Finite Group Databasebeta

← #31901962

Subgroups — #31901962

Order 18176.

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

normal subgroups only

This group has 350 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#22596615C211yesmaximal, characteristic
2#22588475C212yesmaximal
3#22586222C212yesmaximal
4#22586221C211yesmaximal, characteristic
5C71×#1836C211yesmaximal, characteristic
6C2×#14556039C2212yes—
7#17985933—22no—
8#17985932—22no—
9#17985931—22no—
10#17985865C2212yes—
11#17985761—22no—
12#17985761—22no—
13C2×#97×C71C2211yescharacteristic
14#384×C71C2211yescharacteristic
15#382×C71C2211yescharacteristic
16#340×C71—22no—
17C22×C71⋊C8—22no—
18C2×C4⋊Dic71—22no—
19C2×C4⋊Dic71D412yes—
20C2×C4⋊Dic71D412yes—
21C2×Dic71⋊C4—22no—
22C2×Dic71⋊C4—22no—
23#14556039—22no—
24#14556039Q814yes—
25C71⋊#95—22no—
26C71⋊#95—22no—
27C71⋊#95—22no—
28C22×C8×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71D412yes—
32C2×C42×C71C2311yescharacteristic
33#97×C71—21no—
34#97×C71Q812yes—
35#95×C71—22no—
36C22×Dic71—22no—
37C22×Dic71—22no—
38C2×C71⋊C8—24no—
39C2×C71⋊C8—42no—
40C2×C71⋊C8—42no—
41C4⋊Dic71—42no—
42C4⋊Dic71—42no—
43C4⋊Dic71—82no—
44Dic71⋊C4—82no—
45Dic71⋊C4—82no—
46C22×C4×C71—22no—
47C22×C4×C71—41no—
48C22×C4×C71C2×D412yes—
49C22×C4×C71C2×D411yescharacteristic
50C2×C8×C71—22no—
51C2×C8×C71—41no—
52C2×C8×C71—41no—
53C4⋊C4×C71—42no—
54C4⋊C4×C71—42no—
55C4⋊C4×C71—44no—
56C42×C71C2×Q812yes—
57C42×C71C2×Q811yescharacteristic
58C42×C71#4111yescharacteristic
59C23×C71#12011yescharacteristic
60C2×C4×C71—21no—
61C2×C4×C71—22no—
62C2×C4×C71—21no—
63C2×C4×C71—22no—
64C2×C4×C71—42no—
65C2×C4×C71—42no—
66C2×C4×C71—42no—
67C2×C4×C71—82no—
68C2×C4×C71—81no—
69C2×C4×C71C22⋊Q812yes—
70C2×C4×C71C22⋊Q811yescharacteristic
71C2×C4×C71C4⋊Q811yescharacteristic
72C2×C4×C71C4⋊Q812yes—
73C2×Dic71—42no—
74C2×Dic71—42no—
75C2×Dic71—42no—
76C2×Dic71—42no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79C8×C71—42no—
80C71⋊C8—44no—
81C22×C71#39511yescharacteristic
82C22×C71#45412yes—
83C22×C71#45411yescharacteristic
84C22×C71#45512yes—
85C22×C71#45511yescharacteristic
86C4×C71—22no—
87C4×C71—21no—
88C4×C71—22no—
89C4×C71—21no—
90C4×C71—82no—
91C4×C71—161no—
92Dic71—82no—
93Dic71—82no—
94#10586—711nomaximal, Sylow
95C2×C71#165811yescharacteristic
96C2×C71#166211yescharacteristic
97C2×C71#166212yes—
98C2×C71#206112yes—
99C2×C71#206111yescharacteristic
100#2845—711no—
101#1836—712no—
102#1836D7111yescharacteristic
103#1295—712no—
104#1295—711no—
105C71#1058611yescharacteristic, complemented, Sylow
106C2×#97—712no—
107C2×#97C2×D7111yescharacteristic
108#398—1422no—
109#398—1422no—
110#398—1422no—
111#384C2×D7111yescharacteristic
112#382—712no—
113#382C2×D7111yescharacteristic
114#340—22no—
115#340—1422no—
116#340—1422no—
117C22×C8—21no—
118C22×C8—1422no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—712no—
122C2×C4⋊C4—712no—
123C2×C4⋊C4—1422no—
124C2×C4⋊C4—1422no—
125C2×C4⋊C4—1422no—
126C2×C4⋊C4C22⋊D7112yes—
127C2×C42C22×D7111yescharacteristic
128#97—21no—
129#97—714no—
130#97—1422no—
131#97Dic14212yes—
132#95—22no—
133#95—1422no—
134#95—1422no—
135#95—1422no—
136C22×C4—22no—
137C22×C4—41no—
138C22×C4—1422no—
139C22×C4—1422no—
140C22×C4D4×D7112yes—
141C22×C4C2×C22⋊D7111yescharacteristic
142C2×C8—22no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—2842no—
146C2×C8—2842no—
147C2×C8—1424no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—2842no—
152C4⋊C4—2842no—
153C4⋊C4—5682no—
154C4⋊C4—5682no—
155C4⋊C4—5682no—
156C42Q8×D7112yes—
157C42C2×Dic14211yescharacteristic
158C42#1176586511yescharacteristic
159C23#1455609611yescharacteristic
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—82no—
168C2×C4—81no—
169C2×C4—2842no—
170C2×C4—2842no—
171C2×C4—2842no—
172C2×C4—2842no—
173C2×C4—2842no—
174C2×C4—2842no—
175C2×C4#1455610612yes—
176C2×C4#1455610711yescharacteristic
177C2×C4#1455612611yescharacteristic
178C2×C4#1455612712yes—
179C8—42no—
180C8—2844no—
181C22#1798591911yescharacteristic
182C22#1798617012yes—
183C22#1798617111yescharacteristic
184C22#1798617512yes—
185C22#1798617611yescharacteristic
186C4—22no—
187C4—22no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—161no—
192C4—5682no—
193C4—5682no—
194C2#2258764111yescharacteristic
195C2#2258766111yescharacteristic
196C2#2258766212yes—
197C2#2258968812yes—
198C2#2258969011yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1836C2
socle, prime radicalC23×C71#120
hypercenter#1836D71
hypocenterC71#10586
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
105. C71#10586semidirect

Maximal subgroups

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

subgroupindex
#225966152
#225884752
#225862222
#225862212
C71×#18362
#1058671