Finite Group Databasebeta

← #31886682

Subgroups — #31886682

Order 18176.

410 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 95 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#22589883C211yesmaximal, characteristic
2#22589816C212yesmaximal
3#22586166C211yesmaximal, characteristic
4#22586163C212yesmaximal
5C71×#2734C211yesmaximal, characteristic
6C2×#14556059—22no—
7C2×#14556059—21no—
8C2×#14556052—21no—
9#17985869C2211yescharacteristic
10#17985761—22no—
11#17985754C2211yescharacteristic
12#17985754C2212yes—
13C2×C4⋊Q8×C71—21no—
14#398×C71—21no—
15#395×C71—22no—
16#391×C71C2211yescharacteristic
17#389×C71C2212yes—
18#387×C71—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556059—42no—
24#14556059—44no—
25#14556052—42no—
26C22×Q8×C71—21no—
27C22×Q8×C71D411yescharacteristic
28C4⋊Q8×C71—42no—
29C4⋊Q8×C71—42no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71D412yes—
34C2×C4⋊C4×C71D412yes—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C42×C71—21no—
37C2×C42×C71C2311yescharacteristic
38#95×C71—21no—
39#95×C71—22no—
40#95×C71—22no—
41#95×C71—22no—
42C22×Dic71—41no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8—24no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—41no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—41no—
49C4⋊Dic71—41no—
50C4⋊Dic71—41no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C2×Q8×C71—21no—
54C2×Q8×C71—42no—
55C2×Q8×C71—41no—
56C2×Q8×C71—41no—
57C2×Q8×C71—81no—
58C2×Q8×C71—81no—
59C2×Q8×C71SD1612yes—
60C22×C4×C71—22no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—22no—
64C22×C4×C71C2×D411yescharacteristic
65C22×C4×C71C2×D412yes—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—44no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—81no—
73C4⋊C4×C71SD1614yes—
74C42×C71—41no—
75C42×C71—42no—
76C23×C71#12011yescharacteristic
77Q8×C71—42no—
78Q8×C71—81no—
79Q8×C71—84no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—21no—
84C2×C4×C71—24no—
85C2×C4×C71—22no—
86C2×C4×C71—42no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—41no—
90C2×C4×C71—42no—
91C2×C4×C71—41no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—42no—
95C2×C4×C71C2×SD1612yes—
96C2×C4×C71C2×SD1611yescharacteristic
97C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
98C2×C4×C71#13712yes—
99C2×Dic71—82no—
100C2×Dic71—81no—
101C71⋊C8—44no—
102C71⋊C8—42no—
103C22×C71#44812yes—
104C22×C71#45011yescharacteristic
105C22×C71#45311yescharacteristic
106C22×C71#45612yes—
107C22×C71#45811yescharacteristic
108C4×C71—22no—
109C4×C71—22no—
110C4×C71—21no—
111C4×C71—21no—
112C4×C71—42no—
113C4×C71—44no—
114C4×C71—82no—
115C4×C71—81no—
116Dic71—161no—
117#7817—711nomaximal, Sylow
118C2×C71#161112yes—
119C2×C71#161311yescharacteristic
120C2×C71#162311yescharacteristic
121C2×C71#165112yes—
122C2×C71#166011yescharacteristic
123#2734D7111yescharacteristic
124#2096—711no—
125#2084—712no—
126#1272—711no—
127#1271—712no—
128C71#781711yescharacteristic, complemented, Sylow
129C2×C4⋊Q8—21no—
130C2×#106—1422no—
131C2×#106—1421no—
132C2×#103—1421no—
133#398—21no—
134#395—22no—
135#391C2×D7111yescharacteristic
136#389C2×D7112yes—
137#387—22no—
138#384—711no—
139#340—1422no—
140#336—712no—
141#336—711no—
142C22×Q8—21no—
143C22×Q8C22⋊D7111yescharacteristic
144C22×C8—1422no—
145C22×C8—1421no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—1422no—
152C2×C4⋊C4—1421no—
153C2×C4⋊C4C22⋊D7112yes—
154C2×C4⋊C4C22⋊D7112yes—
155C2×C4⋊C4C22⋊D7111yescharacteristic
156C2×C42—21no—
157C2×C42C22×D7111yescharacteristic
158#106—2842no—
159#106—2844no—
160#103—2842no—
161#95—21no—
162#95—22no—
163#95—22no—
164#95—22no—
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×Q8#1176585312yes—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—2841no—
177C22×C4C2×C22⋊D7112yes—
178C22×C4C2×C22⋊D7111yescharacteristic
179C2×C8—2842no—
180C2×C8—2841no—
181C2×C8—2842no—
182C2×C8—2841no—
183C2×C8—1422no—
184C2×C8—1424no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—2841no—
193C4⋊C4—2841no—
194C4⋊C4—2842no—
195C4⋊C4—2842no—
196C4⋊C4#1176585214yes—
197C42—41no—
198C42—42no—
199C23#1455609711yescharacteristic
200Q8—42no—
201Q8—81no—
202Q8—84no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—5682no—
219C2×C4—5681no—
220C2×C4C2×#1176585212yes—
221C2×C4C2×#1176585311yescharacteristic
222C2×C4#1455615411yescharacteristic
223C2×C4#1455615912yes—
224C8—2844no—
225C8—2842no—
226C22#1798613812yes—
227C22#1798615011yescharacteristic
228C22#1798616711yescharacteristic
229C22#1798618212yes—
230C22#1798618911yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—44no—
236C4—42no—
237C4—81no—
238C4—82no—
239C4—11361no—
240C2#2258734812yes—
241C2#2258736011yescharacteristic
242C2#2258741811yescharacteristic
243C2#2258760312yes—
244C2#2258765111yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2734C2
socle, prime radicalC23×C71#120
hypercenter#2734D71
hypocenterC71#7817
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
128. C71#7817semidirect

Maximal subgroups

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

subgroupindex
#225898832
#225898162
#225861662
#225861632
C71×#27342
#781771