Finite Group Databasebeta

← #13249804

Subgroups — #13249804

Order 1792.

458 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C7×#2921C211yesmaximal, characteristic
2#11739709C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11728261C212yesmaximal, complemented
5#11725519C212yesmaximal, complemented
6C2×C7×#117C2211yescharacteristic
7C2×#6432C2211yescharacteristic
8C7×#433—24no—
9C7×#423C2212yes—
10C7×#407—21no—
11C4×C7×M16—21no—
12#89984—21no—
13#89758—21no—
14#89751C412yes—
15#89721—21no—
16#89719C412yes—
17#89708C2211yescharacteristic
18#89700—21no—
19#89697C2212yes—
20#89323—21no—
21#89320—21no—
22#89276—24no—
23#9192—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C22×Dic14C2×C412yes—
26C2×C7×M16—22no—
27C7×#117—22no—
28C7×#117—22no—
29C2×C7×C4⋊C4C2×C411yescharacteristic
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C22⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4D411yescharacteristic
33C2×C42×C7D411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C2×C22⋊Dic7—22no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×Dic7⋊C4—22no—
40#6432—22no—
41#6432—22no—
42C7×C4⋊C8—22no—
43C7×C4⋊C8D414yes—
44C7×C22⋊C8—24no—
45C7×#97—22no—
46C4×C7×C8—22no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—21no—
50C7⋊#95C2×C412yes—
51C22⋊(C7⋊C8)—22no—
52C4⋊(C7⋊C8)—22no—
53C4×C7⋊C8—22no—
54#2921D711yescharacteristic
55#2628—71no—
56#1548—72no—
57#1458—71no—
58#1286—72no—
59C24×C7C22⋊C411yescharacteristic
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×Dic7—22no—
71C22×Dic7—22no—
72C2×Dic14—24no—
73C2×Dic14—82no—
74C7×M16—42no—
75C7×M16—42no—
76C2×C7×C8—24no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—22no—
79C7×C22⋊C4—22no—
80C7×C22⋊C4—22no—
81C42×C7—21no—
82C42×C7—21no—
83C42×C7C2×D412yes—
84C42×C7#4112yes—
85C22⋊Dic7—82no—
86C4⋊Dic7—41no—
87C4⋊Dic7—42no—
88C4⋊Dic7—41no—
89C4⋊Dic7—42no—
90Dic7⋊C4—82no—
91C2×C7⋊C8—42no—
92C2×C4⋊Q8—71no—
93C2×#117C2×D711yescharacteristic
94#433—24no—
95#431—72no—
96#423C2×D712yes—
97#407—21no—
98C4×M16—21no—
99#400—141no—
100#399—141no—
101#395—72no—
102#393—72no—
103#389—141no—
104#382—71no—
105#380—141no—
106#333—144no—
107#324—141no—
108#323—141no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7C2×C22⋊C411yescharacteristic
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7C4×D412yes—
127C2×C4×C7#12011yescharacteristic
128C2×C4×C7#12112yes—
129C2×C4×C7#12311yescharacteristic
130C2×C4×C7#13512yes—
131C2×C4×C7#13712yes—
132C2×Dic7—42no—
133C2×Dic7—42no—
134C2×Dic7—42no—
135C2×Dic7—42no—
136C2×Dic7—42no—
137C2×Dic7—42no—
138Dic14—84no—
139C7×C8—44no—
140C7⋊C8—82no—
141C22×Q8—72no—
142C23×C4C22⋊D711yescharacteristic
143C2×M16—22no—
144C4⋊Q8—142no—
145C4⋊Q8—142no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—141no—
149C2×C4⋊C4—141no—
150C2×C4⋊C4—141no—
151C2×C4⋊C4—142no—
152C2×C4⋊C4C4×D711yescharacteristic
153C2×C4⋊C4D2811yescharacteristic
154C2×C22⋊C4—142no—
155C2×C22⋊C4C4×D711yescharacteristic
156C2×C22⋊C4D2811yescharacteristic
157C2×C42C22⋊D711yescharacteristic
158C2×C42C22×D711yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—142no—
161C4⋊C8C22⋊D714yes—
162C22⋊C8—24no—
163C22⋊C8—142no—
164#97—22no—
165C4×C8—22no—
166C4×C8—142no—
167#95—72no—
168#95—142no—
169#95—142no—
170#95—141no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—42no—
174C22×C7—41no—
175C22×C7—41no—
176C22×C7—41no—
177C22×C7—81no—
178C22×C7#38611yescharacteristic
179C22×C7#41611yescharacteristic
180C22×C7#41911yescharacteristic
181C4×C7—22no—
182C4×C7—22no—
183C4×C7—41no—
184C4×C7—41no—
185C4×C7—41no—
186C4×C7#45212yes—
187C4×C7#48512yes—
188Dic7—82no—
189Dic7—82no—
190C24(C2×D7)⋊C411yescharacteristic
191C2×Q8—144no—
192C2×Q8—562no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—142no—
198C22×C4—142no—
199C22×C4(C2×D7)⋊C411yescharacteristic
200C22×C4(C2×D7)⋊C411yescharacteristic
201C22×C4(C2×D7)⋊C411yescharacteristic
202C22×C4C2×C4×D711yescharacteristic
203C22×C4C2×D2811yescharacteristic
204C22×C4C2×C22⋊D711yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—24no—
208C2×C8—282no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—281no—
212C4⋊C4—282no—
213C4⋊C4—282no—
214C4⋊C4—281no—
215C4⋊C4—562no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—562no—
219C42—21no—
220C42—21no—
221C42#115712yes—
222C42C2×C22⋊D712yes—
223C2×C7—21no—
224C2×C7—21no—
225C2×C7—81no—
226C2×C7#187311yescharacteristic
227C2×C7#190911yescharacteristic
228C2×C7#191411yescharacteristic
229C23—21no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23C2×(C2×D7)⋊C411yescharacteristic
235Q8—564no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—282no—
248C2×C4—282no—
249C2×C4—282no—
250C2×C4—282no—
251C2×C4—282no—
252C2×C4—282no—
253C2×C4C4×C22⋊D712yes—
254C2×C4#649211yescharacteristic
255C2×C4#649312yes—
256C2×C4#651312yes—
257C2×C4#651412yes—
258C2×C4#651611yescharacteristic
259C8—44no—
260C8—562no—
261C7#919211yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#8977411yescharacteristic
270C22#8977611yescharacteristic
271C22#8998511yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—562no—
278C4—562no—
279C4#8976812yes—
280C4#9001112yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#1172933911yescharacteristic
285C2#1172977811yescharacteristic
286C2#1172978411yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2921C2
socleC22×C7#386
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2921D7
hypocenterC7#9192
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
4. #11728261C2semidirect
5. #11725519C2semidirect
261. C7#9192semidirect

Maximal subgroups

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

subgroupindex
C7×#29212
#117397092
#117293252
#117282612
#117255192
#91927