Finite Group Databasebeta

← #13265312

Subgroups — #13265312

Order 1792.

292 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 69 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#11739354C211yesmaximal, characteristic
2#11727057C211yesmaximal, characteristic, complemented
3#11727053C211yesmaximal, characteristic, complemented
4C7×#1366C211yesmaximal, characteristic, complemented
5#11725606C211yesmaximal, characteristic, complemented
6#11725377C211yesmaximal, characteristic, complemented
7#11725377C211yesmaximal, characteristic, complemented
8#90389C412yes—
9C7×#499C2211yescharacteristic
10#89839C412yes—
11D7×M32—21no—
12#89655—21no—
13#89644—21no—
14#89641—21no—
15#89637—21no—
16#89467C2211yescharacteristic
17#89461C2211yescharacteristic
18C7×#364C2211yescharacteristic
19C7×#364C2211yescharacteristic
20#89309—21no—
21#89307—21no—
22#89306—21no—
23#89258C2211yescharacteristic
24#89258C2211yescharacteristic
25#11989—71nomaximal, Sylow
26C7×#125C2×C412yes—
27C2×D56—21no—
28C2×D56—21no—
29C2×#1131D411yescharacteristic
30C2×C8×D7D411yescharacteristic
31#6458—22no—
32#6458—42no—
33#6457—42no—
34#6456—22no—
35#6437C2×C412yes—
36#6435C2×C411yescharacteristic
37C7×M32—21no—
38C7×M32—21no—
39C7×#107—21no—
40C7×#106—21no—
41C4×C7×C8C2311yescharacteristic
42#6389D411yescharacteristic
43C8×Dic7D411yescharacteristic
44#6386—21no—
45#6386—21no—
46#6383—22no—
47#6383—22no—
48#6377C2×C411yescharacteristic
49#6372—41no—
50D7×C16—41no—
51#3176—71no—
52#2166—71no—
53#2153—71no—
54#1366—71no—
55#1366D711yescharacteristic, complemented
56#1357—71no—
57#1357—71no—
58C2×D28—22no—
59C2×D28—42no—
60C2×D28—42no—
61C2×C4×D7—42no—
62C2×C4×D7C22⋊C411yescharacteristic
63C2×C7×C8C2×D411yescharacteristic
64C2×C7×C8C2×D411yescharacteristic
65C7×C4⋊C4—22no—
66C7×C4⋊C4—42no—
67C42×C7C22×C411yescharacteristic
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—42no—
70(C2×D7)⋊C4—42no—
71Dic7⋊C4—21no—
72C4×Dic7C22⋊C411yescharacteristic
73C2×C7⋊C8C22⋊C411yescharacteristic
74C2×C7⋊C8C22⋊C411yescharacteristic
75D56—41no—
76D56—41no—
77D56—41no—
78D56—41no—
79#1131—21no—
80#1131—21no—
81C8×D7—21no—
82C8×D7—21no—
83C7×C16—41no—
84C7×C16—41no—
85C7⋊C16—41no—
86C7⋊C16—41no—
87#540—72no—
88C2×M32—141no—
89#499C2×D711yescharacteristic
90#493—71no—
91#490—72no—
92#469—141no—
93#466—141no—
94#442—141no—
95C4×D8—141no—
96#405—71no—
97#364—71no—
98#364—71no—
99#364—141no—
100#364C2×D711yescharacteristic
101#364C2×D711yescharacteristic
102#349—141no—
103#347—141no—
104C22×D7—42no—
105C22×D7#10011yescharacteristic
106C2×C4×C7—21no—
107C2×C4×C7—42no—
108C2×C4×C7C2×C22⋊C411yescharacteristic
109C2×Dic7—21no—
110C2×Dic7#10011yescharacteristic
111D28—42no—
112D28—42no—
113D28—162no—
114D28—162no—
115C4×D7—21no—
116C4×D7—21no—
117C4×D7—162no—
118C7×C8#13511yescharacteristic
119C7×C8#13511yescharacteristic
120C7×C8AGL(1,ℤ/8)11yescharacteristic
121C7×C8#13711yescharacteristic
122C7⋊C8—21no—
123C7⋊C8—21no—
124C2×D8—141no—
125C2×D8—141no—
126C2×M16—71no—
127C22×C8—71no—
128#127—72no—
129#125C4×D712yes—
130#123—282no—
131#121—142no—
132#121—282no—
133C4×D4—142no—
134#117—71no—
135M32—21no—
136M32—21no—
137M32—141no—
138M32—141no—
139M32—281no—
140C2×C16—281no—
141#107—21no—
142#106—21no—
143#102—142no—
144#102—142no—
145#97—71no—
146#97—71no—
147C4×C8—71no—
148C4×C8C22×D711yescharacteristic
149C22×C7C2×#10011yescharacteristic
150C2×D7—21no—
151C2×D7—41no—
152C2×D7—82no—
153C2×D7—82no—
154C2×D7—82no—
155C4×C7—41no—
156C4×C7—82no—
157C4×C7#41611yescharacteristic
158C4×C7#41711yescharacteristic
159Dic7—21no—
160Dic7—41no—
161C2×D4—142no—
162C2×D4—282no—
163C2×D4—282no—
164C22×C4—71no—
165C22×C4—282no—
166D8—281no—
167D8—281no—
168D8—281no—
169D8—281no—
170M16—141no—
171M16—141no—
172C2×C8—71no—
173C2×C8—71no—
174C2×C8—141no—
175C2×C8—141no—
176C2×C8D4×D711yescharacteristic
177C2×C8D4×D711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—141no—
181C22⋊C4—141no—
182C22⋊C4—282no—
183C22⋊C4—282no—
184C42—71no—
185C42C2×C4×D711yescharacteristic
186C16—41no—
187C16—41no—
188C16—281no—
189C16—281no—
190C2×C7—21no—
191C2×C7#155111yescharacteristic
192D7—41no—
193D7—162no—
194C23—71no—
195C23—282no—
196D4—282no—
197D4—282no—
198D4—1122no—
199D4—1122no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—71no—
203C2×C4—141no—
204C2×C4—141no—
205C2×C4—141no—
206C2×C4—1122no—
207C2×C4D7×C22⋊C411yescharacteristic
208C8—141no—
209C8—141no—
210C8#647711yescharacteristic
211C8#647911yescharacteristic
212C8#648111yescharacteristic
213C8#648211yescharacteristic
214C7#1198911yescharacteristic, complemented, Sylow
215C22—141no—
216C22—281no—
217C22—562no—
218C22—562no—
219C22—562no—
220C22D7×#10011yescharacteristic
221C4—41no—
222C4—82no—
223C4—141no—
224C4—281no—
225C4#8957511yescharacteristic
226C4#8957611yescharacteristic
227C2—21no—
228C2—281no—
229C2—1122no—
230C2#1172623911yescharacteristic
subgroupG/H
centerC2#11726239
commutatorC42×C7C22×C4
FrattiniC4×C8C22×D7
FittingC7×#1366C2
socleC2×C7#1551
prime radical#11739354C2
cosocleC4×C7×C8C23
hypercenter#1366D7
hypocenterC7#11989
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
2. #11727057C2semidirect
3. #11727053C2semidirect
4. C7×#1366C2semidirect
5. #11725606C2semidirect
6. #11725377C2semidirect
7. #11725377C2semidirect
55. #1366D7semidirect
214. C7#11989semidirect

Maximal subgroups

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

subgroupindex
#117393542
#117270572
#117270532
C7×#13662
#117256062
#117253772
#119897