Finite Group Databasebeta

← #13264499

Subgroups — #13264499

Order 1792.

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

normal subgroups only

This group has 388 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#11739707C211yesmaximal, characteristic
2C7×#1822C211yesmaximal, characteristic
3#11729326C211yesmaximal, characteristic
4#11725301C214yesmaximal, complemented
5C2×C7×#117C2211yescharacteristic
6C2×#6487—21no—
7C2×#6432C411yescharacteristic
8C7×#407—21no—
9#89755—21no—
10#89755C2212yes—
11#89725—21no—
12#89725C412yes—
13#89708C411yescharacteristic
14#89701—21no—
15#89698—21no—
16C7×#343—21no—
17C7×#343C2212yes—
18#89323C2212yes—
19#89320—21no—
20#11807—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C22×Dic14—21no—
23C2×C7×M16—22no—
24C7×#117—41no—
25C7×#117—41no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C7×C22⋊C4C2×C411yescharacteristic
29C2×C7×C22⋊C4D411yescharacteristic
30C2×C42×C7C2×C411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C22⋊Dic7—22no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7C2×C412yes—
36#6487—44no—
37C2×Dic7⋊C4—22no—
38#6432—41no—
39#6432—41no—
40C7×C22⋊C8—41no—
41C7×C22⋊C8—41no—
42C7⋊#95—22no—
43C7⋊#95—21no—
44C7⋊#95—22no—
45C7⋊#95C2×C412yes—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—41no—
48#6377—41no—
49#2621—71no—
50#1822D711yescharacteristic
51#1464—71no—
52#1346—74no—
53C24×C7C22⋊C411yescharacteristic
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7—41no—
58C22×C4×C7C22⋊C411yescharacteristic
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22×C411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×Dic7—22no—
65C22×Dic7—22no—
66C2×Dic14—42no—
67C2×Dic14—42no—
68C2×Dic14—41no—
69C2×Dic14—41no—
70C7×M16—42no—
71C7×M16—42no—
72C2×C7×C8—42no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C7×C22⋊C4—41no—
76C7×C22⋊C4—41no—
77C42×C7—21no—
78C42×C7—21no—
79C42×C7—21no—
80C42×C7—21no—
81C22⋊Dic7—44no—
82C4⋊Dic7—42no—
83C4⋊Dic7—42no—
84C4⋊Dic7—42no—
85Dic7⋊C4—44no—
86C2×C7⋊C8—42no—
87C2×C4⋊Q8—71no—
88C2×C22⋊Q8—141no—
89C2×#117C2×D711yescharacteristic
90#432—141no—
91#407—21no—
92#398—72no—
93#398—141no—
94#397—72no—
95#397—141no—
96#391—141no—
97#382—71no—
98#343—21no—
99#343C2×D712yes—
100#324—72no—
101#323—141no—
102C23×C7—21no—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—41no—
106C23×C7—41no—
107C23×C7C2×C22⋊C411yescharacteristic
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7#9911yescharacteristic
123C2×C4×C7#9911yescharacteristic
124C2×C4×C7AGL(1,ℤ/8)12yes—
125C2×Dic7—42no—
126C2×Dic7—42no—
127C2×Dic7—42no—
128C2×Dic7—42no—
129C2×Dic7—42no—
130C2×Dic7—42no—
131Dic14—82no—
132Dic14—82no—
133C7×C8—82no—
134C7⋊C8—82no—
135C22×Q8—141no—
136C23×C4C22⋊D711yescharacteristic
137C2×M16—22no—
138C4⋊Q8—281no—
139C4⋊Q8—281no—
140C22⋊Q8—284no—
141#117—41no—
142#117—41no—
143C2×C4⋊C4—72no—
144C2×C4⋊C4—141no—
145C2×C4⋊C4—141no—
146C2×C4⋊C4—142no—
147C2×C4⋊C4C22⋊D711yescharacteristic
148C2×C4⋊C4C22×D711yescharacteristic
149C2×C22⋊C4—142no—
150C2×C22⋊C4C4×D711yescharacteristic
151C2×C22⋊C4D2811yescharacteristic
152C2×C42C4×D711yescharacteristic
153C2×C42D2811yescharacteristic
154C4⋊C8—281no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—142no—
158#97—281no—
159#95—72no—
160#95—141no—
161#95—142no—
162#95—142no—
163C22×C7—21no—
164C22×C7—21no—
165C22×C7—42no—
166C22×C7—41no—
167C22×C7—41no—
168C22×C7—41no—
169C22×C7—81no—
170C22×C7C2×#9911yescharacteristic
171C22×C7#41811yescharacteristic
172C22×C7#42111yescharacteristic
173C4×C7—22no—
174C4×C7—21no—
175C4×C7—21no—
176C4×C7—41no—
177C4×C7—41no—
178C4×C7—41no—
179C4×C7—41no—
180Dic7—82no—
181Dic7—82no—
182C24(C2×D7)⋊C411yescharacteristic
183C2×Q8—282no—
184C2×Q8—282no—
185C2×Q8—281no—
186C2×Q8—281no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—142no—
192C22×C4—142no—
193C22×C4(C2×D7)⋊C411yescharacteristic
194C22×C4(C2×D7)⋊C411yescharacteristic
195C22×C4(C2×D7)⋊C411yescharacteristic
196C22×C4C2×C4×D711yescharacteristic
197C22×C4C2×D2811yescharacteristic
198C22×C4C2×C22⋊D711yescharacteristic
199M16—42no—
200M16—42no—
201C2×C8—42no—
202C2×C8—282no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—282no—
206C4⋊C4—282no—
207C4⋊C4—282no—
208C4⋊C4—284no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—284no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C2×C7—21no—
217C2×C7—21no—
218C2×C7—81no—
219C2×C7#151011yescharacteristic
220C2×C7#212311yescharacteristic
221C2×C7#213511yescharacteristic
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23C2×(C2×D7)⋊C411yescharacteristic
228Q8—562no—
229Q8—562no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—282no—
245C2×C4—282no—
246C2×C4—282no—
247C2×C4—282no—
248C2×C4—282no—
249C2×C4—282no—
250C2×C4#638011yescharacteristic
251C2×C4#638011yescharacteristic
252C2×C4#651212yes—
253C8—82no—
254C8—562no—
255C7#1180711yescharacteristic, complemented, Sylow
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22C2×#638011yescharacteristic
264C22#8977311yescharacteristic
265C22#8977711yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—562no—
274C4—562no—
275C2—21no—
276C2—21no—
277C2—81no—
278C2#1172728511yescharacteristic
279C2#1172790111yescharacteristic
280C2#1172790411yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1822C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1822D7
hypocenterC7#11807
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
4. #11725301C2semidirect
255. C7#11807semidirect

Maximal subgroups

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

subgroupindex
#117397072
C7×#18222
#117293262
#117253012
#118077