Finite Group Databasebeta

← #31909468

Subgroups — #31909468

Order 18176.

212 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 67 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#22590202C211yesmaximal, characteristic
2#22587253C211yesmaximal, characteristic
3#22587249C211yesmaximal, characteristic, complemented
4#22586405C211yesmaximal, characteristic, complemented
5#22586308C211yesmaximal, characteristic
6#22586262C211yesmaximal, characteristic, complemented
7C71×#1334C211yesmaximal, characteristic, complemented
8C2×#14556071C411yescharacteristic
9#17986130—21no—
10C4×D568C2211yescharacteristic
11#17986076—21no—
12#17985824C411yescharacteristic
13#17985803C411yescharacteristic
14#17985799C411yescharacteristic
15#17985775C2211yescharacteristic
16#17985750C2211yescharacteristic
17Q8⋊(C71⋊C8)—21no—
18#17985726—21no—
19#17985720C2211yescharacteristic
20C4×Q16×C71C2211yescharacteristic
21#346×C71C2211yescharacteristic
22#334×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—21no—
51C2×C71⋊C8—41no—
52C4⋊Dic71—21no—
53(C2×D71)⋊C4—41no—
54C71⋊C16—21no—
55C71⋊C16M1612yescomplemented
56C2×Q8×C71—21no—
57Q16×C71—21no—
58Q16×C71—21no—
59C2×C8×C71—41no—
60C2×C8×C71C22×C411yescharacteristic
61C2×C8×C71#4111yescharacteristic
62C4⋊C4×C71—21no—
63C4⋊C4×C71—41no—
64C42×C71—41no—
65C42×C71C2×D411yescharacteristic
66C16×C71—41no—
67C22×D71—41no—
68Q8×C71—41no—
69Q8×C71—41no—
70C2×C4×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71C4×D411yescharacteristic
73C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
74C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
75C2×Dic71—41no—
76D284—41no—
77D284—41no—
78C4×D71—81no—
79C4×D71—81no—
80C8×C71—21no—
81C8×C71—81no—
82C8×C71C2×M1611yescharacteristic
83C8×C71#13111yescharacteristic
84C71⋊C8—41no—
85C71⋊C8—41no—
86C71⋊C8—81no—
87C22×C71#44211yescharacteristic
88C2×D71—81no—
89C2×D71—81no—
90C2×D71—81no—
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—81no—
94C4×C71—81no—
95C4×C71#43611yescharacteristic
96C4×C71#44411yescharacteristic
97Dic71—81no—
98#12087—711nomaximal, Sylow
99C2×C71#159411yescharacteristic
100C2×C71#218011yescharacteristic
101C2×C71AGL(1,ℤ/16)11yescharacteristic
102D71—161no—
103#2175—711no—
104#1594—711no—
105#1593—711no—
106#1368—711no—
107#1334D7111yescharacteristic, complemented
108#1333—711no—
109#1313—711no—
110C71#1208711yescharacteristic, complemented, Sylow
111C2×SD32—711no—
112#446—1421no—
113C4×Q16C2×D7111yescharacteristic
114C4×D8—711no—
115#436—1421no—
116#366—711no—
117#358—711no—
118#357—711no—
119#346—711no—
120#346C2×D7111yescharacteristic
121#334—711no—
122#334C2×D7111yescharacteristic
123Q8⋊C8—1421no—
124D4⋊C8—1421no—
125#322—711no—
126C2×Q16C4×D7111yescharacteristic
127C2×D8—711no—
128C2×M16—2841no—
129C4×Q8—21no—
130C4×D4—1421no—
131SD32—1422no—
132C2×C16—21no—
133C2×C16—711no—
134C2×C16—711no—
135#107—711no—
136#107C4×D7111yescharacteristic
137C4⋊C8—21no—
138C4⋊C8—2841no—
139C4⋊C8—1421no—
140#103—21no—
141#102—1421no—
142C22⋊C8—2841no—
143#97—2841no—
144C4×C8—711no—
145C4×C8—711no—
146C4×C8C22×D7111yescharacteristic
147C2×Q8—21no—
148C2×D4—1421no—
149C22×C4—2841no—
150Q16—21no—
151Q16—21no—
152D8—1421no—
153D8—1421no—
154M16—5682no—
155C2×C8—41no—
156C2×C8—2841no—
157C2×C8—1421no—
158C2×C8—1421no—
159C2×C8C2×C4×D7111yescharacteristic
160C2×C8#1176586411yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—1421no—
164C22⋊C4—2841no—
165C42—41no—
166C42D4×D7111yescharacteristic
167C16—41no—
168C16—712no—
169C16—1421no—
170C23—2841no—
171Q8—41no—
172Q8—41no—
173D4—2841no—
174D4—2841no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—2841no—
178C2×C4—5681no—
179C2×C4—5681no—
180C2×C4(C22⋊D71)⋊C411yescharacteristic
181C2×C4#1455615211yescharacteristic
182C2×C4#1455615511yescharacteristic
183C8—21no—
184C8—81no—
185C8—2841no—
186C8—2841no—
187C8—5681no—
188C8M16×D7111yescharacteristic
189C8#1455613411yescharacteristic
190C22—5681no—
191C22—5681no—
192C22—5681no—
193C22#1798610811yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—81no—
198C4—5681no—
199C4#1798607411yescharacteristic
200C4#1798611811yescharacteristic
201C2—11361no—
202C2#2258725111yescharacteristic
203C2#2259023411yescharacteristic
204C2#2259023511yescharacteristic
subgroupG/H
centerC22#17986108
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1334C2
socleC22×C71#442
prime radicalC2×D568C2×C4
cosocleC4×C8×C71C23
hypercenter#1334D71
hypocenterC71#12087
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1334

Complements

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

NH ≅ G/Ntype
3. #22587249C2semidirect
4. #22586405C2semidirect
6. #22586262C2semidirect
7. C71×#1334C2semidirect
55. C71⋊C16M16semidirect
107. #1334D71semidirect
110. C71#12087semidirect

Maximal subgroups

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

subgroupindex
#225902022
#225872532
#225872492
#225864052
#225863082
#225862622
C71×#13342
#1208771