Finite Group Databasebeta

← #31906961

Subgroups — #31906961

Order 18176.

262 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 71 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#22597277C211yesmaximal, characteristic
2#22590153C212yesmaximal
3#22586292C212yesmaximal
4#22586291C211yesmaximal, characteristic
5C71×#1327C211yesmaximal, characteristic
6#17986556C411yescharacteristic
7#17986128—22no—
8#17986128—22no—
9#17986027C411yescharacteristic
10#17986027C2211yescharacteristic
11#17985957C411yescharacteristic
12#17985957C2211yescharacteristic
13C2×#14556039C411yescharacteristic
14#17985819—22no—
15#17985781C2212yes—
16#17985776—22no—
17#423×C71C2211yescharacteristic
18#349×C71C2212yes—
19C22×C71⋊C8C2×C411yescharacteristic
20C22×C71⋊C8C2×C411yescharacteristic
21#14556128—21no—
22#14556120—21no—
23#14556107—22no—
24C4×Dic142—22no—
25#14556087C2×C411yescharacteristic
26#14556087C2×C411yescharacteristic
27#14556068—22no—
28Dic71⋊C8—22no—
29Dic71⋊C8—22no—
30#14556039—21no—
31#14556039—21no—
32#14556038—21no—
33#14556038D412yes—
34C8×Dic71—21no—
35C8×Dic71D412yes—
36C2×M16×C71C2×C411yescharacteristic
37C2×M16×C71C2311yescharacteristic
38C2×C42×C71C2×C411yescharacteristic
39M32×C71—22no—
40C4⋊C8×C71—22no—
41C2×Dic142—22no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—21no—
44C2×C71⋊C8—41no—
45C2×C71⋊C8C22⋊C411yescharacteristic
46C2×C71⋊C8C22⋊C411yescharacteristic
47C4⋊Dic71—22no—
48C4⋊Dic71—21no—
49C4⋊Dic71—21no—
50Dic71⋊C4—24no—
51C22⋊Dic71—21no—
52C22⋊Dic71M1612yes—
53C4×Dic71—21no—
54C4×Dic71C22⋊C412yes—
55C71⋊C16—42no—
56C22×C4×C71—21no—
57C22×C4×C71C22×C411yescharacteristic
58M16×C71—41no—
59M16×C71—41no—
60C2×C8×C71—21no—
61C2×C8×C71C2×D412yes—
62C42×C71—21no—
63C42×C71M1611yescharacteristic
64C42×C71M1611yescharacteristic
65C16×C71—42no—
66C23×C71C2×M1611yescharacteristic
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—81no—
72C2×C4×C71C2×C22⋊C411yescharacteristic
73C2×C4×C71C2×M1611yescharacteristic
74C2×Dic71—22no—
75C2×Dic71—22no—
76Dic142—82no—
77C8×C71—22no—
78C8×C71—41no—
79C71⋊C8—21no—
80C71⋊C8—41no—
81C71⋊C8#10112yes—
82C22×C71—21no—
83C22×C71—41no—
84C22×C71#40711yescharacteristic
85C4×C71—21no—
86C4×C71—41no—
87C4×C71—41no—
88C4×C71#41011yescharacteristic
89C4×C71C2×#10111yescharacteristic
90Dic71—42no—
91Dic71—42no—
92#11540—711nomaximal, Sylow
93C2×C71—21no—
94C2×C71—41no—
95C2×C71#146711yescharacteristic
96#2961—711no—
97#2166—712no—
98#1328—712no—
99#1327—711no—
100#1327D7111yescharacteristic
101C71#1154011yescharacteristic, complemented, Sylow
102#533—711no—
103#446—1422no—
104#446—1422no—
105#424—711no—
106#424—711no—
107#423C2×D7111yescharacteristic
108#405—711no—
109#405—711no—
110C2×#97—711no—
111#364—1422no—
112#349—712no—
113#349C2×D7112yes—
114#347—1422no—
115C2×M16C4×D7111yescharacteristic
116C2×M16C22×D7111yescharacteristic
117C22×C8—711no—
118C22×C8—711no—
119C4⋊Q8—1421no—
120#125—1421no—
121C22⋊Q8—1422no—
122C4×Q8—1422no—
123#117—711no—
124#117—711no—
125C2×C42C4×D7111yescharacteristic
126M32—22no—
127M32—1422no—
128C4⋊C8—22no—
129C4⋊C8—1422no—
130C4⋊C8—1422no—
131#97—712no—
132#97—1421no—
133#97—1421no—
134#97—1421no—
135C4×C8—712no—
136C4×C8—1421no—
137C2×Q8—1422no—
138C22×C4—21no—
139C22×C4C2×C4×D7111yescharacteristic
140M16—41no—
141M16—41no—
142C2×C8—21no—
143C2×C8—2841no—
144C2×C8—711no—
145C2×C8—711no—
146C2×C8—1421no—
147C2×C8—1422no—
148C2×C8D4×D7112yes—
149C4⋊C4—1424no—
150C4⋊C4—1422no—
151C4⋊C4—1421no—
152C4⋊C4—1421no—
153C22⋊C4—712no—
154C22⋊C4—1421no—
155C42—21no—
156C42—712no—
157C42—1421no—
158C42#1176584811yescharacteristic
159C42#1176584811yescharacteristic
160C16—42no—
161C16—2842no—
162C23M16×D7111yescharacteristic
163Q8—5682no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—81no—
169C2×C4—1422no—
170C2×C4—1422no—
171C2×C4C22⋊C4×D7111yescharacteristic
172C2×C4C2×#1176584811yescharacteristic
173C8—22no—
174C8—41no—
175C8—2841no—
176C8—712no—
177C8—1421no—
178C22—21no—
179C22—41no—
180C22#1798596111yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—2842no—
185C4—2842no—
186C4#1798597111yescharacteristic
187C4#101×D7111yescharacteristic
188C2—21no—
189C2—41no—
190C2#2258668211yescharacteristic
subgroupG/H
centerC4#17985971
commutatorC22×C4×C71C22×C4
FrattiniC2×M16C22×D71
FittingC71×#1327C2
socleC2×C71#1467
prime radicalC23×C71C2×M16
cosocleC2×M16×C71C23
hypercenter#1327D71
hypocenterC71#11540
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
101. C71#11540semidirect

Maximal subgroups

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

subgroupindex
#225972772
#225901532
#225862922
#225862912
C71×#13272
#1154071