Finite Group Databasebeta

← #31900534

Subgroups — #31900534

Order 18176.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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#22596615C211yesmaximal, characteristic
2#22589313C212yesmaximal
3#22589159C211yesmaximal, characteristic
4#22586222C212yesmaximal
5C71×#1295C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7C2×#14556051—22no—
8C2×#14556039C2211yescharacteristic
9#17985933—22no—
10#17985932—22no—
11#17985931—22no—
12#17985865C2212yes—
13#17985762—22no—
14#17985762—22no—
15C2×#97×C71C2212yes—
16#382×C71C2211yescharacteristic
17C22×Dic142—21no—
18C22×C71⋊C8—21no—
19#14556128—22no—
20#14556128—21no—
21#14556128Q812yes—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71D412yes—
25C2×C4⋊Dic71D412yes—
26C2×Dic71⋊C4—22no—
27C2×Dic71⋊C4—22no—
28#14556051—44no—
29#14556039—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C71⋊#95—22no—
33C22×C8×C71—22no—
34C2×C4⋊C4×C71D412yes—
35C2×C42×C71C2311yescharacteristic
36#97×C71—22no—
37#97×C71—22no—
38#95×C71—22no—
39C2×Dic142—22no—
40C2×Dic142—42no—
41C2×Dic142—42no—
42C2×Dic142—41no—
43C22×Dic71—22no—
44C22×Dic71—22no—
45C22×Dic71—41no—
46C2×C71⋊C8—22no—
47C2×C71⋊C8—41no—
48C2×C71⋊C8—41no—
49C4⋊Dic71—24no—
50C4⋊Dic71—42no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C4⋊Dic71—82no—
54Dic71⋊C4—82no—
55Dic71⋊C4—82no—
56C22×C4×C71—22no—
57C22×C4×C71C2×D412yes—
58C22×C4×C71C2×D411yescharacteristic
59C2×C8×C71—24no—
60C2×C8×C71—42no—
61C2×C8×C71—42no—
62C4⋊C4×C71—42no—
63C42×C71C2×Q811yescharacteristic
64C42×C71#4111yescharacteristic
65C42×C71#4112yes—
66C23×C71#12011yescharacteristic
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71C22⋊Q812yes—
75C2×C4×C71C22⋊Q811yescharacteristic
76C2×C4×C71#12412yes—
77C2×C4×C71#12411yescharacteristic
78C2×Dic71—42no—
79C2×Dic71—42no—
80C2×Dic71—42no—
81C2×Dic71—42no—
82C2×Dic71—42no—
83C2×Dic71—42no—
84C2×Dic71—42no—
85C2×Dic71—81no—
86C2×Dic71—81no—
87Dic142—44no—
88Dic142—82no—
89C8×C71—44no—
90C71⋊C8—42no—
91C22×C71#39311yescharacteristic
92C22×C71#45311yescharacteristic
93C22×C71#45411yescharacteristic
94C22×C71#45512yes—
95C22×C71#45612yes—
96C4×C71—21no—
97C4×C71—21no—
98C4×C71—22no—
99C4×C71—22no—
100C4×C71—82no—
101Dic71—82no—
102Dic71—82no—
103Dic71—82no—
104#10361—711nomaximal, Sylow
105C2×C71#166112yes—
106C2×C71#166211yescharacteristic
107C2×C71#166311yescharacteristic
108C2×C71#202711yescharacteristic
109C2×C71#202812yes—
110#2845—711no—
111#1993—712no—
112#1960—711no—
113#1295—712no—
114#1295D7111yescharacteristic
115C71#1036111yescharacteristic, complemented, Sylow
116C2×C4⋊Q8—711no—
117C2×#103—1422no—
118C2×#97—711no—
119C2×#97C2×D7112yes—
120#398—1422no—
121#398—1422no—
122#398—1422no—
123#382—712no—
124#382C2×D7111yescharacteristic
125#340—1422no—
126#340—1422no—
127C22×Q8—1421no—
128C22×C8—22no—
129C22×C8—1421no—
130C4⋊Q8—712no—
131C4⋊Q8—1421no—
132C4⋊Q8—1422no—
133C2×C4⋊C4—712no—
134C2×C4⋊C4—712no—
135C2×C4⋊C4—1422no—
136C2×C4⋊C4—1422no—
137C2×C4⋊C4—1422no—
138C2×C4⋊C4—1422no—
139C2×C4⋊C4D28412yes—
140C2×C42C22×D7111yescharacteristic
141#103—2844no—
142#97—22no—
143#97—22no—
144#97—1422no—
145#95—22no—
146#95—1422no—
147#95—1422no—
148#95—1422no—
149C2×Q8—2842no—
150C2×Q8—2842no—
151C2×Q8—2841no—
152C2×Q8—1422no—
153C22×C4—22no—
154C22×C4—2841no—
155C22×C4—1422no—
156C22×C4—1422no—
157C22×C4D4×D7112yes—
158C22×C4C2×D28411yescharacteristic
159C2×C8—24no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—2841no—
163C2×C8—2841no—
164C2×C8—1422no—
165C4⋊C4—42no—
166C4⋊C4—2842no—
167C4⋊C4—2842no—
168C4⋊C4—2842no—
169C4⋊C4—5682no—
170C4⋊C4—5682no—
171C4⋊C4—5682no—
172C4⋊C4—1424no—
173C42Q8×D7111yescharacteristic
174C42#1176586411yescharacteristic
175C42#1176586512yes—
176C23#1455609811yescharacteristic
177Q8—2844no—
178Q8—5682no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—2842no—
187C2×C4—2842no—
188C2×C4—2842no—
189C2×C4—2842no—
190C2×C4—2842no—
191C2×C4—2842no—
192C2×C4—2842no—
193C2×C4—5681no—
194C2×C4—5681no—
195C2×C4#1455610612yes—
196C2×C4#1455610811yescharacteristic
197C2×C4#1455611412yes—
198C2×C4#1455611611yescharacteristic
199C8—44no—
200C8—2842no—
201C22#1798591211yescharacteristic
202C22#1798616911yescharacteristic
203C22#1798617411yescharacteristic
204C22#1798617812yes—
205C22#1798618312yes—
206C4—21no—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—5682no—
212C4—5682no—
213C4—5682no—
214C2#2258765812yes—
215C2#2258766411yescharacteristic
216C2#2258766911yescharacteristic
217C2#2258949711yescharacteristic
218C2#2258950012yes—
subgroupG/H
centerC23#14556098
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socle, prime radicalC23×C71#120
hypercenter#1295D71
hypocenterC71#10361
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
115. C71#10361semidirect

Maximal subgroups

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

subgroupindex
#225966152
#225893132
#225891592
#225862222
C71×#12952
#1036171