Finite Group Databasebeta

← #31891018

Subgroups — #31891018

Order 18176.

498 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 131 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#22588332C211yesmaximal, characteristic
2#22586661C211yesmaximal, characteristic
3#22586243C212yesmaximal, complemented
4#22586198C212yesmaximal, complemented
5C71×#2349C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556045—21no—
9#17985768—21no—
10#17985723—22no—
11C2×C4×D4×C71—21no—
12C2×C4×D4×C71C2211yescharacteristic
13#389×C71—21no—
14#385×C71—22no—
15#382×C71C2211yescharacteristic
16#380×C71C412yes—
17C4×C4⋊C4×C71C2211yescharacteristic
18C2×#95×C71C412yes—
19C2×#11765849—22no—
20C4⋊(C71⋊C8)—22no—
21C4⋊(C71⋊C8)—22no—
22C4⋊(C71⋊C8)—22no—
23#14556045—42no—
24C22⋊(C71⋊C8)—22no—
25C4×C71⋊C8—22no—
26C22×D4×C71D411yescharacteristic
27C23×C4×C71—22no—
28C23×C4×C71C2×C412yes—
29C4×D4×C71—22no—
30C4×D4×C71—22no—
31C4×D4×C71—42no—
32C4×D4×C71—42no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C22⋊C4×C71—22no—
38C2×C22⋊C4×C71D412yes—
39C2×C42×C71—21no—
40C2×C42×C71C2311yescharacteristic
41#95×C71—22no—
42#95×C71—21no—
43#95×C71C2×C411yescharacteristic
44#95×C71C2×C412yes—
45#95×C71C2×C411yescharacteristic
46#11765849—42no—
47#11765849—42no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C24×C71C22⋊C412yes—
51C2×D4×C71—22no—
52C2×D4×C71—22no—
53C2×D4×C71D812yes—
54C2×D4×C71SD1612yes—
55C22×C4×C71—21no—
56C22×C4×C71—22no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—21no—
60C22×C4×C71—44no—
61C22×C4×C71—44no—
62C22×C4×C71—82no—
63C22×C4×C71—82no—
64C22×C4×C71C22⋊C412yes—
65C22×C4×C71C22×C411yescharacteristic
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C4⋊C4×C71—22no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—82no—
73C22⋊C4×C71—24no—
74C22⋊C4×C71—44no—
75C42×C71—41no—
76C42×C71—41no—
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—22no—
89C2×C4×C71—41no—
90C2×C4×C71—41no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—42no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—82no—
101C2×C4×C71—82no—
102C2×C4×C71—88no—
103C2×C4×C71#11711yescharacteristic
104C2×C4×C71#11711yescharacteristic
105C2×C4×C71#12312yes—
106C2×C4×C71#12312yes—
107C2×C4×C71C2×D811yescharacteristic
108C2×C4×C71C2×SD1611yescharacteristic
109C2×C4×C71#13712yes—
110C71⋊C8—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—42no—
127C4×C71—81no—
128C4×C71—82no—
129C4×C71#48012yes—
130C4×C71#48312yes—
131#8764—711nomaximal, Sylow
132C2×C71—21no—
133C2×C71—21no—
134C2×C71—44no—
135C2×C71#182311yescharacteristic
136C2×C71#182711yescharacteristic
137C2×C71#182911yescharacteristic
138#2349D7111yescharacteristic
139#1800—711no—
140#1460—711no—
141#1303—712no—
142#1286—712no—
143C71#876411yescharacteristic, complemented, Sylow
144C2×C4×D4—21no—
145C2×C4×D4C2×D7111yescharacteristic
146#431—712no—
147#423—712no—
148C2×#100—1421no—
149#389—21no—
150#385—22no—
151#382C2×D7111yescharacteristic
152#380Dic7112yes—
153C4×C4⋊C4C2×D7111yescharacteristic
154C2×#95Dic7112yes—
155#343—1421no—
156#323—1422no—
157C22×D4C22⋊D7111yescharacteristic
158C23×C4—22no—
159C23×C4C2×Dic7112yes—
160C2×M16—1422no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊D7111yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22⋊D7112yes—
171C2×C42—21no—
172C2×C42C22×D7111yescharacteristic
173C4⋊C8—1422no—
174C4⋊C8—1422no—
175C4⋊C8—1422no—
176#100—2842no—
177C22⋊C8—1422no—
178C4×C8—1422no—
179#95—22no—
180#95—21no—
181#95C2×Dic7111yescharacteristic
182#95C2×Dic7112yes—
183#95C2×Dic7111yescharacteristic
184C24C22⋊Dic7112yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4#1176585012yes—
188C2×D4#1176585312yes—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4C22⋊Dic7112yes—
199C22×C4C22×Dic7111yescharacteristic
200C22×C4C2×C22⋊D7111yescharacteristic
201C22×C4C2×C22⋊D7111yescharacteristic
202M16—2842no—
203M16—2842no—
204C2×C8—2842no—
205C2×C8—2842no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C22⋊C4—24no—
212C22⋊C4—44no—
213C42—41no—
214C42—41no—
215C42#1176586512yes—
216C42#1176586512yes—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23#1455604914yes—
222C23C2×C22⋊Dic7111yescharacteristic
223D4—24no—
224D4—24no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—88no—
241C2×C4#1455608711yescharacteristic
242C2×C4#1455608711yescharacteristic
243C2×C4#1455611112yes—
244C2×C4#1455611112yes—
245C2×C4C2×#1176585011yescharacteristic
246C2×C4C2×#1176585311yescharacteristic
247C2×C4#1455615912yes—
248C8—5682no—
249C8—5682no—
250C22—21no—
251C22—21no—
252C22—24no—
253C22—44no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22#1798586211yescharacteristic
258C22C2×#1455604911yescharacteristic
259C22#1798600811yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—82no—
266C4—81no—
267C4#1798631212yes—
268C4#1798632912yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#2258842611yescharacteristic
273C2#2258844411yescharacteristic
274C2#2258845111yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2349C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2349D71
hypocenterC71#8764
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #22586243C2semidirect
4. #22586198C2semidirect
143. C71#8764semidirect

Maximal subgroups

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

subgroupindex
#225883322
#225866612
#225862432
#225861982
C71×#23492
#876471