Finite Group Databasebeta

← #31909402

Subgroups — #31909402

Order 18176.

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

normal subgroups only

This group has 220 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#22590206C211yesmaximal, characteristic
2#22587248C211yesmaximal, characteristic, complemented
3#22587163C211yesmaximal, characteristic
4#22586309C211yesmaximal, characteristic, complemented
5#22586306C211yesmaximal, characteristic
6#22586262C211yesmaximal, characteristic, complemented
7C71×#1368C211yesmaximal, characteristic, complemented
8C2×#14556073C411yescharacteristic
9#17986126—21no—
10C4×Dic284C2211yescharacteristic
11C4×D568C2211yescharacteristic
12#17986064—21no—
13#17985825C411yescharacteristic
14#17985804C411yescharacteristic
15#17985801C411yescharacteristic
16#17985775C2211yescharacteristic
17#17985729—21no—
18#17985726—21no—
19#17985720C2211yescharacteristic
20#346×C71C2211yescharacteristic
21#334×C71C2211yescharacteristic
22#334×C71C2211yescharacteristic
23C2×Dic284C2×C411yescharacteristic
24C2×D568C2×C411yescharacteristic
25C2×C8×D71—41no—
26C4×Dic142—21no—
27C4×D284—21no—
28#14556073—22no—
29C2×C71⋊C16—21no—
30#14556062C2×C411yescharacteristic
31#14556062C2×C411yescharacteristic
32Dic71⋊C8—41no—
33#14556053—21no—
34#14556050—21no—
35(C2×D71)⋊C8—41no—
36#14556038—41no—
37C4×C71⋊C8D411yescharacteristic
38C4×C71⋊C8D411yescharacteristic
39C2×C16×C71C2×C411yescharacteristic
40C2×C16×C71C2×C411yescharacteristic
41C4⋊C8×C71—21no—
42C4⋊C8×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×Dic142—21no—
45C2×D284—21no—
46C2×C4×D71—41no—
47Dic284—21no—
48Dic284—21no—
49D568—21no—
50D568M1612yes—
51C2×C71⋊C8—21no—
52C2×C71⋊C8—21no—
53C8×D71—82no—
54C4⋊Dic71—21no—
55C4⋊Dic71—21no—
56Dic71⋊C4—41no—
57(C2×D71)⋊C4—41no—
58C4×Dic71—41no—
59C71⋊C16—22no—
60C2×C8×C71—41no—
61C2×C8×C71—41no—
62C2×C8×C71C22×C411yescharacteristic
63C2×C8×C71#4111yescharacteristic
64C42×C71C2×D411yescharacteristic
65C16×C71—21no—
66C16×C71—21no—
67C22×D71—41no—
68C2×C4×C71C4×D411yescharacteristic
69C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
70C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
71C2×Dic71—41no—
72C2×Dic71—41no—
73C2×Dic71—41no—
74D284—22no—
75D284—41no—
76C4×D71—81no—
77C4×D71—81no—
78Dic142—41no—
79Dic142—41no—
80C8×C71—21no—
81C8×C71—81no—
82C8×C71—81no—
83C8×C71C2×M1611yescharacteristic
84C8×C71#13111yescharacteristic
85C71⋊C8—41no—
86C71⋊C8—41no—
87C22×C71#44211yescharacteristic
88C2×D71—42no—
89C2×D71—81no—
90C2×D71—81no—
91C4×C71—21no—
92C4×C71—21no—
93C4×C71#43511yescharacteristic
94C4×C71#44311yescharacteristic
95Dic71—81no—
96Dic71—81no—
97Dic71—81no—
98#12077—711nomaximal, Sylow
99C2×C71#158611yescharacteristic
100C2×C71#218011yescharacteristic
101C2×C71AGL(1,ℤ/16)11yescharacteristic
102D71—82no—
103#2175—711no—
104#1593—711no—
105#1580—711no—
106#1368D7111yescharacteristic, complemented
107#1334—711no—
108#1333—711no—
109#1313—711no—
110C71#1207711yescharacteristic, complemented, Sylow
111C2×SD32—711no—
112#446—1421no—
113C4×Q16—711no—
114C4×D8—711no—
115C8×D4—1421no—
116#366—711no—
117#358—711no—
118#357—711no—
119#346—711no—
120#346C2×D7111yescharacteristic
121#334C2×D7111yescharacteristic
122#334C2×D7111yescharacteristic
123Q8⋊C8—1421no—
124D4⋊C8—1421no—
125#322—711no—
126C2×Q16—711no—
127C2×D8—711no—
128C22×C8—2841no—
129C4×Q8—1421no—
130C4×D4—1421no—
131SD32—1422no—
132C2×C16—1421no—
133C2×C16C4×D7111yescharacteristic
134C2×C16C4×D7111yescharacteristic
135#107—711no—
136#107—711no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—2841no—
140#103—1421no—
141#102—1421no—
142C22⋊C8—2841no—
143#97—2841no—
144C4×C8—711no—
145C4×C8—711no—
146C4×C8C22×D7111yescharacteristic
147C2×Q8—1421no—
148C2×D4—1421no—
149C22×C4—2841no—
150Q16—1421no—
151Q16—1421no—
152D8—712no—
153D8—1421no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—5682no—
157C2×C8—1421no—
158C2×C8—1421no—
159C2×C8C2×C4×D7111yescharacteristic
160C2×C8#1176586611yescharacteristic
161C4⋊C4—2841no—
162C4⋊C4—1421no—
163C4⋊C4—1421no—
164C22⋊C4—2841no—
165C42—2841no—
166C42D4×D7111yescharacteristic
167C16—21no—
168C16—21no—
169C16—1422no—
170C23—2841no—
171Q8—2841no—
172Q8—2841no—
173D4—2841no—
174D4—1422no—
175C2×C4—2841no—
176C2×C4—2841no—
177C2×C4—2841no—
178C2×C4—5681no—
179C2×C4—5681no—
180C2×C4#1455609011yescharacteristic
181C2×C4#1455615511yescharacteristic
182C2×C4#1455615511yescharacteristic
183C8—21no—
184C8—81no—
185C8—81no—
186C8—2841no—
187C8—2841no—
188C8M16×D7111yescharacteristic
189C8#1455613411yescharacteristic
190C22—2842no—
191C22—5681no—
192C22—5681no—
193C22#1798611011yescharacteristic
194C4—21no—
195C4—21no—
196C4—5681no—
197C4—5681no—
198C4—5681no—
199C4#1798606911yescharacteristic
200C4#1798611511yescharacteristic
201C2—5682no—
202C2#2258720311yescharacteristic
203C2#2259023011yescharacteristic
204C2#2259023711yescharacteristic
subgroupG/H
centerC22#17986110
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1368C2
socleC22×C71#442
prime radicalC2×D568C2×C4
cosocleC4×C8×C71C23
hypercenter#1368D71
hypocenterC71#12077
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1368

Complements

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

NH ≅ G/Ntype
2. #22587248C2semidirect
4. #22586309C2semidirect
6. #22586262C2semidirect
7. C71×#1368C2semidirect
106. #1368D71semidirect
110. C71#12077semidirect

Maximal subgroups

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

subgroupindex
#225902062
#225872482
#225871632
#225863092
#225863062
#225862622
C71×#13682
#1207771