C4—2842no—
245C4⋊C4—1424no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42—44no—
249C23—22no—
250C23—22no—
251C23C2×(C2×D71)⋊C411yescharacteristic
252C23#1455609711yescharacteristic
253C23#1455611111yescharacteristic
254C23C2×#1176585311yescharacteristic
255C23C2×#1176585511yescharacteristic
256C23#1455615411yescharacteristic
257C23#1455615911yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—81no—
275C2×C4—2842no—
276C2×C4—2842no—
277C2×C4—2844no—
278C2×C4—2842no—
279C2×C4—2842no—
280C2×C4—5682no—
281C2×C4—5682no—
282C2×C4—5682no—
283C2×C4—5682no—
284C2×C4#1455605114yes—
285C2×C4C4×C22⋊D7112yes—
286C2×C4#1455610212yes—
287C8—5682no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1798587811yescharacteristic
297C22C2×#1455605111yescharacteristic
298C22#1798599811yescharacteristic
299C22#1798615011yescharacteristic
300C22#1798615511yescharacteristic
301C22#1798617212yes—
302C22#1798618912yes—
303C22#1798632911yescharacteristic
304C22#1798633411yescharacteristic
305C4—22no—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—84no—
310C4—5682no—
311C4—5682no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#2258737712yes—
317C2#2258738212yes—
318C2#2258864711yescharacteristic
319C2#2258874311yescharacteristic
320C2#2258875311yescharacteristic
subgroupG/H
centerC23#14556097
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2316C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2316D71
hypocenterC71#9023
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
9. #17985938C4semidirect
166. C71#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#179859382
#225884282
C2×#179857252
#225862182
C71×#23162
#902371