Finite Group Databasebeta

← #31908598

Subgroups — #31908598

Order 18176.

440 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 99 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#22592657C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22586368C214yesmaximal, complemented
4C71×#1459C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic
6#17986023—21no—
7C2×#14556045—21no—
8#17985938C2212yes—
9#17985935C412yes—
10#17985897—21no—
11#17985879—21no—
12#17985871—21no—
13#17985859—21no—
14#17985859C411yescharacteristic
15#17985768C2212yes—
16C2×C4×D4×C71C2211yescharacteristic
17C2×C22⋊C8×C71—21no—
18#324×C71C2212yes—
19#323×C71—21no—
20C23×Dic71—21no—
21C2×#11765849—22no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—41no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×C22⋊Dic71—21no—
29C2×C22⋊Dic71—22no—
30C4⋊(C71⋊C8)—41no—
31C4⋊(C71⋊C8)—41no—
32#14556045—42no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95C2×C412yes—
36C22×D4×C71D411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C22×C8×C71—41no—
40C4×D4×C71—41no—
41C4×D4×C71—41no—
42C2×C4⋊C4×C71C2311yescharacteristic
43C2×C22⋊C4×C71C2×C411yescharacteristic
44C2×C22⋊C4×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C22⋊C8×C71—22no—
47C22⋊C8×C71—41no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50C22×Dic71—42no—
51C22×Dic71—82no—
52#11765849—42no—
53#11765849—42no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—42no—
56C4⋊Dic71—41no—
57C4⋊Dic71—41no—
58C4⋊Dic71—42no—
59Dic71⋊C4—82no—
60C22⋊Dic71—42no—
61C22⋊Dic71—82no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C22⋊C411yescharacteristic
64C2×D4×C71—41no—
65C2×D4×C71—41no—
66C2×D4×C71SD1612yes—
67C2×D4×C71Q1612yes—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—42no—
73C22×C4×C71—42no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C2×C8×C71—42no—
80C2×C8×C71—82no—
81C4⋊C4×C71—41no—
82C22⋊C4×C71—41no—
83C22⋊C4×C71—41no—
84C42×C71—21no—
85C42×C71—21no—
86C23×C71—21no—
87C23×C71—21no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—41no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71—41no—
94C23×C71#10312yes—
95C23×C71#10312yes—
96C23×C71C2×C22⋊C411yescharacteristic
97D4×C71—42no—
98D4×C71—42no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—22no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—41no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—81no—
112C2×C4×C71#9911yescharacteristic
113C2×C4×C71#9911yescharacteristic
114C2×C4×C71C2×SD1611yescharacteristic
115C2×C4×C71C2×Q1611yescharacteristic
116C2×Dic71—42no—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—42no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—84no—
123C8×C71—82no—
124C71⋊C8—82no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—22no—
128C22×C71—22no—
129C22×C71—42no—
130C22×C71—42no—
131C22×C71—42no—
132C22×C71—41no—
133C22×C71—41no—
134C22×C71—41no—
135C22×C71—41no—
136C22×C71—81no—
137C22×C71C2×#9911yescharacteristic
138C22×C71C2×#10311yescharacteristic
139C22×C71#42111yescharacteristic
140C4×C71—21no—
141C4×C71—21no—
142C4×C71—21no—
143C4×C71—41no—
144C4×C71—41no—
145C4×C71—41no—
146C4×C71#35212yes—
147Dic71—82no—
148Dic71—82no—
149#11903—711nomaximal, Sylow
150C2×C71—21no—
151C2×C71—21no—
152C2×C71—42no—
153C2×C71—42no—
154C2×C71#152011yescharacteristic
155C2×C71C2×#35211yescharacteristic
156C2×C71#213411yescharacteristic
157#2487—711no—
158#1800—711no—
159#1459D7111yescharacteristic
160#1353—714no—
161C71#1190311yescharacteristic, complemented, Sylow
162C2×C4×D4C2×D7111yescharacteristic
163C22×C4⋊C4—711no—
164#423—1421no—
165C2×#100—1421no—
166C2×C22⋊C8—21no—
167#400—712no—
168#399—712no—
169#389—1421no—
170#386—1421no—
171#385—1421no—
172#380—711no—
173#380—1421no—
174#343—712no—
175#324C2×D7112yes—
176#323—21no—
177C22×D4D28411yescharacteristic
178C23×C4—1421no—
179C23×C4C4×D7111yescharacteristic
180C23×C4C22⋊D7111yescharacteristic
181C2×M16—1422no—
182C22×C8—41no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—2841no—
186C2×C4⋊C4—2841no—
187C2×C4⋊C4—712no—
188C2×C4⋊C4—1421no—
189C2×C4⋊C4—1421no—
190C2×C4⋊C4—1422no—
191C2×C4⋊C4C22×D7111yescharacteristic
192C2×C22⋊C4—1421no—
193C2×C22⋊C4—1422no—
194C2×C22⋊C4C4×D7111yescharacteristic
195C2×C22⋊C4C22⋊D7111yescharacteristic
196C2×C42D28411yescharacteristic
197C4⋊C8—2841no—
198C4⋊C8—2841no—
199#100—2842no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202#95—712no—
203#95—1422no—
204#95—1422no—
205C24(C2×D71)⋊C411yescharacteristic
206C24(C2×D71)⋊C411yescharacteristic
207C2×D4—41no—
208C2×D4—41no—
209C2×D4#1176585412yes—
210C2×D4Dic28412yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—2842no—
218C22×C4—5682no—
219C22×C4—1422no—
220C22×C4—1422no—
221C22×C4(C2×D71)⋊C411yescharacteristic
222C22×C4(C2×D71)⋊C411yescharacteristic
223C22×C4C2×C4×D7111yescharacteristic
224C22×C4C2×C22⋊D7111yescharacteristic
225C22×C4C2×D28411yescharacteristic
226M16—2842no—
227M16—2842no—
228C2×C8—42no—
229C2×C8—82no—
230C2×C8—2842no—
231C4⋊C4—41no—
232C4⋊C4—2842no—
233C4⋊C4—2842no—
234C4⋊C4—2841no—
235C4⋊C4—2841no—
236C4⋊C4—5682no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—2842no—
240C22⋊C4—5682no—
241C42—21no—
242C42—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1455605312yes—
252C23#1455605312yes—
253C23C2×(C2×D71)⋊C411yescharacteristic
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—81no—
269C2×C4—2842no—
270C2×C4—2842no—
271C2×C4—2842no—
272C2×C4—2842no—
273C2×C4—2842no—
274C2×C4—2842no—
275C2×C4—5684no—
276C2×C4#1455604211yescharacteristic
277C2×C4#1455604211yescharacteristic
278C2×C4C2×#1176585411yescharacteristic
279C2×C4C2×Dic28411yescharacteristic
280C8—82no—
281C8—5682no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22C2×#1455604211yescharacteristic
295C22C2×#1455605311yescharacteristic
296C22#1798601511yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—5682no—
304C4—5682no—
305C4#1798578812yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#2258691311yescharacteristic
311C2C2×#1798578811yescharacteristic
312C2#2259003111yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1459C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C4⋊C4×C71C23
hypercenter#1459D71
hypocenterC71#11903
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #22586368C2semidirect
161. C71#11903semidirect

Maximal subgroups

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

subgroupindex
#225926572
#225883322
#225863682
C71×#14592
#1190371