Finite Group Databasebeta

← #31910720

Subgroups — #31910720

Order 18176.

310 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 71 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#22590535C212yesmaximal
2#22589828C211yesmaximal, characteristic
3#22586352C212yesmaximal
4#22586280C211yesmaximal, characteristic
5C71×#3249C211yesmaximal, characteristic
6C2×#14556062C2211yescharacteristic
7#17985897—21no—
8#17985826—22no—
9#17985803—22no—
10C22⋊(C71⋊C16)C2212yes—
11#17985762—21no—
12#17985762C2211yescharacteristic
13#533×C71—21no—
14#477×C71—22no—
15#467×C71—21no—
16#467×C71C2212yes—
17#462×C71—22no—
18C2×C4⋊C8×C71C2211yescharacteristic
19C2×C4⋊Dic71—21no—
20C2×C4⋊Dic71—21no—
21C2×Dic71⋊C4—41no—
22C2×C71⋊C16—22no—
23#14556062—22no—
24C71⋊#95—41no—
25C2×Q16×C71—21no—
26C2×Q16×C71D412yes—
27C22×C8×C71D411yescharacteristic
28C22×C8×C71C2311yescharacteristic
29C4⋊Q8×C71—21no—
30C4⋊Q8×C71—21no—
31#125×C71—41no—
32C22⋊Q8×C71—22no—
33C22⋊Q8×C71—41no—
34C4×Q8×C71—22no—
35C4×Q8×C71—41no—
36#117×C71—41no—
37C2×C42×C71D411yescharacteristic
38#107×C71—21no—
39#107×C71D412yes—
40C4⋊C8×C71—22no—
41#103×C71—22no—
42#103×C71—22no—
43C22×Dic71—41no—
44C22×Dic71—41no—
45C4⋊Dic71—22no—
46C4⋊Dic71—41no—
47C4⋊Dic71—41no—
48C4⋊Dic71—41no—
49Dic71⋊C4—82no—
50C71⋊C16—42no—
51C2×Q8×C71—22no—
52C2×Q8×C71—41no—
53C22×C4×C71D811yescharacteristic
54C22×C4×C71D811yescharacteristic
55C22×C4×C71C2×D411yescharacteristic
56Q16×C71—42no—
57Q16×C71—81no—
58C2×C8×C71—21no—
59C2×C8×C71—41no—
60C2×C8×C71—41no—
61C2×C8×C71C2×D412yes—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—42no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—42no—
68C22⋊C4×C71—42no—
69C42×C71—21no—
70C42×C71—21no—
71C42×C71—42no—
72C23×C71C2×D811yescharacteristic
73Q8×C71—42no—
74Q8×C71—42no—
75Q8×C71—82no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—41no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71SD3212yes—
85C2×C4×C71#12011yescharacteristic
86C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
87C2×Dic71—42no—
88C2×Dic71—81no—
89C2×Dic71—81no—
90C2×Dic71—81no—
91C2×Dic71—81no—
92C2×Dic71—81no—
93C8×C71—22no—
94C8×C71—41no—
95C22×C71—21no—
96C22×C71—21no—
97C22×C71#44711yescharacteristic
98C22×C71C2×SD3211yescharacteristic
99C22×C71#51011yescharacteristic
100C4×C71—21no—
101C4×C71—22no—
102C4×C71—41no—
103C4×C71—82no—
104C4×C71—82no—
105C4×C71#45611yescharacteristic
106C4×C71#45811yescharacteristic
107Dic71—82no—
108Dic71—161no—
109#12289—711nomaximal, Sylow
110C2×C71—21no—
111C2×C71—21no—
112C2×C71#161011yescharacteristic
113C2×C71#218511yescharacteristic
114C2×C71#219111yescharacteristic
115#3249D7111yescharacteristic
116#2235—712no—
117#2086—711no—
118#1348—712no—
119#1322—711no—
120C71#1228911yescharacteristic, complemented, Sylow
121#533—21no—
122#477—22no—
123#467—21no—
124#467C2×D7112yes—
125#462—22no—
126C2×#107—711no—
127C2×C4⋊C8C2×D7111yescharacteristic
128#389—1421no—
129#367—1422no—
130#358—1422no—
131C22⋊C16—712no—
132#340—711no—
133#340—1421no—
134C2×Q16—21no—
135C2×Q16C22⋊D7112yes—
136C22×C8C22⋊D7111yescharacteristic
137C22×C8C22×D7111yescharacteristic
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140#125—41no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C4×Q8—22no—
144C4×Q8—41no—
145#117—41no—
146C2×C4⋊C4—2841no—
147C2×C4⋊C4—1421no—
148C2×C4⋊C4—1421no—
149C2×C42C22⋊D7111yescharacteristic
150C2×C16—1422no—
151#107—21no—
152#107—1422no—
153#107C22⋊D7112yes—
154C4⋊C8—22no—
155#103—22no—
156#103—22no—
157#95—2841no—
158C2×Q8—22no—
159C2×Q8—41no—
160C22×C4—2841no—
161C22×C4—2841no—
162C22×C4#1176585011yescharacteristic
163C22×C4#1176585011yescharacteristic
164C22×C4C2×C22⋊D7111yescharacteristic
165Q16—42no—
166Q16—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8C2×C22⋊D7112yes—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—2841no—
178C4⋊C4—2841no—
179C4⋊C4—2841no—
180C4⋊C4—5682no—
181C4⋊C4—1422no—
182C22⋊C4—42no—
183C42—21no—
184C42—21no—
185C42—42no—
186C16—2842no—
187C23C2×#1176585011yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—2842no—
200C2×C4—5681no—
201C2×C4—5681no—
202C2×C4—5681no—
203C2×C4—5681no—
204C2×C4—5681no—
205C2×C4#1455607112yes—
206C2×C4#1455609711yescharacteristic
207C2×C4#1455615411yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#1798613311yescharacteristic
213C22C2×#1455607111yescharacteristic
214C22#1798644011yescharacteristic
215C4—22no—
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—5682no—
221C4—11361no—
222C4#1798618211yescharacteristic
223C4#1798618911yescharacteristic
224C2—21no—
225C2—21no—
226C2#2258734311yescharacteristic
227C2#2259025711yescharacteristic
228C2#2259029111yescharacteristic
subgroupG/H
centerC22#17986133
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC71×#3249C2
socleC22×C71#447
prime radicalC23×C71C2×D8
hypercenter#3249D71
hypocenterC71#12289
LCSG ▷ C22×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
120. C71#12289semidirect

Maximal subgroups

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

subgroupindex
#225905352
#225898282
#225863522
#225862802
C71×#32492
#1228971