Finite Group Databasebeta

← #31886582

Subgroups — #31886582

Order 18176.

366 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 83 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#22596328C211yesmaximal, characteristic
2#22589812C212yesmaximal
3#22586166C211yesmaximal, characteristic
4#22586162C212yesmaximal
5C71×#2096C211yesmaximal, characteristic
6C2×#14556061—22no—
7#17985938—21no—
8#17985931—22no—
9#17985906—21no—
10#17985897C2212yes—
11#17985886—22no—
12#17985761—22no—
13#17985754C2212yes—
14C2×#106×C71—21no—
15C2×#103×C71—21no—
16#391×C71C2211yescharacteristic
17#384×C71C2211yescharacteristic
18#336×C71C2211yescharacteristic
19C22×C71⋊C8—22no—
20C2×C4⋊Dic71—21no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71D412yes—
23C2×C4⋊Dic71D412yes—
24C2×Dic71⋊C4—22no—
25C2×C4×Dic71—21no—
26#14556061—44no—
27C71⋊#95—21no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C71⋊#95—22no—
31C22×Q8×C71D411yescharacteristic
32C22×C8×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C42×C71C2311yescharacteristic
37#106×C71—42no—
38#103×C71—42no—
39#95×C71—22no—
40C22×Dic71—22no—
41C22×Dic71—22no—
42C2×C71⋊C8—24no—
43C2×C71⋊C8—42no—
44C2×C71⋊C8—42no—
45C4⋊Dic71—22no—
46C4⋊Dic71—22no—
47C4⋊Dic71—42no—
48C4⋊Dic71—81no—
49C4⋊Dic71—81no—
50Dic71⋊C4—42no—
51Dic71⋊C4—42no—
52C4×Dic71—81no—
53C2×Q8×C71—21no—
54C2×Q8×C71—41no—
55C2×Q8×C71—41no—
56C2×Q8×C71Q1612yes—
57C22×C4×C71—21no—
58C22×C4×C71—21no—
59C22×C4×C71—41no—
60C22×C4×C71C2×D412yes—
61C22×C4×C71C2×D411yescharacteristic
62C2×C8×C71—22no—
63C2×C8×C71—41no—
64C2×C8×C71—41no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C42×C71—41no—
71C23×C71#12011yescharacteristic
72Q8×C71—42no—
73Q8×C71—81no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—41no—
80C2×C4×C71—41no—
81C2×C4×C71—41no—
82C2×C4×C71—42no—
83C2×C4×C71—82no—
84C2×C4×C71—81no—
85C2×C4×C71C2×Q1611yescharacteristic
86C2×C4×C71#13512yes—
87C2×C4×C71AGL(1,ℤ/8)12yes—
88C2×C4×C71#13711yescharacteristic
89C2×Dic71—42no—
90C2×Dic71—42no—
91C2×Dic71—42no—
92C2×Dic71—42no—
93C2×Dic71—42no—
94C2×Dic71—42no—
95C8×C71—42no—
96C71⋊C8—44no—
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—81no—
108C4×C71—161no—
109Dic71—82no—
110Dic71—82no—
111#7802—711nomaximal, Sylow
112C2×C71#160912yes—
113C2×C71#161411yescharacteristic
114C2×C71#162411yescharacteristic
115C2×C71#164812yes—
116C2×C71#166011yescharacteristic
117#2809—711no—
118#2096D7111yescharacteristic
119#2083—712no—
120#1272—711no—
121#1271—712no—
122C71#780211yescharacteristic, complemented, Sylow
123C2×#107—1422no—
124C2×#106—21no—
125C2×#103—21no—
126#400—1421no—
127#398—1422no—
128#391—1421no—
129#391C2×D7111yescharacteristic
130#389—712no—
131#387—1422no—
132#384C2×D7111yescharacteristic
133#340—1422no—
134#336—712no—
135#336C2×D7111yescharacteristic
136C22×Q8C22⋊D7111yescharacteristic
137C22×C8—21no—
138C22×C8—1422no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—712no—
142C2×C4⋊C4—712no—
143C2×C4⋊C4—1422no—
144C2×C4⋊C4—1421no—
145C2×C4⋊C4—1421no—
146C2×C4⋊C4C22⋊D7111yescharacteristic
147C2×C42—1421no—
148C2×C42C22×D7111yescharacteristic
149#107—2844no—
150#106—42no—
151#103—42no—
152#95—22no—
153#95—1421no—
154#95—1422no—
155#95—1422no—
156#95—1422no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8#1176585512yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—1422no—
165C22×C4—1422no—
166C22×C4D4×D7112yes—
167C22×C4C2×C22⋊D7111yescharacteristic
168C2×C8—22no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—2842no—
172C2×C8—2842no—
173C2×C8—1424no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—2842no—
180C4⋊C4—2842no—
181C4⋊C4—2842no—
182C4⋊C4—5681no—
183C4⋊C4—5681no—
184C4⋊C4—1422no—
185C4⋊C4—1422no—
186C42—41no—
187C42—5681no—
188C23#1455609611yescharacteristic
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—41no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—2842no—
203C2×C4—2842no—
204C2×C4—2842no—
205C2×C4—2842no—
206C2×C4—2842no—
207C2×C4—2842no—
208C2×C4C2×#1176585511yescharacteristic
209C2×C4#1455614912yes—
210C2×C4#1455615512yes—
211C2×C4#1455615811yescharacteristic
212C8—42no—
213C8—2844no—
214C22#1798614312yes—
215C22#1798615411yescharacteristic
216C22#1798616611yescharacteristic
217C22#1798618012yes—
218C22#1798618811yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—81no—
225C4—161no—
226C4—5682no—
227C4—5682no—
228C2#2258733612yes—
229C2#2258736411yescharacteristic
230C2#2258742111yescharacteristic
231C2#2258758112yes—
232C2#2258764911yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2096C2
socle, prime radicalC23×C71#120
hypercenter#2096D71
hypocenterC71#7802
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

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
#225963282
#225898122
#225861662
#225861622
C71×#20962
#780271