Finite Group Databasebeta

← #31909451

Subgroups — #31909451

Order 18176.

220 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×#14556071C211yesmaximal, characteristic
2#22587237C211yesmaximal, characteristic, complemented
3#22587224C211yesmaximal, characteristic
4#22586411C211yesmaximal, characteristic, complemented
5#22586302C211yesmaximal, characteristic
6#22586256C211yesmaximal, characteristic, complemented
7C71×#1332C211yesmaximal, characteristic, complemented
8C2×#14556071C411yescharacteristic
9#17986130—21no—
10C4×D568C2211yescharacteristic
11#17986076—21no—
12#17985826C411yescharacteristic
13#17985803C411yescharacteristic
14#17985799C411yescharacteristic
15C4×C71⋊C16C2211yescharacteristic
16#17985752C2211yescharacteristic
17Q8⋊(C71⋊C8)—21no—
18#17985726—21no—
19C8×C71⋊C8C2211yescharacteristic
20C4×Q16×C71C2211yescharacteristic
21C4×C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×D568C2×C411yescharacteristic
24C2×#11765848—41no—
25C4×D284—21no—
26#14556071—22no—
27C2×C71⋊C16C2×C411yescharacteristic
28C2×C71⋊C16C2×C411yescharacteristic
29#14556062C2×C411yescharacteristic
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)—41no—
32#14556050—21no—
33(C2×D71)⋊C8—41no—
34#14556039—41no—
35C4×C71⋊C8D411yescharacteristic
36C4×C71⋊C8D411yescharacteristic
37C2×Q16×C71C2×C411yescharacteristic
38C4×Q8×C71—21no—
39C2×C16×C71—21no—
40#107×C71C2×C411yescharacteristic
41C4⋊C8×C71—21no—
42#103×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×D284—21no—
45C2×C4×D71—41no—
46D568—21no—
47D568—21no—
48#11765848—82no—
49C2×C71⋊C8—21no—
50C2×C71⋊C8—41no—
51C2×C71⋊C8D811yescharacteristic
52C2×C71⋊C8D811yescharacteristic
53C4⋊Dic71—21no—
54(C2×D71)⋊C4—41no—
55C71⋊C16—21no—
56C71⋊C16M1612yescomplemented
57C2×Q8×C71—21no—
58Q16×C71—21no—
59Q16×C71—21no—
60C2×C8×C71—41no—
61C2×C8×C71C22×C411yescharacteristic
62C2×C8×C71#4111yescharacteristic
63C4⋊C4×C71—21no—
64C4⋊C4×C71—41no—
65C42×C71—41no—
66C42×C71C2×D411yescharacteristic
67C16×C71—22no—
68C22×D71—41no—
69Q8×C71—41no—
70Q8×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71—41no—
73C2×C4×C71C4×D411yescharacteristic
74C2×C4×C71C2×D811yescharacteristic
75C2×C4×C71#13511yescharacteristic
76C2×Dic71—41no—
77D284—41no—
78D284—41no—
79C4×D71—81no—
80C4×D71—81no—
81C8×C71—21no—
82C8×C71—81no—
83C8×C71C2×M1611yescharacteristic
84C8×C71#13111yescharacteristic
85C71⋊C8—21no—
86C71⋊C8—21no—
87C71⋊C8—41no—
88C71⋊C8—81no—
89C22×C71C4×D811yescharacteristic
90C2×D71—81no—
91C2×D71—81no—
92C2×D71—81no—
93C4×C71—21no—
94C4×C71—81no—
95C4×C71—81no—
96C4×C71#43611yescharacteristic
97C4×C71#44311yescharacteristic
98C4×C71#50911yescharacteristic
99C4×C71#51011yescharacteristic
100Dic71—81no—
101#12084—711nomaximal, Sylow
102C2×C71#158911yescharacteristic
103C2×C71#217511yescharacteristic
104C2×C71#217911yescharacteristic
105D71—161no—
106C4×SD32—711no—
107#1591—711no—
108#1589—711no—
109#1371—711no—
110#1332D7111yescharacteristic, complemented
111#1331—711no—
112#1311—711no—
113C71#1208411yescharacteristic, complemented, Sylow
114C2×SD32—711no—
115#446—1421no—
116C4×Q16C2×D7111yescharacteristic
117C4×D8—711no—
118#436—1421no—
119#367—711no—
120#358—711no—
121#357—711no—
122C4×C16—711no—
123C4×C16C2×D7111yescharacteristic
124#335—711no—
125#335C2×D7111yescharacteristic
126Q8⋊C8—1421no—
127D4⋊C8—1421no—
128C82—711no—
129C2×Q16C4×D7111yescharacteristic
130C2×D8—711no—
131C2×M16—2841no—
132C4×Q8—21no—
133C4×D4—1421no—
134SD32—1422no—
135C2×C16—21no—
136C2×C16—711no—
137C2×C16—711no—
138#107—711no—
139#107C4×D7111yescharacteristic
140C4⋊C8—21no—
141C4⋊C8—2841no—
142C4⋊C8—1421no—
143#103—21no—
144#102—1421no—
145C22⋊C8—2841no—
146#97—2841no—
147C4×C8—711no—
148C4×C8—711no—
149C4×C8C22×D7111yescharacteristic
150C2×Q8—21no—
151C2×D4—1421no—
152C22×C4—2841no—
153Q16—21no—
154Q16—21no—
155D8—1421no—
156D8—1421no—
157M16—5682no—
158C2×C8—41no—
159C2×C8—2841no—
160C2×C8—711no—
161C2×C8—711no—
162C2×C8—1421no—
163C2×C8C2×C4×D7111yescharacteristic
164C2×C8#1176586411yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—41no—
167C4⋊C4—1421no—
168C22⋊C4—2841no—
169C42—41no—
170C42D4×D7111yescharacteristic
171C16—22no—
172C16—712no—
173C16—1421no—
174C23—2841no—
175Q8—41no—
176Q8—41no—
177D4—2841no—
178D4—2841no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—2841no—
182C2×C4—5681no—
183C2×C4—5681no—
184C2×C4(C22⋊D71)⋊C411yescharacteristic
185C2×C4D8×D7111yescharacteristic
186C2×C4#1455614911yescharacteristic
187C8—21no—
188C8—81no—
189C8—2841no—
190C8—5681no—
191C8—1421no—
192C8—1421no—
193C8M16×D7111yescharacteristic
194C8#1455613411yescharacteristic
195C22—5681no—
196C22—5681no—
197C22—5681no—
198C22#1798607911yescharacteristic
199C4—21no—
200C4—81no—
201C4—81no—
202C4—5681no—
203C4#1798607411yescharacteristic
204C4#1798611311yescharacteristic
205C4#1798643611yescharacteristic
206C4#1798644111yescharacteristic
207C2—11361no—
208C2#2258722211yescharacteristic
209C2#2259020711yescharacteristic
210C2#2259022311yescharacteristic
subgroupG/H
centerC2×C4D8×D71
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1332C2
socleC22×C71C4×D8
prime radicalC2×D568C2×C4
cosocleC4×C8×C71C23
hypercenter#1332D71
hypocenterC71#12084
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
2. #22587237C2semidirect
4. #22586411C2semidirect
6. #22586256C2semidirect
7. C71×#1332C2semidirect
56. C71⋊C16M16semidirect
110. #1332D71semidirect
113. C71#12084semidirect

Maximal subgroups

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

subgroupindex
C4×#145560712
#225872372
#225872242
#225864112
#225863022
#225862562
C71×#13322
#1208471