Finite Group Databasebeta

← #31895296

Subgroups — #31895296

Order 18176.

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

normal subgroups only

This group has 474 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#22593702C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic
3#22586770C212yesmaximal
4#22586199C212yesmaximal
5C71×#2982C211yesmaximal, characteristic
6C2×C4×Dic142C2211yescharacteristic
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985938C412yes—
10#17985919C412yes—
11#17985906—21no—
12#17985895—21no—
13#17985883—22no—
14#17985865C2211yescharacteristic
15#17985768—21no—
16#17985731—24no—
17C2×C4×Q8×C71C2211yescharacteristic
18C8×Q8×C71—24no—
19#431×C71—22no—
20#431×C71C2212yes—
21C22×Dic142C2×C411yescharacteristic
22C2×#11765849—22no—
23C4×Dic142—22no—
24C4×Dic142—22no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—21no—
29C2×C4×Dic71—21no—
30C4⋊(C71⋊C8)—22no—
31C4⋊(C71⋊C8)—22no—
32#14556047—42no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—21no—
37C71⋊#95C2×C412yes—
38C22×Q8×C71D411yescharacteristic
39C4×Q8×C71—22no—
40C4×Q8×C71—22no—
41C2×C4⋊C4×C71C2×C412yes—
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C42×C71D412yes—
44C2×C42×C71C2311yescharacteristic
45C4⋊C8×C71—24no—
46C4⋊C8×C71—22no—
47C22⋊C8×C71—22no—
48C4×C8×C71—24no—
49C4×C8×C71D414yes—
50C2×Dic142—22no—
51C2×Dic142—42no—
52C22×Dic71—21no—
53C22×Dic71—22no—
54C22×Dic71—21no—
55#11765849—42no—
56#11765849—42no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—22no—
59C4⋊Dic71—42no—
60Dic71⋊C4—44no—
61Dic71⋊C4—41no—
62Dic71⋊C4—41no—
63C4×Dic71—42no—
64C2×Q8×C71—22no—
65C2×Q8×C71—22no—
66C2×Q8×C71SD1612yes—
67C2×Q8×C71Q1612yes—
68C22×C4×C71C22⋊C412yes—
69C22×C4×C71C22⋊C412yes—
70C22×C4×C71C22×C411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C2×C8×C71—24no—
74C4⋊C4×C71—24no—
75C4⋊C4×C71—22no—
76C42×C71—22no—
77C42×C71—22no—
78C42×C71C2×D412yes—
79C42×C71#4112yes—
80C23×C71C2×C22⋊C411yescharacteristic
81Q8×C71—24no—
82Q8×C71—24no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71#10314yes—
90C2×C4×C71C4×D412yes—
91C2×C4×C71#12011yescharacteristic
92C2×C4×C71#12112yes—
93C2×C4×C71#12311yescharacteristic
94C2×C4×C71C2×SD1611yescharacteristic
95C2×C4×C71C2×Q1611yescharacteristic
96C2×C4×C71#13512yes—
97C2×Dic71—22no—
98C2×Dic71—41no—
99C2×Dic71—41no—
100C2×Dic71—42no—
101C2×Dic71—42no—
102C2×Dic71—42no—
103C2×Dic71—41no—
104C2×Dic71—42no—
105Dic142—24no—
106Dic142—42no—
107C8×C71—44no—
108C71⋊C8—82no—
109C22×C71—21no—
110C22×C71—21no—
111C22×C71#38611yescharacteristic
112C22×C71C2×#10311yescharacteristic
113C22×C71#41611yescharacteristic
114C4×C71—22no—
115C4×C71—24no—
116C4×C71—21no—
117C4×C71—21no—
118C4×C71—42no—
119C4×C71#46512yes—
120C4×C71#46712yes—
121Dic71—42no—
122Dic71—82no—
123Dic71—81no—
124#9484—711nomaximal, Sylow
125C2×C71—21no—
126C2×C71—21no—
127C2×C71#189111yescharacteristic
128C2×C71#189211yescharacteristic
129C2×C71#191211yescharacteristic
130#2982D7111yescharacteristic
131#2581—711no—
132#1801—711no—
133#1487—712no—
134#1286—712no—
135C71#948411yescharacteristic, complemented, Sylow
136C2×C4×Q8—711no—
137C2×C4×Q8C2×D7111yescharacteristic
138C8×Q8—24no—
139#431—22no—
140#431C2×D7112yes—
141#423—712no—
142C2×#101—1421no—
143#400—712no—
144#395—712no—
145#391—1421no—
146#389—1421no—
147#387—1422no—
148#382—711no—
149#343—1421no—
150Q8⋊C8—1424no—
151C22×Q8—711no—
152C22×Q8D28411yescharacteristic
153C2×M16—1422no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1422no—
157C4×Q8—1422no—
158C2×C4⋊C4—711no—
159C2×C4⋊C4—1422no—
160C2×C4⋊C4—1421no—
161C2×C4⋊C4—1421no—
162C2×C4⋊C4C4×D7112yes—
163C2×C4⋊C4D28411yescharacteristic
164C2×C42—1421no—
165C2×C42C22⋊D7112yes—
166C2×C42C22×D7111yescharacteristic
167C4⋊C8—24no—
168C4⋊C8—22no—
169C4⋊C8—1422no—
170C4⋊C8—1422no—
171#101—2842no—
172C22⋊C8—22no—
173C4×C8—24no—
174C4×C8C22⋊D7114yes—
175#95—712no—
176#95—1422no—
177#95—1422no—
178#95—1422no—
179#95—1421no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—2842no—
183C2×Q8—1422no—
184C2×Q8#1176585412yes—
185C2×Q8Dic28412yes—
186C22×C4—1422no—
187C22×C4—1421no—
188C22×C4—1421no—
189C22×C4(C2×D71)⋊C412yes—
190C22×C4(C2×D71)⋊C412yes—
191C22×C4C2×C4×D7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×D28411yescharacteristic
194M16—2842no—
195M16—2842no—
196C2×C8—24no—
197C2×C8—2842no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—2844no—
201C4⋊C4—2841no—
202C4⋊C4—2841no—
203C4⋊C4—2842no—
204C4⋊C4—1422no—
205C42—22no—
206C42—22no—
207C42—2842no—
208C42C2×C22⋊D7112yes—
209C42#1176586512yes—
210C23C2×(C2×D71)⋊C411yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—2842no—
214Q8—1424no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—2841no—
222C2×C4—2841no—
223C2×C4—2842no—
224C2×C4—2842no—
225C2×C4—2842no—
226C2×C4—2841no—
227C2×C4—2842no—
228C2×C4—1422no—
229C2×C4#1455605314yes—
230C2×C4C4×C22⋊D7112yes—
231C2×C4#1455609711yescharacteristic
232C2×C4#1455610212yes—
233C2×C4#1455611111yescharacteristic
234C2×C4C2×#1176585411yescharacteristic
235C2×C4C2×Dic28411yescharacteristic
236C2×C4#1455615012yes—
237C8—44no—
238C8—5682no—
239C22—21no—
240C22—21no—
241C22#1798587811yescharacteristic
242C22C2×#1455605311yescharacteristic
243C22#1798599611yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—2842no—
250C4—5682no—
251C4—5681no—
252C4#1798622812yes—
253C4#1798624012yes—
254C2—21no—
255C2—21no—
256C2#2258875811yescharacteristic
257C2#2258876311yescharacteristic
258C2#2258887711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2982C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2982D71
hypocenterC71#9484
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
135. C71#9484semidirect

Maximal subgroups

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

subgroupindex
#225937022
#225883362
#225867702
#225861992
C71×#29822
#948471