Finite Group Databasebeta

← #13267160

Subgroups — #13267160

Order 1792.

262 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C7×#3261C211yesmaximal, characteristic
2#11728844C212yesmaximal
3#11728293C211yesmaximal, characteristic
4#11725587C212yesmaximal
5#11725525C211yesmaximal, characteristic
6C7×#541—21no—
7C7×#479—22no—
8C7×#470C2212yes—
9C7×#467—21no—
10C7×#462—22no—
11C7×#424C2211yescharacteristic
12#89897C2211yescharacteristic
13#89360—22no—
14C7⋊#338C2211yescharacteristic
15#89348C2212yes—
16#89347—21no—
17#89328—21no—
18#89282—22no—
19#12305—71nomaximal, Sylow
20C2×C7×Q16—21no—
21C2×C7×Q16D412yes—
22C2×C7×M16C2311yescharacteristic
23C22×C7×C8D411yescharacteristic
24C7×C4⋊Q8—21no—
25C7×#126—42no—
26C7×#125—21no—
27C7×#125—42no—
28C7×C22⋊Q8—22no—
29C4×C7×Q8—22no—
30C7×#117D411yescharacteristic
31C2×#1136—21no—
32C2×#1136—21no—
33C7×#107—21no—
34C7×#106D412yes—
35C7×C4⋊C8—22no—
36C7×#103—22no—
37C7×#103—22no—
38#6393—22no—
39#6386—22no—
40#3261D711yescharacteristic
41#2239—72no—
42#2103—71no—
43#1342—72no—
44#1328—71no—
45C2×C7×Q8—22no—
46C22×C4×C7C2×D411yescharacteristic
47C7×Q16—42no—
48C7×Q16—81no—
49C7×M16—41no—
50C2×C7×C8—21no—
51C2×C7×C8—41no—
52C2×C7×C8C2×D412yes—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—22no—
55C7×C4⋊C4—42no—
56C7×C4⋊C4—42no—
57C7×C4⋊C4—42no—
58C7×C4⋊C4—42no—
59C7×C22⋊C4—42no—
60C7×C22⋊C4SD1612yes—
61C42×C7—21no—
62C42×C7—42no—
63#1136—41no—
64#1136—41no—
65#1136—41no—
66#1136—41no—
67C2×C7⋊C8—41no—
68C2×C7⋊C8—41no—
69C7⋊C16—42no—
70#541—21no—
71#479—22no—
72#470C2×D712yes—
73#467—21no—
74#462—22no—
75#430—71no—
76#424C2×D711yescharacteristic
77#365—142no—
78#362—142no—
79#349—72no—
80#344—141no—
81#341—141no—
82#338—71no—
83C23×C7C2×SD1611yescharacteristic
84C7×Q8—42no—
85C7×Q8—42no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—22no—
89C2×C4×C7—42no—
90C2×C4×C7—42no—
91C2×C4×C7#12011yescharacteristic
92C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
93C7×C8—22no—
94C7×C8—41no—
95C7⋊C8—81no—
96C7⋊C8—81no—
97C2×Q16—21no—
98C2×Q16C22⋊D712yes—
99C2×M16—141no—
100C2×M16—141no—
101C2×M16C22×D711yescharacteristic
102C22×C8C22⋊D711yescharacteristic
103C4⋊Q8—21no—
104#126—42no—
105#125—21no—
106#125—42no—
107C22⋊Q8—22no—
108C4×Q8—22no—
109#117C22⋊D711yescharacteristic
110M32—142no—
111#108—142no—
112#107—21no—
113#106C22⋊D712yes—
114C4⋊C8—22no—
115#103—22no—
116#103—22no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7#45011yescharacteristic
120C4×C7—21no—
121C4×C7—42no—
122C4×C7—82no—
123C4×C7—82no—
124C4×C7#45411yescharacteristic
125C4×C7#45811yescharacteristic
126C2×Q8—22no—
127C22×C4C2×C22⋊D711yescharacteristic
128Q16—42no—
129Q16—81no—
130M16—41no—
131M16—281no—
132M16—281no—
133M16—281no—
134M16—281no—
135C2×C8—21no—
136C2×C8—41no—
137C2×C8—281no—
138C2×C8—281no—
139C2×C8C2×C22⋊D712yes—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4#114212yes—
148C42—21no—
149C42—42no—
150C16—282no—
151C2×C7—21no—
152C2×C7—41no—
153C2×C7#161611yescharacteristic
154C23C2×#114211yescharacteristic
155Q8—42no—
156Q8—42no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4#649511yescharacteristic
163C2×C4#651611yescharacteristic
164C8—22no—
165C8—41no—
166C8—561no—
167C8—561no—
168C7#1230511yescharacteristic, complemented, Sylow
169C22—21no—
170C22—41no—
171C22#8999011yescharacteristic
172C4—21no—
173C4—42no—
174C4—82no—
175C4—82no—
176C4#9001511yescharacteristic
177C4#9001711yescharacteristic
178C2—21no—
179C2—41no—
180C2#1172962911yescharacteristic
subgroupG/H
centerC2#11729629
commutator, cosocleC2×C7×M16C23
FrattiniC2×M16C22×D7
FittingC7×#3261C2
socleC2×C7#1616
prime radicalC23×C7C2×SD16
hypercenter#3261D7
hypocenterC7#12305
LCSG ▷ C2×C7×M16 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3261

Complements

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

NH ≅ G/Ntype
168. C7#12305semidirect

Maximal subgroups

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

subgroupindex
C7×#32612
#117288442
#117282932
#117255872
#117255252
#123057