Finite Group Databasebeta

← #13247363

Subgroups — #13247363

Order 1792.

498 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 123 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#11739778C211yesmaximal, characteristic
2C7×#1460C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11725552C212yesmaximal, complemented
5#11725544C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×D4×Dic7—21no—
8C7×#431C2212yes—
9C2×#6407—21no—
10#89984C412yes—
11C2×C7⋊#95C412yes—
12#89757—21no—
13#89722—22no—
14#89708C2211yescharacteristic
15C4×C4⋊Dic7C2211yescharacteristic
16#89696C2212yes—
17C7×#323—22no—
18#89325—21no—
19#8764—71nomaximal, Sylow
20C22×C7×D4D411yescharacteristic
21C23×C4×C7C2×C412yes—
22C23×Dic7—22no—
23C4×C7×D4—22no—
24C4×C7×D4—22no—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C7×C22⋊C4D412yes—
27C2×C42×C7C2311yescharacteristic
28C2×C22⋊Dic7—22no—
29D4×Dic7—42no—
30D4×Dic7—42no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—22no—
34C2×C4×Dic7—21no—
35C2×#1136—22no—
36C7×C4⋊C8—22no—
37C7×C22⋊C8—22no—
38C4×C7×C8—22no—
39#6407—42no—
40C7⋊#95—22no—
41C7⋊#95—21no—
42C7⋊#95C2×C411yescharacteristic
43C7⋊#95C2×C411yescharacteristic
44C7⋊#95C2×C412yes—
45C4⋊(C7⋊C8)—22no—
46C4⋊(C7⋊C8)—22no—
47#2349—71no—
48#1800—71no—
49#1460D711yescharacteristic
50#1303—72no—
51#1286—72no—
52C24×C7C22⋊C412yes—
53C2×C7×D4—22no—
54C2×C7×D4—22no—
55C2×C7×D4D812yes—
56C2×C7×D4SD1612yes—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—82no—
60C22×C4×C7C22⋊C412yes—
61C22×C4×C7C22×C411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×Dic7—21no—
65C22×Dic7—22no—
66C22×Dic7—21no—
67C22×Dic7—44no—
68C22×Dic7—44no—
69C22×Dic7—82no—
70C2×C7×C8—42no—
71C7×C4⋊C4—22no—
72C7×C22⋊C4—24no—
73C42×C7#4112yes—
74C42×C7#4112yes—
75C22⋊Dic7—44no—
76C4⋊Dic7—42no—
77C4⋊Dic7—41no—
78C4⋊Dic7—41no—
79C4⋊Dic7—82no—
80C4×Dic7—41no—
81C4×Dic7—41no—
82#1136—42no—
83#1136—42no—
84C2×C7⋊C8—42no—
85C2×C4×D4—141no—
86C2×C4×D4C2×D711yescharacteristic
87#431C2×D712yes—
88#423—72no—
89C2×#100—141no—
90#389—141no—
91#385—142no—
92#382—71no—
93#380—72no—
94C4×C4⋊C4—71no—
95C2×#95—72no—
96#343—141no—
97#323—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—42no—
101C23×C7—42no—
102C23×C7#10214yes—
103C23×C7C2×C22⋊C411yescharacteristic
104C7×D4—24no—
105C7×D4—24no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—42no—
110C2×C4×C7—82no—
111C2×C4×C7—82no—
112C2×C4×C7#11711yescharacteristic
113C2×C4×C7#11711yescharacteristic
114C2×C4×C7#12312yes—
115C2×C4×C7#12312yes—
116C2×C4×C7C2×D811yescharacteristic
117C2×C4×C7C2×SD1611yescharacteristic
118C2×C4×C7#13712yes—
119C2×Dic7—22no—
120C2×Dic7—41no—
121C2×Dic7—41no—
122C2×Dic7—41no—
123C2×Dic7—41no—
124C2×Dic7—42no—
125C2×Dic7—42no—
126C2×Dic7—41no—
127C2×Dic7—42no—
128C2×Dic7—88no—
129C7×C8—82no—
130C7⋊C8—82no—
131C22×D4D2811yescharacteristic
132C23×C4—142no—
133C23×C4C4×D712yes—
134C2×M16—142no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—282no—
138C4×D4—282no—
139C2×C4⋊C4—141no—
140C2×C4⋊C4—141no—
141C2×C4⋊C4—142no—
142C2×C4⋊C4D2811yescharacteristic
143C2×C22⋊C4—142no—
144C2×C22⋊C4C22⋊D712yes—
145C2×C42—141no—
146C2×C42C22×D711yescharacteristic
147C4⋊C8—22no—
148C4⋊C8—142no—
149C4⋊C8—142no—
150#100—282no—
151C22⋊C8—22no—
152C4×C8—22no—
153#95—71no—
154#95—72no—
155#95—71no—
156#95—142no—
157#95—141no—
158C22×C7—21no—
159C22×C7—21no—
160C22×C7—24no—
161C22×C7—44no—
162C22×C7—44no—
163C22×C7—42no—
164C22×C7—42no—
165C22×C7#38111yescharacteristic
166C22×C7C2×#10211yescharacteristic
167C22×C7#41911yescharacteristic
168C4×C7—22no—
169C4×C7—21no—
170C4×C7—21no—
171C4×C7—42no—
172C4×C7#48012yes—
173C4×C7#48312yes—
174Dic7—42no—
175Dic7—81no—
176Dic7—82no—
177C24(C2×D7)⋊C412yes—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4#113212yes—
181C2×D4D5612yes—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—82no—
185C22×C4—141no—
186C22×C4—142no—
187C22×C4—141no—
188C22×C4—284no—
189C22×C4—284no—
190C22×C4—562no—
191C22×C4(C2×D7)⋊C412yes—
192C22×C4C2×C4×D711yescharacteristic
193C22×C4C2×D2811yescharacteristic
194C22×C4C2×C22⋊D711yescharacteristic
195M16—282no—
196M16—282no—
197C2×C8—42no—
198C2×C8—282no—
199C4⋊C4—22no—
200C4⋊C4—282no—
201C4⋊C4—281no—
202C4⋊C4—281no—
203C4⋊C4—562no—
204C22⋊C4—24no—
205C22⋊C4—284no—
206C42—281no—
207C42—281no—
208C42#115912yes—
209C42#115912yes—
210C2×C7—21no—
211C2×C7—21no—
212C2×C7—44no—
213C2×C7#182311yescharacteristic
214C2×C7#182711yescharacteristic
215C2×C7#182911yescharacteristic
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23#639514yes—
221C23C2×(C2×D7)⋊C411yescharacteristic
222D4—24no—
223D4—24no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—142no—
231C2×C4—281no—
232C2×C4—281no—
233C2×C4—281no—
234C2×C4—281no—
235C2×C4—282no—
236C2×C4—282no—
237C2×C4—281no—
238C2×C4—282no—
239C2×C4—568no—
240C2×C4#644011yescharacteristic
241C2×C4#644011yescharacteristic
242C2×C4#644912yes—
243C2×C4C2×#113211yescharacteristic
244C2×C4C2×D5611yescharacteristic
245C2×C4#647212yes—
246C2×C4#649812yes—
247C8—82no—
248C8—562no—
249C7#876411yescharacteristic, complemented, Sylow
250C22—21no—
251C22—21no—
252C22—24no—
253C22—44no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22#8971511yescharacteristic
258C22C2×#639511yescharacteristic
259C22#8989211yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—282no—
265C4—562no—
266C4—561no—
267C4#8949312yes—
268C4#8950812yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#1172767811yescharacteristic
273C2#1172779311yescharacteristic
274C2#1172782111yescharacteristic
subgroupG/H
centerC22#89715
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1460C2
socleC22×C7#381
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#1460D7
hypocenterC7#8764
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
4. #11725552C2semidirect
5. #11725544C2semidirect
249. C7#8764semidirect

Maximal subgroups

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

subgroupindex
#117397782
C7×#14602
#117293712
#117255522
#117255442
#87647