Finite Group Databasebeta

← #13274613

Subgroups — #13274613

Order 1792.

200 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 51 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#11729254C211yesmaximal, characteristic, complemented
2#11729247C211yesmaximal, characteristic, complemented
3#11727127C211yesmaximal, characteristic
4#11727111C211yesmaximal, characteristic, complemented
5C7×#1420C211yesmaximal, characteristic, complemented
6#11725405C211yesmaximal, characteristic, complemented
7#11725389C211yesmaximal, characteristic, complemented
8C2×C7×Q32C2211yescharacteristic
9#89963—21no—
10C2×#6402—21no—
11#89676C2211yescharacteristic
12C2×D112C2211yescharacteristic
13#89667—21no—
14C7×M64C2211yescharacteristic
15C7×#368C2211yescharacteristic
16#89317—21no—
17#89316—21no—
18C16×Dic7C2211yescharacteristic
19#89294C2211yescharacteristic
20#89292—21no—
21#89244—21no—
22D224—21no—
23#13691—71nomaximal, Sylow
24C2×C7×Q16—21no—
25#6510—41no—
26C2×#1143—41no—
27#6470—41no—
28#6467—21no—
29C2×D56—21no—
30C7×Q32—21no—
31C7×Q32—21no—
32C2×C7×C16C2311yescharacteristic
33C7×#108—21no—
34#6402—42no—
35#6399—41no—
36C8×Dic7D411yescharacteristic
37C2×C7⋊C16D411yescharacteristic
38Dic56D411yescharacteristic
39#6374—21no—
40D112—21no—
41D112—21no—
42D112D411yescharacteristic
43C7×C32—21no—
44C7⋊C32—21no—
45#2264—71no—
46#2263—71no—
47#2249—71no—
48#2178—71no—
49#1422—71no—
50#1420D711yescharacteristic, complemented
51#1419—71no—
52C2×C7×Q8—41no—
53#1157—41no—
54C2×D28—41no—
55C7×Q16—41no—
56C7×Q16—41no—
57C7×M16—41no—
58C2×C7×C8C2×D411yescharacteristic
59#1143—81no—
60#1143—81no—
61(C2×D7)⋊C4—82no—
62C4×Dic7D811yescharacteristic
63C2×C7⋊C8D811yescharacteristic
64Dic28—21no—
65D56—21no—
66D56—41no—
67D56—41no—
68#1132—41no—
69#1131—81no—
70C8×D7—81no—
71C7×C16C2×D411yescharacteristic
72C7×C16#4111yescharacteristic
73C7⋊C16—21no—
74#508—71no—
75C2×Q32C2×D711yescharacteristic
76C2×SD32—141no—
77C2×D16—71no—
78#495—141no—
79#443—141no—
80Q64—141no—
81SD64—141no—
82SD64—141no—
83D32—141no—
84M64—71no—
85M64C2×D711yescharacteristic
86#368C2×D711yescharacteristic
87#359—141no—
88C4×C16—71no—
89C22×D7—81no—
90C7×Q8—81no—
91C7×Q8—81no—
92C2×C4×C7—81no—
93C2×C4×C7C2×D811yescharacteristic
94C22⋊D7—81no—
95C2×Dic7—21no—
96D28—41no—
97D28—81no—
98D28—81no—
99C4×D7—81no—
100Dic14—41no—
101C7×C8—81no—
102C7×C8#12111yescharacteristic
103C7×C8AGL(1,ℤ/8)11yescharacteristic
104C7⋊C8D1611yescharacteristic
105C7⋊C8D1611yescharacteristic
106#135—141no—
107C2×Q16—21no—
108C2×SD16—281no—
109C2×D8—141no—
110#131—281no—
111#124—281no—
112Q32—21no—
113Q32—21no—
114Q32—71no—
115SD32—141no—
116SD32—282no—
117D16—71no—
118D16—141no—
119D16—141no—
120C2×C16—71no—
121C2×C16C22×D711yescharacteristic
122#108—21no—
123#104—281no—
124C4×C8—71no—
125C32—21no—
126C32—141no—
127C22×C7#50911yescharacteristic
128C2×D7—81no—
129C2×D7—162no—
130C2×D7—161no—
131C4×C7—161no—
132C4×C7#45911yescharacteristic
133C4×C7C2×D1611yescharacteristic
134Dic7—41no—
135Dic7—81no—
136#41—281no—
137C2×Q8—41no—
138C2×D4—281no—
139Q16—41no—
140Q16—41no—
141Q16—141no—
142SD16—281no—
143SD16—561no—
144SD16—561no—
145D8—141no—
146D8—281no—
147D8—281no—
148M16—41no—
149M16—561no—
150C2×C8—71no—
151C2×C8—561no—
152C2×C8D4×D711yescharacteristic
153C22⋊C4—562no—
154C42—71no—
155C16—141no—
156C16#115711yescharacteristic
157C16D4×D711yescharacteristic
158C2×C7—21no—
159C2×C7#220611yescharacteristic
160D7—161no—
161D7—321no—
162C23—561no—
163Q8—81no—
164Q8—81no—
165Q8—281no—
166D4—281no—
167D4—561no—
168D4—561no—
169D4—561no—
170C2×C4—81no—
171C2×C4—141no—
172C2×C4—561no—
173C2×C4D7×D811yescharacteristic
174C8—71no—
175C8—71no—
176C8—81no—
177C8#644711yescharacteristic
178C8#647711yescharacteristic
179C7#1369111yescharacteristic, complemented, Sylow
180C22—561no—
181C22—1122no—
182C22—1121no—
183C22#8968611yescharacteristic
184C4—161no—
185C4—281no—
186C4—561no—
187C4#8955811yescharacteristic
188C4D7×D1611yescharacteristic
189C2—21no—
190C2—1121no—
191C2—2241no—
192C2#1172695611yescharacteristic
subgroupG/H
centerC2#11726956
commutator, cosocleC2×C7×C16C23
FrattiniC2×C16C22×D7
FittingC7×#1420C2
socleC2×C7#2206
prime radical#11727127C2
hypercenter#1420D7
hypocenterC7#13691
LCSG ▷ C2×C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1420

Complements

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

NH ≅ G/Ntype
1. #11729254C2semidirect
2. #11729247C2semidirect
4. #11727111C2semidirect
5. C7×#1420C2semidirect
6. #11725405C2semidirect
7. #11725389C2semidirect
50. #1420D7semidirect
179. C7#13691semidirect

Maximal subgroups

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

subgroupindex
#117292542
#117292472
#117271272
#117271112
C7×#14202
#117254052
#117253892
#136917