Finite Group Databasebeta

← #31890855

Subgroups — #31890855

Order 18176.

482 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 115 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#22590738C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic
3#22586222C212yesmaximal, complemented
4#22586222C212yesmaximal, complemented
5C71×#1749C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)—21no—
7C2×#14556039C2212yes—
8#17985867—21no—
9#17985865C2211yescharacteristic
10#17985865C2211yescharacteristic
11#17985862—21no—
12#17985859C411yescharacteristic
13#17985859C411yescharacteristic
14C4×C4⋊Dic71C411yescharacteristic
15C4×C4⋊Dic71C411yescharacteristic
16C4×C22⋊Dic71—21no—
17#17985845—21no—
18#17985754—21no—
19C22×C42×C71C2211yescharacteristic
20C2×C22⋊C8×C71—21no—
21C2×#97×C71C2212yes—
22#336×C71—21no—
23C22×C71⋊C8—22no—
24C2×C4⋊Dic71C2×C411yescharacteristic
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C22⋊Dic71—21no—
29C2×C22⋊Dic71—21no—
30C2×C4×Dic71—21no—
31C2×C4×Dic71—21no—
32C22⋊(C71⋊C8)—24no—
33#14556039—24no—
34C71⋊#95—21no—
35C71⋊#95—21no—
36C71⋊#95—21no—
37C71⋊#95—21no—
38C71⋊#95—21no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C23×C4×C71C2×C411yescharacteristic
43C23×C4×C71D411yescharacteristic
44C23×C4×C71D411yescharacteristic
45C22×C8×C71—22no—
46C2×C42×C71—24no—
47C2×C42×C71C2×C411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C2×C42×C71C2311yescharacteristic
51C22⋊C8×C71—24no—
52#97×C71—24no—
53C22×Dic71—21no—
54C22×Dic71—21no—
55C22×Dic71—21no—
56C22×Dic71—21no—
57C2×C71⋊C8—24no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—41no—
61C4⋊Dic71—41no—
62C4⋊Dic71—41no—
63C4⋊Dic71—41no—
64C22⋊Dic71—42no—
65C22⋊Dic71—42no—
66C4×Dic71—42no—
67C4×Dic71—42no—
68C24×C71C22⋊C411yescharacteristic
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C2×C8×C71—24no—
85C2×C8×C71—42no—
86C2×C8×C71—42no—
87C42×C71—22no—
88C42×C71—22no—
89C42×C71—22no—
90C42×C71#4114yes—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71#10412yes—
98C23×C71C2×C22⋊C411yescharacteristic
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—21no—
112C2×C4×C71—21no—
113C2×C4×C71—44no—
114C2×C4×C71—44no—
115C2×C4×C71—44no—
116C2×C4×C71#10412yes—
117C2×C4×C71#11712yes—
118C2×C4×C71#12312yes—
119C2×C4×C71#12312yes—
120C2×Dic71—41no—
121C2×Dic71—41no—
122C2×Dic71—41no—
123C2×Dic71—41no—
124C2×Dic71—41no—
125C2×Dic71—41no—
126C2×Dic71—41no—
127C2×Dic71—41no—
128C2×Dic71—41no—
129C2×Dic71—41no—
130C2×Dic71—41no—
131C2×Dic71—41no—
132C8×C71—44no—
133C71⋊C8—44no—
134C22×C71—22no—
135C22×C71—22no—
136C22×C71—22no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71#38111yescharacteristic
142C22×C71#41311yescharacteristic
143C22×C71#41311yescharacteristic
144C22×C71C2×#10411yescharacteristic
145C22×C71C2×#10411yescharacteristic
146C22×C71#42111yescharacteristic
147C22×C71#42111yescharacteristic
148C4×C71—22no—
149C4×C71—22no—
150C4×C71—22no—
151C4×C71—22no—
152C4×C71—41no—
153C4×C71—41no—
154Dic71—81no—
155Dic71—81no—
156Dic71—81no—
157Dic71—81no—
158#8720—711nomaximal, Sylow
159C2×C71—42no—
160C2×C71#152411yescharacteristic
161C2×C71#152911yescharacteristic
162C2×C71#153211yescharacteristic
163C2×C71#153511yescharacteristic
164C2×C71#182211yescharacteristic
165C2×C71#182611yescharacteristic
166C2×C71#182611yescharacteristic
167#2283—711no—
168#1749—711no—
169#1749D7111yescharacteristic
170#1295—712no—
171#1295—712no—
172C71#872011yescharacteristic, complemented, Sylow
173C22×C42C2×D7111yescharacteristic
174C2×C22⋊C8—21no—
175C2×C22⋊C8—1421no—
176C2×#97—712no—
177C2×#97C2×D7112yes—
178#383—1421no—
179#382—711no—
180#382—711no—
181#381—1421no—
182#380—711no—
183#380—711no—
184C4×C4⋊C4—711no—
185C4×C4⋊C4—711no—
186C4×C22⋊C4—1421no—
187#376—1421no—
188#336—21no—
189#336—1421no—
190C23×C4C4×D7111yescharacteristic
191C23×C4D28411yescharacteristic
192C23×C4C22⋊D7111yescharacteristic
193C22×C8—22no—
194C22×C8—1422no—
195C2×C4⋊C4—711no—
196C2×C4⋊C4—711no—
197C2×C4⋊C4—711no—
198C2×C4⋊C4—711no—
199C2×C22⋊C4—1421no—
200C2×C22⋊C4—1421no—
201C2×C42—24no—
202C2×C42—1421no—
203C2×C42—1421no—
204C2×C42C4×D7111yescharacteristic
205C2×C42D28411yescharacteristic
206C2×C42C22⋊D7111yescharacteristic
207C2×C42C22×D7111yescharacteristic
208C22⋊C8—24no—
209C22⋊C8—1424no—
210#97—24no—
211#97—1424no—
212#95—1421no—
213#95—1421no—
214#95—1421no—
215#95—1421no—
216#95—1421no—
217#95—1421no—
218#95—1421no—
219#95—1421no—
220C24(C2×D71)⋊C411yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—1421no—
231C22×C4—1421no—
232C22×C4—1421no—
233C22×C4—1421no—
234C22×C4(C2×D71)⋊C411yescharacteristic
235C22×C4(C2×D71)⋊C411yescharacteristic
236C22×C4(C2×D71)⋊C411yescharacteristic
237C22×C4C2×C4×D7111yescharacteristic
238C22×C4C2×C22⋊D7111yescharacteristic
239C22×C4C2×D28411yescharacteristic
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—2842no—
244C2×C8—2842no—
245C2×C8—1424no—
246C4⋊C4—2841no—
247C4⋊C4—2841no—
248C4⋊C4—2841no—
249C4⋊C4—2841no—
250C22⋊C4—2842no—
251C22⋊C4—2842no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—2842no—
256C42—2842no—
257C42#1176586414yes—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#1455605412yes—
265C23C2×(C2×D71)⋊C411yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—2841no—
284C2×C4—2841no—
285C2×C4—2841no—
286C2×C4—2841no—
287C2×C4—2841no—
288C2×C4—2841no—
289C2×C4—2841no—
290C2×C4—2841no—
291C2×C4—2841no—
292C2×C4—2841no—
293C2×C4—2841no—
294C2×C4—2841no—
295C2×C4#1455605612yes—
296C2×C4#1455608412yes—
297C2×C4#1455610912yes—
298C2×C4#1455611212yes—
299C8—44no—
300C8—2844no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#1798586011yescharacteristic
309C22#1798598111yescharacteristic
310C22#1798598111yescharacteristic
311C22C2×#1455605411yescharacteristic
312C22C2×#1455605611yescharacteristic
313C22#1798601511yescharacteristic
314C22#1798601711yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—5681no—
322C4—5681no—
323C4—5681no—
324C4—5681no—
325C2—42no—
326C2#2258692611yescharacteristic
327C2#2258694511yescharacteristic
328C2#2258696111yescharacteristic
329C2#2258696911yescharacteristic
330C2#2258842111yescharacteristic
331C2#2258843711yescharacteristic
332C2#2258844011yescharacteristic
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#8720
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. #22586222C2semidirect
4. #22586222C2semidirect
172. C71#8720semidirect

Maximal subgroups

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

subgroupindex
#225907382
#225881022
#225862222
C71×#17492
#872071