Finite Group Databasebeta

← #31895297

Subgroups — #31895297

Order 18176.

474 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 135 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#22597421C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic
3#22586769C212yesmaximal
4#22586198C212yesmaximal
5C71×#2581C211yesmaximal, characteristic
6Q8×C71⋊C8—24no—
7#17986052—22no—
8#17986052C2212yes—
9#17986023C2212yes—
10C2×#14556047—21no—
11#17985768—21no—
12Q8⋊(C71⋊C8)—24no—
13C2×C4×Q8×C71C2211yescharacteristic
14C2×C4×Q8×C71C2211yescharacteristic
15#400×C71C412yes—
16#395×C71C412yes—
17#391×C71—21no—
18#389×C71—21no—
19#387×C71—22no—
20#382×C71C2211yescharacteristic
21C2×#11765849—22no—
22C4⋊(C71⋊C8)—24no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556047—42no—
27C22⋊(C71⋊C8)—22no—
28C4×C71⋊C8—24no—
29C4×C71⋊C8D414yes—
30C22×Q8×C71C2×C411yescharacteristic
31C22×Q8×C71D411yescharacteristic
32C4×Q8×C71—22no—
33C4×Q8×C71—22no—
34C4×Q8×C71—22no—
35C4×Q8×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C412yes—
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C42×C71—21no—
43C2×C42×C71D412yes—
44C2×C42×C71C2311yescharacteristic
45#95×C71—22no—
46#95×C71—22no—
47#95×C71—22no—
48#95×C71—21no—
49#95×C71C2×C412yes—
50#11765849—42no—
51#11765849—42no—
52C2×C71⋊C8—24no—
53C2×C71⋊C8—42no—
54C2×Q8×C71—22no—
55C2×Q8×C71—22no—
56C2×Q8×C71—22no—
57C2×Q8×C71—42no—
58C2×Q8×C71SD1612yes—
59C2×Q8×C71Q1612yes—
60C22×C4×C71—21no—
61C22×C4×C71—22no—
62C22×C4×C71—21no—
63C22×C4×C71C22⋊C412yes—
64C22×C4×C71C22⋊C412yes—
65C22×C4×C71C22×C411yescharacteristic
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C4⋊C4×C71—24no—
69C4⋊C4×C71—22no—
70C4⋊C4×C71—22no—
71C4⋊C4×C71—44no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—42no—
75C42×C71—22no—
76C42×C71—22no—
77C42×C71—42no—
78C42×C71C2×D412yes—
79C42×C71#4112yes—
80C23×C71C2×C22⋊C411yescharacteristic
81Q8×C71—24no—
82Q8×C71—24no—
83Q8×C71—24no—
84Q8×C71—42no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
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—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71#10314yes—
100C2×C4×C71C4×D412yes—
101C2×C4×C71#12011yescharacteristic
102C2×C4×C71#12112yes—
103C2×C4×C71#12311yescharacteristic
104C2×C4×C71C2×SD1611yescharacteristic
105C2×C4×C71C2×Q1611yescharacteristic
106C2×C4×C71#13512yes—
107C71⋊C8—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—42no—
120C4×C71—82no—
121C4×C71—81no—
122C4×C71#46512yes—
123C4×C71#46712yes—
124#9484—711nomaximal, Sylow
125C2×C71—21no—
126C2×C71—21no—
127C2×C71#189111yescharacteristic
128C2×C71#189211yescharacteristic
129C2×C71#191211yescharacteristic
130#2982—711no—
131#2581D7111yescharacteristic
132#1801—711no—
133#1487—712no—
134#1286—712no—
135C71#948411yescharacteristic, complemented, Sylow
136C2×C4×Q8C2×D7111yescharacteristic
137C2×C4×Q8C2×D7111yescharacteristic
138C8×Q8—1424no—
139#431—712no—
140#431—1422no—
141#423—712no—
142C2×#101—1421no—
143#400Dic7112yes—
144#395Dic7112yes—
145#391—21no—
146#389—21no—
147#387—22no—
148#382C2×D7111yescharacteristic
149#343—1421no—
150Q8⋊C8—1424no—
151C22×Q8C2×Dic7111yescharacteristic
152C22×Q8C22⋊D7111yescharacteristic
153C2×M16—1422no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4C2×Dic7111yescharacteristic
162C2×C4⋊C4C2×Dic7112yes—
163C2×C4⋊C4C22⋊D7111yescharacteristic
164C2×C42—21no—
165C2×C42C22⋊D7112yes—
166C2×C42C22×D7111yescharacteristic
167C4⋊C8—1424no—
168C4⋊C8—1422no—
169C4⋊C8—1422no—
170C4⋊C8—1422no—
171#101—2842no—
172C22⋊C8—1422no—
173C4×C8—714no—
174C4×C8—1424no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—21no—
179#95C2×Dic7112yes—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8#1176585212yes—
185C2×Q8#1176585512yes—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4C22⋊Dic7112yes—
190C22×C4C22⋊Dic7112yes—
191C22×C4C22×Dic7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×C22⋊D7111yescharacteristic
194M16—2842no—
195M16—2842no—
196C2×C8—2842no—
197C2×C8—1424no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C42—22no—
206C42—22no—
207C42—42no—
208C42D4×D7112yes—
209C42#1176586412yes—
210C23C2×C22⋊Dic7111yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—24no—
214Q8—42no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4#1455605214yes—
230C2×C4D4×Dic7112yes—
231C2×C4#1455609611yescharacteristic
232C2×C4#1455610012yes—
233C2×C4#1455610911yescharacteristic
234C2×C4C2×#1176585211yescharacteristic
235C2×C4C2×#1176585511yescharacteristic
236C2×C4#1455614812yes—
237C8—2844no—
238C8—5682no—
239C22—21no—
240C22—21no—
241C22#1798587911yescharacteristic
242C22C2×#1455605211yescharacteristic
243C22#1798599511yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—81no—
252C4#1798623012yes—
253C4#1798623812yes—
254C2—21no—
255C2—21no—
256C2#2258875911yescharacteristic
257C2#2258876411yescharacteristic
258C2#2258887811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2581C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2581D71
hypocenterC71#9484
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

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
#225974212
#225883362
#225867692
#225861982
C71×#25812
#948471