Finite Group Databasebeta

← #31894113

Subgroups — #31894113

Order 18176.

422 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 107 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#22599288C211yesmaximal, characteristic
2#22597277C211yesmaximal, characteristic
3#22589314C211yesmaximal, characteristic
4#22586905C211yesmaximal, characteristic
5#22586200C211yesmaximal, characteristic
6#22586170C211yesmaximal, characteristic
7C71×#1549C211yesmaximal, characteristic
8#17986556C2211yescharacteristic
9#17986378C412yescomplemented
10#17986349C412yescomplemented
11#17986206—22no—
12#17986155—22no—
13#17986128—22no—
14#17986128—22no—
15#17986052C2211yescharacteristic
16#17986027—21no—
17#17985957—21no—
18C2×#14556039C2211yescharacteristic
19#17985865C2211yescharacteristic
20#17985761—21no—
21#17985761—21no—
22#17985737—22no—
23C2×C4⋊Q8×C71C2211yescharacteristic
24#431×C71C2211yescharacteristic
25#423×C71C2211yescharacteristic
26#332×C71—22no—
27C2×#11765855—24no—
28C22×C71⋊C8D411yescharacteristic
29C22×C71⋊C8D411yescharacteristic
30#14556128C2×C411yescharacteristic
31#14556128C2×C411yescharacteristic
32#14556120—21no—
33#14556107—21no—
34#14556107D412yes—
35C4×Dic142—21no—
36C4×Dic142D412yes—
37#14556087—21no—
38C2×C4⋊Dic71—21no—
39C4⋊(C71⋊C8)—21no—
40Dic71⋊C8—24no—
41#14556052—24no—
42C22⋊(C71⋊C8)—21no—
43#14556039—21no—
44#14556039—21no—
45#14556038—22no—
46C4×C71⋊C8C2×C412yes—
47C8×Dic71—22no—
48C71⋊#95—21no—
49C71⋊#95—21no—
50C22×Q8×C71—21no—
51C2×M16×C71—21no—
52C4⋊Q8×C71—21no—
53C4⋊Q8×C71—21no—
54C4⋊Q8×C71—21no—
55C4⋊Q8×C71C2×C412yes—
56C2×C4⋊C4×C71—21no—
57C2×C4⋊C4×C71—21no—
58C2×C42×C71C2311yescharacteristic
59C4⋊C8×C71—21no—
60C4⋊C8×C71—21no—
61C4⋊C8×C71D412yes—
62C22⋊C8×C71—21no—
63C4×C8×C71—21no—
64C2×Dic142—21no—
65C2×Dic142C22⋊C412yes—
66C22×Dic71—41no—
67#11765855—84no—
68C2×C71⋊C8—21no—
69C2×C71⋊C8—21no—
70C2×C71⋊C8—22no—
71C2×C71⋊C8—21no—
72C2×C71⋊C8—21no—
73C2×C71⋊C8—21no—
74C2×C71⋊C8Q1612yescomplemented
75C4⋊Dic71—21no—
76C4⋊Dic71—21no—
77C4⋊Dic71—21no—
78C4⋊Dic71—41no—
79C4⋊Dic71—41no—
80C4⋊Dic71C22⋊C412yes—
81Dic71⋊C4—22no—
82Dic71⋊C4—22no—
83C22⋊Dic71—22no—
84C4×Dic71—22no—
85C2×Q8×C71—22no—
86C2×Q8×C71—41no—
87C2×Q8×C71—82no—
88C22×C4×C71—41no—
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71#4111yescharacteristic
92M16×C71—41no—
93M16×C71—41no—
94C2×C8×C71—22no—
95C2×C8×C71—41no—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C42×C71C22×C411yescharacteristic
101C42×C71C2×D411yescharacteristic
102C42×C71C2×D411yescharacteristic
103C42×C71#4111yescharacteristic
104C23×C71#12111yescharacteristic
105Q8×C71—44no—
106Q8×C71—82no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—42no—
110C2×C4×C71—82no—
111C2×C4×C71C2×C22⋊C411yescharacteristic
112C2×C4×C71C4×D411yescharacteristic
113C2×C4×C71C4×D411yescharacteristic
114C2×C4×C71#12011yescharacteristic
115C2×C4×C71#12111yescharacteristic
116C2×C4×C71#12311yescharacteristic
117C2×C4×C71C2×Q1611yescharacteristic
118C2×C4×C71#13511yescharacteristic
119C2×C4×C71#13711yescharacteristic
120C2×C4×C71#13711yescharacteristic
121C2×Dic71—22no—
122C2×Dic71—22no—
123C2×Dic71—81no—
124C2×Dic71—81no—
125C2×Dic71—81no—
126Dic142—42no—
127Dic142—81no—
128C8×C71—42no—
129C8×C71—81no—
130C71⋊C8—22no—
131C71⋊C8—42no—
132C71⋊C8—41no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71#38611yescharacteristic
136C22×C71#46211yescharacteristic
137C22×C71#46411yescharacteristic
138C4×C71—21no—
139C4×C71—21no—
140C4×C71—21no—
141C4×C71—21no—
142C4×C71—82no—
143C4×C71C4×Q1611yescharacteristic
144C4×C71#44111yescharacteristic
145C4×C71#45111yescharacteristic
146C4×C71#45211yescharacteristic
147Dic71—42no—
148Dic71—42no—
149Dic71—161no—
150#9290—711nomaximal, Sylow
151C2×C71—21no—
152C2×C71—21no—
153C2×C71#187811yescharacteristic
154C2×C71#190211yescharacteristic
155C2×C71#191011yescharacteristic
156#3209—711no—
157#2961—711no—
158#1993—711no—
159#1549D7111yescharacteristic
160#1518—711no—
161#1286—711no—
162#1275—711no—
163C71#929011yescharacteristic, complemented, Sylow
164#533—711no—
165C2×C4⋊Q8C2×D7111yescharacteristic
166#494—712no—
167#487—712no—
168#462—1422no—
169#451—1422no—
170#446—1422no—
171#446—1422no—
172#431—711no—
173#431C2×D7111yescharacteristic
174#424—1421no—
175#423C2×D7111yescharacteristic
176#405—1421no—
177C2×#97—711no—
178#382—711no—
179#340—1421no—
180#340—1421no—
181#332—22no—
182#328—1422no—
183C22×Q8—21no—
184C2×Q16—1424no—
185C2×M16—21no—
186C22×C8—711no—
187C22×C8—711no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—711no—
192C4⋊Q8—711no—
193C4⋊Q8C4×D7112yes—
194#125—1421no—
195C22⋊Q8—712no—
196C22⋊Q8—1421no—
197C4×Q8—712no—
198C4×Q8—1421no—
199#117—1421no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—1421no—
203C2×C42C22×D7111yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1424no—
207C4⋊C8—1421no—
208C4⋊C8C22⋊D7112yes—
209#103—1424no—
210C22⋊C8—21no—
211C22⋊C8—1421no—
212#97—1422no—
213#97—1421no—
214#97—1421no—
215C4×C8—21no—
216C4×C8—712no—
217C4×C8—1422no—
218#95—1421no—
219#95—1421no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—712no—
224C2×Q8—1421no—
225C22×C4—41no—
226C22×C4—2841no—
227C22×C4D4×D7111yescharacteristic
228C22×C4D4×D7111yescharacteristic
229C22×C4#1176586511yescharacteristic
230Q16—5684no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—712no—
236C2×C8—1422no—
237C2×C8—1421no—
238C2×C8—1421no—
239C2×C8—1421no—
240C2×C8—1421no—
241C2×C8—1421no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—2841no—
247C4⋊C4—2841no—
248C4⋊C4—712no—
249C4⋊C4—1421no—
250C4⋊C4—1421no—
251C4⋊C4—1421no—
252C4⋊C4—1422no—
253C4⋊C4—1422no—
254C22⋊C4—1422no—
255C42—1422no—
256C42C2×C4×D7111yescharacteristic
257C42D4×D7111yescharacteristic
258C42C2×C22⋊D7111yescharacteristic
259C42#1176586411yescharacteristic
260C23#1455610111yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—2842no—
264Q8—5681no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—5681no—
270C2×C4—5681no—
271C2×C4—5681no—
272C2×C4—1422no—
273C2×C4—1422no—
274C2×C4C22⋊C4×D7111yescharacteristic
275C2×C4(C22⋊D71)⋊C411yescharacteristic
276C2×C4C4×C22⋊D7111yescharacteristic
277C2×C4#1455609611yescharacteristic
278C2×C4#1455609911yescharacteristic
279C2×C4#1455611011yescharacteristic
280C2×C4Q16×D7111yescharacteristic
281C2×C4#1455614911yescharacteristic
282C2×C4#1455615711yescharacteristic
283C2×C4#1455616011yescharacteristic
284C8—42no—
285C8—81no—
286C8—2842no—
287C8—2841no—
288C8—1422no—
289C22—21no—
290C22—21no—
291C22#1798587511yescharacteristic
292C22#1798620811yescharacteristic
293C22#1798622411yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—2842no—
300C4—2842no—
301C4—11361no—
302C4#1798609111yescharacteristic
303C4#1798610311yescharacteristic
304C4#1798615311yescharacteristic
305C4#1798615811yescharacteristic
306C2—21no—
307C2—21no—
308C2#2258867811yescharacteristic
309C2#2258881611yescharacteristic
310C2#2258886411yescharacteristic
subgroupG/H
centerC22#17985875
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1549C2
socleC22×C71#386
prime radicalC23×C71#121
cosocleC2×C42×C71C23
hypercenter#1549D71
hypocenterC71#9290
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
9. #17986378C4semidirect
10. #17986349C4semidirect
74. C2×C71⋊C8Q16semidirect
163. C71#9290semidirect

Maximal subgroups

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

subgroupindex
#225992882
#225972772
#225893142
#225869052
#225862002
#225861702
C71×#15492
#929071