Finite Group Databasebeta

← #31901587

Subgroups — #31901587

Order 18176.

386 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 77 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#22588514C211yesmaximal, characteristic, complemented
2#22587062C212yesmaximal, complemented
3#22586951C212yesmaximal, complemented
4#22586251C211yesmaximal, characteristic, complemented
5C71×#3253C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×#14556063—22no—
8#17986023C2211yescharacteristic
9C2×#14556039C2211yescharacteristic
10#17985745—22no—
11#17985743—22no—
12#17985739—24no—
13#532×C71C2212yes—
14#476×C71—22no—
15#469×C71—22no—
16#459×C71—22no—
17#423×C71C2211yescharacteristic
18C2×#11765849—21no—
19C22×C71⋊C8—21no—
20#14556063—44no—
21C4⋊(C71⋊C8)—22no—
22C4⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24#14556039—22no—
25#14556039—21no—
26#14556039D412yes—
27C2×#41×C71—22no—
28C2×D8×C71—22no—
29C2×M16×C71D412yes—
30C4⋊Q8×C71D412yes—
31#127×C71D412yes—
32#124×C71—22no—
33#121×C71—22no—
34#121×C71—22no—
35C4×D4×C71—22no—
36C4×D4×C71—22no—
37C2×C42×C71C2311yescharacteristic
38#106×C71—22no—
39C4⋊C8×C71—22no—
40#102×C71—24no—
41#11765849—42no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8—41no—
46C2×C71⋊C8—42no—
47#41×C71—84no—
48C2×Q8×C71—22no—
49C2×D4×C71—22no—
50C2×D4×C71—22no—
51C2×D4×C71—22no—
52C2×D4×C71—42no—
53C22×C4×C71—42no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71#4112yes—
56D8×C71—82no—
57M16×C71—42no—
58C2×C8×C71—22no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—22no—
61C22⋊C4×C71—42no—
62C22⋊C4×C71—42no—
63C42×C71C2×D411yescharacteristic
64C42×C71C2×D412yes—
65C42×C71#4111yescharacteristic
66C23×C71—42no—
67C23×C71#12411yescharacteristic
68Q8×C71—82no—
69D4×C71—42no—
70D4×C71—42no—
71D4×C71—82no—
72D4×C71—82no—
73D4×C71—84no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—84no—
79C2×C4×C71#12011yescharacteristic
80C2×C4×C71#12111yescharacteristic
81C2×C4×C71#12112yes—
82C2×C4×C71#12312yes—
83C8×C71—42no—
84C71⋊C8—24no—
85C71⋊C8—82no—
86C71⋊C8—81no—
87C22×C71—22no—
88C22×C71—21no—
89C22×C71—82no—
90C22×C71—82no—
91C22×C71—82no—
92C22×C71#39411yescharacteristic
93C4×C71—22no—
94C4×C71—22no—
95C4×C71—21no—
96C4×C71—82no—
97C4×C71#45311yescharacteristic
98C4×C71#45611yescharacteristic
99#10524—711nomaximal, Sylow
100C2×C71—41no—
101C2×C71—162no—
102C2×C71#203911yescharacteristic
103C2×C71#204812yes—
104#3253D7111yescharacteristic
105#1849—711no—
106#1557—712no—
107#1530—712no—
108#1307—711no—
109C71#1052411yescharacteristic, complemented, Sylow
110#532C2×D7112yes—
111#476—22no—
112#469—22no—
113#459—22no—
114#432—712no—
115C2×#108—1422no—
116#423—711no—
117#423C2×D7111yescharacteristic
118C2×#97—711no—
119#332—1422no—
120#331—1422no—
121#329—1424no—
122C2×#41—22no—
123C2×D8—22no—
124C2×M16—1421no—
125C2×M16C22⋊D7112yes—
126C22×C8—1421no—
127C4⋊Q8C22⋊D7112yes—
128#127C22⋊D7112yes—
129#124—22no—
130#121—22no—
131#121—22no—
132C4×D4—22no—
133C4×D4—22no—
134C2×C42C22×D7111yescharacteristic
135#108—2844no—
136#106—22no—
137C4⋊C8—22no—
138C4⋊C8—1422no—
139C4⋊C8—1422no—
140#102—24no—
141C22⋊C8—1422no—
142#97—712no—
143#97—1422no—
144#97—1421no—
145#41—84no—
146C2×Q8—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—42no—
151C22×C4—42no—
152C22×C4C2×C22⋊D7111yescharacteristic
153C22×C4#1176586412yes—
154D8—82no—
155M16—42no—
156M16—2842no—
157C2×C8—22no—
158C2×C8—2841no—
159C2×C8—2842no—
160C2×C8—1422no—
161C2×C8—1422no—
162C2×C8—1422no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C42D4×D7111yescharacteristic
168C42C2×C22⋊D7112yes—
169C42#1176586411yescharacteristic
170C23—42no—
171C23#1455611511yescharacteristic
172Q8—82no—
173D4—42no—
174D4—42no—
175D4—82no—
176D4—82no—
177D4—84no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4#1455609711yescharacteristic
184C2×C4#1455609911yescharacteristic
185C2×C4#1455610012yes—
186C2×C4#1455610912yes—
187C8—42no—
188C8—5682no—
189C8—5681no—
190C8—1424no—
191C22—21no—
192C22—22no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22#1798591311yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—82no—
201C4#1798616711yescharacteristic
202C4#1798618211yescharacteristic
203C2—41no—
204C2—162no—
205C2#2258956111yescharacteristic
206C2#2258961012yes—
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#3253C2
socleC22×C71#394
hypercenter#3253D71
hypocenterC71#10524
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #22588514C2semidirect
2. #22587062C2semidirect
3. #22586951C2semidirect
4. #22586251C2semidirect
109. C71#10524semidirect

Maximal subgroups

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

subgroupindex
#225885142
#225870622
#225869512
#225862512
C71×#32532
#1052471