Finite Group Databasebeta

← #31891088

Subgroups — #31891088

Order 18176.

466 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#14556043C211yesmaximal, characteristic
2#22586678C211yesmaximal, characteristic
3#22586226C212yesmaximal, complemented
4#22586192C212yesmaximal, complemented
5C71×#2357C211yesmaximal, characteristic
6C2×#14556043—21no—
7#17985963C2212yes—
8C4×C22⋊Dic71C2212yes—
9#17985766—21no—
10#17985725—21no—
11#17985723—21no—
12C2×C4×D4×C71C2211yescharacteristic
13#390×C71—21no—
14#388×C71—21no—
15#388×C71—21no—
16#382×C71C411yescharacteristic, complemented
17#381×C71C411yescharacteristic, complemented
18#381×C71C2211yescharacteristic
19C4×C4⋊C4×C71—21no—
20C4×C4⋊C4×C71C411yescharacteristic, complemented
21C4×C22⋊C4×C71C411yescharacteristic, complemented
22C4×C22⋊C4×C71C2211yescharacteristic
23C2×#11765849—22no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—22no—
26#14556043—24no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C71⋊#95—22no—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71C2311yescharacteristic
33C4×D4×C71—22no—
34C4×D4×C71—22no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—21no—
39C2×C22⋊C4×C71—21no—
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71D411yescharacteristic
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C42×C71—21no—
45C2×C42×C71—21no—
46C2×C42×C71—21no—
47C2×C42×C71C2×C411yescharacteristic
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71C2×C411yescharacteristic
52#95×C71C2×C411yescharacteristic
53#95×C71C2×C411yescharacteristic
54#95×C71C2×C411yescharacteristic
55C22×Dic71—42no—
56#11765849—82no—
57C2×C71⋊C8—42no—
58C22⋊Dic71—24no—
59C22⋊Dic71—42no—
60C4×Dic71—82no—
61C24×C71C22⋊C411yescharacteristic
62C24×C71C2×D411yescharacteristic
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71—41no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—81no—
73C22×C4×C71—81no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71#4112yes—
80C22×C4×C71#4112yes—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—42no—
84C22⋊C4×C71—22no—
85C22⋊C4×C71—22no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—81no—
89C22⋊C4×C71—81no—
90C42×C71—21no—
91C42×C71—21no—
92C42×C71—42no—
93C42×C71—81no—
94C42×C71—81no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71C2×C22⋊C411yescharacteristic
105C23×C71#12312yes—
106D4×C71—44no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—22no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—81no—
130C2×C4×C71—81no—
131C2×C4×C71#10412yes—
132C2×C4×C71#11711yescharacteristic
133C2×C4×C71#11711yescharacteristic
134C2×C4×C71#12312yes—
135C2×Dic71—44no—
136C2×Dic71—82no—
137C2×Dic71—82no—
138C71⋊C8—82no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—22no—
142C22×C71—22no—
143C22×C71—22no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—81no—
148C22×C71—81no—
149C22×C71—81no—
150C22×C71—81no—
151C22×C71#38111yescharacteristic
152C22×C71C2×#10411yescharacteristic
153C22×C71#42111yescharacteristic
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—21no—
157C4×C71—21no—
158C4×C71—42no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—81no—
162C4×C71—81no—
163C4×C71—81no—
164Dic71—84no—
165#8778—711nomaximal, Sylow
166C2×C71—21no—
167C2×C71—21no—
168C2×C71—42no—
169C2×C71—81no—
170C2×C71#182611yescharacteristic
171C2×C71#182711yescharacteristic
172C2×C71#182811yescharacteristic
173#2357D7111yescharacteristic
174C4×#99—711no—
175#1465—711no—
176#1297—712no—
177#1284—712no—
178C71#877811yescharacteristic, complemented, Sylow
179C2×C4×D4C2×D7111yescharacteristic
180C2×#99—1421no—
181#407—712no—
182#390—21no—
183#388—21no—
184#388—21no—
185#382Dic7111yescharacteristic, complemented
186#381Dic7111yescharacteristic, complemented
187#381C2×D7111yescharacteristic
188C4×C4⋊C4—21no—
189C4×C4⋊C4Dic7111yescharacteristic, complemented
190C4×C22⋊C4—712no—
191C4×C22⋊C4Dic7111yescharacteristic, complemented
192C4×C22⋊C4C2×D7111yescharacteristic
193#342—1421no—
194#324—1421no—
195#323—1421no—
196C22×D4C22⋊D7111yescharacteristic
197C23×C4C2×Dic7111yescharacteristic
198C23×C4C22×D7111yescharacteristic
199C2×M16—1422no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22⋊D7111yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—1422no—
210C2×C22⋊C4C22⋊D7111yescharacteristic
211C2×C22⋊C4C22⋊D7111yescharacteristic
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—1422no—
216C2×C42C2×Dic7111yescharacteristic
217#99—1424no—
218C22⋊C8—1422no—
219C22⋊C8—1422no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—1422no—
224#95C2×Dic7111yescharacteristic
225#95C2×Dic7111yescharacteristic
226#95C2×Dic7111yescharacteristic
227#95C2×Dic7111yescharacteristic
228C24C22⋊Dic7111yescharacteristic
229C24C2×C22⋊D7111yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—2842no—
242C22×C4C22⋊Dic7111yescharacteristic
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22×Dic7111yescharacteristic
246C22×C4C2×C22⋊D7111yescharacteristic
247C22×C4#1176586512yes—
248C22×C4#1176586512yes—
249M16—5682no—
250C2×C8—2842no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—2842no—
261C22⋊C4—1424no—
262C42—21no—
263C42—21no—
264C42—42no—
265C42—81no—
266C42—81no—
267C42—5682no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊Dic7111yescharacteristic
278C23#1455611112yes—
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—2844no—
305C2×C4—5682no—
306C2×C4—5682no—
307C2×C4#1455605512yes—
308C2×C4#1455608711yescharacteristic
309C2×C4#1455608711yescharacteristic
310C2×C4#1455611112yes—
311C8—5682no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1798586211yescharacteristic
325C22C2×#1455605511yescharacteristic
326C22#1798601611yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—5684no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2258843911yescharacteristic
343C2#2258844411yescharacteristic
344C2#2258844811yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2357C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2357D71
hypocenterC71#8778
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
3. #22586226C2semidirect
4. #22586192C2semidirect
16. #382×C71C4semidirect
17. #381×C71C4semidirect
20. C4×C4⋊C4×C71C4semidirect
21. C4×C22⋊C4×C71C4semidirect
178. C71#8778semidirect
185. #382Dic71semidirect
186. #381Dic71semidirect
189. C4×C4⋊C4Dic71semidirect
191. C4×C22⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866782
#225862262
#225861922
C71×#23572
#877871