Finite Group Databasebeta

← #31917771

Subgroups — #31917771

Order 18176.

458 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590005C212yesmaximal, complemented
2#22586503C214yesmaximal, complemented
3C71×#3151C211yesmaximal, characteristic
4#17985973—22no—
5#17985794—22no—
6#17985792C2214yescomplemented
7C2×AGL(1,ℤ/8)×C71—22no—
8#532×C71C2211yescharacteristic
9C2×#127×C71C411yescharacteristic, complemented
10#496×C71—22no—
11#493×C71C2212yes—
12#492×C71C412yescomplemented
13C4×M16×C71C411yescharacteristic, complemented
14#14556087—42no—
15#14556043—24no—
16#14556043—42no—
17C2×#41×C71D412yes—
18C22×D4×C71—21no—
19C22×D4×C71—41no—
20AGL(1,ℤ/8)×C71—44no—
21AGL(1,ℤ/8)×C71—44no—
22C2×SD16×C71—24no—
23C2×D8×C71—24no—
24C2×M16×C71—21no—
25C4⋊Q8×C71C2×C411yescharacteristic
26#127×C71—21no—
27#127×C71—41no—
28#127×C71C2×C412yes—
29#127×C71C2311yescharacteristic
30#124×C71D412yes—
31#121×C71—41no—
32C4×D4×C71—41no—
33C2×C42×C71C2×C411yescharacteristic
34#97×C71—21no—
35C4×C8×C71C2×C412yes—
36C4⋊Dic71—82no—
37C22⋊Dic71—44no—
38C4×Dic71—82no—
39C24×C71—41no—
40#41×C71—42no—
41#41×C71—42no—
42C2×Q8×C71C22⋊C412yes—
43C2×D4×C71—22no—
44C2×D4×C71—22no—
45C2×D4×C71—42no—
46C2×D4×C71—42no—
47C2×D4×C71—41no—
48C2×D4×C71—42no—
49C2×D4×C71—42no—
50C2×D4×C71—41no—
51C2×D4×C71—84no—
52C2×D4×C71C2×D412yes—
53C22×C4×C71—21no—
54C22×C4×C71—22no—
55C22×C4×C71C22⋊C411yescharacteristic
56SD16×C71—48no—
57D8×C71—48no—
58M16×C71—24no—
59C2×C8×C71—22no—
60C4⋊C4×C71—41no—
61C22⋊C4×C71—42no—
62C42×C71—21no—
63C42×C71C22⋊C411yescharacteristic
64C42×C71C22×C411yescharacteristic
65C23×C71—22no—
66C23×C71—42no—
67C23×C71—81no—
68C23×C71—81no—
69C23×C71—84no—
70C23×C71#9911yescharacteristic
71Q8×C71—42no—
72D4×C71—42no—
73D4×C71—44no—
74D4×C71—82no—
75D4×C71—84no—
76D4×C71—82no—
77D4×C71—84no—
78D4×C71—82no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71#9911yescharacteristic
87C2×C4×C71C2×C22⋊C411yescharacteristic
88C2×Dic71—84no—
89C8×C71—24no—
90C22×C71—21no—
91C22×C71—41no—
92C22×C71—42no—
93C22×C71—82no—
94C22×C71—84no—
95C22×C71—82no—
96C22×C71—162no—
97C22×C71—162no—
98C22×C71C2×#9911yescharacteristic
99C4×C71—22no—
100C4×C71—21no—
101C4×C71—41no—
102C4×C71—42no—
103C4×C71#35312yes—
104Dic71—164no—
105#13585—711nomaximal, Sylow
106C2×C71—21no—
107C2×C71—41no—
108C2×C71—82no—
109C2×C71—162no—
110C2×C71C2×#35311yescharacteristic
111#3151D7111yescharacteristic
112#2126—712no—
113#1408—714no—
114C71#1358511yescharacteristic, complemented, Sylow
115C2×AGL(1,ℤ/8)—22no—
116#532C2×D7111yescharacteristic
117C2×#127Dic7111yescharacteristic, complemented
118#496—22no—
119#493C2×D7112yes—
120#492Dic7112yescomplemented
121#410—1422no—
122C4×M16Dic7111yescharacteristic, complemented
123#354—1422no—
124#353—714no—
125C2×#41C22⋊D7112yes—
126C22×D4—21no—
127C22×D4—41no—
128AGL(1,ℤ/8)—44no—
129AGL(1,ℤ/8)—44no—
130C2×SD16—24no—
131C2×D8—24no—
132C2×M16—21no—
133C4⋊Q8C2×Dic7111yescharacteristic
134#127—21no—
135#127—41no—
136#127C2×Dic7112yes—
137#127C22×D7111yescharacteristic
138#124C22⋊D7112yes—
139#121—41no—
140C4×D4—41no—
141#117—2842no—
142C2×C42C2×Dic7111yescharacteristic
143#99—2842no—
144#99—1424no—
145#97—21no—
146C4×C8C2×Dic7112yes—
147C24—41no—
148#41—42no—
149#41—42no—
150C2×Q8C22⋊Dic7112yes—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—84no—
160C2×D4C2×C22⋊D7112yes—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4C22⋊Dic7111yescharacteristic
164SD16—48no—
165D8—48no—
166M16—24no—
167C2×C8—22no—
168C4⋊C4—41no—
169C4⋊C4—5682no—
170C22⋊C4—42no—
171C22⋊C4—2844no—
172C42—21no—
173C42—5682no—
174C42C22⋊Dic7111yescharacteristic
175C42C22×Dic7111yescharacteristic
176C23—22no—
177C23—42no—
178C23—81no—
179C23—81no—
180C23—84no—
181C23#1455604311yescharacteristic
182Q8—42no—
183D4—42no—
184D4—44no—
185D4—84no—
186D4—82no—
187D4—82no—
188D4—84no—
189D4—82no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—5684no—
198C2×C4#1455604311yescharacteristic
199C2×C4C2×C22⋊Dic7111yescharacteristic
200C8—24no—
201C22—21no—
202C22—41no—
203C22—42no—
204C22—82no—
205C22—84no—
206C22—82no—
207C22—162no—
208C22—162no—
209C22C2×#1455604311yescharacteristic
210C4—22no—
211C4—21no—
212C4—41no—
213C4—42no—
214C4—11364no—
215C4#1798579212yes—
216C2—21no—
217C2—41no—
218C2—82no—
219C2—162no—
220C2C2×#1798579211yescharacteristic
subgroupG/H
centerC2C2×#17985792
commutatorC42×C71C22×C4
Frattini#127C22×D71
Fitting, prime radicalC71×#3151C2
socleC2×C71C2×#353
cosocle#127×C71C23
hypercenter#3151D71
hypocenterC71#13585
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #22590005C2semidirect
2. #22586503C2semidirect
6. #17985792C22semidirect
9. C2×#127×C71C4semidirect
12. #492×C71C4semidirect
13. C4×M16×C71C4semidirect
114. C71#13585semidirect
117. C2×#127Dic71semidirect
120. #492Dic71semidirect
122. C4×M16Dic71semidirect

Maximal subgroups

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

subgroupindex
#225900052
#225865032
C71×#31512
#1358571