Finite Group Databasebeta

← #31895294

Subgroups — #31895294

Order 18176.

474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 127 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#22593702C211yesmaximal, characteristic
3#22586768C212yesmaximal
4#22586200C212yesmaximal
5C71×#1801C211yesmaximal, characteristic
6C2×C4×Dic142C2211yescharacteristic
7Q8×C71⋊C8—24no—
8#17986052—22no—
9#17986052C2212yes—
10#17985938C412yes—
11#17985919C412yes—
12#17985906—21no—
13#17985895—21no—
14#17985883—22no—
15#17985865C2211yescharacteristic
16#17985729—24no—
17C2×C4×Q8×C71C2211yescharacteristic
18#423×C71C2212yes—
19C2×#101×C71—21no—
20#343×C71—21no—
21C22×Dic142C2×C411yescharacteristic
22C4×Dic142—22no—
23C4×Dic142—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×Dic71⋊C4—22no—
27C2×Dic71⋊C4—21no—
28C2×C4×Dic71—21no—
29C4⋊(C71⋊C8)—24no—
30C4⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C4×C71⋊C8—24no—
33C4×C71⋊C8D414yes—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—21no—
38C71⋊#95C2×C412yes—
39C22×Q8×C71D411yescharacteristic
40C2×M16×C71—22no—
41C4×Q8×C71—22no—
42C4×Q8×C71—22no—
43C2×C4⋊C4×C71C2×C412yes—
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C42×C71D412yes—
46C2×C42×C71C2311yescharacteristic
47C4⋊C8×C71—22no—
48C4⋊C8×C71—22no—
49#101×C71—42no—
50C2×Dic142—22no—
51C2×Dic142—42no—
52C22×Dic71—21no—
53C22×Dic71—22no—
54C22×Dic71—21no—
55C2×C71⋊C8—24no—
56C4⋊Dic71—22no—
57C4⋊Dic71—42no—
58Dic71⋊C4—44no—
59Dic71⋊C4—41no—
60Dic71⋊C4—41no—
61C4×Dic71—42no—
62C2×Q8×C71—22no—
63C2×Q8×C71—22no—
64C2×Q8×C71SD1612yes—
65C2×Q8×C71Q1612yes—
66C22×C4×C71C22⋊C412yes—
67C22×C4×C71C22⋊C412yes—
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71M16×C71—42no—
72M16×C71—42no—
73C2×C8×C71—42no—
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—82no—
108C71⋊C8—44no—
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#2982—711no—
131#2581—711no—
132#1801D7111yescharacteristic
133#1487—712no—
134#1286—712no—
135C71#948411yescharacteristic, complemented, Sylow
136C2×C4×Q8—711no—
137C2×C4×Q8C2×D7111yescharacteristic
138C8×Q8—1424no—
139#431—712no—
140#431—1422no—
141#423C2×D7112yes—
142C2×#101—21no—
143#400—712no—
144#395—712no—
145#391—1421no—
146#389—1421no—
147#387—1422no—
148#382—711no—
149#343—21no—
150Q8⋊C8—1424no—
151C22×Q8—711no—
152C22×Q8C22⋊D7111yescharacteristic
153C2×M16—22no—
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⋊C4C22⋊D7111yescharacteristic
164C2×C42—1421no—
165C2×C42D28412yes—
166C2×C42C22×D7111yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—1424no—
170C4⋊C8—1422no—
171#101—42no—
172C22⋊C8—1422no—
173C4×C8—714no—
174C4×C8—1424no—
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#1176585312yes—
185C2×Q8#1176585512yes—
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—42no—
195M16—42no—
196C2×C8—42no—
197C2×C8—1424no—
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—
208C42D4×D7112yes—
209C42#1176586412yes—
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#1455605114yes—
230C2×C4(C22⋊D71)⋊C412yes—
231C2×C4#1455609811yescharacteristic
232C2×C4#1455610012yes—
233C2×C4#1455611211yescharacteristic
234C2×C4C2×#1176585311yescharacteristic
235C2×C4C2×#1176585511yescharacteristic
236C2×C4#1455614812yes—
237C8—82no—
238C8—2844no—
239C22—21no—
240C22—21no—
241C22#1798587611yescharacteristic
242C22C2×#1455605111yescharacteristic
243C22#1798599311yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—2842no—
250C4—5682no—
251C4—5681no—
252C4#1798622612yes—
253C4#1798623812yes—
254C2—21no—
255C2—21no—
256C2#2258875611yescharacteristic
257C2#2258876111yescharacteristic
258C2#2258887511yescharacteristic
subgroupG/H
centerC22#17985876
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1801C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1801D71
hypocenterC71#9484
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

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
#225937022
#225867682
#225862002
C71×#18012
#948471