Finite Group Databasebeta

← #13250475

Subgroups — #13250475

Order 1792.

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

normal subgroups only

This group has 426 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#11736716C211yesmaximal, characteristic
2#11736490C211yesmaximal, characteristic
3#11728260C211yesmaximal, characteristic
4#11727306C211yesmaximal, characteristic
5#11727300C211yesmaximal, characteristic
6C7×#1286C211yesmaximal, characteristic
7#11725546C211yesmaximal, characteristic
8#90161C2211yescharacteristic
9C2×#6432C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C7×#382C2211yescharacteristic
13#89894—21no—
14#89891—21no—
15C2×#6384—21no—
16C2×#6384—21no—
17#89697C2211yescharacteristic
18C2×#6377C2211yescharacteristic
19#89623—22no—
20#89600—22no—
21#89600—22no—
22#89493—22no—
23#89453C412yescomplemented
24#89452C412yescomplemented
25#89277—22no—
26#89247—22no—
27#9292—71nomaximal, Sylow
28C22×Dic14—21no—
29C2×C7×M16—21no—
30C2×C7×C4⋊C4—21no—
31C2×C42×C7C2311yescharacteristic
32#6489—21no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35#6487—21no—
36#6487D412yes—
37C22×C7⋊C8D411yescharacteristic
38C22×C7⋊C8D411yescharacteristic
39#6433—21no—
40#6432—21no—
41#6432—21no—
42#6432—21no—
43#6432C2×C412yes—
44#6432C2×C411yescharacteristic
45#6432C2×C411yescharacteristic
46C4×Dic14—21no—
47C4×Dic14D412yes—
48C7×C4⋊C8—21no—
49C7×C4⋊C8—21no—
50C7×C4⋊C8D412yes—
51C7×C22⋊C8—21no—
52C4×C7×C8C2×C412yes—
53C7×#95—21no—
54C7×#95—21no—
55C22⋊(C7⋊C8)—21no—
56#6391—24no—
57#6390—24no—
58#6389—22no—
59Dic7⋊C8—24no—
60C8×Dic7—22no—
61#6384—42no—
62#6384—42no—
63C4⋊(C7⋊C8)—21no—
64#6377—21no—
65#6377—21no—
66C4×C7⋊C8—21no—
67#3229—71no—
68#2961—71no—
69#1960—71no—
70#1546—71no—
71#1518—71no—
72#1286D711yescharacteristic
73#1275—71no—
74C22×C4×C7—41no—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7#4111yescharacteristic
78C22×Dic7—41no—
79C2×Dic14—22no—
80C2×Dic14—21no—
81C2×Dic14—41no—
82C2×Dic14—41no—
83C2×Dic14—41no—
84C2×Dic14—41no—
85C2×Dic14—41no—
86C2×Dic14C22⋊C412yes—
87C7×M16—41no—
88C7×M16—41no—
89C2×C7×C8—22no—
90C2×C7×C8—22no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C42×C7C22×C411yescharacteristic
94C42×C7C2×D411yescharacteristic
95C42×C7C2×D411yescharacteristic
96C42×C7#4111yescharacteristic
97C22⋊Dic7—22no—
98C4⋊Dic7—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7—22no—
102C4⋊Dic7—22no—
103C4⋊Dic7—41no—
104C4⋊Dic7—41no—
105C4⋊Dic7C22⋊C412yes—
106Dic7⋊C4—22no—
107Dic7⋊C4—22no—
108C4×Dic7—22no—
109C2×C7⋊C8—21no—
110C2×C7⋊C8—21no—
111C2×C7⋊C8—21no—
112C2×C7⋊C8—21no—
113C2×C7⋊C8—21no—
114C2×C7⋊C8—41no—
115C2×C7⋊C8SD1612yes—
116#533—71no—
117C2×C4⋊Q8—71no—
118#498—72no—
119#487—72no—
120#483—142no—
121#475—142no—
122#446—142no—
123#446—142no—
124#431—71no—
125#431C2×D711yescharacteristic
126#424—141no—
127#423C2×D711yescharacteristic
128C2×#103—141no—
129C2×#103—141no—
130#405—141no—
131C2×#97—71no—
132#382C2×D711yescharacteristic
133#332—142no—
134#328—142no—
135C23×C7#12111yescharacteristic
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—81no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7C2×C22⋊C411yescharacteristic
142C2×C4×C7C4×D411yescharacteristic
143C2×C4×C7C4×D411yescharacteristic
144C2×C4×C7#12011yescharacteristic
145C2×C4×C7#12111yescharacteristic
146C2×C4×C7#12311yescharacteristic
147C2×C4×C7C2×SD1611yescharacteristic
148C2×C4×C7#13511yescharacteristic
149C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C7#13711yescharacteristic
151C2×Dic7—22no—
152C2×Dic7—22no—
153C2×Dic7—42no—
154C2×Dic7—81no—
155C2×Dic7—81no—
156Dic14—42no—
157Dic14—42no—
158Dic14—42no—
159Dic14—81no—
160Dic14—81no—
161Dic14—81no—
162C7×C8—42no—
163C7×C8—42no—
164C7⋊C8—22no—
165C7⋊C8—41no—
166C7⋊C8—81no—
167C22×Q8—141no—
168C2×M16—21no—
169C22×C8—71no—
170C22×C8—71no—
171C4⋊Q8—71no—
172C4⋊Q8—71no—
173C4⋊Q8—72no—
174C4⋊Q8—141no—
175C4⋊Q8—141no—
176C4⋊Q8—141no—
177#125—141no—
178C22⋊Q8—72no—
179C22⋊Q8—141no—
180C4×Q8—72no—
181C4×Q8—141no—
182#117—141no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—141no—
185C2×C4⋊C4—141no—
186C2×C42C22×D711yescharacteristic
187#106—144no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—144no—
191C4⋊C8—141no—
192C4⋊C8D2812yes—
193#103—144no—
194#103—282no—
195#103—282no—
196C22⋊C8—21no—
197C22⋊C8—141no—
198#97—142no—
199#97—141no—
200#97—141no—
201C4×C8—142no—
202C4×C8—141no—
203C4×C8C4×D712yes—
204#95—21no—
205#95—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7#38611yescharacteristic
209C22×C7#46011yescharacteristic
210C22×C7#46411yescharacteristic
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—161no—
216C4×C7C4×SD1611yescharacteristic
217C4×C7#44011yescharacteristic
218C4×C7#44811yescharacteristic
219C4×C7#44911yescharacteristic
220Dic7—42no—
221Dic7—42no—
222Dic7—82no—
223C2×Q8—72no—
224C2×Q8—141no—
225C2×Q8—142no—
226C2×Q8—281no—
227C2×Q8—281no—
228C2×Q8—281no—
229C2×Q8—281no—
230C2×Q8—281no—
231C22×C4—41no—
232C22×C4—281no—
233C22×C4#115711yescharacteristic
234C22×C4D4×D711yescharacteristic
235C22×C4D4×D711yescharacteristic
236M16—41no—
237M16—41no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—72no—
241C2×C8—141no—
242C2×C8—141no—
243C2×C8—141no—
244C2×C8—141no—
245C2×C8—141no—
246C2×C8—281no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—72no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C4⋊C4—142no—
256C4⋊C4—142no—
257C4⋊C4—281no—
258C4⋊C4—281no—
259C22⋊C4—142no—
260C42—142no—
261C42C2×C4×D711yescharacteristic
262C42C2×D2811yescharacteristic
263C42D4×D711yescharacteristic
264C42#116111yescharacteristic
265C2×C7—21no—
266C2×C7—21no—
267C2×C7#188011yescharacteristic
268C2×C7#190211yescharacteristic
269C2×C7#190911yescharacteristic
270C23#644711yescharacteristic
271Q8—282no—
272Q8—282no—
273Q8—282no—
274Q8—561no—
275Q8—561no—
276Q8—561no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—142no—
283C2×C4—142no—
284C2×C4—282no—
285C2×C4—561no—
286C2×C4—561no—
287C2×C4C4×D2811yescharacteristic
288C2×C4D7×C22⋊C411yescharacteristic
289C2×C4#644511yescharacteristic
290C2×C4#645611yescharacteristic
291C2×C4#645711yescharacteristic
292C2×C4#645811yescharacteristic
293C2×C4#647411yescharacteristic
294C2×C4D7×SD1611yescharacteristic
295C2×C4#647911yescharacteristic
296C2×C4#648111yescharacteristic
297C8—42no—
298C8—42no—
299C8—142no—
300C8—281no—
301C8—561no—
302C7#929211yescharacteristic, complemented, Sylow
303C22—21no—
304C22—21no—
305C22#8943711yescharacteristic
306C22#8953411yescharacteristic
307C22#8957011yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—161no—
313C4—282no—
314C4—282no—
315C4—562no—
316C4#8955311yescharacteristic
317C4#8957811yescharacteristic
318C4#8962411yescharacteristic
319C4#8966111yescharacteristic
320C2—21no—
321C2—21no—
322C2#1172646611yescharacteristic
323C2#1172666311yescharacteristic
324C2#1172679111yescharacteristic
subgroupG/H
centerC22#89437
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1286C2
socleC22×C7#386
prime radicalC23×C7#121
cosocleC2×C42×C7C23
hypercenter#1286D7
hypocenterC7#9292
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
23. #89453C4semidirect
24. #89452C4semidirect
302. C7#9292semidirect

Maximal subgroups

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

subgroupindex
#117367162
#117364902
#117282602
#117273062
#117273002
C7×#12862
#117255462
#92927