Finite Group Databasebeta

← #31886550

Subgroups — #31886550

Order 18176.

478 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 95 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#22589868C211yesmaximal, characteristic
2#22589811C212yesmaximal, complemented
3#22586163C212yesmaximal, complemented
4#22586161C211yesmaximal, characteristic
5C71×#2683C211yesmaximal, characteristic
6C2×#14556061—22no—
7C2×#14556061—21no—
8C2×#14556049—21no—
9#17985869C2211yescharacteristic
10#17985761—22no—
11#17985754C2212yes—
12#17985754C2211yescharacteristic
13C2×C22⋊Q8×C71—22no—
14#396×C71—22no—
15#395×C71—21no—
16#390×C71C2211yescharacteristic
17#389×C71C2212yes—
18#386×C71—21no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556061—42no—
24#14556061—44no—
25#14556049—42no—
26C22×Q8×C71—21no—
27C22×D4×C71D411yescharacteristic
28C23×C4×C71—21no—
29C22⋊Q8×C71—44no—
30C22⋊Q8×C71—44no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—22no—
33C2×C4⋊C4×C71D412yes—
34C2×C4⋊C4×C71D412yes—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C22⋊C4×C71—22no—
37C2×C22⋊C4×C71—22no—
38C2×C22⋊C4×C71—21no—
39C2×C42×C71C2311yescharacteristic
40#95×C71—21no—
41#95×C71—22no—
42C22×Dic71—41no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8—24no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—41no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—41no—
49C4⋊Dic71—41no—
50C4⋊Dic71—41no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C24×C71—21no—
54C2×Q8×C71—44no—
55C2×Q8×C71—81no—
56C2×D4×C71—21no—
57C2×D4×C71—41no—
58C2×D4×C71—41no—
59C2×D4×C71Q1612yes—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—21no—
63C22×C4×C71—44no—
64C22×C4×C71—81no—
65C22×C4×C71C2×D412yes—
66C22×C4×C71C2×D411yescharacteristic
67C4⋊C4×C71—22no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—44no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71D814yes—
74C22⋊C4×C71—44no—
75C22⋊C4×C71—42no—
76C22⋊C4×C71—42no—
77C22⋊C4×C71—81no—
78C42×C71—41no—
79C23×C71—22no—
80C23×C71—42no—
81C23×C71—41no—
82C23×C71—42no—
83C23×C71—41no—
84C23×C71#12011yescharacteristic
85Q8×C71—84no—
86D4×C71—42no—
87D4×C71—81no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—24no—
93C2×C4×C71—42no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—84no—
101C2×C4×C71C2×D812yes—
102C2×C4×C71C2×Q1611yescharacteristic
103C2×C4×C71AGL(1,ℤ/8)12yes—
104C2×C4×C71#13711yescharacteristic
105C2×Dic71—82no—
106C2×Dic71—81no—
107C71⋊C8—44no—
108C71⋊C8—42no—
109C22×C71—42no—
110C22×C71—44no—
111C22×C71—82no—
112C22×C71—81no—
113C22×C71—81no—
114C22×C71#44712yes—
115C22×C71#45111yescharacteristic
116C22×C71#45311yescharacteristic
117C22×C71#45612yes—
118C22×C71#45811yescharacteristic
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—21no—
122C4×C71—21no—
123C4×C71—44no—
124C4×C71—82no—
125C4×C71—81no—
126Dic71—161no—
127#7797—711nomaximal, Sylow
128C2×C71—82no—
129C2×C71#161012yes—
130C2×C71#161411yescharacteristic
131C2×C71#161911yescharacteristic
132C2×C71#164912yes—
133C2×C71#166011yescharacteristic
134#2683D7111yescharacteristic
135#2093—711no—
136#2083—712no—
137#1271—712no—
138#1270—711no—
139C71#779711yescharacteristic, complemented, Sylow
140C2×C22⋊Q8—22no—
141C2×#107—1422no—
142C2×#107—1421no—
143C2×#102—1421no—
144#396—22no—
145#395—21no—
146#390C2×D7111yescharacteristic
147#389C2×D7112yes—
148#386—21no—
149#384—711no—
150#340—1422no—
151#336—712no—
152#336—711no—
153C22×Q8—21no—
154C22×D4C22⋊D7111yescharacteristic
155C23×C4—21no—
156C22×C8—1422no—
157C22×C8—1421no—
158C22⋊Q8—44no—
159C22⋊Q8—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—1422no—
163C2×C4⋊C4—1421no—
164C2×C4⋊C4C22⋊D7112yes—
165C2×C4⋊C4C22⋊D7112yes—
166C2×C4⋊C4C22⋊D7111yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C42C22×D7111yescharacteristic
171#107—2842no—
172#107—2844no—
173#102—2842no—
174#95—21no—
175#95—22no—
176C24—21no—
177C2×Q8—44no—
178C2×Q8—81no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1176585512yes—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—44no—
187C22×C4—81no—
188C22×C4—2841no—
189C22×C4C2×C22⋊D7112yes—
190C22×C4C2×C22⋊D7111yescharacteristic
191C2×C8—2842no—
192C2×C8—2841no—
193C2×C8—2842no—
194C2×C8—2841no—
195C2×C8—1422no—
196C2×C8—1424no—
197C4⋊C4—22no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—2841no—
204C4⋊C4—2841no—
205C4⋊C4—2842no—
206C4⋊C4—2842no—
207C4⋊C4#1176585014yes—
208C22⋊C4—44no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—81no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1455609711yescharacteristic
219Q8—84no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—5682no—
236C2×C4—5681no—
237C2×C4C2×#1176585012yes—
238C2×C4C2×#1176585511yescharacteristic
239C2×C4#1455615412yes—
240C2×C4#1455615911yescharacteristic
241C8—2844no—
242C8—2842no—
243C22—44no—
244C22—42no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22#1798613312yes—
249C22#1798615511yescharacteristic
250C22#1798616711yescharacteristic
251C22#1798618212yes—
252C22#1798618911yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—44no—
258C4—81no—
259C4—82no—
260C4—11361no—
261C2—82no—
262C2#2258734312yes—
263C2#2258736511yescharacteristic
264C2#2258739411yescharacteristic
265C2#2258758912yes—
266C2#2258765111yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2683C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2683D71
hypocenterC71#7797
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
2. #22589811C2semidirect
3. #22586163C2semidirect
139. C71#7797semidirect

Maximal subgroups

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

subgroupindex
#225898682
#225898112
#225861632
#225861612
C71×#26832
#779771