Finite Group Databasebeta

← #31908501

Subgroups — #31908501

Order 18176.

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

normal subgroups only

This group has 460 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#22592648C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586345C214yesmaximal, complemented
4C71×#1800C211yesmaximal, characteristic
5C2×#14556109—21no—
6C22×C4⋊Dic71C411yescharacteristic
7C2×C22⋊(C71⋊C8)—21no—
8#17985929C2212yes—
9#17985924C412yes—
10#17985897—21no—
11#17985860—21no—
12#17985859—21no—
13#17985859C411yescharacteristic
14#17985725C2212yes—
15#17985723—21no—
16C2×C4×D4×C71C2211yescharacteristic
17#423×C71—21no—
18C2×#100×C71—21no—
19#343×C71C2212yes—
20C23×Dic71—21no—
21C22×C71⋊C8—41no—
22#14556109—42no—
23#14556109—42no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—41no—
27C2×C4⋊Dic71—41no—
28C2×C4⋊Dic71C2×C412yes—
29C2×Dic71⋊C4—22no—
30C2×C22⋊Dic71—22no—
31C2×C22⋊Dic71—21no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)—41no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95C2×C412yes—
37C22×D4×C71D411yescharacteristic
38C23×C4×C71C2×C411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C2×M16×C71—22no—
41C4×D4×C71—41no—
42C4×D4×C71—41no—
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C22⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C4⋊C8×C71—41no—
48C4⋊C8×C71—41no—
49#100×C71—42no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52C22×Dic71—42no—
53C22×Dic71—42no—
54C22×Dic71—42no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—82no—
57C4⋊Dic71—42no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60C4⋊Dic71—42no—
61Dic71⋊C4—42no—
62Dic71⋊C4—42no—
63C22⋊Dic71—42no—
64C22⋊Dic71—44no—
65C24×C71C22⋊C411yescharacteristic
66C24×C71C22⋊C411yescharacteristic
67C2×D4×C71—41no—
68C2×D4×C71—41no—
69C2×D4×C71D812yes—
70C2×D4×C71SD1612yes—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82M16×C71—42no—
83M16×C71—42no—
84C2×C8×C71—42no—
85C4⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—41no—
88C42×C71—21no—
89C42×C71—21no—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—41no—
95C23×C71—41no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71#10212yes—
99C23×C71#10212yes—
100C23×C71C2×C22⋊C411yescharacteristic
101D4×C71—42no—
102D4×C71—42no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—22no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—81no—
116C2×C4×C71#9911yescharacteristic
117C2×C4×C71#9911yescharacteristic
118C2×C4×C71C2×D811yescharacteristic
119C2×C4×C71C2×SD1611yescharacteristic
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—84no—
127C8×C71—82no—
128C71⋊C8—82no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—22no—
132C22×C71—22no—
133C22×C71—42no—
134C22×C71—42no—
135C22×C71—42no—
136C22×C71—41no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—81no—
141C22×C71C2×#9911yescharacteristic
142C22×C71C2×#10211yescharacteristic
143C22×C71#42111yescharacteristic
144C4×C71—21no—
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71#35212yes—
151Dic71—82no—
152Dic71—82no—
153#11881—711nomaximal, Sylow
154C2×C71—21no—
155C2×C71—21no—
156C2×C71—42no—
157C2×C71—42no—
158C2×C71#151911yescharacteristic
159C2×C71C2×#35211yescharacteristic
160C2×C71#213511yescharacteristic
161#2486—711no—
162#1800D7111yescharacteristic
163#1459—711no—
164#1346—714no—
165C71#1188111yescharacteristic, complemented, Sylow
166C2×#123—1421no—
167C2×C4×D4C2×D7111yescharacteristic
168C22×C4⋊C4—711no—
169#423—21no—
170C2×#100—21no—
171C2×C22⋊C8—1421no—
172#398—712no—
173#397—712no—
174#389—1421no—
175#381—1421no—
176#380—711no—
177#380—1421no—
178#343C2×D7112yes—
179#324—712no—
180#323—1421no—
181C22×D4C22⋊D7111yescharacteristic
182C23×C4—1421no—
183C23×C4C4×D7111yescharacteristic
184C23×C4D28411yescharacteristic
185C2×M16—22no—
186C22×C8—2841no—
187#123—2842no—
188#123—2842no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—2841no—
192C2×C4⋊C4—2841no—
193C2×C4⋊C4—712no—
194C2×C4⋊C4—1421no—
195C2×C4⋊C4—1421no—
196C2×C4⋊C4—1422no—
197C2×C4⋊C4C22×D7111yescharacteristic
198C2×C22⋊C4—1421no—
199C2×C22⋊C4—1422no—
200C2×C22⋊C4C4×D7111yescharacteristic
201C2×C22⋊C4D28411yescharacteristic
202C2×C42C22⋊D7111yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205#100—42no—
206C22⋊C8—2841no—
207C22⋊C8—1422no—
208#95—712no—
209#95—1422no—
210#95—1422no—
211C24(C2×D71)⋊C411yescharacteristic
212C24(C2×D71)⋊C411yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#1176585012yes—
216C2×D4#1176585212yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—2842no—
224C22×C4—2842no—
225C22×C4—2842no—
226C22×C4—1422no—
227C22×C4—1422no—
228C22×C4(C2×D71)⋊C411yescharacteristic
229C22×C4(C2×D71)⋊C411yescharacteristic
230C22×C4C2×C4×D7111yescharacteristic
231C22×C4C2×C22⋊D7111yescharacteristic
232C22×C4C2×D28411yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—2842no—
237C2×C8—5682no—
238C4⋊C4—41no—
239C4⋊C4—2842no—
240C4⋊C4—2842no—
241C4⋊C4—2841no—
242C4⋊C4—2841no—
243C4⋊C4—2842no—
244C4⋊C4—2842no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—2842no—
248C22⋊C4—2844no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1455604812yes—
260C23#1455604812yes—
261C23C2×(C2×D71)⋊C411yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—2842no—
278C2×C4—2842no—
279C2×C4—2842no—
280C2×C4—2842no—
281C2×C4—2842no—
282C2×C4—2842no—
283C2×C4—5684no—
284C2×C4#1455604211yescharacteristic
285C2×C4#1455604211yescharacteristic
286C2×C4C2×#1176585011yescharacteristic
287C2×C4C2×#1176585211yescharacteristic
288C8—82no—
289C8—5682no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1455604211yescharacteristic
303C22C2×#1455604811yescharacteristic
304C22#1798601711yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—5682no—
312C4—5682no—
313C4#1798579012yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2258690711yescharacteristic
319C2C2×#1798579011yescharacteristic
320C2#2259003411yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1800C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C4⋊C4×C71C23
hypercenter#1800D71
hypocenterC71#11881
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #22586345C2semidirect
165. C71#11881semidirect

Maximal subgroups

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

subgroupindex
#225926482
#225866582
#225863452
C71×#18002
#1188171