Finite Group Databasebeta

← #31891019

Subgroups — #31891019

Order 18176.

498 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 123 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#22591250C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22586242C212yesmaximal, complemented
4#22586199C212yesmaximal, complemented
5C71×#1460C211yesmaximal, characteristic
6C2×D4×Dic71—21no—
7#17986023C2212yes—
8C2×#14556045—21no—
9#17985898—21no—
10#17985874—22no—
11#17985865C2211yescharacteristic
12#17985859C412yes—
13C4×C4⋊Dic71C2211yescharacteristic
14C2×C71⋊#95C412yes—
15#17985768—21no—
16C2×C4×D4×C71C2211yescharacteristic
17#431×C71C2212yes—
18#323×C71—22no—
19C23×Dic71—22no—
20C2×#11765849—22no—
21D4×Dic71—42no—
22D4×Dic71—42no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C22⋊Dic71—22no—
27C2×C4×Dic71—21no—
28C4⋊(C71⋊C8)—22no—
29C4⋊(C71⋊C8)—22no—
30#14556045—42no—
31C71⋊#95—22no—
32C71⋊#95—21no—
33C71⋊#95C2×C411yescharacteristic
34C71⋊#95C2×C412yes—
35C71⋊#95C2×C411yescharacteristic
36C22×D4×C71D411yescharacteristic
37C23×C4×C71C2×C412yes—
38C4×D4×C71—22no—
39C4×D4×C71—22no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71D412yes—
42C2×C42×C71C2311yescharacteristic
43C4⋊C8×C71—22no—
44C22⋊C8×C71—22no—
45C4×C8×C71—22no—
46C22×Dic71—21no—
47C22×Dic71—22no—
48C22×Dic71—21no—
49C22×Dic71—44no—
50C22×Dic71—44no—
51C22×Dic71—82no—
52#11765849—42no—
53#11765849—42no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—42no—
56C4⋊Dic71—41no—
57C4⋊Dic71—41no—
58C4⋊Dic71—82no—
59C22⋊Dic71—44no—
60C4×Dic71—41no—
61C4×Dic71—41no—
62C24×C71C22⋊C412yes—
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71D812yes—
66C2×D4×C71SD1612yes—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—82no—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—42no—
75C4⋊C4×C71—22no—
76C22⋊C4×C71—24no—
77C42×C71#4112yes—
78C42×C71#4112yes—
79C23×C71—22no—
80C23×C71—22no—
81C23×C71—42no—
82C23×C71—42no—
83C23×C71#10214yes—
84C23×C71C2×C22⋊C411yescharacteristic
85D4×C71—24no—
86D4×C71—24no—
87C2×C4×C71—22no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—42no—
91C2×C4×C71—82no—
92C2×C4×C71—82no—
93C2×C4×C71#11711yescharacteristic
94C2×C4×C71#11711yescharacteristic
95C2×C4×C71#12312yes—
96C2×C4×C71#12312yes—
97C2×C4×C71C2×D811yescharacteristic
98C2×C4×C71C2×SD1611yescharacteristic
99C2×C4×C71#13712yes—
100C2×Dic71—22no—
101C2×Dic71—41no—
102C2×Dic71—41no—
103C2×Dic71—41no—
104C2×Dic71—41no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—41no—
108C2×Dic71—42no—
109C2×Dic71—88no—
110C8×C71—82no—
111C71⋊C8—82no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71—24no—
115C22×C71—44no—
116C22×C71—44no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71#38111yescharacteristic
120C22×C71C2×#10211yescharacteristic
121C22×C71#41911yescharacteristic
122C4×C71—22no—
123C4×C71—21no—
124C4×C71—21no—
125C4×C71—42no—
126C4×C71#48012yes—
127C4×C71#48312yes—
128Dic71—42no—
129Dic71—81no—
130Dic71—82no—
131#8764—711nomaximal, Sylow
132C2×C71—21no—
133C2×C71—21no—
134C2×C71—44no—
135C2×C71#182311yescharacteristic
136C2×C71#182711yescharacteristic
137C2×C71#182911yescharacteristic
138#2349—711no—
139#1800—711no—
140#1460D7111yescharacteristic
141#1303—712no—
142#1286—712no—
143C71#876411yescharacteristic, complemented, Sylow
144C2×C4×D4—1421no—
145C2×C4×D4C2×D7111yescharacteristic
146#431C2×D7112yes—
147#423—712no—
148C2×#100—1421no—
149#389—1421no—
150#385—1422no—
151#382—711no—
152#380—712no—
153C4×C4⋊C4—711no—
154C2×#95—712no—
155#343—1421no—
156#323—22no—
157C22×D4D28411yescharacteristic
158C23×C4—1422no—
159C23×C4C4×D7112yes—
160C2×M16—1422no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—2842no—
164C4×D4—2842no—
165C2×C4⋊C4—1421no—
166C2×C4⋊C4—1421no—
167C2×C4⋊C4—1422no—
168C2×C4⋊C4D28411yescharacteristic
169C2×C22⋊C4—1422no—
170C2×C22⋊C4C22⋊D7112yes—
171C2×C42—1421no—
172C2×C42C22×D7111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—1422no—
175C4⋊C8—1422no—
176#100—2842no—
177C22⋊C8—22no—
178C4×C8—22no—
179#95—711no—
180#95—712no—
181#95—711no—
182#95—1422no—
183#95—1421no—
184C24(C2×D71)⋊C412yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4D56812yes—
188C2×D4#1176585412yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—82no—
192C22×C4—2844no—
193C22×C4—2844no—
194C22×C4—5682no—
195C22×C4—1422no—
196C22×C4—1421no—
197C22×C4—1421no—
198C22×C4(C2×D71)⋊C412yes—
199C22×C4C2×C4×D7111yescharacteristic
200C22×C4C2×C22⋊D7111yescharacteristic
201C22×C4C2×D28411yescharacteristic
202M16—2842no—
203M16—2842no—
204C2×C8—42no—
205C2×C8—2842no—
206C4⋊C4—22no—
207C4⋊C4—2842no—
208C4⋊C4—2841no—
209C4⋊C4—2841no—
210C4⋊C4—5682no—
211C22⋊C4—24no—
212C22⋊C4—2844no—
213C42—2841no—
214C42—2841no—
215C42#1176586412yes—
216C42#1176586412yes—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23#1455605014yes—
222C23C2×(C2×D71)⋊C411yescharacteristic
223D4—24no—
224D4—24no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—2841no—
232C2×C4—2841no—
233C2×C4—2841no—
234C2×C4—2841no—
235C2×C4—2842no—
236C2×C4—2842no—
237C2×C4—2841no—
238C2×C4—2842no—
239C2×C4—5688no—
240C2×C4—1422no—
241C2×C4#1455608411yescharacteristic
242C2×C4#1455608411yescharacteristic
243C2×C4#1455610912yes—
244C2×C4#1455611212yes—
245C2×C4C2×D56811yescharacteristic
246C2×C4C2×#1176585411yescharacteristic
247C2×C4#1455616112yes—
248C8—82no—
249C8—5682no—
250C22—21no—
251C22—21no—
252C22—24no—
253C22—44no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22#1798586011yescharacteristic
258C22C2×#1455605011yescharacteristic
259C22#1798600911yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—2842no—
265C4—5682no—
266C4—5681no—
267C4#1798631312yes—
268C4#1798633012yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#2258842711yescharacteristic
273C2#2258844111yescharacteristic
274C2#2258844911yescharacteristic
subgroupG/H
centerC22#17985860
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1460C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#1460D71
hypocenterC71#8764
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #22586242C2semidirect
4. #22586199C2semidirect
143. C71#8764semidirect

Maximal subgroups

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

subgroupindex
#225912502
#225883322
#225862422
#225861992
C71×#14602
#876471