Finite Group Databasebeta

← #13250726

Subgroups — #13250726

Order 1792.

444 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C7×#2320C211yesmaximal, characteristic
2#11739893C211yesmaximal, characteristic
3#11729377C211yesmaximal, characteristic
4#11727247C211yesmaximal, characteristic
5#11727187C211yesmaximal, characteristic
6#11725427C211yesmaximal, characteristic
7#11725421C211yesmaximal, characteristic
8C2×C4×C7×Q8C2211yescharacteristic
9#90161C2211yescharacteristic
10C7×#391—21no—
11C7×#389—21no—
12C7×#387—22no—
13C7×#383—21no—
14C7×#382—21no—
15C4×C7×C4⋊C4C2211yescharacteristic
16C7×#376C2211yescharacteristic
17#90000—21no—
18#89803C412yescomplemented
19#89802—22no—
20#89792C412yescomplemented
21#89743—21no—
22#89698C2211yescharacteristic
23#89698C2211yescharacteristic
24C4×#1136C2211yescharacteristic
25Q8⋊(C7⋊C8)—22no—
26#89318—22no—
27#89247—21no—
28#89247—21no—
29#9328—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C4×C7×Q8—21no—
32C4×C7×Q8—21no—
33C4×C7×Q8C2×C412yes—
34C4×C7×Q8D412yes—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C42×C7—21no—
40C2×C42×C7—21no—
41C2×C42×C7D412yes—
42C2×C42×C7C2311yescharacteristic
43C2×#1144—22no—
44#6489—21no—
45#6487—21no—
46#6487—21no—
47C2×#1136—21no—
48#6433—21no—
49#6432C2×C411yescharacteristic
50#6432C2×C411yescharacteristic
51C4×Dic14—21no—
52C4×Dic14—21no—
53C7×#95—22no—
54C7×#95—22no—
55C7×#95—22no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95D411yescharacteristic
59C7×#95D411yescharacteristic
60#6409—22no—
61C22⋊(C7⋊C8)—21no—
62C22⋊(C7⋊C8)—21no—
63#6384—22no—
64#6382—22no—
65C4⋊(C7⋊C8)—21no—
66C4⋊(C7⋊C8)C2×C412yes—
67#6377—21no—
68#6377—21no—
69#6377—21no—
70C4×C7⋊C8—21no—
71#3113—71no—
72#2320D711yescharacteristic
73#1986—71no—
74#1506—71no—
75#1491—71no—
76#1300—71no—
77#1285—71no—
78C2×C7×Q8—21no—
79C2×C7×Q8—42no—
80C2×C7×Q8C22⋊C412yes—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—22no—
85C22×C4×C7—21no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C2×Dic14—21no—
90C2×Dic14—21no—
91C7×C4⋊C4—22no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—82no—
96C7×C4⋊C4C22⋊C412yes—
97C42×C7—22no—
98C42×C7—22no—
99C42×C7—42no—
100C42×C7—42no—
101C42×C7C22×C411yescharacteristic
102C42×C7C2×D411yescharacteristic
103C42×C7#4111yescharacteristic
104C42×C7#4111yescharacteristic
105C22⋊Dic7—41no—
106#1144—44no—
107C4⋊Dic7—21no—
108C4⋊Dic7—21no—
109C4⋊Dic7—21no—
110C4⋊Dic7—21no—
111Dic7⋊C4—41no—
112Dic7⋊C4—41no—
113C4×Dic7—41no—
114#1136—22no—
115#1136—41no—
116C2×C7⋊C8—22no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8—41no—
119#533—71no—
120C2×C4×Q8C2×D711yescharacteristic
121#483—141no—
122#475—72no—
123#462—72no—
124#451—141no—
125#441—142no—
126#432—71no—
127#432—71no—
128C4×M16—71no—
129#391—21no—
130#389—21no—
131#387—22no—
132#383—21no—
133#382—21no—
134C4×C4⋊C4C2×D711yescharacteristic
135#376C2×D711yescharacteristic
136#339—142no—
137#328—141no—
138#328—141no—
139Q8⋊C8—142no—
140C23×C7#12011yescharacteristic
141C7×Q8—22no—
142C7×Q8—22no—
143C7×Q8—42no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—42no—
156C2×C4×C7—41no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—42no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7C2×C22⋊C411yescharacteristic
163C2×C4×C7C4×D411yescharacteristic
164C2×C4×C7C4×D411yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12111yescharacteristic
167C2×C4×C7#12311yescharacteristic
168C2×Dic7—41no—
169C2×Dic7—41no—
170Dic14—42no—
171Dic14—81no—
172C7⋊C8—42no—
173C7⋊C8—42no—
174C7⋊C8—81no—
175C22×Q8C22⋊D711yescharacteristic
176C2×Q16—142no—
177C2×M16—141no—
178C4⋊Q8—71no—
179C4⋊Q8—71no—
180#125—141no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—141no—
186C4×Q8—141no—
187C4×Q8C4×D712yes—
188C4×Q8D2812yes—
189#117—141no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊D711yescharacteristic
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C22⋊D712yes—
197C2×C42C22×D711yescharacteristic
198#106—142no—
199C4⋊C8—72no—
200C4⋊C8—141no—
201#103—142no—
202#103—142no—
203C22⋊C8—141no—
204C22⋊C8—141no—
205#97—141no—
206#97—141no—
207#97—141no—
208C4×C8—141no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95C22⋊D711yescharacteristic
215#95C22⋊D711yescharacteristic
216C22×C7—21no—
217C22×C7—21no—
218C22×C7#38611yescharacteristic
219C22×C7#45511yescharacteristic
220C22×C7#45611yescharacteristic
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—42no—
227C4×C7—42no—
228C4×C7—81no—
229C4×C7—82no—
230C4×C7—81no—
231C4×C7#42112yes—
232Dic7—81no—
233Dic7—81no—
234C2×Q8—21no—
235C2×Q8—42no—
236C2×Q8—141no—
237C2×Q8—141no—
238C2×Q8(C2×D7)⋊C412yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4C2×C22⋊D711yescharacteristic
245C22×C4C2×C22⋊D711yescharacteristic
246C22×C4C2×C22⋊D711yescharacteristic
247Q16—284no—
248M16—142no—
249M16—281no—
250C2×C8—142no—
251C2×C8—281no—
252C2×C8—281no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—82no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—281no—
263C4⋊C4—281no—
264C4⋊C4(C2×D7)⋊C412yes—
265C22⋊C4—281no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—42no—
270C42—281no—
271C42C2×C4×D711yescharacteristic
272C42C2×D2811yescharacteristic
273C42#115711yescharacteristic
274C42#115711yescharacteristic
275C2×C7—21no—
276C2×C7—21no—
277C2×C7#188211yescharacteristic
278C2×C7#189711yescharacteristic
279C2×C7#190511yescharacteristic
280C23#651611yescharacteristic
281Q8—22no—
282Q8—22no—
283Q8—42no—
284Q8—282no—
285Q8—561no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—281no—
305C2×C4—281no—
306C2×C4C2×(C2×D7)⋊C411yescharacteristic
307C2×C4C4×C22⋊D711yescharacteristic
308C2×C4C4×C22⋊D711yescharacteristic
309C2×C4#649211yescharacteristic
310C2×C4#649311yescharacteristic
311C2×C4#649311yescharacteristic
312C8—282no—
313C8—282no—
314C8—561no—
315C7#932811yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22#8998511yescharacteristic
319C22#9001411yescharacteristic
320C22#9001811yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—82no—
330C4—81no—
331C4—561no—
332C4—561no—
333C4#8977712yes—
334C2—21no—
335C2—21no—
336C2#1172934511yescharacteristic
337C2#1172941611yescharacteristic
338C2#1172942011yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2320C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#2320D7
hypocenterC7#9328
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2320

Complements

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

NH ≅ G/Ntype
18. #89803C4semidirect
20. #89792C4semidirect
315. C7#9328semidirect

Maximal subgroups

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

subgroupindex
C7×#23202
#117398932
#117293772
#117272472
#117271872
#117254272
#117254212
#93287