Finite Group Databasebeta

← #13264010

Subgroups — #13264010

Order 1792.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#11739707C211yesmaximal, characteristic
2C7×#1742C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11725301C214yesmaximal, complemented
5C22×C7×M16—21no—
6C2×C7×#117C2211yescharacteristic
7C2×#6487—21no—
8C2×#6432C411yescharacteristic
9#89755—21no—
10#89755C2212yes—
11#89725—21no—
12#89725C412yes—
13#89708C411yescharacteristic
14#89701—21no—
15#89697—21no—
16C7×#343—21no—
17C7×#343C2212yes—
18#89323C2212yes—
19#89320—21no—
20#11700—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C22×Dic14—21no—
23C2×C7×M16—24no—
24C2×C7×M16—22no—
25C2×C7×M16—41no—
26C2×C7×M16—42no—
27C22×C7×C8—41no—
28C7×#117—41no—
29C7×#117—41no—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C4⋊C4C2311yescharacteristic
32C2×C7×C22⋊C4C2×C411yescharacteristic
33C2×C7×C22⋊C4D411yescharacteristic
34C2×C42×C7C2×C411yescharacteristic
35C2×C42×C7D411yescharacteristic
36C2×C22⋊Dic7—22no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7C2×C412yes—
40#6487—42no—
41#6487—42no—
42C2×Dic7⋊C4—22no—
43#6432—41no—
44#6432—41no—
45C7⋊#95—22no—
46C7⋊#95—21no—
47C7⋊#95—22no—
48C7⋊#95C2×C412yes—
49C22⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—41no—
51C4×C7⋊C8—41no—
52#2621—71no—
53#1742D711yescharacteristic
54#1458—71no—
55#1346—74no—
56C24×C7C22⋊C411yescharacteristic
57C22×C4×C7—21no—
58C22×C4×C7—21no—
59C22×C4×C7—22no—
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7D812yes—
64C22×C4×C7SD1612yes—
65C22×C4×C7C22×C411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×Dic7—22no—
69C22×Dic7—22no—
70C2×Dic14—41no—
71C2×Dic14—41no—
72C2×Dic14—42no—
73C2×Dic14—42no—
74C7×M16—24no—
75C7×M16—44no—
76C7×M16—42no—
77C2×C7×C8—42no—
78C2×C7×C8—44no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—41no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C22⋊Dic7—44no—
88C4⋊Dic7—42no—
89C4⋊Dic7—42no—
90C4⋊Dic7—42no—
91Dic7⋊C4—42no—
92Dic7⋊C4—42no—
93C2×C7⋊C8—42no—
94C22×M16—21no—
95C2×C4⋊Q8—71no—
96C2×C22⋊Q8—141no—
97C2×#117C2×D711yescharacteristic
98#431—141no—
99#398—72no—
100#398—141no—
101#397—72no—
102#397—141no—
103#391—141no—
104#382—71no—
105#343—21no—
106#343C2×D712yes—
107#324—72no—
108#323—141no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7#10212yes—
114C23×C7C2×C22⋊C411yescharacteristic
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7#9911yescharacteristic
130C2×C4×C7#9911yescharacteristic
131C2×C4×C7#10212yes—
132C2×C4×C7#10412yes—
133C2×C4×C7C2×D811yescharacteristic
134C2×C4×C7C2×SD1611yescharacteristic
135C2×Dic7—42no—
136C2×Dic7—42no—
137C2×Dic7—42no—
138C2×Dic7—42no—
139C2×Dic7—42no—
140C2×Dic7—42no—
141Dic14—82no—
142Dic14—82no—
143C7×C8—44no—
144C7⋊C8—82no—
145C22×Q8—141no—
146C23×C4C22⋊D711yescharacteristic
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—41no—
150C2×M16—42no—
151C22×C8—41no—
152C4⋊Q8—281no—
153C4⋊Q8—281no—
154C22⋊Q8—282no—
155C22⋊Q8—282no—
156#117—41no—
157#117—41no—
158C2×C4⋊C4—72no—
159C2×C4⋊C4—141no—
160C2×C4⋊C4—141no—
161C2×C4⋊C4—142no—
162C2×C4⋊C4C22⋊D711yescharacteristic
163C2×C4⋊C4C22×D711yescharacteristic
164C2×C22⋊C4—142no—
165C2×C22⋊C4C4×D711yescharacteristic
166C2×C22⋊C4D2811yescharacteristic
167C2×C42C4×D711yescharacteristic
168C2×C42D2811yescharacteristic
169C4⋊C8—281no—
170C22⋊C8—142no—
171C4×C8—281no—
172#95—72no—
173#95—141no—
174#95—142no—
175#95—142no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—22no—
179C22×C7—42no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—81no—
183C22×C7C2×#9911yescharacteristic
184C22×C7C2×#10211yescharacteristic
185C22×C7C2×#10411yescharacteristic
186C4×C7—21no—
187C4×C7—21no—
188C4×C7—41no—
189C4×C7—41no—
190C4×C7—41no—
191C4×C7—41no—
192C4×C7#32712yes—
193C4×C7#32712yes—
194Dic7—82no—
195Dic7—82no—
196C24(C2×D7)⋊C411yescharacteristic
197C2×Q8—282no—
198C2×Q8—281no—
199C2×Q8—281no—
200C2×Q8—282no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—142no—
205C22×C4—142no—
206C22×C4(C2×D7)⋊C411yescharacteristic
207C22×C4(C2×D7)⋊C411yescharacteristic
208C22×C4(C2×D7)⋊C411yescharacteristic
209C22×C4#114112yes—
210C22×C4#114312yes—
211C22×C4C2×C4×D711yescharacteristic
212C22×C4C2×D2811yescharacteristic
213C22×C4C2×C22⋊D711yescharacteristic
214M16—24no—
215M16—44no—
216M16—42no—
217C2×C8—42no—
218C2×C8—44no—
219C2×C8—282no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—282no—
223C4⋊C4—282no—
224C4⋊C4—282no—
225C4⋊C4—282no—
226C4⋊C4—282no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—284no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C2×C7—21no—
235C2×C7—21no—
236C2×C7—42no—
237C2×C7C2×#32711yescharacteristic
238C2×C7#212311yescharacteristic
239C2×C7#213511yescharacteristic
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23#638312yes—
245C23C2×(C2×D7)⋊C411yescharacteristic
246Q8—562no—
247Q8—562no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—282no—
263C2×C4—282no—
264C2×C4—282no—
265C2×C4—282no—
266C2×C4—282no—
267C2×C4—282no—
268C2×C4#637912yes—
269C2×C4#638011yescharacteristic
270C2×C4#638011yescharacteristic
271C2×C4#638312yes—
272C2×C4C2×#114111yescharacteristic
273C2×C4C2×#114311yescharacteristic
274C8—44no—
275C8—562no—
276C7#1170011yescharacteristic, complemented, Sylow
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×#637911yescharacteristic
285C22C2×#638011yescharacteristic
286C22C2×#638311yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—562no—
294C4—562no—
295C4#8924612yes—
296C4#8924612yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#8924611yescharacteristic
301C2#1172790111yescharacteristic
302C2#1172790411yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1742C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1742D7
hypocenterC7#11700
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
4. #11725301C2semidirect
276. C7#11700semidirect

Maximal subgroups

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

subgroupindex
#117397072
C7×#17422
#117293252
#117253012
#117007