Finite Group Databasebeta

← #31906920

Subgroups — #31906920

Order 18176.

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

normal subgroups only

This group has 596 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#22586678C212yesmaximal, complemented
2#22586286C212yesmaximal, complemented
3#22586272C212yesmaximal, complemented
4C4×#127×C71C211yesmaximal, characteristic
5#17985963—22no—
6#17985725C2214yes—
7#17985723—22no—
8C2×#127×C71C411yescharacteristic
9C2×C4×D4×C71—22no—
10C2×C4×D4×C71C411yescharacteristic
11C2×C4×D4×C71C2211yescharacteristic
12#390×C71—22no—
13#390×C71C411yescharacteristic
14#390×C71C2211yescharacteristic
15#384×C71C2211yescharacteristic
16C43×C71C411yescharacteristic
17C2×#11765849—42no—
18C22⋊(C71⋊C8)—24no—
19C22⋊(C71⋊C8)—42no—
20C22×D4×C71—22no—
21C22×D4×C71C2×C411yescharacteristic
22C22×D4×C71C2×C411yescharacteristic
23C23×C4×C71—21no—
24C23×C4×C71D412yes—
25#127×C71—22no—
26#127×C71—22no—
27C4×D4×C71—41no—
28C4×D4×C71—41no—
29C4×D4×C71—48no—
30C4×D4×C71—41no—
31C4×D4×C71—41no—
32C2×C4⋊C4×C71—22no—
33C2×C4⋊C4×C71C2×C411yescharacteristic
34C2×C4⋊C4×C71C2311yescharacteristic
35C2×C22⋊C4×C71—22no—
36C2×C22⋊C4×C71—21no—
37C2×C22⋊C4×C71—22no—
38C2×C22⋊C4×C71D412yes—
39C2×C42×C71—21no—
40C2×C42×C71—21no—
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71C2×C411yescharacteristic
44#11765849—84no—
45C2×C71⋊C8—44no—
46C24×C71—21no—
47C24×C71C22⋊C412yes—
48C2×D4×C71—28no—
49C2×D4×C71—21no—
50C2×D4×C71—21no—
51C2×D4×C71—21no—
52C2×D4×C71—41no—
53C2×D4×C71—41no—
54C2×D4×C71—41no—
55C2×D4×C71—41no—
56C2×D4×C71—44no—
57C2×D4×C71—44no—
58C2×D4×C71M1612yescomplemented
59C22×C4×C71—21no—
60C22×C4×C71—21no—
61C22×C4×C71—22no—
62C22×C4×C71—22no—
63C22×C4×C71—21no—
64C22×C4×C71—44no—
65C22×C4×C71—44no—
66C22×C4×C71—41no—
67C22×C4×C71—41no—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22×C411yescharacteristic
71C22×C4×C71C2×D412yes—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—44no—
75C22⋊C4×C71—42no—
76C22⋊C4×C71—44no—
77C22⋊C4×C71—44no—
78C22⋊C4×C71—42no—
79C42×C71—24no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—41no—
84C42×C71—42no—
85C42×C71M1612yescomplemented
86C23×C71—22no—
87C23×C71—22no—
88C23×C71—22no—
89C23×C71—22no—
90C23×C71—42no—
91C23×C71—42no—
92C23×C71—42no—
93C23×C71—42no—
94C23×C71—41no—
95C23×C71—41no—
96C23×C71C2×C22⋊C411yescharacteristic
97D4×C71—42no—
98D4×C71—416no—
99D4×C71—42no—
100D4×C71—42no—
101D4×C71—42no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—21no—
110C2×C4×C71—22no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—44no—
114C2×C4×C71—41no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—41no—
118C2×C4×C71—82no—
119C2×C4×C71—82no—
120C2×C4×C71—82no—
121C2×C4×C71#9911yescharacteristic
122C2×C4×C71#9911yescharacteristic
123C2×C4×C71C2×M1611yescharacteristic
124C2×C4×C71C2×M1611yescharacteristic
125C71⋊C8—84no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—44no—
129C22×C71—44no—
130C22×C71—42no—
131C22×C71—42no—
132C22×C71—42no—
133C22×C71—42no—
134C22×C71—82no—
135C22×C71—82no—
136C22×C71—81no—
137C22×C71—81no—
138C22×C71#40711yescharacteristic
139C22×C71C2×#9911yescharacteristic
140C22×C71#41311yescharacteristic
141C4×C71—24no—
142C4×C71—21no—
143C4×C71—21no—
144C4×C71—21no—
145C4×C71—42no—
146C4×C71—42no—
147C4×C71—41no—
148C4×C71—81no—
149C4×C71#35312yes—
150#11529—711nomaximal, Sylow
151C2×C71—21no—
152C2×C71—21no—
153C2×C71—82no—
154C2×C71—82no—
155C2×C71#146511yescharacteristic
156C2×C71C2×#35311yescharacteristic
157C2×C71#213211yescharacteristic
158C4×#127D7111yescharacteristic
159#1465—712no—
160#1325—712no—
161#1318—712no—
162C71#1152911yescharacteristic, complemented, Sylow
163C2×#127Dic7111yescharacteristic
164C2×C4×D4—22no—
165C2×C4×D4Dic7111yescharacteristic
166C2×C4×D4C2×D7111yescharacteristic
167#407—1422no—
168#390—22no—
169#390Dic7111yescharacteristic
170#390C2×D7111yescharacteristic
171#384C2×D7111yescharacteristic
172C43Dic7111yescharacteristic
173#324—714no—
174#323—1422no—
175C22×D4—22no—
176C22×D4C2×Dic7111yescharacteristic
177C22×D4C2×Dic7111yescharacteristic
178C23×C4—21no—
179C23×C4C22⋊D7112yes—
180C2×M16—2842no—
181#127—22no—
182#127—22no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—48no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C2×Dic7111yescharacteristic
190C2×C4⋊C4C22×D7111yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C22⋊D7112yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2×Dic7111yescharacteristic
198C2×C42C2×Dic7111yescharacteristic
199C2×C42C2×Dic7111yescharacteristic
200C22⋊C8—2842no—
201C22⋊C8—1424no—
202C24—21no—
203C24C22⋊Dic7112yes—
204C2×D4—28no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4#1176584912yescomplemented
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C22⋊Dic7111yescharacteristic
225C22×C4C22⋊Dic7111yescharacteristic
226C22×C4C22×Dic7111yescharacteristic
227C22×C4C2×C22⋊D7112yes—
228M16—5684no—
229C2×C8—2844no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—42no—
237C42—24no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—41no—
242C42—42no—
243C42#1176584912yescomplemented
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23C2×C22⋊Dic7111yescharacteristic
255D4—42no—
256D4—416no—
257D4—42no—
258D4—42no—
259D4—42no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4#1455604311yescharacteristic
280C2×C4#1455604311yescharacteristic
281C2×C4C2×#1176584911yescharacteristic
282C2×C4C2×#1176584911yescharacteristic
283C8—5684no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—82no—
294C22—81no—
295C22—81no—
296C22#1798596311yescharacteristic
297C22C2×#1455604311yescharacteristic
298C22#1798598311yescharacteristic
299C4—24no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—42no—
305C4—41no—
306C4—81no—
307C4#1798579212yes—
308C2—21no—
309C2—21no—
310C2—82no—
311C2—82no—
312C2#2258667811yescharacteristic
313C2C2×#1798579211yescharacteristic
314C2#2259002511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC4×#127×C71C2
socleC22×C71C2×#99
prime radicalC2×#127×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenterC4×#127D71
hypocenterC71#11529
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. #22586678C2semidirect
2. #22586286C2semidirect
3. #22586272C2semidirect
58. C2×D4×C71M16semidirect
85. C42×C71M16semidirect
162. C71#11529semidirect
214. C2×D4#11765849semidirect
243. C42#11765849semidirect

Maximal subgroups

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

subgroupindex
#225866782
#225862862
#225862722
C4×#127×C712
#1152971