Finite Group Databasebeta

← #31912626

Subgroups — #31912626

Order 18176.

628 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#17985792C212yesmaximal
2C4×#14556043C212yesmaximal, complemented
3#22586465C212yesmaximal, complemented
4C71×#2328C211yesmaximal, characteristic
5C2×#14556043C2214yes—
6C4×C22⋊Dic71—22no—
7#17985792C418yescomplemented
8#17985766—22no—
9C2×#127×C71C2211yescharacteristic
10C2×C4×D4×C71C411yescharacteristic, complemented
11C2×C4×D4×C71C2211yescharacteristic
12#390×C71C411yescharacteristic, complemented
13#390×C71C2211yescharacteristic
14#386×C71—22no—
15#386×C71—22no—
16#382×C71C411yescharacteristic, complemented
17#376×C71C411yescharacteristic, complemented
18C2×C22⋊Dic71—24no—
19C2×C4×Dic71—42no—
20#14556043—24no—
21#14556043C2×C418yescomplemented
22C71⋊#95—42no—
23C22×D4×C71—22no—
24C22×D4×C71C2×C411yescharacteristic
25C22×D4×C71C2311yescharacteristic
26C23×C4×C71—21no—
27C23×C4×C71D412yes—
28#127×C71—22no—
29#127×C71C2×C414yes—
30C4×D4×C71—42no—
31C4×D4×C71—42no—
32C2×C4⋊C4×C71C2×C411yescharacteristic
33C2×C4⋊C4×C71C2×C411yescharacteristic
34C2×C22⋊C4×C71—21no—
35C2×C22⋊C4×C71—22no—
36C2×C22⋊C4×C71—22no—
37C2×C22⋊C4×C71D412yes—
38C2×C42×C71C2×C411yescharacteristic
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71C2×C411yescharacteristic
41#95×C71—21no—
42#95×C71—21no—
43#95×C71—22no—
44C22×Dic71—44no—
45C22⋊Dic71—28no—
46C22⋊Dic71—44no—
47C4×Dic71—84no—
48C24×C71—21no—
49C24×C71C2×D412yes—
50C2×D4×C71—28no—
51C2×D4×C71—21no—
52C2×D4×C71—41no—
53C2×D4×C71—41no—
54C2×D4×C71—41no—
55C2×D4×C71—41no—
56C2×D4×C71—82no—
57C2×D4×C71—82no—
58C2×D4×C71C4212yescomplemented
59C2×D4×C71C22×C412yes—
60C2×D4×C71#4112yes—
61C22×C4×C71—21no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—42no—
67C22×C4×C71—82no—
68C22×C4×C71—82no—
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22×C411yescharacteristic
73C4⋊C4×C71—41no—
74C4⋊C4×C71—41no—
75C22⋊C4×C71—42no—
76C22⋊C4×C71—42no—
77C22⋊C4×C71—82no—
78C22⋊C4×C71—82no—
79C42×C71—22no—
80C42×C71—21no—
81C42×C71—41no—
82C42×C71C4212yescomplemented
83C23×C71—22no—
84C23×C71—22no—
85C23×C71—22no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71—42no—
89C23×C71—42no—
90C23×C71—41no—
91C23×C71—41no—
92C23×C71C2×C22⋊C411yescharacteristic
93C23×C71C4×D414yes—
94D4×C71—42no—
95D4×C71—42no—
96D4×C71—42no—
97D4×C71—42no—
98D4×C71—88no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—22no—
102C2×C4×C71—21no—
103C2×C4×C71—22no—
104C2×C4×C71—41no—
105C2×C4×C71—44no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—42no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—42no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—82no—
116C2×C4×C71—84no—
117C2×C4×C71#9912yes—
118C2×C4×C71C2×C4211yescharacteristic
119C2×C4×C71#11711yescharacteristic
120C2×Dic71—48no—
121C2×Dic71—84no—
122C2×Dic71—84no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—24no—
126C22×C71—44no—
127C22×C71—44no—
128C22×C71—42no—
129C22×C71—42no—
130C22×C71—42no—
131C22×C71—82no—
132C22×C71—82no—
133C22×C71—81no—
134C22×C71—81no—
135C22×C71C4×C22⋊C411yescharacteristic
136C22×C71C2×#9911yescharacteristic
137C22×C71#41011yescharacteristic
138C4×C71—22no—
139C4×C71—21no—
140C4×C71—21no—
141C4×C71—42no—
142C4×C71—42no—
143C4×C71—41no—
144C4×C71—81no—
145C4×C71—81no—
146Dic71—88no—
147#12628—711nomaximal, Sylow
148C2×C71—21no—
149C2×C71—21no—
150C2×C71—44no—
151C2×C71—82no—
152C2×C71C4×#9911yescharacteristic
153C2×C71#212711yescharacteristic
154C2×C71#212811yescharacteristic
155#2328D7111yescharacteristic
156C2×#353—712no—
157C4×#99—712no—
158#1393—712no—
159C71#1262811yescharacteristic, complemented, Sylow
160C2×#127C2×D7111yescharacteristic
161C2×C4×D4Dic7111yescharacteristic, complemented
162C2×C4×D4C2×D7111yescharacteristic
163C2×#99—714no—
164#390Dic7111yescharacteristic, complemented
165#390C2×D7111yescharacteristic
166#386—22no—
167#386—22no—
168#382Dic7111yescharacteristic, complemented
169C4×C22⋊C4—1422no—
170#376Dic7111yescharacteristic, complemented
171#353—718no—
172#342—1422no—
173C22×D4—22no—
174C22×D4C2×Dic7111yescharacteristic
175C22×D4C22×D7111yescharacteristic
176C23×C4—21no—
177C23×C4C22⋊D7112yes—
178#127—22no—
179#127C4×D7114yes—
180C4×D4—42no—
181C4×D4—42no—
182C2×C4⋊C4C2×Dic7111yescharacteristic
183C2×C4⋊C4C2×Dic7111yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—1424no—
188C2×C22⋊C4C22⋊D7112yes—
189C2×C42—2842no—
190C2×C42C2×Dic7111yescharacteristic
191C2×C42C2×Dic7111yescharacteristic
192C2×C42C2×Dic7111yescharacteristic
193#99—718no—
194#99—1424no—
195#95—21no—
196#95—22no—
197#95—21no—
198#95—2842no—
199C24—21no—
200C24C2×C22⋊D7112yes—
201C2×D4—28no—
202C2×D4—21no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—82no—
208C2×D4—82no—
209C2×D4C4×Dic7112yescomplemented
210C2×D4C2×C4×D7112yes—
211C2×D4#1176586512yes—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—2844no—
221C22×C4C22⋊Dic7111yescharacteristic
222C22×C4C22⋊Dic7112yes—
223C22×C4C22⋊Dic7111yescharacteristic
224C22×C4C22×Dic7111yescharacteristic
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—2844no—
232C22⋊C4—1428no—
233C42—22no—
234C42—21no—
235C42—41no—
236C42—5684no—
237C42C4×Dic7112yescomplemented
238C23—22no—
239C23—22no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊Dic7111yescharacteristic
248C23C4×C22⋊D7114yes—
249D4—42no—
250D4—42no—
251D4—42no—
252D4—42no—
253D4—88no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—44no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—2848no—
273C2×C4—5684no—
274C2×C4—5684no—
275C2×C4#1455604312yes—
276C2×C4C2×C4×Dic7111yescharacteristic
277C2×C4#1455608711yescharacteristic
278C22—21no—
279C22—21no—
280C22—24no—
281C22—44no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—82no—
287C22—82no—
288C22—81no—
289C22—81no—
290C22C4×C22⋊Dic7111yescharacteristic
291C22C2×#1455604311yescharacteristic
292C22#1798597311yescharacteristic
293C4—22no—
294C4—21no—
295C4—21no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4—5688no—
302C2—21no—
303C2—21no—
304C2—44no—
305C2—82no—
306C2C4×#1455604311yescharacteristic
307C2#2259000911yescharacteristic
308C2#2259001211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC2×C4×C71C2×C42
FrattiniC22×D4C22×D71
FittingC71×#2328C2
socleC22×C71C2×#99
prime radicalC2×#127×C71C22
cosocleC22×D4×C71C23
hypercenter#2328D71
hypocenterC71#12628
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. C4×#14556043C2semidirect
3. #22586465C2semidirect
7. #17985792C4semidirect
10. C2×C4×D4×C71C4semidirect
12. #390×C71C4semidirect
16. #382×C71C4semidirect
17. #376×C71C4semidirect
21. #14556043C2×C4semidirect
58. C2×D4×C71C42semidirect
82. C42×C71C42semidirect
159. C71#12628semidirect
161. C2×C4×D4Dic71semidirect
164. #390Dic71semidirect
168. #382Dic71semidirect
170. #376Dic71semidirect
209. C2×D4C4×Dic71semidirect
237. C42C4×Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179857922
C4×#145560432
#225864652
C71×#23282
#1262871