Finite Group Databasebeta

← #13273481

Subgroups — #13273481

Order 1792.

416 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 79 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#11734049C211yesmaximal, characteristic
2C7×#1857C211yesmaximal, characteristic
3#11729412C212yesmaximal
4#11725505C212yesmaximal
5#11725450C211yesmaximal, characteristic
6C7×#519C2211yescharacteristic
7C7×#428—21no—
8C2×C7×#101C2211yescharacteristic
9C7×#391C2211yescharacteristic
10#90002C2212yes—
11C2×#6410C2212yes—
12#89767—22no—
13#89751—21no—
14#89701—21no—
15#89428—22no—
16#89418—22no—
17C7×#339—21no—
18#89327—22no—
19#13464—71nomaximal, Sylow
20C22×C7×Q8C2311yescharacteristic
21C22×Dic14—21no—
22C22×Dic14—21no—
23C2×C7×M16—21no—
24C4×C7×Q8—41no—
25C4×C7×Q8—41no—
26C7×#117—21no—
27C7×#117—22no—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7D412yes—
32C2×Dic7⋊C4—22no—
33C2×C4×Dic7D412yes—
34C2×#1136—22no—
35C7×#107—41no—
36C7×#106—41no—
37C7×#101—21no—
38C7×#101Q812yescomplemented
39C7×#95—22no—
40#6410—22no—
41#6410Q814yescomplemented
42C7⋊#95—21no—
43C7⋊#95—22no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46#6382—42no—
47#6381—42no—
48#2811—71no—
49#1857—72no—
50#1857D711yescharacteristic
51#1394—72no—
52#1394—71no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8C2×Q811yescharacteristic
56C2×C7×Q8C2×Q812yes—
57C2×C7×Q8#4111yescharacteristic
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7C2×D412yes—
61C22×C4×C7C2×D411yescharacteristic
62C22×Dic7—22no—
63C22×Dic7—22no—
64C2×Dic14—24no—
65C2×Dic14—81no—
66C2×Dic14—81no—
67C2×Dic14—82no—
68C2×Dic14—81no—
69C7×M16—22no—
70C7×M16—41no—
71C2×C7×C8—41no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—41no—
77C7×C22⋊C4—41no—
78C42×C7—41no—
79C42×C7—42no—
80C42×C7—41no—
81C4⋊Dic7—42no—
82C4⋊Dic7—81no—
83Dic7⋊C4—82no—
84C4×Dic7—42no—
85#1136—24no—
86#1136—42no—
87C2×C7⋊C8—42no—
88#519C2×D711yescharacteristic
89#428—21no—
90#428—142no—
91C2×#101—72no—
92C2×#101C2×D711yescharacteristic
93#395—142no—
94#395—141no—
95#391—72no—
96#391—142no—
97#391—141no—
98#391C2×D711yescharacteristic
99#339—21no—
100#339—142no—
101C23×C7#12011yescharacteristic
102C7×Q8—41no—
103C7×Q8—42no—
104C7×Q8—41no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—22no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7—41no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7C22⋊Q812yes—
117C2×C4×C7C22⋊Q811yescharacteristic
118C2×C4×C7C4⋊Q812yes—
119C2×C4×C7C4⋊Q811yescharacteristic
120C2×Dic7—24no—
121C2×Dic7—42no—
122C2×Dic7—42no—
123C2×Dic7—42no—
124C2×Dic7—42no—
125C2×Dic7—42no—
126Dic14—84no—
127Dic14—84no—
128C7×C8—42no—
129C7⋊C8—44no—
130C22×Q8—141no—
131C22×Q8—141no—
132C22×Q8C22×D711yescharacteristic
133C2×M16—21no—
134C2×M16—142no—
135C4×Q8—41no—
136C4×Q8—41no—
137#117—22no—
138#117—21no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4—142no—
141C2×C4⋊C4—141no—
142C2×C4⋊C4C22⋊D711yescharacteristic
143C2×C42—72no—
144C2×C42C22⋊D711yescharacteristic
145#107—41no—
146#107—282no—
147#106—41no—
148#106—282no—
149#101—21no—
150#101—74no—
151#101—142no—
152#101Dic1412yescomplemented
153#95—22no—
154#95—142no—
155#95—141no—
156#95—142no—
157#95—142no—
158C22×C7—21no—
159C22×C7—21no—
160C22×C7#39511yescharacteristic
161C22×C7B(4,𝔽2)11yescharacteristic
162C22×C7#45811yescharacteristic
163C4×C7—22no—
164C4×C7—21no—
165C4×C7—22no—
166C4×C7—21no—
167C4×C7—42no—
168C4×C7—81no—
169C4×C7—81no—
170C4×C7—81no—
171Dic7—44no—
172Dic7—82no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—144no—
176C2×Q8—561no—
177C2×Q8—561no—
178C2×Q8—562no—
179C2×Q8—561no—
180C2×Q8C2×Dic1411yescharacteristic
181C2×Q8#115711yescharacteristic
182C2×Q8Q8×D712yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—142no—
186C22×C4—142no—
187C22×C4D4×D712yes—
188C22×C4C2×C22⋊D711yescharacteristic
189M16—22no—
190M16—41no—
191M16—144no—
192M16—282no—
193C2×C8—41no—
194C2×C8—282no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—282no—
201C4⋊C4—562no—
202C4⋊C4—561no—
203C22⋊C4—41no—
204C42—41no—
205C42—41no—
206C42—42no—
207C42—282no—
208C2×C7—21no—
209C2×C7—21no—
210C2×C7#206211yescharacteristic
211C2×C7#220211yescharacteristic
212C2×C7#220511yescharacteristic
213C23#650011yescharacteristic
214Q8—41no—
215Q8—41no—
216Q8—42no—
217Q8—564no—
218Q8—564no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—144no—
231C2×C4—282no—
232C2×C4—282no—
233C2×C4—282no—
234C2×C4—282no—
235C2×C4—282no—
236C2×C4#645112yes—
237C2×C4#645912yes—
238C2×C4#648711yescharacteristic
239C2×C4#650711yescharacteristic
240C8—42no—
241C8—284no—
242C7#1346411yescharacteristic, complemented, Sylow
243C22—21no—
244C22—21no—
245C22#8975111yescharacteristic
246C22#8980411yescharacteristic
247C22#8982511yescharacteristic
248C4—21no—
249C4—22no—
250C4—22no—
251C4—21no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—284no—
257C4—562no—
258C2—21no—
259C2—21no—
260C2#1172734811yescharacteristic
261C2#1172874111yescharacteristic
262C2#1172874911yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC22×C7×Q8C23
FrattiniC22×Q8C22×D7
FittingC7×#1857C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#1857D7
hypocenterC7#13464
LCSG ▷ C22×C7×Q8 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
38. C7×#101Q8semidirect
41. #6410Q8semidirect
152. #101Dic14semidirect
242. C7#13464semidirect

Maximal subgroups

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

subgroupindex
#117340492
C7×#18572
#117294122
#117255052
#117254502
#134647