Finite Group Databasebeta

← #13257731

Subgroups — #13257731

Order 1792.

372 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 81 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#11740815C211yesmaximal, characteristic
2C7×#1561C211yesmaximal, characteristic
3#11729579C211yesmaximal, characteristic
4#11728276C211yesmaximal, characteristic
5#11728261C211yesmaximal, characteristic
6#11728249C211yesmaximal, characteristic
7#11725516C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×#6432C2211yescharacteristic
10C7×#423C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C2×#6398—22no—
13#89865—21no—
14#89864—21no—
15#89822—22no—
16#89803—21no—
17#89803—21no—
18#89697C2211yescharacteristic
19#89697C2211yescharacteristic
20C4×#1136C2211yescharacteristic
21C7×#332—21no—
22C7×#332—21no—
23Q8⋊(C7⋊C8)—22no—
24#89276—22no—
25#89275—21no—
26#89274—21no—
27#10493—71nomaximal, Sylow
28C22×Dic14D412yes—
29C2×C7×M16—21no—
30C2×C7×M16—21no—
31C7×C4⋊Q8D411yescharacteristic
32C7×C4⋊Q8D411yescharacteristic
33C7×#125—21no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C4×C7×Q8—21no—
37C4×C7×Q8—21no—
38C7×#117—21no—
39C2×C42×C7C2311yescharacteristic
40C2×#1144—22no—
41C2×#1144—22no—
42C2×C4⋊Dic7—21no—
43C2×C4⋊Dic7—21no—
44C2×#1136—21no—
45#6432—21no—
46#6432—21no—
47#6432—21no—
48#6432—21no—
49C7×C4⋊C8—21no—
50C7×C4⋊C8—21no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C22⋊(C7⋊C8)—21no—
54C22⋊(C7⋊C8)—21no—
55#6398—44no—
56#6384—22no—
57#6384—22no—
58C4⋊(C7⋊C8)D411yescharacteristic
59C4⋊(C7⋊C8)D411yescharacteristic
60C4⋊(C7⋊C8)D412yes—
61#6377—21no—
62C4×C7⋊C8—21no—
63C4×C7⋊C8—21no—
64C4×C7⋊C8—21no—
65#3209—71no—
66#1966—71no—
67#1574—71no—
68#1561D711yescharacteristic
69#1548—71no—
70#1485—71no—
71#1299—71no—
72C2×C7×Q8—21no—
73C2×C7×Q8—21no—
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7#4111yescharacteristic
76C22×C4×C7#4111yescharacteristic
77C22×Dic7—22no—
78C2×Dic14—22no—
79C2×Dic14—22no—
80C2×Dic14—42no—
81C2×Dic14—42no—
82C7×M16—42no—
83C7×M16—81no—
84C2×C7×C8—41no—
85C2×C7×C8—41no—
86C7×C4⋊C4—21no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C22⋊C4—41no—
93C42×C7—41no—
94C42×C7C2×D411yescharacteristic
95C42×C7C2×D411yescharacteristic
96C42×C7C2×D411yescharacteristic
97C42×C7#4111yescharacteristic
98#1144—82no—
99#1144—82no—
100C4⋊Dic7—41no—
101C4⋊Dic7—41no—
102C4⋊Dic7—41no—
103C4⋊Dic7—41no—
104#1136—22no—
105#1136—41no—
106C2×C7⋊C8—21no—
107C2×C7⋊C8—21no—
108C2×C7⋊C8—22no—
109C2×C7⋊C8—41no—
110#533C2×D711yescharacteristic
111C2×C4⋊Q8—71no—
112#494—141no—
113#487—141no—
114#462—141no—
115#462—141no—
116#451—142no—
117#431—71no—
118#431—71no—
119#423C2×D711yescharacteristic
120#423C2×D711yescharacteristic
121C2×#101—142no—
122C4×M16—71no—
123#335—141no—
124#334—141no—
125#333—142no—
126#332—21no—
127#332—21no—
128Q8⋊C8—142no—
129C23×C7#12311yescharacteristic
130C7×Q8—41no—
131C7×Q8—41no—
132C7×Q8—41no—
133C7×Q8—41no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—22no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7#12011yescharacteristic
142C2×C4×C7#12111yescharacteristic
143C2×C4×C7#12111yescharacteristic
144C2×C4×C7#12111yescharacteristic
145C2×C4×C7#12311yescharacteristic
146C2×C4×C7#12411yescharacteristic
147C2×Dic7—44no—
148C2×Dic7—42no—
149Dic14—42no—
150Dic14—42no—
151Dic14—82no—
152C7×C8—81no—
153C7×C8—81no—
154C7⋊C8—42no—
155C7⋊C8—41no—
156C7⋊C8—41no—
157C7⋊C8—42no—
158C22×Q8—72no—
159C2×Q16—142no—
160C2×Q16—142no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—141no—
164C4⋊Q8—141no—
165C4⋊Q8—141no—
166C4⋊Q8—141no—
167C4⋊Q8—141no—
168C4⋊Q8C22⋊D711yescharacteristic
169C4⋊Q8C22⋊D711yescharacteristic
170#125—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175#117—21no—
176C2×C4⋊C4—141no—
177C2×C4⋊C4—141no—
178C2×C42C22×D711yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—72no—
184C4⋊C8—71no—
185C4⋊C8—71no—
186#103—142no—
187#103—142no—
188#101—284no—
189C22⋊C8—141no—
190C22⋊C8—141no—
191#97—141no—
192C4×C8—141no—
193C4×C8—141no—
194C4×C8—141no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7#39411yescharacteristic
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—81no—
203C4×C7—81no—
204C4×C7#45311yescharacteristic
205C4×C7#45511yescharacteristic
206C4×C7#47111yescharacteristic
207C4×C7#47311yescharacteristic
208C4×C7#47312yes—
209Dic7—82no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—142no—
213C2×Q8—142no—
214C2×Q8—282no—
215C2×Q8—282no—
216C22×C4—142no—
217C22×C4#115711yescharacteristic
218C22×C4D4×D711yescharacteristic
219C22×C4#115911yescharacteristic
220Q16—562no—
221Q16—562no—
222M16—42no—
223M16—81no—
224M16—142no—
225M16—281no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—142no—
229C2×C8—141no—
230C2×C8—141no—
231C2×C8—281no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—281no—
239C4⋊C4—281no—
240C4⋊C4—281no—
241C4⋊C4—281no—
242C22⋊C4—41no—
243C42—41no—
244C42#115711yescharacteristic
245C42D4×D711yescharacteristic
246C42D4×D711yescharacteristic
247C42C2×C22⋊D711yescharacteristic
248C2×C7—41no—
249C2×C7#203711yescharacteristic
250C2×C7#204711yescharacteristic
251C2×C7#205211yescharacteristic
252C23#644611yescharacteristic
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—282no—
258Q8—282no—
259Q8—562no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—284no—
268C2×C4—282no—
269C2×C4#644711yescharacteristic
270C2×C4#644711yescharacteristic
271C2×C4#644811yescharacteristic
272C2×C4#649211yescharacteristic
273C2×C4#650011yescharacteristic
274C2×C4#650211yescharacteristic
275C8—81no—
276C8—81no—
277C8—282no—
278C8—281no—
279C8—281no—
280C8—282no—
281C7#1049311yescharacteristic, complemented, Sylow
282C22—21no—
283C22—21no—
284C22—21no—
285C22#8973111yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—81no—
290C4—81no—
291C4—562no—
292C4#8952712yes—
293C4#8997011yescharacteristic
294C4#8997211yescharacteristic
295C4#8997411yescharacteristic
296C4#8997611yescharacteristic
297C2—41no—
298C2#1172787411yescharacteristic
299C2#1172806011yescharacteristic
300C2#1172806411yescharacteristic
subgroupG/H
centerC22#89731
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1561C2
socleC22×C7#394
prime radicalC23×C7#123
hypercenter#1561D7
hypocenterC7#10493
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
281. C7#10493semidirect

Maximal subgroups

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

subgroupindex
#117408152
C7×#15612
#117295792
#117282762
#117282612
#117282492
#117255162
#104937