Finite Group Databasebeta

← #31891029

Subgroups — #31891029

Order 18176.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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#22591261C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic
3#22586242C212yesmaximal
4#22586199C212yesmaximal
5C71×#1461C211yesmaximal, characteristic
6C2×Q8×Dic71—21no—
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985898—21no—
10#17985898—22no—
11#17985865C412yes—
12#17985865C2211yescharacteristic
13C4×C4⋊Dic71C412yes—
14C4×C4⋊Dic71C2211yescharacteristic
15#17985768—21no—
16C2×C4×Q8×C71C2211yescharacteristic
17#431×C71C2212yes—
18#324×C71—22no—
19C2×#11765849—22no—
20Q8×Dic71—42no—
21Q8×Dic71—42no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4×Dic71—22no—
27C2×C4×Dic71—21no—
28C4⋊(C71⋊C8)—22no—
29C4⋊(C71⋊C8)—22no—
30#14556047—42no—
31C71⋊#95—22no—
32C71⋊#95—21no—
33C71⋊#95C2×C411yescharacteristic
34C71⋊#95C2×C412yes—
35C71⋊#95C2×C411yescharacteristic
36C22×Q8×C71D411yescharacteristic
37C4×Q8×C71—22no—
38C4×Q8×C71—22no—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71D412yes—
41C2×C42×C71C2×C412yes—
42C2×C42×C71C2311yescharacteristic
43C4⋊C8×C71—22no—
44C22⋊C8×C71—22no—
45C4×C8×C71—22no—
46C22×Dic71—21no—
47C22×Dic71—21no—
48C22×Dic71—22no—
49#11765849—42no—
50#11765849—42no—
51C2×C71⋊C8—42no—
52C4⋊Dic71—42no—
53C4⋊Dic71—44no—
54C4⋊Dic71—44no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57C4×Dic71—44no—
58C4×Dic71—41no—
59C4×Dic71—41no—
60C2×Q8×C71—22no—
61C2×Q8×C71—22no—
62C2×Q8×C71D812yes—
63C2×Q8×C71SD1612yes—
64C22×C4×C71C22⋊C412yes—
65C22×C4×C71C22⋊C412yes—
66C22×C4×C71C22×C411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69C2×C8×C71—42no—
70C4⋊C4×C71—24no—
71C4⋊C4×C71—22no—
72C42×C71—22no—
73C42×C71—22no—
74C42×C71#4112yes—
75C42×C71#4112yes—
76C23×C71C2×C22⋊C411yescharacteristic
77Q8×C71—24no—
78Q8×C71—24no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—41no—
83C2×C4×C71—41no—
84C2×C4×C71—42no—
85C2×C4×C71#10214yes—
86C2×C4×C71#11711yescharacteristic
87C2×C4×C71#11711yescharacteristic
88C2×C4×C71#12312yes—
89C2×C4×C71#12312yes—
90C2×C4×C71C2×D811yescharacteristic
91C2×C4×C71C2×SD1611yescharacteristic
92C2×C4×C71#13712yes—
93C2×Dic71—22no—
94C2×Dic71—41no—
95C2×Dic71—41no—
96C2×Dic71—41no—
97C2×Dic71—42no—
98C2×Dic71—41no—
99C2×Dic71—42no—
100C2×Dic71—42no—
101C2×Dic71—41no—
102C8×C71—82no—
103C71⋊C8—82no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71#38111yescharacteristic
107C22×C71C2×#10211yescharacteristic
108C22×C71#41911yescharacteristic
109C4×C71—24no—
110C4×C71—22no—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71—42no—
114C4×C71#48012yes—
115C4×C71#48312yes—
116Dic71—42no—
117Dic71—82no—
118Dic71—81no—
119#8766—711nomaximal, Sylow
120C2×C71—21no—
121C2×C71—21no—
122C2×C71#182311yescharacteristic
123C2×C71#182711yescharacteristic
124C2×C71#183011yescharacteristic
125#2351—711no—
126#1801—711no—
127#1461D7111yescharacteristic
128#1303—712no—
129#1286—712no—
130C71#876611yescharacteristic, complemented, Sylow
131C2×C4×Q8—1421no—
132C2×C4×Q8C2×D7111yescharacteristic
133#431C2×D7112yes—
134#423—712no—
135C2×#101—1421no—
136#389—1421no—
137#389—1422no—
138#382—712no—
139#382—711no—
140C4×C4⋊C4—712no—
141C4×C4⋊C4—711no—
142#343—1421no—
143#324—22no—
144C22×Q8D28411yescharacteristic
145C2×M16—1422no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—2842no—
149C4×Q8—2842no—
150C2×C4⋊C4—1422no—
151C2×C4⋊C4—1422no—
152C2×C4⋊C4—1421no—
153C2×C4⋊C4—1421no—
154C2×C4⋊C4D28411yescharacteristic
155C2×C4⋊C4C22⋊D7112yes—
156C2×C42—1422no—
157C2×C42—1421no—
158C2×C42C4×D7112yes—
159C2×C42C22×D7111yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—1422no—
162C4⋊C8—1422no—
163#101—2842no—
164C22⋊C8—22no—
165C4×C8—22no—
166#95—711no—
167#95—712no—
168#95—711no—
169#95—1422no—
170#95—1421no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8D56812yes—
174C2×Q8#1176585412yes—
175C22×C4—1421no—
176C22×C4—1422no—
177C22×C4—1421no—
178C22×C4(C2×D71)⋊C412yes—
179C22×C4(C2×D71)⋊C412yes—
180C22×C4C2×C4×D7111yescharacteristic
181C22×C4C2×C22⋊D7111yescharacteristic
182C22×C4C2×D28411yescharacteristic
183M16—2842no—
184M16—2842no—
185C2×C8—42no—
186C2×C8—2842no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—2842no—
190C4⋊C4—2844no—
191C4⋊C4—2844no—
192C4⋊C4—2841no—
193C4⋊C4—2841no—
194C42—22no—
195C42—22no—
196C42—2844no—
197C42—2841no—
198C42—2841no—
199C42#1176586412yes—
200C42#1176586412yes—
201C23C2×(C2×D71)⋊C411yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—2841no—
211C2×C4—2841no—
212C2×C4—2841no—
213C2×C4—2842no—
214C2×C4—2841no—
215C2×C4—2842no—
216C2×C4—2842no—
217C2×C4—2841no—
218C2×C4—1422no—
219C2×C4#1455605014yes—
220C2×C4#1455608411yescharacteristic
221C2×C4#1455608411yescharacteristic
222C2×C4#1455610912yes—
223C2×C4#1455611212yes—
224C2×C4C2×D56811yescharacteristic
225C2×C4C2×#1176585411yescharacteristic
226C2×C4#1455616112yes—
227C8—82no—
228C8—5682no—
229C22—21no—
230C22—21no—
231C22#1798586011yescharacteristic
232C22C2×#1455605011yescharacteristic
233C22#1798600911yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—2842no—
240C4—5682no—
241C4—5681no—
242C4#1798631312yes—
243C4#1798633012yes—
244C2—21no—
245C2—21no—
246C2#2258842711yescharacteristic
247C2#2258844111yescharacteristic
248C2#2258845211yescharacteristic
subgroupG/H
centerC22#17985860
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1461C2
socleC22×C71#381
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1461D71
hypocenterC71#8766
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
130. C71#8766semidirect

Maximal subgroups

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

subgroupindex
#225912612
#225883362
#225862422
#225861992
C71×#14612
#876671