Finite Group Databasebeta

← #31907826

Subgroups — #31907826

Order 18176.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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×#14556045C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586367C212yesmaximal, complemented
4#22586346C212yesmaximal, complemented
5C71×#2395C211yesmaximal, characteristic
6C2×#14556045—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C4×#11765849—21no—
9#17985768C2212yes—
10#17985725C2212yes—
11#17985723—21no—
12C2×C4×D4×C71C2211yescharacteristic
13C2×#117×C71C411yescharacteristic
14#400×C71C2211yescharacteristic
15#399×C71C411yescharacteristic
16#398×C71C2211yescharacteristic
17#397×C71—21no—
18#397×C71C411yescharacteristic
19#394×C71—21no—
20C4×C4⋊C4×C71—21no—
21C4×C22⋊C4×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic
23C2×#11765849—22no—
24C22×C71⋊C8—41no—
25#14556045—24no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—41no—
28#14556039—41no—
29C4×C71⋊C8—41no—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C4×D4×C71—41no—
34C4×D4×C71—41no—
35#117×C71—41no—
36#117×C71—41no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C42×C71—21no—
49C2×C42×C71—21no—
50C2×C42×C71D411yescharacteristic
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71C2×C411yescharacteristic
56#95×C71C2×C411yescharacteristic
57#11765849—24no—
58#11765849—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—82no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C22⋊C411yescharacteristic
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71—41no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
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
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—42no—
87C22⋊C4×C71—41no—
88C22⋊C4×C71—41no—
89C22⋊C4×C71—42no—
90C22⋊C4×C71—42no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—42no—
96C42×C71—41no—
97C42×C71—41no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71#10412yes—
108C23×C71C2×C22⋊C411yescharacteristic
109D4×C71—44no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—42no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
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—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—81no—
135C2×C4×C71#9911yescharacteristic
136C2×C4×C71#9911yescharacteristic
137C2×C4×C71#13512yes—
138C71⋊C8—44no—
139C71⋊C8—82no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—22no—
143C22×C71—42no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71—81no—
150C22×C71—81no—
151C22×C71—81no—
152C22×C71C2×#9911yescharacteristic
153C22×C71#41611yescharacteristic
154C22×C71C2×#10411yescharacteristic
155C4×C71—21no—
156C4×C71—22no—
157C4×C71—42no—
158C4×C71—41no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—81no—
162C4×C71—81no—
163C4×C71—81no—
164C4×C71#35212yes—
165#11735—711nomaximal, Sylow
166C2×C71—21no—
167C2×C71—21no—
168C2×C71—42no—
169C2×C71—81no—
170C2×C71#150111yescharacteristic
171C2×C71C2×#35211yescharacteristic
172C2×C71#213211yescharacteristic
173#2395D7111yescharacteristic
174C4×#100—711no—
175#1459—711no—
176#1353—712no—
177#1346—712no—
178C71#1173511yescharacteristic, complemented, Sylow
179C2×C4×D4C2×D7111yescharacteristic
180C2×#117Dic7111yescharacteristic
181C2×#100—1421no—
182C2×C22⋊C8—1421no—
183C4×M16—1421no—
184#400C2×D7111yescharacteristic
185#399Dic7111yescharacteristic
186#398C2×D7111yescharacteristic
187#397—21no—
188#397Dic7111yescharacteristic
189#394—21no—
190C4×C4⋊C4—21no—
191C4×C22⋊C4—21no—
192C4×C22⋊C4Dic7111yescharacteristic
193#343—712no—
194#324—712no—
195#323—1421no—
196C22×D4C22⋊D7111yescharacteristic
197C23×C4C2×Dic7111yescharacteristic
198C23×C4C22⋊D7111yescharacteristic
199C2×M16—1422no—
200C22×C8—2841no—
201C4×D4—41no—
202C4×D4—41no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×Dic7111yescharacteristic
208C2×C4⋊C4C2×Dic7111yescharacteristic
209C2×C4⋊C4C22×D7111yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C2×Dic7111yescharacteristic
215C2×C22⋊C4C22⋊D7111yescharacteristic
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C22⋊D7111yescharacteristic
219#100—1424no—
220C22⋊C8—2841no—
221C22⋊C8—1422no—
222#97—2841no—
223C4×C8—2841no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95C2×Dic7111yescharacteristic
229#95C2×Dic7111yescharacteristic
230C24C22⋊Dic7111yescharacteristic
231C24C22⋊Dic7111yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4C22⋊Dic7111yescharacteristic
246C22×C4C22⋊Dic7111yescharacteristic
247C22×C4C22×Dic7111yescharacteristic
248C22×C4C2×C22⋊D7111yescharacteristic
249C22×C4C2×C22⋊D7111yescharacteristic
250M16—2842no—
251M16—1424no—
252C2×C8—2842no—
253C2×C8—2842no—
254C2×C8—5682no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—21no—
267C42—21no—
268C42—42no—
269C42—41no—
270C42—41no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1455605512yes—
281C23C2×C22⋊Dic7111yescharacteristic
282D4—44no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4#1455604311yescharacteristic
309C2×C4#1455604311yescharacteristic
310C2×C4#1455614812yes—
311C8—2844no—
312C8—5682no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C2×#1455604311yescharacteristic
326C22#1798599511yescharacteristic
327C22C2×#1455605511yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4#1798578912yes—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2258683111yescharacteristic
343C2C2×#1798578911yescharacteristic
344C2#2259002511yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2395C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C4⋊C4×C71C23
hypercenter#2395D71
hypocenterC71#11735
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2395

Complements

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

NH ≅ G/Ntype
3. #22586367C2semidirect
4. #22586346C2semidirect
178. C71#11735semidirect

Maximal subgroups

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

subgroupindex
C4×#145560452
#225866582
#225863672
#225863462
C71×#23952
#1173571