Finite Group Databasebeta

← #31884166

Subgroups — #31884166

Order 18176.

212 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 49 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#22589663C211yesmaximal, characteristic
2#22586280C211yesmaximal, characteristic
3C71×#1292C211yesmaximal, characteristic
4C22×Dic284C411yescharacteristic
5C2×#14556053—21no—
6#17985905—21no—
7C22⋊(C71⋊C16)—21no—
8#17985762C411yescharacteristic
9#423×C71—21no—
10C2×C4⋊C8×C71C2211yescharacteristic
11C22×Dic142—21no—
12C2×Dic284—41no—
13C2×Dic284—42no—
14C2×C4⋊Dic71—21no—
15C2×C71⋊C16—41no—
16#14556053—42no—
17C71⋊#95—41no—
18C71⋊#95—41no—
19C2×M16×C71—41no—
20C22×C8×C71C2×C411yescharacteristic
21C22×C8×C71D411yescharacteristic
22C2×C42×C71D411yescharacteristic
23C4⋊C8×C71—22no—
24C4⋊C8×C71—41no—
25C4⋊C8×C71—41no—
26C2×Dic142—22no—
27C2×Dic142—41no—
28C2×Dic142—41no—
29C2×Dic142—41no—
30C2×Dic142—41no—
31C2×Dic142—41no—
32C22×Dic71—41no—
33C22×Dic71—41no—
34Dic284—42no—
35Dic284—42no—
36C4⋊Dic71—41no—
37C4⋊Dic71—41no—
38C71⋊C16—81no—
39C22×C4×C71C22⋊C411yescharacteristic
40C22×C4×C71D811yescharacteristic
41C22×C4×C71SD1611yescharacteristic
42M16×C71—81no—
43M16×C71—81no—
44C2×C8×C71—21no—
45C2×C8×C71—21no—
46C2×C8×C71—41no—
47C2×C8×C71—41no—
48C2×C8×C71—42no—
49C42×C71—21no—
50C42×C71—21no—
51C23×C71#10211yescharacteristic
52C2×C4×C71—21no—
53C2×C4×C71—21no—
54C2×C4×C71—21no—
55C2×C4×C71—21no—
56C2×C4×C71—21no—
57C2×C4×C71—41no—
58C2×C4×C71#9911yescharacteristic
59C2×C4×C71#10411yescharacteristic
60C2×C4×C71SD3211yescharacteristic
61C2×C4×C71Q3211yescharacteristic
62C2×Dic71—42no—
63C2×Dic71—81no—
64C2×Dic71—81no—
65C2×Dic71—81no—
66C2×Dic71—81no—
67C2×Dic71—81no—
68Dic142—42no—
69Dic142—42no—
70Dic142—81no—
71Dic142—81no—
72C8×C71—41no—
73C8×C71—41no—
74C8×C71—82no—
75C22×C71—21no—
76C22×C71—21no—
77C22×C71#32711yescharacteristic
78C22×C71#35811yescharacteristic
79C22×C71#36011yescharacteristic
80C4×C71—21no—
81C4×C71—21no—
82C4×C71—21no—
83C4×C71—41no—
84C4×C71AΣL(1,𝔽16)11yescharacteristic
85C4×C71#35511yescharacteristic
86Dic71—82no—
87Dic71—161no—
88#7090—711nomaximal, Sylow
89C2×C71—21no—
90C2×C71—21no—
91C2×C71#134211yescharacteristic
92C2×C71#134511yescharacteristic
93C2×C71#134811yescharacteristic
94#2056—711no—
95#1322—711no—
96#1292D7111yescharacteristic
97C71#709011yescharacteristic, complemented, Sylow
98C22×Q16—711no—
99#423—21no—
100C2×C4⋊C8C2×D7111yescharacteristic
101C2×#103—1421no—
102#391—1421no—
103C22⋊C16—1421no—
104#340—711no—
105C22×Q8—1421no—
106C2×Q16—2841no—
107C2×Q16—2842no—
108C2×M16—41no—
109C22×C8C4×D7111yescharacteristic
110C22×C8D28411yescharacteristic
111C2×C4⋊C4—1421no—
112C2×C42C22⋊D7111yescharacteristic
113C2×C16—2841no—
114C4⋊C8—22no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117#103—2842no—
118#95—2841no—
119#95—2841no—
120C2×Q8—2841no—
121C2×Q8—2841no—
122C2×Q8—2841no—
123C2×Q8—2841no—
124C2×Q8—2841no—
125C2×Q8—1422no—
126C22×C4—2841no—
127C22×C4—2841no—
128C22×C4(C2×D71)⋊C411yescharacteristic
129C22×C4#1176585011yescharacteristic
130C22×C4#1176585211yescharacteristic
131Q16—2842no—
132Q16—2842no—
133M16—81no—
134M16—81no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—42no—
140C4⋊C4—2841no—
141C4⋊C4—2841no—
142C42—21no—
143C42—21no—
144C16—5681no—
145C23#1455604811yescharacteristic
146Q8—2842no—
147Q8—2842no—
148Q8—5681no—
149Q8—5681no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—2842no—
157C2×C4—5681no—
158C2×C4—5681no—
159C2×C4—5681no—
160C2×C4—5681no—
161C2×C4—5681no—
162C2×C4#1455604211yescharacteristic
163C2×C4#1455605611yescharacteristic
164C2×C4#1455607211yescharacteristic
165C2×C4#1455607411yescharacteristic
166C8—41no—
167C8—41no—
168C8—82no—
169C22—21no—
170C22—21no—
171C22#1798573211yescharacteristic
172C22#1798580211yescharacteristic
173C22#1798580811yescharacteristic
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—5682no—
179C4—11361no—
180C4#1798578711yescharacteristic
181C4#1798579511yescharacteristic
182C2—21no—
183C2—21no—
184C2#2258633311yescharacteristic
185C2#2258634211yescharacteristic
186C2#2258635111yescharacteristic
subgroupG/H
centerC22#17985732
commutatorC22×C8×C71C2×C4
FrattiniC2×C4⋊C8C2×D71
FittingC71×#1292C2
socleC22×C71#327
prime radicalC23×C71#102
cosocleC2×C4⋊C8×C71C22
hypercenter#1292D71
hypocenterC71#7090
LCSG ▷ C22×C8×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1292

Complements

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

NH ≅ G/Ntype
97. C71#7090semidirect

Maximal subgroups

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

subgroupindex
#225896632
#225862802
C71×#12922
#709071