Finite Group Databasebeta

← #31884776

Subgroups — #31884776

Order 18176.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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#22588411C211yesmaximal, characteristic
2#22586674C211yesmaximal, characteristic
3C71×#1464C211yesmaximal, characteristic
4#17986056—21no—
5C2×C22⋊(C71⋊C8)C411yescharacteristic
6C2×C22⋊(C71⋊C8)C411yescharacteristic
7#17985754—21no—
8#17985754—21no—
9#17985725C412yes—
10#17985723—21no—
11C2×#117×C71C2211yescharacteristic
12#432×C71—21no—
13#324×C71C412yes—
14#323×C71—21no—
15C22×C71⋊C8—21no—
16C22×C71⋊C8—21no—
17C4⋊(C71⋊C8)—41no—
18C22⋊(C71⋊C8)—22no—
19C22⋊(C71⋊C8)—41no—
20C22⋊(C71⋊C8)—41no—
21#14556039—41no—
22C23×C4×C71C2×C411yescharacteristic
23#117×C71—41no—
24#117×C71—41no—
25C2×C4⋊C4×C71C2×C411yescharacteristic
26C2×C4⋊C4×C71C2×C411yescharacteristic
27C2×C22⋊C4×C71D411yescharacteristic
28C2×C22⋊C4×C71Q811yescharacteristic
29C2×C42×C71D411yescharacteristic
30C2×C42×C71D411yescharacteristic
31C4⋊C8×C71—41no—
32C22⋊C8×C71—22no—
33#97×C71—41no—
34C2×C71⋊C8—42no—
35C2×C71⋊C8—41no—
36C2×C71⋊C8—41no—
37C2×C71⋊C8—41no—
38C2×C71⋊C8—41no—
39C2×C71⋊C8—41no—
40C2×C71⋊C8—41no—
41C24×C71C4⋊C411yescharacteristic
42C22×C4×C71—21no—
43C22×C4×C71—21no—
44C22×C4×C71—41no—
45C22×C4×C71—41no—
46C22×C4×C71C4211yescharacteristic
47C22×C4×C71C22⋊C411yescharacteristic
48C22×C4×C71C22⋊C411yescharacteristic
49C22×C4×C71C22⋊C411yescharacteristic
50C22×C4×C71C4⋊C411yescharacteristic
51C22×C4×C71C4⋊C411yescharacteristic
52C2×C8×C71—42no—
53C4⋊C4×C71—41no—
54C4⋊C4×C71—41no—
55C22⋊C4×C71—41no—
56C22⋊C4×C71—41no—
57C42×C71—21no—
58C42×C71—21no—
59C42×C71—21no—
60C42×C71—21no—
61C23×C71—21no—
62C23×C71—21no—
63C23×C71—21no—
64C23×C71—41no—
65C23×C71—41no—
66C23×C71#9511yescharacteristic
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—41no—
79C2×C4×C71—41no—
80C2×C4×C71—41no—
81C2×C4×C71—41no—
82C2×C4×C71#9911yescharacteristic
83C2×C4×C71#9911yescharacteristic
84C8×C71—82no—
85C71⋊C8—81no—
86C71⋊C8—81no—
87C71⋊C8—82no—
88C22×C71—21no—
89C22×C71—21no—
90C22×C71—41no—
91C22×C71—41no—
92C22×C71—41no—
93C22×C71—81no—
94C22×C71—81no—
95C22×C71#34211yescharacteristic
96C22×C71#34411yescharacteristic
97C22×C71#34411yescharacteristic
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—41no—
101C4×C71—41no—
102C4×C71—41no—
103C4×C71—41no—
104C4×C71—41no—
105#7327—711nomaximal, Sylow
106C2×C71—21no—
107C2×C71—21no—
108C2×C71—81no—
109C2×C71#130611yescharacteristic
110C2×C71#139511yescharacteristic
111C2×C71#139511yescharacteristic
112#1818—711no—
113#1464—711no—
114#1464D7111yescharacteristic
115C71#732711yescharacteristic, complemented, Sylow
116C2×#117C2×D7111yescharacteristic
117#432—21no—
118#432—1421no—
119C2×C22⋊C8—711no—
120C2×C22⋊C8—711no—
121#336—1421no—
122#336—1421no—
123#324—712no—
124#324Dic7112yes—
125#323—21no—
126#323—1421no—
127C23×C4C4×D7111yescharacteristic
128C22×C8—1421no—
129C22×C8—1421no—
130#117—41no—
131#117—41no—
132C2×C4⋊C4C4×D7111yescharacteristic
133C2×C4⋊C4C2×Dic7111yescharacteristic
134C2×C22⋊C4Dic14211yescharacteristic
135C2×C22⋊C4C22⋊D7111yescharacteristic
136C2×C42D28411yescharacteristic
137C2×C42C22⋊D7111yescharacteristic
138C4⋊C8—41no—
139C4⋊C8—2841no—
140C22⋊C8—22no—
141C22⋊C8—2841no—
142C22⋊C8—2841no—
143C22⋊C8—1422no—
144#97—41no—
145#97—2841no—
146C24Dic71⋊C411yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4C4×Dic7111yescharacteristic
152C22×C4(C2×D71)⋊C411yescharacteristic
153C22×C4(C2×D71)⋊C411yescharacteristic
154C22×C4C22⋊Dic7111yescharacteristic
155C22×C4Dic71⋊C411yescharacteristic
156C22×C4C4⋊Dic7111yescharacteristic
157C2×C8—42no—
158C2×C8—2842no—
159C2×C8—2841no—
160C2×C8—2841no—
161C2×C8—2841no—
162C2×C8—2841no—
163C2×C8—2841no—
164C2×C8—2841no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—41no—
177C23—41no—
178C23C71⋊#9511yescharacteristic
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4#1455604211yescharacteristic
195C2×C4#1455604211yescharacteristic
196C8—82no—
197C8—5681no—
198C8—5681no—
199C8—5682no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—81no—
207C22#1798576511yescharacteristic
208C22#1798577011yescharacteristic
209C22#1798577111yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—41no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2258624911yescharacteristic
221C2#2258646811yescharacteristic
222C2#2258647011yescharacteristic
subgroupG/H
centerC22#17985765
commutatorC22×C4×C71C42
FrattiniC2×#117C2×D71
FittingC71×#1464C2
socleC22×C71#342
prime radicalC24×C71C4⋊C4
cosocleC2×#117×C71C22
hypercenter#1464D71
hypocenterC71#7327
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
115. C71#7327semidirect

Maximal subgroups

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

subgroupindex
#225884112
#225866742
C71×#14642
#732771