Finite Group Databasebeta

← #31901953

Subgroups — #31901953

Order 18176.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#22596612C211yesmaximal, characteristic
2#22588509C211yesmaximal, characteristic
3#22588474C212yesmaximal
4#22586222C212yesmaximal
5C71×#1295C211yesmaximal, characteristic
6C2×#14556061—21no—
7C2×#14556059—21no—
8C2×#14556039C2211yescharacteristic
9#17985941—22no—
10#17985938—21no—
11#17985920—22no—
12#17985919—21no—
13#17985869C2211yescharacteristic
14#17985865C2212yes—
15#17985762—22no—
16#17985762—22no—
17C2×#97×C71C2212yes—
18#382×C71C2211yescharacteristic
19C22×Dic142—21no—
20C22×C71⋊C8—21no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71D412yes—
25C2×C4⋊Dic71D412yes—
26C2×Dic71⋊C4—22no—
27#14556061—42no—
28#14556059—42no—
29#14556039—21no—
30#14556039Q812yescomplemented
31C71⋊#95—21no—
32C71⋊#95—22no—
33C71⋊#95—21no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C22×C8×C71—22no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C42×C71C2311yescharacteristic
40#97×C71—22no—
41#97×C71Q814yescomplemented
42#95×C71—22no—
43C2×Dic142—81no—
44C2×Dic142—82no—
45C22×Dic71—22no—
46C22×Dic71—22no—
47C22×Dic71—41no—
48C2×C71⋊C8—22no—
49C2×C71⋊C8—22no—
50C2×C71⋊C8—22no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C4⋊Dic71—41no—
54C4⋊Dic71—41no—
55C4⋊Dic71—44no—
56C4⋊Dic71—81no—
57Dic71⋊C4—82no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71C2×D412yes—
61C22×C4×C71C2×D411yescharacteristic
62C2×C8×C71—24no—
63C2×C8×C71—42no—
64C2×C8×C71—42no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C42×C71C2×Q811yescharacteristic
68C42×C71C2×Q812yes—
69C42×C71#4111yescharacteristic
70C23×C71#12011yescharacteristic
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—41no—
76C2×C4×C71—41no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71C22⋊Q812yes—
80C2×C4×C71C22⋊Q811yescharacteristic
81C2×C4×C71C4⋊Q812yes—
82C2×C4×C71C4⋊Q811yescharacteristic
83C2×Dic71—42no—
84C2×Dic71—42no—
85C2×Dic71—42no—
86C2×Dic71—42no—
87C2×Dic71—42no—
88C2×Dic71—42no—
89C2×Dic71—81no—
90C2×Dic71—82no—
91Dic142—84no—
92C8×C71—44no—
93C71⋊C8—24no—
94C22×C71#39511yescharacteristic
95C22×C71#45311yescharacteristic
96C22×C71#45412yes—
97C22×C71#45512yes—
98C22×C71#45611yescharacteristic
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—22no—
102C4×C71—22no—
103C4×C71—81no—
104C4×C71—81no—
105Dic71—82no—
106Dic71—82no—
107Dic71—161no—
108#10584—711nomaximal, Sylow
109C2×C71#165711yescharacteristic
110C2×C71#165911yescharacteristic
111C2×C71#166412yes—
112C2×C71#206011yescharacteristic
113C2×C71#206112yes—
114#2844—711no—
115#1847—711no—
116#1836—712no—
117#1295—712no—
118#1295D7111yescharacteristic
119C71#1058411yescharacteristic, complemented, Sylow
120C2×#107—1421no—
121C2×#106—1421no—
122C2×#97—711no—
123C2×#97C2×D7112yes—
124#400—1422no—
125#400—1421no—
126#395—1422no—
127#395—1421no—
128#384—711no—
129#382—712no—
130#382C2×D7111yescharacteristic
131#340—1422no—
132#340—1422no—
133C22×Q8—1421no—
134C22×C8—22no—
135C22×C8—1421no—
136C2×C4⋊C4—712no—
137C2×C4⋊C4—712no—
138C2×C4⋊C4—1422no—
139C2×C4⋊C4—1421no—
140C2×C4⋊C4—1422no—
141C2×C4⋊C4—1421no—
142C2×C4⋊C4D28411yescharacteristic
143C2×C4⋊C4D28411yescharacteristic
144C2×C42C22×D7111yescharacteristic
145#107—2842no—
146#106—2842no—
147#97—22no—
148#97—712no—
149#97—1421no—
150#97Dic14214yescomplemented
151#95—22no—
152#95—1422no—
153#95—1421no—
154#95—1421no—
155#95—1422no—
156#95—1422no—
157C2×Q8—5681no—
158C2×Q8—5682no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—2841no—
162C22×C4—1422no—
163C22×C4—1422no—
164C22×C4D4×D7112yes—
165C22×C4C2×D28411yescharacteristic
166C2×C8—24no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—1422no—
170C2×C8—1422no—
171C2×C8—1422no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—2842no—
175C4⋊C4—2842no—
176C4⋊C4—2841no—
177C4⋊C4—2841no—
178C4⋊C4—2844no—
179C4⋊C4—5682no—
180C4⋊C4—5681no—
181C42Q8×D7111yescharacteristic
182C42C2×Dic14212yes—
183C42#1176586411yescharacteristic
184C23#1455609811yescharacteristic
185Q8—5684no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—2842no—
195C2×C4—2842no—
196C2×C4—2842no—
197C2×C4—2842no—
198C2×C4—2842no—
199C2×C4—2842no—
200C2×C4—5681no—
201C2×C4—5682no—
202C2×C4#1455610512yes—
203C2×C4#1455610811yescharacteristic
204C2×C4#1455612712yes—
205C2×C4#1455612811yescharacteristic
206C8—44no—
207C8—1424no—
208C22#1798592011yescharacteristic
209C22#1798616911yescharacteristic
210C22#1798617312yes—
211C22#1798617812yes—
212C22#1798618411yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—5682no—
220C4—5682no—
221C4—11361no—
222C2#2258763911yescharacteristic
223C2#2258764711yescharacteristic
224C2#2258767612yes—
225C2#2258968611yescharacteristic
226C2#2258968912yes—
subgroupG/H
centerC23#14556098
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socle, prime radicalC23×C71#120
hypercenter#1295D71
hypocenterC71#10584
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
30. #14556039Q8semidirect
41. #97×C71Q8semidirect
119. C71#10584semidirect
150. #97Dic142semidirect

Maximal subgroups

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

subgroupindex
#225966122
#225885092
#225884742
#225862222
C71×#12952
#1058471