Finite Group Databasebeta

← #31909409

Subgroups — #31909409

Order 18176.

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

normal subgroups only

This group has 234 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#22590217C211yesmaximal, characteristic
2#22587253C211yesmaximal, characteristic, complemented
3#22587163C211yesmaximal, characteristic, complemented
4#22586306C211yesmaximal, characteristic, complemented
5#22586306C211yesmaximal, characteristic, complemented
6#22586262C211yesmaximal, characteristic, complemented
7C71×#1368C211yesmaximal, characteristic, complemented
8C2×D1136C411yescharacteristic
9C4×D568C2211yescharacteristic
10C4×D568C2211yescharacteristic
11#17986076—21no—
12#17986064—21no—
13#17985827C411yescharacteristic
14#17985801C411yescharacteristic
15#17985801C411yescharacteristic
16#17985775C2211yescharacteristic
17#17985726—21no—
18#17985726—21no—
19#17985720C2211yescharacteristic
20#346×C71C2211yescharacteristic
21#334×C71C2211yescharacteristic
22#334×C71C2211yescharacteristic
23C2×D568C2×C411yescharacteristic
24C2×D568C2×C411yescharacteristic
25C2×#11765848—41no—
26C2×C8×D71—41no—
27C4×D284—21no—
28C4×D284—21no—
29D1136—22no—
30C2×C71⋊C16—21no—
31#14556062C2×C411yescharacteristic
32#14556062C2×C411yescharacteristic
33#14556050—21no—
34#14556050—21no—
35(C2×D71)⋊C8—41no—
36(C2×D71)⋊C8—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×D284—21no—
45C2×D284—21no—
46C2×C4×D71—41no—
47C2×C4×D71—41no—
48D568—21no—
49D568—21no—
50D568—21no—
51D568M1612yescomplemented
52#11765848—82no—
53C2×C71⋊C8—21no—
54C2×C71⋊C8—21no—
55C8×D71—82no—
56C4⋊Dic71—21no—
57C4⋊Dic71—21no—
58(C2×D71)⋊C4—41no—
59(C2×D71)⋊C4—41no—
60C71⋊C16—22no—
61C2×C8×C71—41no—
62C2×C8×C71—41no—
63C2×C8×C71C22×C411yescharacteristic
64C2×C8×C71#4111yescharacteristic
65C42×C71C2×D411yescharacteristic
66C16×C71—21no—
67C16×C71—21no—
68C22×D71—41no—
69C22×D71—41no—
70C2×C4×C71C4×D411yescharacteristic
71C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
72C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
73C2×Dic71—41no—
74C2×Dic71—41no—
75D284—22no—
76D284—41no—
77D284—41no—
78D284—41no—
79C4×D71—81no—
80C4×D71—81no—
81C4×D71—81no—
82C4×D71—81no—
83C8×C71—21no—
84C8×C71—81no—
85C8×C71—81no—
86C8×C71C2×M1611yescharacteristic
87C8×C71#13111yescharacteristic
88C71⋊C8—41no—
89C71⋊C8—41no—
90C22×C71#44211yescharacteristic
91C2×D71—42no—
92C2×D71—81no—
93C2×D71—81no—
94C2×D71—81no—
95C2×D71—81no—
96C2×D71—81no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71#43511yescharacteristic
100C4×C71#44311yescharacteristic
101Dic71—81no—
102Dic71—81no—
103#12078—711nomaximal, Sylow
104C2×C71#158611yescharacteristic
105C2×C71AGL(1,ℤ/16)11yescharacteristic
106C2×C71AGL(1,ℤ/16)11yescharacteristic
107D71—82no—
108D71—161no—
109#2177—711no—
110#1594—711no—
111#1580—711no—
112#1368D7111yescharacteristic, complemented
113#1333—711no—
114#1333—711no—
115#1313—711no—
116C71#1207811yescharacteristic, complemented, Sylow
117C2×D16—711no—
118C4×D8—711no—
119C4×D8—711no—
120#436—1421no—
121C8×D4—1421no—
122#367—711no—
123#357—711no—
124#357—711no—
125#346—711no—
126#346C2×D7111yescharacteristic
127#334C2×D7111yescharacteristic
128#334C2×D7111yescharacteristic
129D4⋊C8—1421no—
130D4⋊C8—1421no—
131#322—711no—
132C2×D8—711no—
133C2×D8—711no—
134C2×M16—2841no—
135C22×C8—2841no—
136C4×D4—1421no—
137C4×D4—1421no—
138D16—1422no—
139C2×C16—1421no—
140C2×C16C4×D7111yescharacteristic
141C2×C16C4×D7111yescharacteristic
142#107—711no—
143#107—711no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146#102—1421no—
147#102—1421no—
148C22⋊C8—2841no—
149C22⋊C8—2841no—
150C4×C8—711no—
151C4×C8—711no—
152C4×C8C22×D7111yescharacteristic
153C2×D4—1421no—
154C2×D4—1421no—
155C22×C4—2841no—
156C22×C4—2841no—
157D8—712no—
158D8—1421no—
159D8—1421no—
160D8—1421no—
161M16—5682no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—5682no—
165C2×C8—1421no—
166C2×C8—1421no—
167C2×C8C2×C4×D7111yescharacteristic
168C2×C8#1176586611yescharacteristic
169C4⋊C4—1421no—
170C4⋊C4—1421no—
171C22⋊C4—2841no—
172C22⋊C4—2841no—
173C42D4×D7111yescharacteristic
174C16—21no—
175C16—21no—
176C16—1422no—
177C23—2841no—
178C23—2841no—
179D4—2841no—
180D4—2841no—
181D4—2841no—
182D4—1422no—
183C2×C4—2841no—
184C2×C4—2841no—
185C2×C4—5681no—
186C2×C4—5681no—
187C2×C4—5681no—
188C2×C4—5681no—
189C2×C4#1455609011yescharacteristic
190C2×C4#1455615511yescharacteristic
191C2×C4#1455615511yescharacteristic
192C8—21no—
193C8—81no—
194C8—81no—
195C8—2841no—
196C8—2841no—
197C8M16×D7111yescharacteristic
198C8#1455613411yescharacteristic
199C22—2842no—
200C22—5681no—
201C22—5681no—
202C22—5681no—
203C22—5681no—
204C22—5681no—
205C22#1798611011yescharacteristic
206C4—21no—
207C4—21no—
208C4—5681no—
209C4—5681no—
210C4#1798606911yescharacteristic
211C4#1798611511yescharacteristic
212C2—5682no—
213C2—11361no—
214C2#2258720311yescharacteristic
215C2#2259023711yescharacteristic
216C2#2259023711yescharacteristic
subgroupG/H
centerC22#17986110
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1368C2
socleC22×C71#442
prime radicalC2×D1136C4
cosocleC4×C8×C71C23
hypercenter#1368D71
hypocenterC71#12078
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. #22587253C2semidirect
3. #22587163C2semidirect
4. #22586306C2semidirect
5. #22586306C2semidirect
6. #22586262C2semidirect
7. C71×#1368C2semidirect
51. D568M16semidirect
112. #1368D71semidirect
116. C71#12078semidirect

Maximal subgroups

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

subgroupindex
#225902172
#225872532
#225871632
#225863062
#225862622
C71×#13682
#1207871