Finite Group Databasebeta

← #31895925

Subgroups — #31895925

Order 18176.

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

normal subgroups only

This group has 326 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#22589316C211yesmaximal, characteristic
2#22588195C211yesmaximal, characteristic
3#22586995C211yesmaximal, characteristic
4#22586742C211yesmaximal, characteristic
5#22586235C211yesmaximal, characteristic
6#22586172C211yesmaximal, characteristic
7C71×#1571C211yesmaximal, characteristic
8#17986556C2211yescharacteristic
9#17986052C2211yescharacteristic
10C2×C4⋊(C71⋊C8)C2211yescharacteristic
11C2×#14556038—21no—
12C42×Dic71C2211yescharacteristic
13#17985760—21no—
14#17985760—21no—
15#17985753—21no—
16#17985747C412yescomplemented
17#17985746C412yescomplemented
18#17985729—22no—
19#431×C71C2211yescharacteristic
20#423×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22#335×C71—22no—
23C22×C71⋊C8—21no—
24#14556128C2×C411yescharacteristic
25#14556128C2×C411yescharacteristic
26#14556120—21no—
27#14556107—21no—
28#14556107—21no—
29C4×Dic142—21no—
30C4×Dic142—21no—
31#14556087—21no—
32C2×C4×Dic71—21no—
33C2×C4×Dic71—21no—
34C2×C4×Dic71D411yescharacteristic
35C2×C4×Dic71D411yescharacteristic
36C4⋊(C71⋊C8)—21no—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)C2×C412yes—
39C22⋊(C71⋊C8)—21no—
40#14556038—42no—
41C4×C71⋊C8—21no—
42C2×M16×C71—21no—
43C22×C8×C71—21no—
44C2×C42×C71C2311yescharacteristic
45C4⋊C8×C71—21no—
46C4⋊C8×C71—21no—
47C4⋊C8×C71—21no—
48C4⋊C8×C71C2×C412yes—
49C4⋊C8×C71D412yes—
50C22⋊C8×C71—21no—
51C4×C8×C71—21no—
52C2×Dic142—21no—
53C2×Dic142—21no—
54C22×Dic71—21no—
55C22×Dic71—21no—
56C2×C71⋊C8—22no—
57C2×C71⋊C8—41no—
58C2×C71⋊C8—41no—
59C2×C71⋊C8—41no—
60C4⋊Dic71—21no—
61C4⋊Dic71—21no—
62C4⋊Dic71—21no—
63C4⋊Dic71—21no—
64Dic71⋊C4—41no—
65Dic71⋊C4—41no—
66C22⋊Dic71—41no—
67C4×Dic71—21no—
68C4×Dic71—21no—
69C4×Dic71—21no—
70C4×Dic71—42no—
71C4×Dic71—42no—
72C4×Dic71—41no—
73C4×Dic71Q1612yes—
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71#4111yescharacteristic
76C22×C4×C71#4111yescharacteristic
77M16×C71—41no—
78M16×C71—41no—
79C2×C8×C71—22no—
80C2×C8×C71—22no—
81C2×C8×C71—41no—
82C2×C8×C71—41no—
83C2×C8×C71—41no—
84C42×C71C22×C411yescharacteristic
85C42×C71C2×D411yescharacteristic
86C42×C71#4111yescharacteristic
87C42×C71#4111yescharacteristic
88C23×C71#12311yescharacteristic
89C2×C4×C71—22no—
90C2×C4×C71—41no—
91C2×C4×C71#11711yescharacteristic
92C2×C4×C71C4×D411yescharacteristic
93C2×C4×C71C4×D411yescharacteristic
94C2×C4×C71#12111yescharacteristic
95C2×C4×C71#12411yescharacteristic
96C2×C4×C71#12611yescharacteristic
97C2×C4×C71C2×Q1611yescharacteristic
98C2×C4×C71#13511yescharacteristic
99C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C71#13711yescharacteristic
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—41no—
104C2×Dic71—41no—
105C2×Dic71—41no—
106C2×Dic71—41no—
107C2×Dic71—41no—
108C2×Dic71—41no—
109Dic142—42no—
110Dic142—81no—
111C8×C71—42no—
112C8×C71—42no—
113C8×C71—81no—
114C71⋊C8—42no—
115C71⋊C8—81no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71#38811yescharacteristic
119C22×C71#48411yescharacteristic
120C22×C71#48511yescharacteristic
121C4×C71—21no—
122C4×C71—21no—
123C4×C71—21no—
124C4×C71C4×Q1611yescharacteristic
125C4×C71#44011yescharacteristic
126C4×C71#44412yes—
127C4×C71#46211yescharacteristic
128C4×C71#46311yescharacteristic
129Dic71—42no—
130Dic71—42no—
131Dic71—81no—
132Dic71—81no—
133#9584—711nomaximal, Sylow
134C2×C71—21no—
135C2×C71—21no—
136C2×C71#192811yescharacteristic
137C2×C71#193211yescharacteristic
138C2×C71#193811yescharacteristic
139#1994—711no—
140#1767—711no—
141#1571D7111yescharacteristic
142#1542—711no—
143#1481—711no—
144#1301—711no—
145#1276—711no—
146C71#958411yescharacteristic, complemented, Sylow
147#533—711no—
148#431—711no—
149#431C2×D7111yescharacteristic
150#423C2×D7111yescharacteristic
151C2×C4⋊C8—711no—
152C2×C4⋊C8C2×D7111yescharacteristic
153C2×#97—1421no—
154C43—711no—
155#339—1421no—
156#339—1421no—
157#336—1421no—
158#335—22no—
159#333—712no—
160#332—712no—
161Q8⋊C8—1422no—
162C2×M16—21no—
163C22×C8—21no—
164C22×C8—1421no—
165C4⋊Q8—711no—
166C4⋊Q8—711no—
167#125—1421no—
168C22⋊Q8—1421no—
169C22⋊Q8—1421no—
170C4×Q8—1421no—
171C4×Q8—1421no—
172#117—1421no—
173C2×C42—711no—
174C2×C42—711no—
175C2×C42—1421no—
176C2×C42—1421no—
177C2×C42C22×D7111yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—712no—
182C4⋊C8—1421no—
183C4⋊C8—1421no—
184C4⋊C8C4×D7112yes—
185C4⋊C8C22⋊D7112yes—
186C22⋊C8—21no—
187C22⋊C8—1421no—
188#97—2842no—
189C4×C8—21no—
190C4×C8—1421no—
191C2×Q8—1421no—
192C2×Q8—1421no—
193C22×C4—1421no—
194C22×C4—1421no—
195C22×C4D4×D7111yescharacteristic
196C22×C4#1176586511yescharacteristic
197C22×C4#1176586611yescharacteristic
198M16—41no—
199M16—41no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—2841no—
206C2×C8—2841no—
207C2×C8—2841no—
208C2×C8—1422no—
209C4⋊C4—2841no—
210C4⋊C4—2841no—
211C4⋊C4—1421no—
212C4⋊C4—1421no—
213C4⋊C4—1421no—
214C4⋊C4—1421no—
215C22⋊C4—2841no—
216C42—2842no—
217C42—2842no—
218C42—2841no—
219C42—712no—
220C42—1421no—
221C42—1421no—
222C42—1421no—
223C42C2×C4×D7111yescharacteristic
224C42C2×C22⋊D7111yescharacteristic
225C42#1176586411yescharacteristic
226C42#1176586611yescharacteristic
227C23#1455611311yescharacteristic
228Q8—2842no—
229Q8—5681no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—2841no—
233C2×C4—2841no—
234C2×C4—2841no—
235C2×C4—2841no—
236C2×C4—2841no—
237C2×C4—2841no—
238C2×C4—1422no—
239C2×C4—1422no—
240C2×C4#1455608611yescharacteristic
241C2×C4C4×C22⋊D7111yescharacteristic
242C2×C4#1455609011yescharacteristic
243C2×C4#1455610011yescharacteristic
244C2×C4#1455611711yescharacteristic
245C2×C4#1455612311yescharacteristic
246C2×C4Q16×D7111yescharacteristic
247C2×C4#1455614711yescharacteristic
248C2×C4#1455615211yescharacteristic
249C2×C4#1455616011yescharacteristic
250C8—42no—
251C8—42no—
252C8—81no—
253C8—2842no—
254C8—5681no—
255C22—21no—
256C22—21no—
257C22#1798588911yescharacteristic
258C22#1798633611yescharacteristic
259C22#1798634211yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—2842no—
264C4—2842no—
265C4—5681no—
266C4—5681no—
267C4#1798609311yescharacteristic
268C4#1798609811yescharacteristic
269C4#1798611912yes—
270C4#1798620711yescharacteristic
271C4#1798621211yescharacteristic
272C2—21no—
273C2—21no—
274C2#2258898011yescharacteristic
275C2#2258900811yescharacteristic
276C2#2258905011yescharacteristic
subgroupG/H
centerC2×C4#14556100
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1571C2
socleC22×C71#388
prime radicalC23×C71#123
cosocleC2×C42×C71C23
hypercenter#1571D71
hypocenterC71#9584
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
16. #17985747C4semidirect
17. #17985746C4semidirect
146. C71#9584semidirect

Maximal subgroups

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

subgroupindex
#225893162
#225881952
#225869952
#225867422
#225862352
#225861722
C71×#15712
#958471