Finite Group Databasebeta

← #31908606

Subgroups — #31908606

Order 18176.

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

normal subgroups only

This group has 372 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
1#22588336C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586370C212yesmaximal
4#22586367C212yesmaximal
5C71×#2581C211yesmaximal, characteristic
6#17986052—21no—
7#17986023—21no—
8C2×#14556047—21no—
9#17985768C2212yes—
10#17985725—21no—
11#17985725C2212yes—
12C2×C4×Q8×C71C411yescharacteristic
13C2×C4×Q8×C71C2211yescharacteristic
14#400×C71C411yescharacteristic
15#400×C71C2211yescharacteristic
16#395×C71C411yescharacteristic
17#395×C71C2211yescharacteristic
18#391×C71—21no—
19#389×C71—21no—
20#387×C71—21no—
21#387×C71—21no—
22#382×C71C411yescharacteristic
23C2×#11765849—22no—
24C4⋊(C71⋊C8)—41no—
25C4⋊(C71⋊C8)—41no—
26C4⋊(C71⋊C8)—41no—
27#14556047—42no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—41no—
30C22×Q8×C71C2×C411yescharacteristic
31C22×Q8×C71D411yescharacteristic
32C4×Q8×C71—41no—
33C4×Q8×C71—41no—
34C4×Q8×C71—41no—
35C4×Q8×C71—41no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C42×C71—21no—
45C2×C42×C71C2×C411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
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—42no—
58#11765849—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C2×Q8×C71—41no—
62C2×Q8×C71—41no—
63C2×Q8×C71—42no—
64C2×Q8×C71—41no—
65C2×Q8×C71SD1612yes—
66C2×Q8×C71Q1612yes—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C4⋊C4×C71—41no—
79C4⋊C4×C71—41no—
80C4⋊C4×C71—41no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—42no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C42×C71—21no—
88C42×C71—21no—
89C42×C71—21no—
90C42×C71—21no—
91C42×C71—21no—
92C42×C71—21no—
93C42×C71—42no—
94C23×C71C2×C22⋊C411yescharacteristic
95Q8×C71—42no—
96Q8×C71—42no—
97Q8×C71—42no—
98Q8×C71—42no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—22no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—42no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71#9911yescharacteristic
124C2×C4×C71#9911yescharacteristic
125C2×C4×C71#10312yes—
126C2×C4×C71#10312yes—
127C2×C4×C71C2×SD1611yescharacteristic
128C2×C4×C71C2×Q1611yescharacteristic
129C71⋊C8—82no—
130C71⋊C8—82no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71C2×#9911yescharacteristic
134C22×C71C2×#10311yescharacteristic
135C22×C71#42111yescharacteristic
136C4×C71—21no—
137C4×C71—21no—
138C4×C71—21no—
139C4×C71—22no—
140C4×C71—22no—
141C4×C71—21no—
142C4×C71—41no—
143C4×C71—41no—
144C4×C71—42no—
145C4×C71—41no—
146C4×C71—81no—
147C4×C71—81no—
148C4×C71—81no—
149#11905—711nomaximal, Sylow
150C2×C71—21no—
151C2×C71—21no—
152C2×C71#152011yescharacteristic
153C2×C71#212011yescharacteristic
154C2×C71#213311yescharacteristic
155#2581D7111yescharacteristic
156#1801—711no—
157#1461—711no—
158#1354—712no—
159#1353—712no—
160C71#1190511yescharacteristic, complemented, Sylow
161C2×C4×Q8Dic7111yescharacteristic
162C2×C4×Q8C2×D7111yescharacteristic
163#431—1421no—
164#423—1421no—
165C2×#101—1421no—
166#400Dic7111yescharacteristic
167#400C2×D7111yescharacteristic
168#395Dic7111yescharacteristic
169#395C2×D7111yescharacteristic
170#391—21no—
171#389—21no—
172#387—21no—
173#387—21no—
174#382Dic7111yescharacteristic
175#343—712no—
176#324—712no—
177#324—1421no—
178C22×Q8C2×Dic7111yescharacteristic
179C22×Q8C22⋊D7111yescharacteristic
180C2×M16—1422no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C2×Dic7111yescharacteristic
190C2×C4⋊C4C2×Dic7111yescharacteristic
191C2×C4⋊C4C22⋊D7111yescharacteristic
192C2×C4⋊C4C22×D7111yescharacteristic
193C2×C42—21no—
194C2×C42C2×Dic7111yescharacteristic
195C2×C42C22⋊D7111yescharacteristic
196C2×C42C22⋊D7111yescharacteristic
197C4⋊C8—2841no—
198C4⋊C8—2841no—
199C4⋊C8—2841no—
200#101—2842no—
201C22⋊C8—1422no—
202C4×C8—2841no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95C2×Dic7111yescharacteristic
211#95C2×Dic7111yescharacteristic
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8#1176585212yes—
217C2×Q8#1176585512yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4C22⋊Dic7111yescharacteristic
223C22×C4C22⋊Dic7111yescharacteristic
224C22×C4C22⋊Dic7111yescharacteristic
225C22×C4C22⋊Dic7111yescharacteristic
226C22×C4C22×Dic7111yescharacteristic
227C22×C4C2×C22⋊D7111yescharacteristic
228C22×C4C2×C22⋊D7111yescharacteristic
229M16—2842no—
230M16—2842no—
231C2×C8—2842no—
232C2×C8—2842no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—42no—
249C23C2×C22⋊Dic7111yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—42no—
253Q8—42no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4#1455604311yescharacteristic
279C2×C4#1455604311yescharacteristic
280C2×C4#1455605212yes—
281C2×C4#1455605212yes—
282C2×C4C2×#1176585211yescharacteristic
283C2×C4C2×#1176585511yescharacteristic
284C8—5682no—
285C8—5682no—
286C22—21no—
287C22—21no—
288C22C2×#1455604311yescharacteristic
289C22C2×#1455605211yescharacteristic
290C22#1798601611yescharacteristic
291C4—21no—
292C4—22no—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C2—21no—
305C2—21no—
306C2#2258691211yescharacteristic
307C2#2258998111yescharacteristic
308C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2581C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2581D71
hypocenterC71#11905
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
160. C71#11905semidirect

Maximal subgroups

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

subgroupindex
#225883362
#225866642
#225863702
#225863672
C71×#25812
#1190571