Finite Group Databasebeta

← #31884882

Subgroups — #31884882

Order 18176.

202 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 43 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#22588554C211yesmaximal, characteristic
2#22586274C211yesmaximal, characteristic
3C71×#1272C211yesmaximal, characteristic
4#17986477C411yescharacteristic
5#17986039—21no—
6C2×#14556047C411yescharacteristic
7#17985759—21no—
8#17985725—21no—
9#391×C71C2211yescharacteristic
10#336×C71—21no—
11C2×#11765849—21no—
12Q8×Dic71—41no—
13Q8×Dic71—41no—
14#14556087—21no—
15#14556087—41no—
16#14556061—41no—
17#14556059—41no—
18#14556047—21no—
19#14556047C812yescomplemented
20C22⋊(C71⋊C8)—41no—
21C22×Q8×C71C2×C411yescharacteristic
22C22×C8×C71—41no—
23C2×C4⋊C4×C71D411yescharacteristic
24C2×C42×C71D411yescharacteristic
25#95×C71—41no—
26#11765849—22no—
27#11765849—41no—
28C2×C71⋊C8—41no—
29C2×C71⋊C8—81no—
30C4⋊Dic71—21no—
31C4⋊Dic71—21no—
32C4⋊Dic71—41no—
33C4⋊Dic71—41no—
34C22⋊Dic71—41no—
35C4×Dic71—41no—
36C4×Dic71—42no—
37C2×Q8×C71—21no—
38C2×Q8×C71—41no—
39C2×Q8×C71—41no—
40C2×Q8×C71C2×C811yescharacteristic
41C2×Q8×C71M1611yescharacteristic
42C22×C4×C71—21no—
43C22×C4×C71—21no—
44C22×C4×C71—21no—
45C22×C4×C71C22⋊C411yescharacteristic
46C2×C8×C71—42no—
47C2×C8×C71—81no—
48C2×C8×C71—81no—
49C4⋊C4×C71—41no—
50C42×C71—41no—
51C23×C71#9911yescharacteristic
52Q8×C71—41no—
53Q8×C71—41no—
54Q8×C71—42no—
55C2×C4×C71—21no—
56C2×C4×C71—21no—
57C2×C4×C71—21no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—21no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71—41no—
66C2×C4×C71—81no—
67C2×C4×C71C22⋊C811yescharacteristic
68C2×C4×C71#10111yescharacteristic
69C2×Dic71—41no—
70C2×Dic71—41no—
71C8×C71—82no—
72C71⋊C8—42no—
73C71⋊C8—161no—
74C22×C71—21no—
75C22×C71—21no—
76C22×C71#32411yescharacteristic
77C22×C71AΣL(1,𝔽16)11yescharacteristic
78C22×C71#35211yescharacteristic
79C4×C71—21no—
80C4×C71—21no—
81C4×C71—41no—
82C4×C71—41no—
83C4×C71—41no—
84C4×C71—41no—
85C4×C71—81no—
86Dic71—81no—
87Dic71—81no—
88#7376—711nomaximal, Sylow
89C2×C71—21no—
90C2×C71—21no—
91C2×C71#132611yescharacteristic
92C2×C71#140411yescharacteristic
93C2×C71#140511yescharacteristic
94#1857—711no—
95#1319—711no—
96#1272D7111yescharacteristic
97C71#737611yescharacteristic, complemented, Sylow
98#519—711no—
99#428—1421no—
100C2×#101—711no—
101#391C2×D7111yescharacteristic
102#339—1421no—
103#336—21no—
104#324—1421no—
105C22×Q8C4×D7111yescharacteristic
106C2×M16—1421no—
107C22×C8—41no—
108C4×Q8—2841no—
109C4×Q8—2841no—
110#117—2841no—
111#117—1421no—
112C2×C4⋊C4D28411yescharacteristic
113C2×C42C22⋊D7111yescharacteristic
114#107—2841no—
115#106—2841no—
116#101—712no—
117#101—1421no—
118C22⋊C8—2841no—
119#95—41no—
120C2×Q8—21no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8C8×D7111yescharacteristic
124C2×Q8#1176584811yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4(C2×D71)⋊C411yescharacteristic
129M16—2841no—
130M16—1422no—
131C2×C8—42no—
132C2×C8—81no—
133C2×C8—81no—
134C2×C8—2841no—
135C2×C8—5681no—
136C4⋊C4—41no—
137C4⋊C4—2841no—
138C4⋊C4—2841no—
139C4⋊C4—1421no—
140C4⋊C4—1421no—
141C22⋊C4—2841no—
142C42—41no—
143C42—2841no—
144C42—2842no—
145C23#1455604211yescharacteristic
146Q8—41no—
147Q8—41no—
148Q8—42no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—81no—
161C2×C4—2841no—
162C2×C4—2841no—
163C2×C4(C2×D71)⋊C811yescharacteristic
164C2×C4#1455604611yescharacteristic
165C8—82no—
166C8—2842no—
167C8—11361no—
168C22—21no—
169C22—21no—
170C22#1798572411yescharacteristic
171C22#1798578711yescharacteristic
172C22#1798578811yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—81no—
180C4—5681no—
181C4—5681no—
182C2—21no—
183C2—21no—
184C2#2258628911yescharacteristic
185C2#2258649211yescharacteristic
186C2#2258649511yescharacteristic
subgroupG/H
centerC22#17985787
commutatorC2×Q8×C71C2×C8
Frattini#391C2×D71
FittingC71×#1272C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC23×C71#99
cosocle#391×C71C22
hypercenter#1272D71
hypocenterC71#7376
LCSG ▷ C2×Q8×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1272

Complements

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

NH ≅ G/Ntype
19. #14556047C8semidirect
97. C71#7376semidirect

Maximal subgroups

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

subgroupindex
#225885542
#225862742
C71×#12722
#737671