Finite Group Databasebeta

← #31906866

Subgroups — #31906866

Order 18176.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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#22586674C212yesmaximal
2#22586286C214yesmaximal, complemented
3C71×#2317C211yesmaximal, characteristic
4#17986056—22no—
5#17985725C2214yes—
6#17985723—22no—
7C2×#117×C71C411yescharacteristic
8C2×#117×C71C2211yescharacteristic
9C22×C4⋊C4×C71C411yescharacteristic
10#389×C71—22no—
11#385×C71—22no—
12#384×C71C2212yes—
13#382×C71C412yes—
14C4⋊(C71⋊C8)—42no—
15C22⋊(C71⋊C8)—24no—
16#14556039—42no—
17C23×C4×C71—21no—
18C23×C4×C71C2×C411yescharacteristic
19#117×C71—24no—
20#117×C71—42no—
21C2×C4⋊C4×C71—21no—
22C2×C4⋊C4×C71—21no—
23C2×C4⋊C4×C71—24no—
24C2×C4⋊C4×C71—42no—
25C2×C4⋊C4×C71C2×C412yes—
26C2×C4⋊C4×C71C2×C411yescharacteristic
27C2×C4⋊C4×C71C2311yescharacteristic
28C2×C22⋊C4×C71—21no—
29C2×C22⋊C4×C71D412yes—
30C2×C42×C71C2×C412yes—
31C2×C42×C71D412yes—
32#95×C71—24no—
33C2×C71⋊C8—44no—
34C24×C71C22⋊C411yescharacteristic
35C22×C4×C71—22no—
36C22×C4×C71—22no—
37C22×C4×C71—22no—
38C22×C4×C71—22no—
39C22×C4×C71—22no—
40C22×C4×C71—42no—
41C22×C4×C71—82no—
42C22×C4×C71C22⋊C412yes—
43C22×C4×C71C22⋊C411yescharacteristic
44C22×C4×C71C22×C411yescharacteristic
45C22×C4×C71C2×D412yes—
46C4⋊C4×C71—24no—
47C4⋊C4×C71—41no—
48C4⋊C4×C71—41no—
49C4⋊C4×C71—42no—
50C4⋊C4×C71—42no—
51C4⋊C4×C71—84no—
52C22⋊C4×C71—24no—
53C22⋊C4×C71—42no—
54C42×C71—22no—
55C42×C71—22no—
56C42×C71—22no—
57C42×C71M1614yes—
58C23×C71—21no—
59C23×C71—21no—
60C23×C71—41no—
61C23×C71—41no—
62C23×C71#10112yes—
63C23×C71C2×C22⋊C411yescharacteristic
64C2×C4×C71—28no—
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—24no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71—44no—
77C2×C4×C71—42no—
78C2×C4×C71—84no—
79C2×C4×C71#9912yes—
80C2×C4×C71C2×M1612yes—
81C71⋊C8—84no—
82C22×C71—21no—
83C22×C71—21no—
84C22×C71—22no—
85C22×C71—42no—
86C22×C71—42no—
87C22×C71—41no—
88C22×C71—41no—
89C22×C71#40711yescharacteristic
90C22×C71C2×#9911yescharacteristic
91C22×C71C2×#10111yescharacteristic
92C4×C71—22no—
93C4×C71—22no—
94C4×C71—44no—
95C4×C71—42no—
96C4×C71—42no—
97C4×C71—82no—
98#11516—711nomaximal, Sylow
99C2×C71—21no—
100C2×C71—21no—
101C2×C71—42no—
102C2×C71#146311yescharacteristic
103C2×C71#212811yescharacteristic
104C2×C71#213511yescharacteristic
105#2317D7111yescharacteristic
106#1464—712no—
107#1325—714no—
108C71#1151611yescharacteristic, complemented, Sylow
109C2×#117Dic7111yescharacteristic
110C2×#117C2×D7111yescharacteristic
111C22×C4⋊C4Dic7111yescharacteristic
112#432—1422no—
113#389—22no—
114#385—22no—
115#384C2×D7112yes—
116#382Dic7112yes—
117#324—714no—
118#323—1422no—
119C23×C4—21no—
120C23×C4C2×Dic7111yescharacteristic
121#117—24no—
122#117—42no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×Dic7112yes—
128C2×C4⋊C4C2×Dic7111yescharacteristic
129C2×C4⋊C4C22×D7111yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4C22⋊D7112yes—
132C2×C42C2×Dic7112yes—
133C2×C42C22⋊D7112yes—
134C4⋊C8—2842no—
135C22⋊C8—1424no—
136#97—2842no—
137#95—24no—
138C24C22⋊Dic7111yescharacteristic
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—82no—
146C22×C4C22⋊Dic7112yes—
147C22×C4C22⋊Dic7111yescharacteristic
148C22×C4C22×Dic7111yescharacteristic
149C22×C4C2×C22⋊D7112yes—
150C2×C8—2844no—
151C4⋊C4—24no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—84no—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42#1176584914yes—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#1455604712yes—
168C23C2×C22⋊Dic7111yescharacteristic
169C2×C4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—84no—
184C2×C4#1455604312yes—
185C2×C4C2×#1176584912yes—
186C8—5684no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22#1798596311yescharacteristic
195C22C2×#1455604311yescharacteristic
196C22C2×#1455604711yescharacteristic
197C4—22no—
198C4—22no—
199C4—44no—
200C4—42no—
201C4—42no—
202C4—82no—
203C2—21no—
204C2—21no—
205C2—42no—
206C2#2258667011yescharacteristic
207C2#2259001211yescharacteristic
208C2#2259003511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2317C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2317D71
hypocenterC71#11516
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #22586286C2semidirect
108. C71#11516semidirect

Maximal subgroups

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

subgroupindex
#225866742
#225862862
C71×#23172
#1151671