Finite Group Databasebeta

← #13248307

Subgroups — #13248307

Order 1792.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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#11736712C211yesmaximal, characteristic
2C7×#1574C211yesmaximal, characteristic
3#11728531C211yesmaximal, characteristic
4#11728268C211yesmaximal, characteristic
5#11728249C211yesmaximal, characteristic
6#11725542C211yesmaximal, characteristic
7#11725542C211yesmaximal, characteristic
8#90161C2211yescharacteristic
9C7×#431C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C42×Dic7C2211yescharacteristic
13#89697C2211yescharacteristic
14#89697C2211yescharacteristic
15#89603—21no—
16#89602—21no—
17#89492—22no—
18#89451—21no—
19C8×Dic14—21no—
20C7×#335—21no—
21C7×#334—21no—
22#89350—22no—
23#89281C411yescharacteristic, complemented
24#89281C411yescharacteristic, complemented
25#89277C411yescharacteristic, complemented
26#89277C411yescharacteristic, complemented
27#89275—21no—
28#89274—21no—
29#8936—71nomaximal, Sylow
30C2×C7×M16—21no—
31C2×C42×C7C2311yescharacteristic
32#6489D412yes—
33#6487—21no—
34#6487—21no—
35C2×C4×Dic7—21no—
36C2×C4×Dic7—21no—
37C2×C4×Dic7—21no—
38#6433Q812yes—
39#6432C2×C411yescharacteristic
40#6432C2×C411yescharacteristic
41C4×Dic14—21no—
42C4×Dic14—21no—
43C7×C4⋊C8—21no—
44C7×C4⋊C8—21no—
45C7×C4⋊C8C2×C411yescharacteristic
46C7×C4⋊C8C2×C411yescharacteristic
47C7×C4⋊C8D411yescharacteristic
48C7×C4⋊C8D411yescharacteristic
49C7×C22⋊C8—21no—
50C7×C22⋊C8—21no—
51C4×C7×C8—21no—
52C4×C7×C8—21no—
53C22⋊(C7⋊C8)—21no—
54C22⋊(C7⋊C8)—21no—
55#6389—22no—
56Dic7⋊C8—22no—
57Dic7⋊C8—22no—
58C8×Dic7—22no—
59C4⋊(C7⋊C8)C2×C411yescharacteristic
60C4⋊(C7⋊C8)C2×C411yescharacteristic
61C4⋊(C7⋊C8)D411yescharacteristic
62C4⋊(C7⋊C8)D411yescharacteristic
63C4×C7⋊C8—21no—
64C4×C7⋊C8—21no—
65#2986—71no—
66#1994—71no—
67#1574—71no—
68#1574D711yescharacteristic
69#1544—71no—
70#1276—71no—
71#1276—71no—
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7#4111yescharacteristic
74C22×C4×C7#4111yescharacteristic
75C22×Dic7—41no—
76C2×Dic14—21no—
77C2×Dic14—21no—
78C7×M16—42no—
79C2×C7×C8—21no—
80C2×C7×C8—21no—
81C2×C7×C8—21no—
82C2×C7×C8—21no—
83C2×C7×C8—41no—
84C42×C7C22×C411yescharacteristic
85C42×C7C2×D411yescharacteristic
86C42×C7C2×D411yescharacteristic
87C42×C7C2×Q811yescharacteristic
88C22⋊Dic7—22no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C4⋊C412yes—
92C4⋊Dic7C4⋊C412yes—
93Dic7⋊C4—22no—
94Dic7⋊C4—22no—
95C4×Dic7—22no—
96C4×Dic7—22no—
97C4×Dic7—22no—
98C4×Dic7—22no—
99C4×Dic7—41no—
100C4×Dic7—41no—
101C4×Dic7—41no—
102C2×C7⋊C8—21no—
103C2×C7⋊C8—21no—
104C2×C7⋊C8—21no—
105C2×C7⋊C8—21no—
106#533—71no—
107#446—141no—
108#446—141no—
109C8×Q8—141no—
110C8×Q8—141no—
111#433—142no—
112#431—71no—
113#431—71no—
114#431C2×D711yescharacteristic
115#431C2×D711yescharacteristic
116#423C2×D711yescharacteristic
117C43—71no—
118#339—142no—
119#335—21no—
120#335—141no—
121#334—21no—
122#334—141no—
123#332—71no—
124#332—71no—
125#332—71no—
126#332—71no—
127C23×C7#12311yescharacteristic
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C2×C4⋊C411yescharacteristic
135C2×C4×C7C4×D411yescharacteristic
136C2×C4×C7C4×D411yescharacteristic
137C2×C4×C7#12011yescharacteristic
138C2×C4×C7C22⋊Q811yescharacteristic
139C2×C4×C7C22⋊Q811yescharacteristic
140C2×Dic7—22no—
141C2×Dic7—22no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7—42no—
145Dic14—41no—
146Dic14—41no—
147Dic14—41no—
148Dic14—41no—
149C7×C8—41no—
150C7×C8—41no—
151C7×C8—41no—
152C7×C8—41no—
153C7×C8—81no—
154C7⋊C8—41no—
155C7⋊C8—41no—
156C7⋊C8—41no—
157C7⋊C8—41no—
158C2×M16—21no—
159C4⋊Q8—71no—
160C4⋊Q8—71no—
161#125—72no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164C4×Q8—141no—
165C4×Q8—141no—
166#117—72no—
167C2×C42—141no—
168C2×C42—141no—
169C2×C42—141no—
170C2×C42C22×D711yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—71no—
174C4⋊C8—71no—
175C4⋊C8—71no—
176C4⋊C8—71no—
177C4⋊C8—142no—
178C4⋊C8—142no—
179C4⋊C8C4×D711yescharacteristic
180C4⋊C8C4×D711yescharacteristic
181C4⋊C8C22⋊D711yescharacteristic
182C4⋊C8C22⋊D711yescharacteristic
183C22⋊C8—21no—
184C22⋊C8—21no—
185C22⋊C8—141no—
186C22⋊C8—141no—
187#97—142no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—142no—
191C4×C8—141no—
192C4×C8—141no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7#38511yescharacteristic
197C4×C7—21no—
198C4×C7—21no—
199C4×C7—21no—
200C4×C7#44311yescharacteristic
201C4×C7#44311yescharacteristic
202C4×C7#44411yescharacteristic
203C4×C7#44411yescharacteristic
204C4×C7#45411yescharacteristic
205C4×C7#45611yescharacteristic
206Dic7—44no—
207Dic7—42no—
208Dic7—42no—
209C2×Q8—141no—
210C2×Q8—141no—
211C22×C4—281no—
212C22×C4#115711yescharacteristic
213C22×C4D4×D711yescharacteristic
214C22×C4#116111yescharacteristic
215M16—42no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—141no—
222C2×C8—141no—
223C2×C8—141no—
224C2×C8—141no—
225C4⋊C4—72no—
226C4⋊C4—72no—
227C4⋊C4—141no—
228C4⋊C4—142no—
229C4⋊C4—141no—
230C4⋊C4—142no—
231C22⋊C4—142no—
232C42—142no—
233C42—142no—
234C42—142no—
235C42—142no—
236C42—281no—
237C42—281no—
238C42—281no—
239C42C2×C4×D711yescharacteristic
240C42D4×D711yescharacteristic
241C42Q8×D711yescharacteristic
242C42C2×C22⋊D711yescharacteristic
243C2×C7—41no—
244C2×C7#185711yescharacteristic
245C2×C7#186611yescharacteristic
246C2×C7#186611yescharacteristic
247C23#645711yescharacteristic
248Q8—281no—
249Q8—281no—
250Q8—281no—
251Q8—281no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—282no—
261C2×C4—282no—
262C2×C4—282no—
263C2×C4D7×C4⋊C411yescharacteristic
264C2×C4#645611yescharacteristic
265C2×C4#645911yescharacteristic
266C2×C4C4×C22⋊D711yescharacteristic
267C2×C4#650011yescharacteristic
268C2×C4#650911yescharacteristic
269C8—41no—
270C8—41no—
271C8—41no—
272C8—41no—
273C8—81no—
274C8—281no—
275C8—281no—
276C8—281no—
277C8—281no—
278C7#893611yescharacteristic, complemented, Sylow
279C22—21no—
280C22—21no—
281C22—21no—
282C22#8976111yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—284no—
287C4—282no—
288C4—282no—
289C4#8966611yescharacteristic
290C4#8966711yescharacteristic
291C4#8991611yescharacteristic
292C4#8991711yescharacteristic
293C4#8997311yescharacteristic
294C4#8997711yescharacteristic
295C2—41no—
296C2#1172829911yescharacteristic
297C2#1172843111yescharacteristic
298C2#1172847611yescharacteristic
subgroupG/H
centerC2×C4#6500
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1574C2
socleC22×C7#385
prime radicalC23×C7#123
cosocleC2×C42×C7C23
hypercenter#1574D7
hypocenterC7#8936
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
23. #89281C4semidirect
24. #89281C4semidirect
25. #89277C4semidirect
26. #89277C4semidirect
278. C7#8936semidirect

Maximal subgroups

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

subgroupindex
#117367122
C7×#15742
#117285312
#117282682
#117282492
#117255422
#89367