Finite Group Databasebeta

← #13250506

Subgroups — #13250506

Order 1792.

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

normal subgroups only

This group has 454 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#11740815C211yesmaximal, characteristic
2#11731599C211yesmaximal, characteristic
3C7×#1518C211yesmaximal, characteristic
4#11729594C211yesmaximal, characteristic
5#11728533C211yesmaximal, characteristic
6#11725543C211yesmaximal, characteristic
7#11725437C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×#6432C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#382C2211yescharacteristic
12#89865C412yescomplemented
13#89864C412yescomplemented
14#89822—22no—
15#89803—22no—
16C4⋊C4×Dic7—21no—
17#89745—21no—
18#89704C2211yescharacteristic
19#89697C2211yescharacteristic
20#89696C2211yescharacteristic
21#89425—22no—
22#89418—22no—
23C7×#328—22no—
24#89327—21no—
25#89327—21no—
26#89276—22no—
27#9297—71nomaximal, Sylow
28C22×Dic14D412yes—
29C7×C4⋊Q8C2×C411yescharacteristic
30C7×C4⋊Q8C2×C411yescharacteristic
31C7×#125—21no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8—21no—
36C7×#117—21no—
37C2×C7×C4⋊C4—21no—
38C2×C42×C7C2311yescharacteristic
39C2×#1144—24no—
40C2×C4⋊Dic7—21no—
41C2×C4⋊Dic7—21no—
42C2×Dic7⋊C4—22no—
43C2×C4×Dic7—22no—
44C2×C4×Dic7D411yescharacteristic
45C2×C4×Dic7D411yescharacteristic
46C2×#1136—21no—
47#6432—21no—
48#6432—21no—
49#6432C2×C412yes—
50#6432D412yes—
51C7×C4⋊C8—21no—
52C7×C22⋊C8—21no—
53C4×C7×C8—21no—
54C7×#95—21no—
55C7×#95D412yes—
56C7⋊#95—22no—
57C7⋊#95—22no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C22⋊(C7⋊C8)—21no—
61#6384—24no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4×C7⋊C8C2×C412yes—
66#3209—71no—
67#2331—71no—
68#1995—71no—
69#1548—71no—
70#1518D711yescharacteristic
71#1286—71no—
72#1274—71no—
73C2×C7×Q8—21no—
74C2×C7×Q8—21no—
75C22×C4×C7—22no—
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×Dic7—21no—
80C22×Dic7—22no—
81C22×Dic7—21no—
82C2×Dic14—22no—
83C2×Dic14—82no—
84C2×Dic14C22⋊C414yes—
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×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C22⋊C4—41no—
95C42×C7—41no—
96C42×C7C22×C411yescharacteristic
97C42×C7C2×D411yescharacteristic
98C42×C7#4111yescharacteristic
99C42×C7#4111yescharacteristic
100#1144—84no—
101C4⋊Dic7—22no—
102C4⋊Dic7—22no—
103C4⋊Dic7—22no—
104C4⋊Dic7—22no—
105Dic7⋊C4—82no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109C4×Dic7—82no—
110C4×Dic7Q1612yescomplemented
111#1136—41no—
112#1136—41no—
113C2×C7⋊C8—22no—
114C2×C7⋊C8—41no—
115#533C2×D711yescharacteristic
116C2×C4⋊Q8—71no—
117#494—72no—
118#487—72no—
119#462—142no—
120#451—142no—
121#431—71no—
122#431C2×D711yescharacteristic
123#423—71no—
124#391—142no—
125#389—141no—
126#387—142no—
127#382C2×D711yescharacteristic
128C4×C4⋊C4—141no—
129#376—71no—
130#339—141no—
131#339—141no—
132#333—142no—
133#328—22no—
134C23×C7#12011yescharacteristic
135C7×Q8—42no—
136C7×Q8—81no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—42no—
144C2×C4×C7C2×C22⋊C411yescharacteristic
145C2×C4×C7C4×D411yescharacteristic
146C2×C4×C7C4×D411yescharacteristic
147C2×C4×C7#12111yescharacteristic
148C2×C4×C7#12111yescharacteristic
149C2×C4×C7#12311yescharacteristic
150C2×C4×C7C2×Q1611yescharacteristic
151C2×C4×C7#13511yescharacteristic
152C2×C4×C7#13711yescharacteristic
153C2×C4×C7#13711yescharacteristic
154C2×Dic7—24no—
155C2×Dic7—22no—
156C2×Dic7—41no—
157C2×Dic7—42no—
158C2×Dic7—42no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163Dic14—44no—
164Dic14—82no—
165C7×C8—81no—
166C7⋊C8—42no—
167C7⋊C8—81no—
168C22×Q8—72no—
169C2×Q16—144no—
170C2×M16—141no—
171C4⋊Q8—72no—
172C4⋊Q8—72no—
173C4⋊Q8—141no—
174C4⋊Q8—141no—
175C4⋊Q8C4×D711yescharacteristic
176C4⋊Q8C4×D711yescharacteristic
177#125—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—141no—
185C2×C4⋊C4—141no—
186C2×C4⋊C4—142no—
187C2×C42—71no—
188C2×C42—71no—
189C2×C42—142no—
190C2×C42C22×D711yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—141no—
193C4⋊C8—141no—
194C4⋊C8—141no—
195#103—144no—
196C22⋊C8—21no—
197C22⋊C8—141no—
198C4×C8—21no—
199C4×C8—72no—
200#95—21no—
201#95—142no—
202#95—142no—
203#95—141no—
204#95—141no—
205#95C22⋊D712yes—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7#38611yescharacteristic
209C22×C7#45111yescharacteristic
210C22×C7#45211yescharacteristic
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—82no—
216C4×C7—81no—
217C4×C7—81no—
218C4×C7C4×Q1611yescharacteristic
219C4×C7#44111yescharacteristic
220C4×C7#46211yescharacteristic
221C4×C7#46411yescharacteristic
222Dic7—42no—
223Dic7—44no—
224Dic7—81no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—74no—
228C2×Q8—142no—
229C2×Q8—562no—
230C22×C4—22no—
231C22×C4—141no—
232C22×C4—142no—
233C22×C4—141no—
234C22×C4D4×D711yescharacteristic
235C22×C4D4×D711yescharacteristic
236C22×C4C2×C22⋊D711yescharacteristic
237Q16—564no—
238M16—281no—
239M16—281no—
240C2×C8—41no—
241C2×C8—142no—
242C2×C8—281no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—142no—
252C4⋊C4—142no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C4⋊C4—562no—
256C22⋊C4—41no—
257C42—41no—
258C42—72no—
259C42—141no—
260C42—141no—
261C42—141no—
262C42—562no—
263C42C2×C4×D711yescharacteristic
264C42#115711yescharacteristic
265C42D4×D711yescharacteristic
266C42#115911yescharacteristic
267C2×C7—21no—
268C2×C7—21no—
269C2×C7#187811yescharacteristic
270C2×C7#190511yescharacteristic
271C2×C7#191111yescharacteristic
272C23#650011yescharacteristic
273Q8—42no—
274Q8—81no—
275Q8—284no—
276Q8—562no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—142no—
285C2×C4—144no—
286C2×C4—281no—
287C2×C4—282no—
288C2×C4—282no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4D7×C22⋊C411yescharacteristic
294C2×C4(C22⋊D7)⋊C411yescharacteristic
295C2×C4#644611yescharacteristic
296C2×C4#644711yescharacteristic
297C2×C4#647811yescharacteristic
298C2×C4#647911yescharacteristic
299C2×C4D7×Q1611yescharacteristic
300C2×C4#648111yescharacteristic
301C2×C4C4×C22⋊D711yescharacteristic
302C2×C4#650111yescharacteristic
303C8—81no—
304C8—282no—
305C8—561no—
306C7#929711yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22#8972811yescharacteristic
310C22#8994511yescharacteristic
311C22#8995811yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—82no—
317C4—81no—
318C4—81no—
319C4—282no—
320C4—284no—
321C4—561no—
322C4#8956111yescharacteristic
323C4#8956211yescharacteristic
324C4#8956511yescharacteristic
325C4#8957011yescharacteristic
326C2—21no—
327C2—21no—
328C2#1172780611yescharacteristic
329C2#1172803511yescharacteristic
330C2#1172816211yescharacteristic
subgroupG/H
centerC22#89728
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1518C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#1518D7
hypocenterC7#9297
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
12. #89865C4semidirect
13. #89864C4semidirect
110. C4×Dic7Q16semidirect
306. C7#9297semidirect

Maximal subgroups

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

subgroupindex
#117408152
#117315992
C7×#15182
#117295942
#117285332
#117255432
#117254372
#92977