Finite Group Databasebeta

← #31908126

Subgroups — #31908126

Order 18176.

502 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588417C211yesmaximal, characteristic, complemented
2#22586674C211yesmaximal, characteristic, complemented
3#22586337C214yesmaximal, complemented
4C71×#2619C211yesmaximal, characteristic
5#17986056—21no—
6#17985862—21no—
7#17985766—21no—
8#17985766C2212yes—
9#17985725—21no—
10#17985723C2212yes—
11C2×#127×C71C411yescharacteristic, complemented
12C2×#121×C71—21no—
13C2×#117×C71C2211yescharacteristic
14#394×C71—21no—
15#394×C71C412yescomplemented
16#392×C71—21no—
17#392×C71C2212yes—
18#390×C71—21no—
19#382×C71C411yescharacteristic, complemented
20C2×C22⋊Dic71—22no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23#14556039—41no—
24C71⋊#95—41no—
25C71⋊#95—41no—
26C22×D4×C71—22no—
27C22×D4×C71—21no—
28C22×D4×C71—21no—
29C22×D4×C71C2×C412yes—
30C23×C4×C71D411yescharacteristic
31#127×C71—41no—
32#127×C71—41no—
33#121×C71—44no—
34#117×C71—41no—
35#117×C71—41no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C22⋊C4×C71—22no—
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71D411yescharacteristic
43C2×C22⋊C4×C71C2311yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46#95×C71—21no—
47#95×C71C2×C412yes—
48C22×Dic71—42no—
49C2×C71⋊C8—42no—
50C22⋊Dic71—42no—
51C22⋊Dic71—42no—
52C24×C71—22no—
53C24×C71C2×D411yescharacteristic
54C2×D4×C71—42no—
55C2×D4×C71—42no—
56C2×D4×C71—42no—
57C2×D4×C71—42no—
58C2×D4×C71—44no—
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61C2×D4×C71—82no—
62C2×D4×C71—82no—
63C2×D4×C71—82no—
64C2×D4×C71—82no—
65C22×C4×C71—22no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—41no—
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22×C411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C4⋊C4×C71—41no—
77C4⋊C4×C71—41no—
78C4⋊C4×C71—42no—
79C22⋊C4×C71—41no—
80C22⋊C4×C71—41no—
81C22⋊C4×C71—44no—
82C22⋊C4×C71—82no—
83C22⋊C4×C71—82no—
84C42×C71—21no—
85C42×C71—21no—
86C42×C71—21no—
87C42×C71—21no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—41no—
91C23×C71—41no—
92C23×C71—42no—
93C23×C71—42no—
94C23×C71—42no—
95C23×C71—42no—
96C23×C71—42no—
97C23×C71—42no—
98C23×C71—42no—
99C23×C71C2×C22⋊C411yescharacteristic
100C23×C71AGL(1,ℤ/8)12yes—
101D4×C71—82no—
102D4×C71—82no—
103D4×C71—84no—
104D4×C71—84no—
105D4×C71—88no—
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—21no—
114C2×C4×C71—21no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71#9911yescharacteristic
125C2×C4×C71#9911yescharacteristic
126C2×Dic71—82no—
127C2×Dic71—82no—
128C2×Dic71—82no—
129C71⋊C8—82no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—22no—
133C22×C71—42no—
134C22×C71—41no—
135C22×C71—41no—
136C22×C71—81no—
137C22×C71—82no—
138C22×C71—82no—
139C22×C71—82no—
140C22×C71—82no—
141C22×C71—82no—
142C22×C71—82no—
143C22×C71—82no—
144C22×C71C2×#9911yescharacteristic
145C22×C71#41811yescharacteristic
146C22×C71#42111yescharacteristic
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—41no—
152C4×C71—41no—
153C4×C71—41no—
154C4×C71—82no—
155Dic71—162no—
156#11800—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—21no—
159C2×C71—42no—
160C2×C71—162no—
161C2×C71#151011yescharacteristic
162C2×C71#212211yescharacteristic
163C2×C71#212811yescharacteristic
164#2619D7111yescharacteristic
165#1820—711no—
166#1464—711no—
167#1343—714no—
168C71#1180011yescharacteristic, complemented, Sylow
169C2×#127Dic7111yescharacteristic, complemented
170C2×#121—21no—
171C2×#117C2×D7111yescharacteristic
172#432—1421no—
173#394—21no—
174#394Dic7112yescomplemented
175#392—21no—
176#392C2×D7112yes—
177#390—21no—
178#382Dic7111yescharacteristic, complemented
179#381—1421no—
180#342—712no—
181#342—1421no—
182#324—1421no—
183#323—712no—
184C22×D4—22no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C2×Dic7112yes—
188C23×C4C22⋊D7111yescharacteristic
189#127—41no—
190#127—41no—
191#121—44no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic7111yescharacteristic
196C2×C4⋊C4C22⋊D7111yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1422no—
201C2×C22⋊C4C22⋊D7111yescharacteristic
202C2×C22⋊C4C22×D7111yescharacteristic
203C2×C42C2×Dic7111yescharacteristic
204C2×C42C22⋊D7111yescharacteristic
205C4⋊C8—2841no—
206C22⋊C8—1422no—
207#97—2841no—
208#95—21no—
209#95—2841no—
210#95—2841no—
211#95C2×Dic7112yes—
212C24—22no—
213C24C2×C22⋊D7111yescharacteristic
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—2842no—
231C22×C4C22⋊Dic7111yescharacteristic
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4C22⋊Dic7111yescharacteristic
234C22×C4C22⋊Dic7111yescharacteristic
235C22×C4C22×Dic7111yescharacteristic
236C22×C4C2×C22⋊D7111yescharacteristic
237C2×C8—2842no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—2842no—
247C22⋊C4—2842no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23C2×C22⋊Dic7111yescharacteristic
264C23#1455615412yes—
265D4—82no—
266D4—82no—
267D4—84no—
268D4—84no—
269D4—88no—
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—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—5682no—
289C2×C4—5682no—
290C2×C4—5682no—
291C2×C4#1455604311yescharacteristic
292C2×C4#1455604311yescharacteristic
293C8—5682no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22C2×#1455604311yescharacteristic
309C22#1798600411yescharacteristic
310C22#1798601611yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4—11362no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—162no—
324C2#2258686911yescharacteristic
325C2#2258999011yescharacteristic
326C2#2259001211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2619C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2619D71
hypocenterC71#11800
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #22588417C2semidirect
2. #22586674C2semidirect
3. #22586337C2semidirect
11. C2×#127×C71C4semidirect
15. #394×C71C4semidirect
19. #382×C71C4semidirect
168. C71#11800semidirect
169. C2×#127Dic71semidirect
174. #394Dic71semidirect
178. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884172
#225866742
#225863372
C71×#26192
#1180071