Finite Group Databasebeta

← #13272905

Subgroups — #13272905

Order 1792.

402 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#89452C211yesmaximal, characteristic, complemented
2C2×#89313C212yesmaximal
3#11727549C211yesmaximal, characteristic
4C7×#1376C211yesmaximal, characteristic, complemented
5#11725622C212yesmaximal, complemented
6C2×#6467C2211yescharacteristic
7C2×#6432—21no—
8C2×C7×M32C2212yes—
9C2×C4×C7×C8C2211yescharacteristic
10#89882—21no—
11C2×#6393C2211yescharacteristic
12C2×#6391—21no—
13C2×#6391C2212yes—
14C2×#6379—21no—
15#89452—24no—
16#89313—24no—
17#13371—71nomaximal, Sylow
18C2×#1157—21no—
19C22×Dic14—41no—
20C22×C7×C8D411yescharacteristic
21C22×C7×C8C2311yescharacteristic
22C2×C42×C7D411yescharacteristic
23C2×C4⋊Dic7—22no—
24C2×C4⋊Dic7—41no—
25C2×#1136—21no—
26C2×Dic28Q811yescharacteristic, complemented
27#6467—22no—
28#6467—22no—
29C2×D56Q811yescharacteristic, complemented
30C2×#1132—21no—
31#6432—24no—
32#6432—42no—
33C7×M32—22no—
34C7×M32—22no—
35C2×C7×C16—22no—
36C4×C7×C8—22no—
37(C2×D7)⋊C8—42no—
38#6393—22no—
39#6391—24no—
40#6391—22no—
41#6391D414yes—
42#6379—42no—
43C2×#498—71no—
44C2×#360—72no—
45#1974—71no—
46#1383—72no—
47#1376D711yescharacteristic, complemented
48C22×C4×C7—21no—
49C22×C4×C7C2×D411yescharacteristic
50C2×C22⋊D7—41no—
51C22×Dic7—42no—
52#1157—41no—
53#1157—41no—
54#1157—41no—
55#1157—41no—
56C2×D28—21no—
57C2×C4×D7—41no—
58C2×Dic14—21no—
59C2×Dic14—44no—
60C2×Dic14—84no—
61C2×C7×C8—21no—
62C2×C7×C8—21no—
63C2×C7×C8SD1612yes—
64C2×C7×C8C2×D412yes—
65C2×C7×C8C2×Q811yescharacteristic
66C2×C7×C8#4112yes—
67C2×C7×C8#4111yescharacteristic
68C42×C7—21no—
69C42×C7D812yes—
70C4⋊Dic7—24no—
71C4⋊Dic7—44no—
72C4⋊Dic7—42no—
73#1136—41no—
74#1136—41no—
75C2×C7⋊C8—41no—
76Dic28—22no—
77D56—22no—
78#1132—42no—
79C7×C16—42no—
80C2×#135—71no—
81C2×C4⋊Q8—141no—
82C2×M32C2×D712yes—
83#498—144no—
84C2×#108—71no—
85C2×#106—72no—
86C2×#106—141no—
87C2×#104—141no—
88#408—141no—
89C2×C4×C8C2×D711yescharacteristic
90#360—144no—
91C23×C7AGL(1,ℤ/8)11yescharacteristic
92C22×D7—41no—
93C2×C4×C7—24no—
94C2×C4×C7—22no—
95C2×C4×C7#12011yescharacteristic
96C2×C4×C7C22⋊Q811yescharacteristic
97C2×C4×C7#12411yescharacteristic
98C2×C4×C7C2×D811yescharacteristic
99C2×C4×C7C2×SD1611yescharacteristic
100C2×C4×C7#13711yescharacteristic
101C22⋊D7—81no—
102C22⋊D7—81no—
103C2×Dic7—44no—
104C2×Dic7—41no—
105C2×Dic7—84no—
106D28—41no—
107D28—41no—
108C4×D7—81no—
109C4×D7—81no—
110Dic14—41no—
111Dic14—41no—
112Dic14—88no—
113C7×C8—22no—
114C7×C8C22⋊Q812yes—
115C7×C8#12412yes—
116C7⋊C8—81no—
117C2×#41—141no—
118C22×Q8—281no—
119#135—142no—
120#135—142no—
121C2×Q16—71no—
122C2×SD16—141no—
123C2×D8—71no—
124C2×M16—141no—
125C22×C8D2811yescharacteristic
126C22×C8C22×D711yescharacteristic
127C4⋊Q8—144no—
128C4⋊Q8—282no—
129C2×C4⋊C4—142no—
130C2×C4⋊C4—281no—
131C2×C42D2811yescharacteristic
132M32—22no—
133M32—22no—
134C2×C16—22no—
135#108—142no—
136#106—74no—
137#106—144no—
138#106—142no—
139#104—282no—
140C22⋊C8—282no—
141C4×C8—22no—
142C22×C7—22no—
143C22×C7#44711yescharacteristic
144C22×C7#47611yescharacteristic
145C22×C7#48911yescharacteristic
146C2×D7—82no—
147C2×D7—81no—
148C4×C7—24no—
149C4×C7#39311yescharacteristic
150C4×C7#44811yescharacteristic
151C4×C7#47411yescharacteristic
152C4×C7#48611yescharacteristic
153Dic7—84no—
154Dic7—81no—
155#41—281no—
156#41—281no—
157#41—281no—
158#41—281no—
159C2×Q8—141no—
160C2×Q8—284no—
161C2×Q8—564no—
162C2×D4—141no—
163C2×D4—281no—
164C22×C4—21no—
165C22×C4—282no—
166C22×C4—281no—
167C22×C4C2×D2811yescharacteristic
168Q16—142no—
169SD16—282no—
170D8—142no—
171M16—281no—
172M16—281no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—281no—
176C2×C8#113212yes—
177C2×C8#115712yes—
178C2×C8D4×D712yes—
179C2×C8#115911yescharacteristic
180C2×C8Q8×D711yescharacteristic
181C4⋊C4—144no—
182C4⋊C4—284no—
183C4⋊C4—282no—
184C42—21no—
185C42D5612yes—
186C16—42no—
187C2×C7—21no—
188C2×C7—21no—
189C2×C7#202311yescharacteristic
190C2×C7#223012yes—
191D7—161no—
192C23—281no—
193C23#647111yescharacteristic
194Q8—281no—
195Q8—281no—
196Q8—568no—
197D4—281no—
198D4—281no—
199D4—561no—
200D4—561no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—281no—
204C2×C4—284no—
205C2×C4—564no—
206C2×C4—561no—
207C2×C4—561no—
208C2×C4#643811yescharacteristic
209C2×C4#644511yescharacteristic
210C2×C4#646011yescharacteristic
211C2×C4C2×#113211yescharacteristic
212C2×C4C2×D5611yescharacteristic
213C2×C4#647211yescharacteristic
214C8—22no—
215C8—561no—
216C8#644812yes—
217C8#645912yes—
218C7#1337111yescharacteristic, complemented, Sylow
219C22—22no—
220C22—561no—
221C22—562no—
222C22#8948611yescharacteristic
223C22#8950411yescharacteristic
224C22#8961411yescharacteristic
225C4—24no—
226C4—564no—
227C4—561no—
228C4#8944511yescharacteristic
229C4#8946611yescharacteristic
230C4#8951011yescharacteristic
231C4#8961811yescharacteristic
232C2—21no—
233C2—21no—
234C2—1121no—
235C2#1172643211yescharacteristic
236C2#1172706112yes—
subgroupG/H
centerC22#89614
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#1376C2
socleC22×C7#476
prime radicalC2×#6467C22
hypercenter#1376D7
hypocenterC7#13371
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
1. C2×#89452C2semidirect
4. C7×#1376C2semidirect
5. #11725622C2semidirect
26. C2×Dic28Q8semidirect
29. C2×D56Q8semidirect
47. #1376D7semidirect
218. C7#13371semidirect

Maximal subgroups

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

subgroupindex
C2×#894522
C2×#893132
#117275492
C7×#13762
#117256222
#133717