Finite Group Databasebeta

← #13267155

Subgroups — #13267155

Order 1792.

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

normal subgroups only

This group has 286 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×#3259C211yesmaximal, characteristic
2#11728842C212yesmaximal, complemented
3#11728293C211yesmaximal, characteristic, complemented
4#11725585C212yesmaximal, complemented
5#11725525C211yesmaximal, characteristic, complemented
6C7×#540—21no—
7C7×#478—22no—
8C7×#469C2212yes—
9C7×#466—21no—
10C7×#459—22no—
11C7×#424C2211yescharacteristic
12#89897C2211yescharacteristic
13#89358—22no—
14C7⋊#338C2211yescharacteristic
15#89348C2212yes—
16#89347—21no—
17#89328—21no—
18#89282—22no—
19#12304—71nomaximal, Sylow
20C2×C7×D8—21no—
21C2×C7×D8D412yes—
22C2×C7×M16C2311yescharacteristic
23C22×C7×C8D411yescharacteristic
24C7×#127—21no—
25C7×#125—21no—
26C7×#123—42no—
27C7×#121—22no—
28C7×#121—42no—
29C4×C7×D4—22no—
30C7×#117D411yescharacteristic
31C2×#1136—21no—
32C2×#1136—21no—
33C7×#107—21no—
34C7×#106D412yes—
35C7×C4⋊C8—22no—
36C7×#102—22no—
37C7×#102—22no—
38#6393—22no—
39#6386—22no—
40#3259D711yescharacteristic
41#2238—72no—
42#2103—71no—
43#1341—72no—
44#1328—71no—
45C2×C7×D4—22no—
46C2×C7×D4—42no—
47C2×C7×D4—42no—
48C22×C4×C7—42no—
49C22×C4×C7C2×D411yescharacteristic
50C7×D8—42no—
51C7×D8—81no—
52C7×M16—41no—
53C2×C7×C8—21no—
54C2×C7×C8—41no—
55C2×C7×C8C2×D412yes—
56C7×C4⋊C4—21no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4—42no—
59C7×C22⋊C4—42no—
60C7×C22⋊C4—42no—
61C7×C22⋊C4SD1612yes—
62C42×C7—21no—
63#1136—41no—
64#1136—41no—
65#1136—41no—
66#1136—41no—
67C2×C7⋊C8—41no—
68C2×C7⋊C8—41no—
69C7⋊C16—42no—
70#540—21no—
71#478—22no—
72#469C2×D712yes—
73#466—21no—
74#459—22no—
75#430—71no—
76#424C2×D711yescharacteristic
77#365—142no—
78#361—142no—
79#349—72no—
80#344—141no—
81#341—141no—
82#338—71no—
83C23×C7—42no—
84C23×C7C2×SD1611yescharacteristic
85C7×D4—42no—
86C7×D4—42no—
87C7×D4—162no—
88C7×D4—162no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—42no—
93C2×C4×C7—162no—
94C2×C4×C7#12011yescharacteristic
95C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
96C7×C8—22no—
97C7×C8—41no—
98C7⋊C8—81no—
99C7⋊C8—81no—
100C2×D8—21no—
101C2×D8C22⋊D712yes—
102C2×M16—141no—
103C2×M16—141no—
104C2×M16C22×D711yescharacteristic
105C22×C8C22⋊D711yescharacteristic
106#127—21no—
107#125—21no—
108#123—42no—
109#121—22no—
110#121—42no—
111C4×D4—22no—
112#117C22⋊D711yescharacteristic
113M32—142no—
114#108—142no—
115#107—21no—
116#106C22⋊D712yes—
117C4⋊C8—22no—
118#102—22no—
119#102—22no—
120C22×C7—21no—
121C22×C7—41no—
122C22×C7—82no—
123C22×C7—82no—
124C22×C7—82no—
125C22×C7#45011yescharacteristic
126C4×C7—21no—
127C4×C7—42no—
128C4×C7—82no—
129C4×C7#45411yescharacteristic
130C4×C7#45811yescharacteristic
131C2×D4—22no—
132C2×D4—42no—
133C2×D4—42no—
134C22×C4—42no—
135C22×C4C2×C22⋊D711yescharacteristic
136D8—42no—
137D8—81no—
138M16—41no—
139M16—281no—
140M16—281no—
141M16—281no—
142M16—281no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—281no—
146C2×C8—281no—
147C2×C8C2×C22⋊D712yes—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4#114212yes—
154C42—21no—
155C16—282no—
156C2×C7—21no—
157C2×C7—41no—
158C2×C7—162no—
159C2×C7#161611yescharacteristic
160C23—42no—
161C23C2×#114211yescharacteristic
162D4—42no—
163D4—42no—
164D4—162no—
165D4—162no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—162no—
171C2×C4#649511yescharacteristic
172C2×C4#651611yescharacteristic
173C8—22no—
174C8—41no—
175C8—561no—
176C8—561no—
177C7#1230411yescharacteristic, complemented, Sylow
178C22—21no—
179C22—41no—
180C22—82no—
181C22—82no—
182C22—82no—
183C22#8999011yescharacteristic
184C4—21no—
185C4—42no—
186C4—82no—
187C4#9001511yescharacteristic
188C4#9001711yescharacteristic
189C2—21no—
190C2—41no—
191C2—162no—
192C2#1172962911yescharacteristic
subgroupG/H
centerC2#11729629
commutator, cosocleC2×C7×M16C23
FrattiniC2×M16C22×D7
Fitting, prime radicalC7×#3259C2
socleC2×C7#1616
hypercenter#3259D7
hypocenterC7#12304
LCSG ▷ C2×C7×M16 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3259

Complements

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

NH ≅ G/Ntype
2. #11728842C2semidirect
3. #11728293C2semidirect
4. #11725585C2semidirect
5. #11725525C2semidirect
177. C7#12304semidirect

Maximal subgroups

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

subgroupindex
C7×#32592
#117288422
#117282932
#117255852
#117255252
#123047