Finite Group Databasebeta

← #13250500

Subgroups — #13250500

Order 1792.

470 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 113 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#11740672C211yesmaximal, characteristic
2#11740668C211yesmaximal, characteristic
3C7×#1546C211yesmaximal, characteristic
4#11728547C211yesmaximal, characteristic
5#11728269C211yesmaximal, characteristic
6#11725543C211yesmaximal, characteristic
7#11725519C211yesmaximal, characteristic
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×Q8×Dic7—21no—
10#90161C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13#90002—21no—
14#89866—22no—
15#89865C412yescomplemented
16#89851C412yescomplemented
17#89753—22no—
18#89753—22no—
19#89743—22no—
20#89708C2211yescharacteristic
21#89704C2211yescharacteristic
22#89697C2211yescharacteristic
23C7×#332—22no—
24#89350—21no—
25#89350—21no—
26#89262—22no—
27#9296—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C2×C7×M16—21no—
30C7×C4⋊Q8—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8C2×C412yes—
33C7×C4⋊Q8D412yes—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4D412yes—
36C2×C42×C7C2311yescharacteristic
37Q8×Dic7—42no—
38Q8×Dic7—42no—
39#6489—21no—
40C2×C4⋊Dic7—22no—
41C2×C4⋊Dic7—21no—
42#6487—21no—
43#6487—21no—
44C2×C4×Dic7—22no—
45C2×C4×Dic7D411yescharacteristic
46C2×C4×Dic7D411yescharacteristic
47#6433—21no—
48#6432C2×C411yescharacteristic
49#6432C2×C411yescharacteristic
50C4×Dic14—21no—
51C4×Dic14—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C22⋊C8—21no—
56C4×C7×C8—21no—
57C7⋊#95—24no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95D412yes—
62C22⋊(C7⋊C8)—21no—
63#6384—24no—
64#6382—24no—
65C4⋊(C7⋊C8)—21no—
66C4×C7⋊C8C2×C412yes—
67#3229—71no—
68#2327—71no—
69#1995—71no—
70#1546D711yescharacteristic
71#1518—71no—
72#1286—71no—
73#1274—71no—
74C2×C7×Q8—22no—
75C2×C7×Q8—22no—
76C2×C7×Q8—41no—
77C2×C7×Q8—41no—
78C2×C7×Q8—41no—
79C2×C7×Q8—41no—
80C22×C4×C7—22no—
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×Dic7—21no—
85C22×Dic7—21no—
86C22×Dic7—22no—
87C2×Dic14—21no—
88C2×Dic14—21no—
89C7×M16—41no—
90C7×M16—41no—
91C2×C7×C8—41no—
92C2×C7×C8—41no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4C22⋊C414yes—
97C42×C7C22×C411yescharacteristic
98C42×C7C2×D411yescharacteristic
99C42×C7#4111yescharacteristic
100C42×C7#4111yescharacteristic
101C22⋊Dic7—41no—
102C4⋊Dic7—21no—
103C4⋊Dic7—21no—
104C4⋊Dic7—21no—
105C4⋊Dic7—21no—
106C4⋊Dic7—44no—
107C4⋊Dic7—41no—
108C4⋊Dic7—41no—
109Dic7⋊C4—41no—
110Dic7⋊C4—41no—
111C4×Dic7—21no—
112C4×Dic7—21no—
113C4×Dic7—21no—
114C4×Dic7—44no—
115C4×Dic7—41no—
116C4×Dic7SD1612yes—
117C2×C7⋊C8—22no—
118#533—71no—
119C2×C4⋊Q8C2×D711yescharacteristic
120C2×C4×Q8—141no—
121#498—72no—
122#487—72no—
123#483—142no—
124#475—142no—
125#431—71no—
126#431C2×D711yescharacteristic
127#423C2×D711yescharacteristic
128#391—141no—
129#387—142no—
130#387—142no—
131#382—71no—
132#376—71no—
133#339—141no—
134#339—141no—
135#332—22no—
136#328—142no—
137C23×C7#12011yescharacteristic
138C7×Q8—42no—
139C7×Q8—42no—
140C7×Q8—42no—
141C7×Q8—42no—
142C2×C4×C7—24no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7C2×C22⋊C411yescharacteristic
148C2×C4×C7C4×D411yescharacteristic
149C2×C4×C7C4×D411yescharacteristic
150C2×C4×C7#12111yescharacteristic
151C2×C4×C7#12111yescharacteristic
152C2×C4×C7#12311yescharacteristic
153C2×C4×C7C2×SD1611yescharacteristic
154C2×C4×C7#13511yescharacteristic
155C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C7#13711yescharacteristic
157C2×Dic7—22no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—42no—
162C2×Dic7—41no—
163C2×Dic7—42no—
164C2×Dic7—41no—
165C2×Dic7—41no—
166C2×Dic7—42no—
167C2×Dic7—41no—
168Dic14—42no—
169Dic14—81no—
170C7×C8—81no—
171C7×C8—81no—
172C7⋊C8—42no—
173C22×Q8—21no—
174C2×M16—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—71no—
178C4⋊Q8—71no—
179C4⋊Q8C4×D712yes—
180C4⋊Q8C22⋊D712yes—
181#125—141no—
182C22⋊Q8—141no—
183C22⋊Q8—141no—
184C4×Q8—141no—
185C4×Q8—141no—
186C4×Q8—282no—
187C4×Q8—282no—
188#117—141no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—142no—
191C2×C4⋊C4—141no—
192C2×C4⋊C4D2812yes—
193C2×C42—71no—
194C2×C42—71no—
195C2×C42—142no—
196C2×C42C22×D711yescharacteristic
197#106—144no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—141no—
202#103—144no—
203C22⋊C8—21no—
204C22⋊C8—141no—
205C4×C8—21no—
206C4×C8—72no—
207#95—72no—
208#95—144no—
209#95—141no—
210#95—141no—
211#95—141no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7#38611yescharacteristic
215C22×C7#45011yescharacteristic
216C22×C7#45211yescharacteristic
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—44no—
222C4×C7C4×SD1611yescharacteristic
223C4×C7#44011yescharacteristic
224C4×C7#46111yescharacteristic
225C4×C7#46311yescharacteristic
226Dic7—42no—
227Dic7—82no—
228Dic7—81no—
229Dic7—81no—
230Dic7—81no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—141no—
238C2×Q8—141no—
239C22×C4—22no—
240C22×C4—141no—
241C22×C4—142no—
242C22×C4—141no—
243C22×C4C2×D2811yescharacteristic
244C22×C4D4×D711yescharacteristic
245C22×C4D4×D711yescharacteristic
246M16—41no—
247M16—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—142no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—284no—
259C4⋊C4—281no—
260C4⋊C4—281no—
261C4⋊C4—281no—
262C4⋊C4—281no—
263C4⋊C4(C2×D7)⋊C414yes—
264C22⋊C4—281no—
265C42—72no—
266C42—141no—
267C42—141no—
268C42—141no—
269C42—284no—
270C42—281no—
271C42C2×C4×D711yescharacteristic
272C42#115911yescharacteristic
273C42#115911yescharacteristic
274C42C2×C22⋊D711yescharacteristic
275C2×C7—21no—
276C2×C7—21no—
277C2×C7#187711yescharacteristic
278C2×C7#190511yescharacteristic
279C2×C7#191211yescharacteristic
280C23#644511yescharacteristic
281Q8—42no—
282Q8—42no—
283Q8—42no—
284Q8—42no—
285Q8—282no—
286Q8—561no—
287C2×C4—24no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—142no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4—282no—
296C2×C4—281no—
297C2×C4—282no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—282no—
302C2×C4—281no—
303C2×C4(C22⋊D7)⋊C411yescharacteristic
304C2×C4(C22⋊D7)⋊C411yescharacteristic
305C2×C4#644911yescharacteristic
306C2×C4#647411yescharacteristic
307C2×C4#647511yescharacteristic
308C2×C4D7×SD1611yescharacteristic
309C2×C4#647811yescharacteristic
310C2×C4C2×(C2×D7)⋊C411yescharacteristic
311C2×C4#650211yescharacteristic
312C2×C4#650211yescharacteristic
313C8—81no—
314C8—81no—
315C8—282no—
316C7#929611yescharacteristic, complemented, Sylow
317C22—21no—
318C22—21no—
319C22#8954811yescharacteristic
320C22#8955511yescharacteristic
321C22#8973011yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—44no—
327C4—282no—
328C4—562no—
329C4—561no—
330C4—561no—
331C4—561no—
332C4#8952911yescharacteristic
333C4#8953011yescharacteristic
334C4#8992511yescharacteristic
335C4#8993411yescharacteristic
336C2—21no—
337C2—21no—
338C2#1172785911yescharacteristic
339C2#1172792511yescharacteristic
340C2#1172808811yescharacteristic
subgroupG/H
centerC22#89730
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1546C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#1546D7
hypocenterC7#9296
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
15. #89865C4semidirect
16. #89851C4semidirect
316. C7#9296semidirect

Maximal subgroups

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

subgroupindex
#117406722
#117406682
C7×#15462
#117285472
#117282692
#117255432
#117255192
#92967