Finite Group Databasebeta

← #31886583

Subgroups — #31886583

Order 18176.

366 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 87 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#22589811C212yesmaximal
3#22586166C211yesmaximal, characteristic
4#22586163C212yesmaximal
5C71×#2809C211yesmaximal, characteristic
6C2×#14556061—22no—
7C2×#14556059—21no—
8C2×#14556052—21no—
9#17985869C2211yescharacteristic
10#17985761—22no—
11#17985754C2211yescharacteristic
12#17985754C2212yes—
13#400×C71—21no—
14#398×C71—22no—
15#391×C71—21no—
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#14556061—44no—
24#14556059—42no—
25#14556052—42no—
26C22×Q8×C71D411yescharacteristic
27C2×C4⋊C4×C71—22no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71D412yes—
31C2×C4⋊C4×C71D412yes—
32C2×C4⋊C4×C71D411yescharacteristic
33C2×C42×C71—21no—
34C2×C42×C71C2311yescharacteristic
35#95×C71—21no—
36#95×C71—22no—
37#95×C71—22no—
38#95×C71—22no—
39#95×C71—22no—
40C22×Dic71—41no—
41C2×C71⋊C8—22no—
42C2×C71⋊C8—24no—
43C2×C71⋊C8—42no—
44C2×C71⋊C8—41no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—41no—
47C4⋊Dic71—41no—
48C4⋊Dic71—41no—
49C4⋊Dic71—42no—
50C4⋊Dic71—42no—
51C2×Q8×C71—21no—
52C2×Q8×C71—41no—
53C2×Q8×C71—41no—
54C2×Q8×C71Q1612yes—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71C2×D411yescharacteristic
60C22×C4×C71C2×D412yes—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—42no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—42no—
66C4⋊C4×C71—42no—
67C4⋊C4×C71—81no—
68C4⋊C4×C71—81no—
69C42×C71—41no—
70C42×C71—81no—
71C23×C71#12011yescharacteristic
72Q8×C71—42no—
73Q8×C71—81no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—41no—
80C2×C4×C71—41no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71—41no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71—42no—
89C2×C4×C71C2×Q1611yescharacteristic
90C2×C4×C71#13512yes—
91C2×C4×C71AGL(1,ℤ/8)12yes—
92C2×C4×C71#13711yescharacteristic
93C2×Dic71—82no—
94C2×Dic71—81no—
95C71⋊C8—44no—
96C71⋊C8—42no—
97C22×C71#44912yes—
98C22×C71#45111yescharacteristic
99C22×C71#45311yescharacteristic
100C22×C71#45612yes—
101C22×C71#45811yescharacteristic
102C4×C71—22no—
103C4×C71—22no—
104C4×C71—21no—
105C4×C71—21no—
106C4×C71—42no—
107C4×C71—82no—
108C4×C71—82no—
109C4×C71—81no—
110Dic71—161no—
111#7802—711nomaximal, Sylow
112C2×C71#160912yes—
113C2×C71#161411yescharacteristic
114C2×C71#162411yescharacteristic
115C2×C71#164812yes—
116C2×C71#166011yescharacteristic
117#2809D7111yescharacteristic
118#2096—711no—
119#2083—712no—
120#1272—711no—
121#1271—712no—
122C71#780211yescharacteristic, complemented, Sylow
123C2×#107—1422no—
124C2×#106—1421no—
125C2×#103—1421no—
126#400—21no—
127#398—22no—
128#391—21no—
129#391C2×D7111yescharacteristic
130#389C2×D7112yes—
131#387—22no—
132#384—711no—
133#340—1422no—
134#336—712no—
135#336—711no—
136C22×Q8C22⋊D7111yescharacteristic
137C22×C8—1422no—
138C22×C8—1421no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—1422no—
143C2×C4⋊C4—1421no—
144C2×C4⋊C4C22⋊D7112yes—
145C2×C4⋊C4C22⋊D7112yes—
146C2×C4⋊C4C22⋊D7111yescharacteristic
147C2×C42—21no—
148C2×C42C22×D7111yescharacteristic
149#107—2844no—
150#106—2842no—
151#103—2842no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8#1176585512yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—2841no—
166C22×C4C2×C22⋊D7111yescharacteristic
167C22×C4C2×C22⋊D7112yes—
168C2×C8—2842no—
169C2×C8—2841no—
170C2×C8—2842no—
171C2×C8—2841no—
172C2×C8—1422no—
173C2×C8—1424no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—2841no—
183C4⋊C4—2841no—
184C4⋊C4—2842no—
185C4⋊C4—2842no—
186C42—41no—
187C42—81no—
188C23#1455609711yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—5682no—
207C2×C4—5681no—
208C2×C4C2×#1176585511yescharacteristic
209C2×C4#1455614812yes—
210C2×C4#1455615312yes—
211C2×C4#1455615911yescharacteristic
212C8—2844no—
213C8—2842no—
214C22#1798614412yes—
215C22#1798615511yescharacteristic
216C22#1798616711yescharacteristic
217C22#1798618212yes—
218C22#1798618911yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4—82no—
227C4—11361no—
228C2#2258733712yes—
229C2#2258736511yescharacteristic
230C2#2258742311yescharacteristic
231C2#2258758212yes—
232C2#2258765111yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2809C2
socle, prime radicalC23×C71#120
hypercenter#2809D71
hypocenterC71#7802
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2809

Complements

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

NH ≅ G/Ntype
122. C71#7802semidirect

Maximal subgroups

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

subgroupindex
#225898832
#225898112
#225861662
#225861632
C71×#28092
#780271