Finite Group Databasebeta

← #31918265

Subgroups — #31918265

Order 18176.

200 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 51 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#22590660C211yesmaximal, characteristic, complemented
2#22590653C211yesmaximal, characteristic
3#22590598C211yesmaximal, characteristic, complemented
4#22590220C211yesmaximal, characteristic, complemented
5#22586537C211yesmaximal, characteristic, complemented
6#22586532C211yesmaximal, characteristic, complemented
7C71×#1419C211yesmaximal, characteristic, complemented
8#17986431C2211yescharacteristic
9C2×Dic568C2211yescharacteristic
10C2×#14556072—21no—
11#17986382—21no—
12#17986115—21no—
13Dic1136—21no—
14#17985838—21no—
15#17985837—21no—
16#17985834—21no—
17#17985833C2211yescharacteristic
18#17985805—21no—
19C16×Dic71C2211yescharacteristic
20C2×D16×C71C2211yescharacteristic
21M64×C71C2211yescharacteristic
22#368×C71C2211yescharacteristic
23#14556150—21no—
24C2×Dic284—21no—
25C2×#11765853—41no—
26#14556136—41no—
27#14556115—41no—
28Dic568—21no—
29Dic568—21no—
30Dic568D411yescharacteristic
31#14556073—21no—
32#14556072—42no—
33D1136D411yescharacteristic
34C2×C71⋊C16D411yescharacteristic
35#14556054—41no—
36C8×Dic71D411yescharacteristic
37C71⋊C32—21no—
38C2×D8×C71—21no—
39D16×C71—21no—
40D16×C71—21no—
41C2×C16×C71C2311yescharacteristic
42#108×C71—21no—
43C32×C71—21no—
44#11765865—41no—
45C2×Dic142—41no—
46Dic284—21no—
47Dic284—41no—
48Dic284—41no—
49#11765854—41no—
50#11765853—81no—
51#11765853—81no—
52D568—21no—
53#11765848—81no—
54C2×C71⋊C8D811yescharacteristic
55C8×D71—81no—
56C22⋊Dic71—82no—
57C4×Dic71D811yescharacteristic
58C71⋊C16—21no—
59C2×D4×C71—41no—
60D8×C71—41no—
61D8×C71—41no—
62M16×C71—41no—
63C2×C8×C71C2×D411yescharacteristic
64C16×C71C2×D411yescharacteristic
65C16×C71#4111yescharacteristic
66C23×C71—81no—
67D4×C71—81no—
68D4×C71—81no—
69C2×C4×C71C2×D811yescharacteristic
70C22⋊D71—81no—
71C2×Dic71—21no—
72C2×Dic71—81no—
73D284—41no—
74C4×D71—81no—
75Dic142—41no—
76Dic142—81no—
77Dic142—81no—
78C8×C71—81no—
79C8×C71#12111yescharacteristic
80C8×C71AGL(1,ℤ/8)11yescharacteristic
81C71⋊C8D1611yescharacteristic
82C71⋊C8D1611yescharacteristic
83C22×C71—162no—
84C22×C71—161no—
85C22×C71#50911yescharacteristic
86C2×D71—81no—
87C4×C71#45911yescharacteristic
88C4×C71C2×D1611yescharacteristic
89Dic71—41no—
90Dic71—81no—
91Dic71—161no—
92#13691—711nomaximal, Sylow
93C2×C71—21no—
94C2×C71—321no—
95C2×C71#220611yescharacteristic
96D71—161no—
97#2264—711no—
98#2263—711no—
99#2249—711no—
100#2178—711no—
101#1422—711no—
102#1420—711no—
103#1419D7111yescharacteristic, complemented
104C71#1369111yescharacteristic, complemented, Sylow
105#508—711no—
106C2×Q32—711no—
107C2×SD32—1421no—
108C2×D16C2×D7111yescharacteristic
109#495—1421no—
110#443—1421no—
111Q64—1421no—
112SD64—1421no—
113SD64—1421no—
114D32—1421no—
115M64—711no—
116M64C2×D7111yescharacteristic
117#368C2×D7111yescharacteristic
118#359—1421no—
119C4×C16—711no—
120#135—1421no—
121C2×Q16—1421no—
122C2×SD16—2841no—
123C2×D8—21no—
124#131—2841no—
125#124—2841no—
126Q32—711no—
127Q32—1421no—
128Q32—1421no—
129SD32—2842no—
130SD32—1421no—
131D16—21no—
132D16—21no—
133D16—711no—
134C2×C16—711no—
135C2×C16C22×D7111yescharacteristic
136#108—21no—
137#104—2841no—
138C4×C8—711no—
139C32—21no—
140C32—1421no—
141#41—2841no—
142C2×Q8—2841no—
143C2×D4—41no—
144Q16—2841no—
145Q16—2841no—
146Q16—1421no—
147SD16—2841no—
148SD16—5681no—
149SD16—5681no—
150D8—41no—
151D8—41no—
152D8—1421no—
153M16—41no—
154M16—5681no—
155C2×C8—5681no—
156C2×C8—711no—
157C2×C8D4×D7111yescharacteristic
158C22⋊C4—5682no—
159C42—711no—
160C16—1421no—
161C16D4×D7111yescharacteristic
162C16#1176586511yescharacteristic
163C23—81no—
164Q8—2841no—
165Q8—5681no—
166Q8—5681no—
167D4—81no—
168D4—81no—
169D4—2841no—
170D4—5681no—
171C2×C4—5681no—
172C2×C4—5681no—
173C2×C4—1421no—
174C2×C4D8×D7111yescharacteristic
175C8—81no—
176C8—711no—
177C8—711no—
178C8#1455610111yescharacteristic
179C8#1455615511yescharacteristic
180C22—162no—
181C22—161no—
182C22—5681no—
183C22#1798643611yescharacteristic
184C4—2841no—
185C4—5681no—
186C4—11361no—
187C4#1798619311yescharacteristic
188C4D16×D7111yescharacteristic
189C2—21no—
190C2—321no—
191C2—11361no—
192C2#2259037411yescharacteristic
subgroupG/H
centerC2#22590374
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1419C2
socleC2×C71#2206
prime radical#22590653C2
hypercenter#1419D71
hypocenterC71#13691
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1419

Complements

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

NH ≅ G/Ntype
1. #22590660C2semidirect
3. #22590598C2semidirect
4. #22590220C2semidirect
5. #22586537C2semidirect
6. #22586532C2semidirect
7. C71×#1419C2semidirect
103. #1419D71semidirect
104. C71#13691semidirect

Maximal subgroups

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

subgroupindex
#225906602
#225906532
#225905982
#225902202
#225865372
#225865322
C71×#14192
#1369171